Can one rule predict how freely released objects move, how clocks at different heights compare, and how light travels?

We will first solve the falling-object problem without geometry or gravitational mass. We will then ask whether the measurements admit one gravitational rule that works for different test objects, and introduce curved spacetime only when a single observational model must describe falling objects, clocks, and light.

1. What does a released object do near Earth?

Choose a vertical position coordinate \(z\) that increases upward. Release a small object in an evacuated chamber and record its position with the chamber’s clocks. Over a sufficiently small height and time, the observations are described by

where \(z_0\) is its position at \(t=0\), \(u_0\) is its initial upward velocity, and \(g\) is a measured positive constant. Differentiating twice gives

This rule already predicts the position of that object. If it is released from rest at height \(h\) above the floor, then \(z_0=h\) and \(u_0=0\). It reaches the floor when

so

Now repeat the evacuated-chamber experiment with objects made from different materials. Give them the same initial position and velocity:

Within experimental precision, the same value of \(g\) describes every tested object. Objects released with the same initial conditions therefore follow the same trajectory, regardless of their material or mass. This conclusion applies to sufficiently small test objects whose own influence on Earth can be neglected.

The original drop problem is solved for every such object. We did not need either inertial mass or gravitational mass. Now ask for a different prediction:

How strong must a spring or floor be to support different objects at different locations?

2. How strong must the support be?

The earlier mechanics construction defined relative inertial mass by comparing the accelerations of two objects under the same controlled interaction:

The subscript \(\mathrm{i}\) labels the inertial mass already calibrated in that separate experiment. Once those masses were calibrated, the same construction defined force by

Suppose those experiments assign object \(A\) an inertial mass of \(1\ \mathrm{kg}\) and object \(B\) an inertial mass of \(2\ \mathrm{kg}\).

Now construct the support experiment. Fix one end of a calibrated spring to the ceiling and attach a test object \(X\) to its lower end. Wait until the object is at rest. Let \(x_X\) be the spring’s downward extension from its natural length.

The drop rule tells us how \(X\) moves after release, but it does not predict the extension \(x_X\) while the spring holds it. Suppose the spring’s calibrated stiffness is \(k=100\ \mathrm{N/m}\). It pulls upward with magnitude \(kx_X\). Because the supported object has zero acceleration, the total force on it is zero. The downward gravitational force must therefore have the same magnitude:

Here \(W_X\) is the object’s weight at that location. This is an operational measurement of gravitational force: the calibrated spring force balances it.

Now repeat the measurement at two locations, \(L_1\) and \(L_2\). The following values illustrate the observed pattern:

There is not one numerical gravitational force shared by both objects. At either location, object \(B\) has twice the weight of object \(A\). But the ratio belonging to the pair of objects is stable:

Experiments with other calibrated springs check that this ratio does not depend on the supporting apparatus. Experiments at other locations check that it does not depend on the local gravitational environment. When those checks succeed, we can assign the stable factor to the objects. Define relative passive gravitational mass by

when both weights are measured at the same location \(L\). Choosing one reference object fixes the overall gravitational-mass unit. “Passive” means that this quantity describes how strongly the test object responds to an existing gravitational environment. It does not yet describe how strongly the object would produce gravity if used as a source.

An object’s gravitational mass stays fixed when we move it to another location, while its weight can change.

Once the object factors have been calibrated, divide them out of the measured weights. At location \(L\), define

The measured stability of the weight ratios makes \(\gamma(L)\) independent of which test object \(X\) is used. It describes the location and source, while \(m_{\mathrm{g},X}\) describes the test object. Every separate weight measurement is now summarized by one reusable rule:

This factorization does predictive work. After measuring \(\gamma(L)\) with one reference object, we can predict the extension of a spring of stiffness \(k\) supporting any object whose passive gravitational mass is known:

A floor supporting the same object must provide the upward force \(m_{\mathrm{g},X}\gamma(L)\). Thus the construction answers the question that free-fall acceleration alone could not answer.

What happens when the support is removed?

Now release \(A\) and \(B\) in the evacuated chamber at \(L_1\). Their measured accelerations are

To connect the weight measurements to the earlier spring calibration, imagine removing gravity while keeping the spring attached and initially stretched by \(x_X\): this returns us to the spring experiment in Rediscovering Mechanics and Energy from Questions. Let \(a_X\) be the initial upward acceleration in that spring-only experiment. The spring calibration gives \(m_{\mathrm{i},X}a_X=kx_X\), which by our definitions of weight and \(\gamma\) is also \(m_{\mathrm{g},X}\gamma\):

and hence

For \(A\) and \(B\), the measured weights and independently calibrated inertial masses give \(a_A=9.8/1=9.8\ \mathrm{m/s^2}\) and \(a_B=19.6/2=9.8\ \mathrm{m/s^2}\). Compare these spring-calibrated accelerations with the separate release measurements in the table: each object’s downward free-fall acceleration has the same magnitude as its upward spring-only acceleration. This agreement must be checked experimentally; it does not follow from the definition of \(\gamma\).

These equal spring-only accelerations imply equal ratios \(m_{\mathrm{g}}/m_{\mathrm{i}}\) at the same location. Repeating both experiments with other objects tests whether the agreement and the mass proportionality persist. Within experimental precision, they hold for every tested object.

Choose the gravitational-mass unit so that one reference object has the same numerical inertial and passive gravitational mass. The universal proportionality then gives

and \(\gamma(L_1)=9.8\ \mathrm{m/s^2}=g\). The larger gravitational force on \(B\) does not give it a larger acceleration because its inertial mass is larger by the same factor. After establishing this result, either mass symbol alone is sufficient for calculations; keeping them separate allowed the equality to be discovered rather than assumed.

3. What does an accelerometer measure while falling?

The balance experiment used a supported spring to measure weight. Now attach the same kind of spring to a box’s ceiling and hang the test mass from it. This physical box, together with its clocks, is what we mean by a laboratory.

We want to predict the spring’s extension while the box is supported and while it falls.

Take downward as positive for this spring calculation. Let \(x_{\mathrm{mass}}\) and \(x_{\mathrm{ceiling}}\) be the positions of the mass and the spring’s ceiling attachment, measured from the same fixed ground origin. The extension is

where \(\ell_0\) is the spring’s constant natural length. Differentiating twice with respect to time gives

Using the earlier force rules, the mass has downward gravitational force \(m\gamma\) and upward spring force \(kx\), so

Here \(m\) is the test mass, with inertial and gravitational mass equal in our chosen units, and \(\gamma=g\) locally. Assume gravity is approximately uniform across the apparatus and the box is much heavier than the test mass, so the spring’s pull has a negligible effect on the box’s motion.

While the box rests on the ground, the ground holds the box up, and the stretched spring supports the test mass. Mark the spring’s extension in this state. This is the earlier support experiment: \(a_{\mathrm{ceiling}}=0\), and the settled mass has \(d^2x/dt^2=0\). Therefore

Now release the entire box. Its ceiling accelerates downward at approximately \(a_{\mathrm{ceiling}}=\gamma\). Substituting into the same relative-motion equation gives

This is the spring equation already solved in Rediscovering Mechanics and Energy from Questions. Set \(t=0\) at release. The mass initially has zero velocity relative to the box and extension \(x_0=m\gamma/k\), so the earlier solution gives

We want the instrument’s reading after it has settled, so we need to ask what happens as \(t\to\infty\). This solution oscillates forever and has no such limit.

To model settling, include a damping force \(-b\,dx/dt\) that opposes the mass’s motion relative to the box. Here \(b>0\) is a calibrated damping coefficient, in \(\mathrm{kg/s}\). This is an additional force model, whose applicability must be checked experimentally. The equation becomes

For positive \(m\), \(k\), and \(b\), every solution of this damped model decays to zero, so

We could display the extension \(x\) directly. However, different springs and test masses would give different extensions under the same conditions: in the supported experiment, doubling the stiffness halves \(x\), while doubling the test mass doubles it. Using the earlier calibration, we display \(kx/m\) so that these instruments agree once settled. This normalization expresses the result in acceleration units, giving our accelerometer the displayed reading

with a positive reading displayed as upward. This conversion applies whether the box is stationary or moving.

This gives a settled reading of \(\gamma=g\) upward while the box is supported and zero during free fall. In a finite drop, the instrument needs time to settle before that zero reading is a good approximation. We have now predicted both readings; the instrument reports acceleration relative to free fall. We must distinguish two meanings of acceleration:

• Coordinate acceleration is calculated by differentiating position twice with respect to time. For our box, it is \(a_{\mathrm{ceiling}}=d^2x_{\mathrm{ceiling}}/dt^2\) relative to the ground, with downward positive.

• Proper acceleration is reported by an accelerometer attached to the object. For our box, its vertical component is the settled reading \(kx/m\), with upward positive.

The direction conventions belong to this calculation; either type of acceleration can point upward or downward. Both quantities above were calculated using the same ground reference frame. A zero accelerometer reading identifies free fall, while a nonzero reading measures how strongly the box is being pushed away from free fall.

4. Can an accelerating rocket reproduce the falling observations?

Consider a rocket far from gravitating bodies. Its engines make the floor accelerate upward with constant proper acceleration \(a\). Place our laboratory box inside it, initially held at rest above the rocket floor, then release the whole box. The box follows an inertial path while the rocket floor accelerates toward it. Take the rocket floor as the origin, with \(z\) increasing upward and \(t\) measured by a clock at the floor. Over a small enough region and duration after release, the box’s position obeys

Set \(a=g\). This is the same local update measured for our released laboratory box near Earth, with height measured above the ground:

An accelerometer fixed to the rocket floor reads upward acceleration \(g\), just as one supported on the ground does. The released box has zero proper acceleration; relative to the rocket floor, it appears to fall. These local observations do not reveal whether the surrounding room is supported near Earth or accelerating in empty space.

Our spring calculation predicts the same behavior in the two freely moving laboratories. We extend this agreement, as a hypothesis, to other local experiments, such as collisions and light propagation. This is the equivalence principle:

In a sufficiently small freely falling laboratory, these experiments follow the same rules as in a freely moving laboratory far from gravity.

Here the laboratory far from gravity is the released box inside the rocket. “Sufficiently small” matters because Earth’s gravitational acceleration changes with position and direction. A single falling laboratory can remove the common acceleration at one place, but it cannot remove differences across an arbitrarily large region.

5. What does the rocket predict for clocks at different heights?

Place a light source at the rocket floor and a receiver a fixed ruler distance \(h>0\) above it. Both remain attached to the rocket. The source emits \(\nu_{\mathrm{lower}}\) wave cycles per second of its own clock. A crest is a peak of the wave; following successive crests lets us compare the instruments’ clock readings.

How many cycles arrive per second of the upper clock, and what does this tell us about the two clocks?

For this calculation, \(t\) and \(z\) are coordinates of one inertial frame in which the rocket is initially at rest, with \(z\) increasing upward. At emission, \(t=0\), the source is at \(z=0\) and the receiver at \(z=h\). The emitted crest follows \(z_{\mathrm{light}}=ct\).

When does the crest arrive?

Let \(a_{\mathrm{upper}}>0\) be the receiver’s constant proper acceleration, in \(\mathrm{m/s^2}\); \(a\) remains the floor’s. Keeping the rocket’s ruler length fixed requires a different acceleration at each height. We first need the receiver’s trajectory in our inertial frame.

Write \(v(t)=dz_{\mathrm{upper}}/dt\) for its velocity. At any chosen instant, set \(w\) equal to that velocity and choose a second inertial frame moving at the fixed velocity \(w\) relative to the first, with the same upward axis. Place its spatial origin at the receiver at that instant and set the clock there to zero. The receiver is momentarily at rest at this origin; afterward, the origin continues at velocity \(w\) while the receiver accelerates.

Write \(F_w(v(t))\) for the receiver’s velocity in this second frame. Section 10 of From Light Signals to Spacetime gives

Let \(\tau_{\mathrm{upper}}(t)\) be the reading of the clock attached to the upper receiver at inertial time \(t\). Its rate is positive, so a derivative with respect to this reading means \(d/d\tau_{\mathrm{upper}}:=(d\tau_{\mathrm{upper}}/dt)^{-1}d/dt\) on records along the receiver’s path. At the chosen instant, the receiver is at rest in the second frame. Its proper acceleration is therefore

Apply the chain rule with \(w\) held fixed during differentiation:

We need the velocity-conversion derivative and the clock-rate factor. Writing \(v\) for \(v(t)\), differentiate the velocity formula and use the ideal-clock rule from Section 8 of the preceding post:

Substitute these factors, then evaluate at \(w=v(t)\):

Integrate both sides from emission at \(t=0\), when \(v(0)=0\), to a later inertial-frame time \(t\). Use \(s\) for the integration time:

Solve for \(v(t)\), then integrate the velocity over the same time interval, starting at \(z_{\mathrm{upper}}(0)=h\):

For \(a_{\mathrm{upper}}t/c\ll1\), expand the square root to obtain

The notation \(O(q^n)\) denotes a remainder bounded by a constant times \(\lvert q\rvert^n\) as \(q\to0\), with other parameters fixed.

Let \(T\) be the reception time in the original inertial frame. The crest and receiver meet then, so their positions are equal. Using the light path \(z_{\mathrm{light}}(t)=ct\) gives

Substitute the receiver’s approximate trajectory into this equation and solve for \(T\):

Define \(\epsilon:=a_{\mathrm{upper}}h/c^2\ll1\). Of the two quadratic roots, choose the smaller:

The larger root has \(a_{\mathrm{upper}}T/c\approx2\), outside the regime of our trajectory approximation. Expand the smaller root using \(\sqrt{1-2\epsilon}=1-\epsilon-\tfrac12\epsilon^2+O(\epsilon^3)\):

Substitute this time into the velocity formula to find the arrival speed:

This also checks the condition \(a_{\mathrm{upper}}T/c\ll1\) used in the trajectory expansion.

How far apart are nearby arrivals?

Let \(t_{\mathrm e}\) be a crest’s emission time and \(t_{\mathrm r}(t_{\mathrm e})\) its reception time, both in the same inertial frame. For the crest considered above, \(t_{\mathrm e}=0\) and \(t_{\mathrm r}(0)=T\). Write \(z_{\mathrm{lower}}(t)\) for the source’s trajectory. Each crest travels at speed \(c\), so

Differentiate with respect to \(t_{\mathrm e}\), using the chain rule for \(t_{\mathrm r}(t_{\mathrm e})\):

At \(t_{\mathrm e}=0\), the source’s velocity is zero and the receiver’s velocity at \(t_{\mathrm r}(0)=T\) is \(v(T)\). Substituting these velocities gives

This derivative is the limiting ratio of nearby emission and reception intervals.

What do the instruments’ own clocks record?

Let \(N\) count wave cycles, including fractions of a cycle. The instantaneous frequency recorded by an instrument is \(\nu=dN/d\tau\), where \(\tau\) is its own clock reading. A measurement over a short interval approximates this rate.

Here \(\tau_{\mathrm{lower}}\) is the reading of the clock attached to the source at the rocket floor, and \(\tau_{\mathrm{upper}}\) is the reading of the clock attached to the receiver above it.

At emission \(t_{\mathrm e}=0\) the source is at rest; at reception \(t_{\mathrm r}=T\) the receiver has velocity \(v(T)\). The ideal-clock rule gives

For each reception reading \(\tau_{\mathrm{upper}}\), let \(\tau_{\mathrm{lower}}(\tau_{\mathrm{upper}})\) be the source-clock reading at the corresponding emission, just as we paired \(t_{\mathrm e}\) and \(t_{\mathrm r}\). Write \(N_{\mathrm{lower}}\) and \(N_{\mathrm{upper}}\) for the cycle counts as functions of the respective clock readings, with matching labels for the same cycles. Then

Differentiate with respect to \(\tau_{\mathrm{upper}}\) using the chain rule:

The source frequency is evaluated at the corresponding emission. Divide by it, then express the clock-reading derivative through the inertial emission and reception times. For the crest emitted at \(t_{\mathrm e}=0\) and received at \(t_{\mathrm r}=T\), substituting the clock rates and interval ratio gives

The square-root factor differs from one only at order \(\epsilon^2\). To first order, the reduction comes from the receiver’s motion during the flight.

Transfer this prediction to a small supported laboratory in a steady gravitational field using the equivalence principle. Set \(a_{\mathrm{upper}}=g\), with \(g\) measured at the receiver, and require negligible field variation across height \(h\). Our earlier definition \(\epsilon=a_{\mathrm{upper}}h/c^2\) then gives \(\epsilon=gh/c^2\). Keeping terms through first order gives

The clock-interval ratio is the reciprocal of the frequency ratio, so

This answers the clock question to first order in \(gh/c^2\): the upper clock records more elapsed time between receptions than the lower clock records between the corresponding emissions. The frequency reduction is called gravitational redshift. It vanishes when \(g=0\) or \(h=0\).

6. What does the rocket predict for light paths?

Now emit a horizontal pulse across width \(L>0\), toward a mark on the opposite wall initially level with the source. Both start at floor height and have proper acceleration \(a\).

How far below the mark does the pulse arrive?

Again use an inertial frame in which the rocket is at rest at emission. Put the source at \((x,z)=(0,0)\) and the mark at \((L,0)\). The light follows \(x_{\mathrm{light}}=ct\) and \(z_{\mathrm{light}}=0\).

The wall remains at \(x=L\), so the pulse arrives at \(T=L/c\). Substituting this time into Section 5’s trajectory, with initial height zero and acceleration \(a\), gives

Thus \(aL/c^2\ll1\) makes the leading gap \(\Delta z\approx aL^2/(2c^2)\) reliable. The mark’s arrival speed is approximately \(aL/c\), so its Lorentz factor differs from one only at order \((aL/c^2)^2\). Converting the gap to a wall-ruler distance therefore preserves the leading result.

Repeating the calculation at intermediate horizontal positions gives the path recorded with the rocket’s rulers, to this accuracy:

Setting \(a=g\) predicts the leading deflection in a sufficiently small supported laboratory near Earth. This answers the horizontal-pulse question. Both rocket predictions were obtained in flat spacetime; a bent path in accelerating coordinates does not establish curvature.

What observation distinguishes Earth’s gravitational field from the effects of accelerating a laboratory in flat spacetime?

7. Why can gravity be removed only locally?

Earlier we treated gravity as giving every released object the same downward acceleration \(g\) throughout a small laboratory. Under that approximation, two objects released from rest keep the same distance apart.

Return to Earth and release two objects from rest side by side. Each accelerates approximately toward Earth’s center, so their paths converge and the distance between them decreases. Now release one object above the other. The lower object is closer to Earth’s center and accelerates downward more strongly, so the distance between them increases. These are the small effects we neglected when we used the same downward acceleration throughout the laboratory.

The earlier result about mass and material still applies: objects given the same position and initial velocity follow the same trajectory. Here the two objects start at different positions. A freely falling observer can measure the changing distance between them using light signals.

To calculate the leading effect, use the ground laboratory’s time \(t\) and approximately Cartesian positions \(\mathbf x_1(t)\) and \(\mathbf x_2(t)\). Let \(\mathbf g(\mathbf x)\) be the smooth, steady acceleration field measured by release experiments. The separation \(\boldsymbol\xi:=\mathbf x_2-\mathbf x_1\) obeys

The second line is the first-order Taylor expansion for a small separation. Spatial indices \(i,j\) run from \(1\) to \(3\); a repeated index is summed. The derivative measures how one acceleration component changes with one position coordinate while the other coordinates remain fixed.

These relative accelerations are called tidal effects. Subtracting the same laboratory acceleration from both objects leaves their difference unchanged. In the rocket, freely released objects have constant relative velocities in an inertial frame. Near Earth, the measured tidal acceleration remains even in a freely falling laboratory.

8. What geometry predicts clocks and free paths together?

The small-laboratory clock and light questions are answered. We now seek a description that can also accommodate the tidal measurements across a larger region.

How can we extend the special-relativistic interval while preserving the clock and light rules in each freely falling laboratory?

Return to the supported laboratory near Earth from Section 5. Fix a light source and its clock to the floor, and hold a receiver and its clock at height \(h\) above it. Send light upward through the evacuated space between them. Assume the apparatus stays in place and Earth’s gravitational field does not change during the comparison.

We now use coordinates fixed to this ground laboratory, replacing the inertial coordinates used for the rocket calculation. Set \(z=0\) at the floor and \(z=h\) at the receiver, with \(z\) increasing upward. Choose the coordinate time \(t\) in seconds to agree with the floor clock there. Each signal repeats the same journey under unchanged conditions, so it has the same coordinate travel time \(D\). If \(t_{\mathrm e}\) and \(t_{\mathrm r}\) now denote emission and reception times in these ground coordinates, then

Thus signals emitted a coordinate-time interval apart arrive that same coordinate-time interval apart. The clocks’ own elapsed times can differ. Write the rate of a clock held at height \(z\) as \(d\tau/dt=A(z)\), where \(A>0\) is dimensionless and \(A(0)=1\) because \(t\) agrees with the floor clock. Using the equal coordinate-time intervals, Section 5’s comparison of the clocks gives

Repeating this comparison at other small heights gives \(A(z)\approx1+gz/c^2\). To express the clock-rate rule using one-forms, we use the pullback construction from Section 8 of From Light Signals to Spacetime.

Let \(\zeta:I\to M\) describe one clock held at a fixed height \(z_0\) in the spacetime region \(M\). Use \(\lambda\in I\subset\mathbb R\) for its parameter, chosen to equal the ground coordinate time in seconds, so \(t(\zeta(\lambda))=\lambda\) and \(z(\zeta(\lambda))=z_0\). Let \(\tau:I\to\mathbb R\) give that clock’s reading in seconds. The measured rate is then \(d\tau/d\lambda=A(z_0)\).

At \(p=\zeta(\lambda)\), the coordinate one-form \(dt_p\) acts on \(T_pM\), while \(d\tau_\lambda\) acts on \(T_\lambda I\). Pulling \(dt\) back along \(\zeta\) puts it on the clock’s parameter interval:

The differential of the clock-reading function therefore satisfies

At each \(\lambda\), both sides are covectors in \(T_\lambda^*I\). Following the preceding post’s ideal-clock rule, we require the line element along this path to satisfy

This is a bilinear form on \(I\) and determines the time coefficient that a spacetime metric must reproduce along the stationary path. Describing moving clocks and light also requires spatial and mixed coefficients. We adopt the same bilinear structure used in special relativity, allowing its coefficients to depend smoothly on the event.

At an event \(p\in M\), the metric is a symmetric bilinear map

As in the preceding post, \(T_pM\) contains tangents to paths through \(p\). Require the metric to be nondegenerate, so its matrix is invertible, with signature \((-,+,+,+)\): in a suitable basis it has one negative and three positive squared directions. This preserves the local distinction between clock, light, and spatial directions.

Choose coordinates \(x^\mu=(ct,x,y,z)\), all in length units. Greek indices run from \(0\) to \(3\) and repeated indices are summed. Write \(\partial_\mu:=\partial/\partial x^\mu\) for the coordinate tangent basis and \(dx^\mu\) for its dual covectors. They satisfy \(dx^\mu(\partial_\nu)=\delta^\mu{}_{\nu}\), where \(\delta^\mu{}_{\nu}\) is one for equal indices and zero otherwise. The metric components and their assembled map are

Here \(g_{\mu\nu}\) are dimensionless metric components; the scalar \(g\) in the earlier drop rule still denotes acceleration. The line element along a path is \(ds^2:=\zeta^*g\). For our stationary clock, \(\zeta^*(dx^0)=c\,d\lambda\) and the three spatial coordinate one-forms pull back to zero. Thus \(\zeta^*g=g_{00}(\zeta(\lambda))c^2\,d\lambda\otimes d\lambda\). Comparing with the clock result fixes \(g_{00}=-A(z)^2\) at each height. The components depend on coordinates, so their variation alone does not diagnose curvature.

What does the metric predict along a recorded path?

Now allow the clock to move. Keep \(\lambda\) equal to laboratory time along its path \(\zeta:I\to M\). The differential of the path, also called its pushforward, is the linear map

It sends the parameter’s unit tangent to the clock’s spacetime tangent:

Each coordinate one-form \(dx^\mu_{\zeta(\lambda)}:T_{\zeta(\lambda)}M\to\mathbb R\) then takes this spacetime tangent to a real number. These numbers are the components of \(V\), and the chain rule identifies them with the coordinate rates:

Write \(g(V,V)\) for the scalar function \(\lambda\mapsto g_{\zeta(\lambda)}(V(\lambda),V(\lambda))\). Evaluating the metric yields

We extend the ideal-clock rule from special relativity by requiring, for a clock path with \(g(V,V)<0\),

For the stationary clock, \(V^\mu=(c,0,0,0)\), recovering \(d\tau/d\lambda=\sqrt{-g_{00}}=A(z_0)\). Integrating the rate predicts the reading along any specified clock path.

For light, we retain the local special-relativistic condition: a nonzero tangent \(K\in T_pM\) must satisfy \(g_p(K,K)=0\). Such a direction is called null. These clock and light rules are physical assumptions of the metric model.

For a smooth metric of this signature, coordinates can be chosen so that \(g_{\mu\nu}=\operatorname{diag}(-1,1,1,1)\) and its first coordinate derivatives vanish at a chosen event. Identifying these with the coordinates of a local freely falling laboratory implements the equivalence principle. Light then has speed \(c\) measured with that observer’s clocks and rulers, although its speed calculated from coordinates covering an extended region can differ from \(c\).

9. How does the metric predict free fall?

Knowing the metric now lets us calculate the time along a given path and identify possible light directions. We still need a rule that selects the path of a released object.

How do we express continued inertial motion when the tangent spaces at successive events are different?

Keep the path \(\zeta:I\to M\) and its clock-reading function \(\tau:I\to\mathbb R\) from Section 8. The tangent \(V\) measures change in the clock’s spacetime position per second of laboratory time. To express that change per second recorded by the moving clock, we first choose the corresponding tangent on \(I\). The clock’s rate \(\tau'(\lambda):=d\tau/d\lambda\) is positive, so \(\tau\) is a smooth coordinate on \(I\). Write \(\partial_\tau:=\partial/\partial\tau\) for its coordinate tangent field, characterized by \(d\tau_\lambda(\left.\partial_\tau\right\rvert_\lambda)=1\). Since \(d\tau=\tau'\,d\lambda\), this gives

The differential of the path maps this tangent into spacetime:

For any smooth scalar record \(q:I\to\mathbb R\), its rate per moving-clock second is the action of this coordinate tangent on \(q\):

In particular, let \(q^\mu:=x^\mu\circ\zeta\) be the four coordinate records. The chain rule relates these rates to the components of \(U\):

The clock rule from Section 8 gives the normalization

How do we differentiate a vector along the path?

We can differentiate each scalar component \(U^\mu\), but tangents at different events belong to different tangent spaces. The metric compares two vectors at the same event; a derivative also needs a rule for comparing vectors along a path. To specify such a rule, we first describe how it acts on vector fields in the surrounding region.

A smooth vector field assigns a tangent vector to each event. Use the coordinates and dual bases from Section 8 to write two such fields:

We want to differentiate \(Y\) along \(X\). First consider each component \(Y^\nu\), which is a smooth real-valued function within the coordinate chart. Its differential is the one-form field

To find the component’s rate of change along \(X\), apply this one-form to \(X\). At an event \(p\), the dual-basis identity \(dx^\mu_p(\left.\partial_\alpha\right\rvert_p)=\delta^\mu{}_\alpha\) gives

The input is a tangent vector and the output is a real number. Doing this at each event defines the scalar function \(d(Y^\nu)(X)=X^\mu\partial_\mu Y^\nu\). Assemble these component rates using the coordinate basis at that event:

This expression differentiates only the component functions. A derivative of \(Y=Y^\nu\partial_\nu\) must also specify how to differentiate the basis vectors. Different coordinate choices change the division between these two contributions.

For any smooth scalar function \(f\), the notation \(df(X)\) has the same meaning: the scalar function whose value at \(p\) is \(df_p(X(p))\). Multiplication of a vector field by a scalar function is defined at each point by \((fX)(p):=f(p)X(p)\).

For vector fields \(X\) and \(Y\), write \(\nabla_XY\) for the covariant derivative of \(Y\) in direction \(X\). Its value at an event \(p\) must be a vector in \(T_pM\), independent of the coordinates used to calculate it. We require this derivative to be additive in both inputs, linear over smooth functions in its direction input, and to obey the product rule in the field being differentiated. For smooth real-valued functions \(f,h\) and another vector field \(Z\), the last two requirements mean

A rule \(\nabla\) with these properties is a connection on the tangent bundle, the collection of tangent spaces over \(M\). These are requirements defining the kind of derivative we seek; they do not yet choose one connection.

Once a connection is chosen, each \(\nabla_{\partial_\mu}\partial_\nu\) is a vector field and therefore has an expansion in the coordinate basis. Define its coefficient functions \(\Gamma^\rho{}_{\mu\nu}\) by

Here \(\mu\) specifies the direction of differentiation, \(\nu\) specifies the basis vector being differentiated, and \(\rho\) specifies the output component. These functions are the connection coefficients. Linearity in the direction input now gives the basis derivative along \(X\):

Apply the product rule to \(Y=Y^\nu\partial_\nu\), then substitute this basis derivative:

The first term is exactly the component rate we calculated above; the second is the contribution from differentiating the basis. Relabel the summed index \(\nu\) as \(\rho\) in the first term and use \(d(Y^\rho)(X)=X^\mu\partial_\mu Y^\rho\) to combine them:

The coefficients \(\Gamma^\rho{}_{\mu\nu}\) themselves are not tensor components: a coordinate change also differentiates the functions relating the two bases. Their transformation makes the complete derivative independent of the coordinate choice.

Return to our clock path. Let \(W(\lambda)=W^\mu(\lambda)\left.\partial_\mu\right\rvert_{\zeta(\lambda)}\) be a smooth assignment of vectors along it. Write \(DW/D\lambda\) for the path derivative induced by the connection in the parameter direction \(\partial/\partial\lambda\). The notation \(\partial_\beta\circ\zeta\) restricts a coordinate basis field to this path. Since its tangent has components \(V^\alpha\), the induced derivative of the basis is

The component functions \(W^\beta\) live on \(I\), where their differentials satisfy \(d(W^\beta)(\partial/\partial\lambda)=dW^\beta/d\lambda\). Applying the same product rule to \(W=W^\beta(\partial_\beta\circ\zeta)\) gives

The second path derivative uses \(\partial_\tau=(\tau')^{-1}\partial/\partial\lambda\) on \(I\). Setting \(W=U\) gives the clock’s acceleration vector \(DU/D\tau\). A vector assignment with zero path derivative is called parallel along the path; solving that equation from an initial vector defines parallel transport.

Which connection does the metric select?

We have defined the derivative once its coefficients are supplied. To calculate those coefficients from the metric, impose two further conditions.

First, require the metric comparison of two parallel vectors to remain constant along a path. For vector fields \(Y,Z\), the notation \(g(Y,Z)\) denotes the scalar function \(p\mapsto g_p(Y(p),Z(p))\). Its differential \(d\bigl(g(Y,Z)\bigr)\) is a one-form field; applying it to \(X\) gives the rate of change of the pairing along \(X\). Metric preservation requires that this derivative account for the change of each vector:

This condition is called metric compatibility. Substitute \(X=\partial_\alpha\), \(Y=\partial_\beta\), and \(Z=\partial_\nu\). Using \(g(\partial_\beta,\partial_\nu)=g_{\beta\nu}\) and the definition of the connection coefficients gives

Second, require \(\nabla_{\partial_\alpha}\partial_\beta=\nabla_{\partial_\beta}\partial_\alpha\) in a coordinate basis, or equivalently \(\Gamma^\rho{}_{\alpha\beta}=\Gamma^\rho{}_{\beta\alpha}\). This is the zero-torsion condition in coordinates. It is an additional choice, not a consequence of metric compatibility. Standard general relativity uses both conditions.

To solve for the coefficients, write metric compatibility with three arrangements of the indices:

Add the first two equations and subtract the third. Symmetry of \(g\) and of the lower connection indices cancels the terms containing \(\Gamma^\rho{}_{\alpha\nu}\) and \(\Gamma^\rho{}_{\beta\nu}\), leaving

Let \(g^{\mu\nu}\) be the inverse matrix, defined by \(g^{\mu\rho}g_{\rho\nu}=\delta^\mu{}_{\nu}\). Multiplying by \(\tfrac12g^{\mu\nu}\) gives

Substitution verifies that these coefficients satisfy both conditions; the calculation shows that any connection satisfying them must have this formula. The resulting unique connection is the Levi-Civita connection of \(g\). Its coordinate coefficients are called Christoffel symbols. They have units of inverse length because the metric components are dimensionless and the coordinates have length units. If those components are constant in a coordinate chart, the formula gives \(\Gamma^\mu{}_{\alpha\beta}=0\) there, and the covariant derivative reduces to differentiating vector components in that basis.

Which paths represent free motion?

We postulate that an ideal test object with no nongravitational forces has \(DU/D\tau=0\). To express this condition in coordinates, first set \(W=U\) in the path derivative defined above, then substitute \(U^\mu=\partial_\tau q^\mu\):

The coordinate basis vectors at \(\zeta(\lambda)\) are linearly independent. Therefore \(DU/D\tau=0\) holds exactly when each coefficient vanishes, giving one equation for each \(\mu=0,1,2,3\):

These paths are called geodesics. In the freely falling coordinates of Section 8, the first metric derivatives and hence \(\Gamma\) vanish at the chosen event. The equation reduces there to the inertial rule \(\partial_\tau(\partial_\tau q^\mu)=0\). For a supported clock, \(DU/D\tau\) instead gives its nonzero proper-acceleration vector; its spatial components in the clock’s rest frame are accelerometer readings.

Light follows null geodesics in the geometric-optics approximation, where its wavelength is small compared with the scales on which the geometry varies. A null ray has no increasing proper-time reading to use as a coordinate on its parameter interval. Give its path a separate name \(\chi:J_{\mathrm{light}}\to M\), where \(J_{\mathrm{light}}\subset\mathbb R\) is an interval with a dimensionless, increasing parameter \(r\). Its coordinate records are \(q_{\mathrm{light}}^\mu:=x^\mu\circ\chi\). Choose \(r\) so that the covariant derivative of its tangent along the ray vanishes:

A parameter for which the first equation has this form is called affine; a constant positive rescaling and shift preserve it. It need not equal laboratory coordinate time. The first equation continues the light direction along the ray, while the second imposes the local light-speed condition.

10. How does the familiar gravitational field appear?

Return to the weak, steady field measured in the ground laboratory. We want to check whether the geodesic rule reproduces the drop acceleration as well as the clock comparison. Use approximately Cartesian spatial coordinates, with the floor at \(\mathbf x=0\) and \(t\) calibrated to its clock.

Assume the measured field \(\mathbf g(\mathbf x)\) has path-independent work per unit mass. Work here in the approximately Euclidean space \(\Omega\) of ground-fixed positions. Choose any smooth spatial path \(\boldsymbol\ell:[0,1]\to\Omega\) from the floor to \(\mathbf x\). As in the conservative-energy construction, define the potential from release measurements:

Here \(u\) is a dimensionless path parameter, \(\ell^j\) are the path’s Cartesian components, and \(\delta_{ij}\) supplies their Euclidean scalar product. In this spatial calculation, \(x^i\) and \(dx^i\) denote coordinates and one-forms on \(\Omega\). The potential has units \(\mathrm{m^2/s^2}\). Path independence gives \(d\Phi=-\delta_{ij}g^i\,dx^j\) on \(\Omega\), or \(g^i=-\delta^{ij}\partial_j\Phi\) in components. Near the floor, \(\mathbf g=(0,0,-g)\), so \(\Phi(z)=gz\).

Allow the stationary clock factor \(A:\Omega\to(0,\infty)\) to depend on all three spatial coordinates. For a tilted light path, Section 5’s arrival-spacing calculation uses the velocity component along the ray. Applying that comparison between nearby supported clocks gives, to first order,

All three expressions are one-forms on the same spatial domain \(\Omega\); \(A^{-1}dA\) means multiplication by the scalar \(1/A\). The chain rule gives \(d(\ln A)=A^{-1}dA\). In the spacetime metric, \(A\) and \(\Phi\) mean these spatial functions composed with the projection \((t,\mathbf x)\mapsto\mathbf x\). Integrating along a spatial path from the floor, where \(A=1\), and expanding for \(\lvert\Phi\rvert/c^2\ll1\) gives

We have fixed this coefficient using the clock result. Now assume a static metric with \(g_{0i}=0\) and spatial components close to \(\delta_{ij}\). Its time derivatives vanish, so Section 9’s connection formula gives, to leading order,

Keep the falling object’s path parameter \(\lambda\) equal to ground coordinate time, so its coordinate records satisfy \(q^0(\lambda)=c\lambda\). For slow motion in the weak field, the clock rule gives \(\tau'\approx1\). The exact conversion for each spatial record is

Multiplying the spatial geodesic equation by \((\tau')^2\) therefore gives

All coefficients here are evaluated at \(\zeta(\lambda)\). For the static metric, \(\Gamma^i{}_{0j}=0\). The terms with two spatial velocities and the clock-rate correction are higher order in speed divided by \(c\): the time component of the geodesic equation gives \(\tau''/\tau'=2\Gamma^0{}_{0j}\,dq^j/d\lambda\). Keeping the two time inputs, with \(dq^0/d\lambda=c\), yields

Since \(q^3=z\circ\zeta\) and \(\lambda\) is ground time, the vertical equation at the floor is precisely the original drop rule \(d^2z(t)/dt^2\approx-g\) for the recorded height. Agreement with the clock formula used to construct \(g_{00}\) is built in; recovering the drop rule also uses the geodesic postulate. Spatial metric corrections are unnecessary at this order for slow objects, but they matter for extended light-path predictions.

11. What remains after the free-fall rule is known?

The metric and geodesic rule now predict clocks and motion for a specified geometry. We still need to express the tidal measurement in that geometry and determine which metric surrounds a given source.

How do the components of one direction change along another?

Section 7 compared neighboring drops. We now want to calculate their relative acceleration using the connection. This will require differentiating both along a clock’s path and across the family of nearby paths. To understand how the two directions enter the calculation, first examine the component changes of one vector field along another.

For smooth vector fields \(X=X^\mu\partial_\mu\) and \(Y=Y^\nu\partial_\nu\), Section 9 derived the one-form \(d(Y^\nu)\) and its scalar evaluation \(d(Y^\nu)(X)\). Reuse the resulting component rates:

This is the first term in our calculation of \(\nabla_XY\), before adding \(Y^\nu\nabla_X\partial_\nu\). By itself it depends on the coordinate choice.

Exchanging the roles of the two fields gives the component changes of \(X\) along \(Y\):

Their difference defines the Lie bracket in these coordinates:

This construction takes two vector fields and produces a vector field in the chosen chart. We still need to check that their difference gives the same vector field when we change coordinates, even though either term separately need not.

To check this, apply the differential of an arbitrary smooth scalar function \(f\) to the result. Since \(df(\partial_\nu)=\partial_\nu f\), we obtain

We can recover this expression by differentiating \(f\) in two orders. The first rates are \(df(Y)=Y^\nu\partial_\nu f\) and \(df(X)=X^\mu\partial_\mu f\). Differentiate each rate along the other field and use the product rule:

Subtract the second equation from the first. Relabel the summed index \(\mu\) as \(\nu\) in its first term; the second derivatives cancel because coordinate partial derivatives commute. The result is

The left side is defined using differentials acting on vector fields, independently of any coordinates. A vector is determined by these evaluations: taking the coordinate functions as \(f\) recovers each of its components. Since the identity holds for every smooth \(f\), it determines \([X,Y]\) independently of the chosen chart. The bracket therefore measures the difference between the two orders of directional differentiation, without a metric or connection. It depends on derivatives of the input fields as well as their values at the event. For coordinate basis fields, the components are constant, so \([\partial_\mu,\partial_\nu]=0\).

Which part of this difference measures curvature?

We can now compare the bracket with the covariant derivatives using the same component rates. Subtract the two versions of Section 9’s formula and exchange the summed indices in the second connection term:

The last equality uses the symmetry of the lower indices of the Levi-Civita coefficients. Thus each covariant derivative contains a basis contribution, but those contributions cancel in their difference.

Now compare the two orders of covariant differentiation of a vector field \(Z\). Expanding \(X=X^\mu\partial_\mu\) and \(Y=Y^\nu\partial_\nu\) with the connection’s product rule gives

Subtract the expression with \(X\) and \(Y\) exchanged. The terms differentiating their components give \(\nabla_{[X,Y]}Z\) by the bracket formula just derived:

To isolate the remaining difference, define the connection’s curvature by

Here \(X,Y\) specify the two directions of differentiation and \(Z\) is the vector field being differentiated. This equation fixes our sign convention.

To calculate the result, use the coordinate formula from Section 9 twice. For \(Z=Z^\rho\partial_\rho\), it gives

Differentiating this field along \(\partial_\mu\) gives

Subtract the expression with \(\mu\) and \(\nu\) exchanged. The second derivatives of \(Z^\rho\) cancel, and the two terms containing first derivatives of \(Z\) cancel against their counterparts. Since the coordinate directions have zero bracket, the remaining terms give

Setting \(Z=\partial_\sigma\) identifies these as the curvature components: \(\mathcal R(\partial_\mu,\partial_\nu)\partial_\sigma=R^\rho{}_{\sigma\mu\nu}\partial_\rho\). The indices \(\mu,\nu\) label the two directions, \(\sigma\) labels the vector being differentiated, and \(\rho\) labels the output component. Substituting into the expression for general directions gives

Only the values of \(X,Y,Z\) at each event remain. The result is linear in each of those three vectors, so it defines a tensor at \(p\):

For the Levi-Civita connection this is the Riemann curvature tensor of the metric. Its components have units of inverse length squared. A flat metric in inertial coordinates has \(\Gamma=0\) throughout the chart, giving \(\mathcal R=0\). Setting \(\Gamma=0\) only at one event leaves its derivatives in the curvature formula.

How does curvature predict the separation of free paths?

To compare neighboring drops, label the clocks by a dimensionless number \(b\) in an interval \(B\) containing zero, with \(b=0\) for the reference clock. Set their initial zero readings at simultaneous events in the reference clock’s local rest frame. Define a smooth map \(F:J\times B\to M\), where \(J\) is a common interval of clock readings in seconds: \(F(s,b)\) is the event where clock \(b\) reads \(s\). Here \(s\) is the clock reading itself; this family is specified directly by those readings rather than by laboratory time.

The differential of \(F\) gives two vectors at that same event:

The first follows a clock, while the second varies which clock we follow at a fixed reading. It is a tangent vector, not a subtraction of spacetime points. Let \(D_s\) and \(D_b\) denote the connection’s derivatives along these two parameter directions, defined as for a single path in Section 9. The parameter directions commute, so zero torsion gives \(D_s\xi=D_bU\). The curvature definition gives \((D_sD_b-D_bD_s)U=\mathcal R(U,\xi)U\). Since each clock falls freely, \(D_sU=0\), and therefore

This is the geodesic-deviation rule: curvature predicts the relative acceleration of neighboring free paths. For a small label difference \(\Delta b\), the spatial components of \(\Delta b\,\xi(s,0)\) in the reference clock’s rest frame give their separation to first order. Unlike Section 7’s Euclidean difference \(\mathbf x_2-\mathbf x_1\), \(\xi\) is a vector at the reference event; it provides the infinitesimal separation needed in curved spacetime.

A coordinate change can set the connection coefficients to zero at one event. It cannot make a nonzero curvature tensor vanish there. The accelerating rocket has zero spacetime curvature; Earth’s tidal measurements require a nonzero value.

What selects the metric around Earth?

How do the source’s measured energy, momentum, and stress determine the geometry?

Organize these source measurements in a symmetric bilinear map \(T_p:T_pM\times T_pM\to\mathbb R\), called the stress-energy tensor. At each event choose the unit time and spatial axes of a freely falling laboratory. Hats label components in this orthonormal basis, where the metric is \(\operatorname{diag}(-1,1,1,1)\):

All entries have energy-density units and can be calibrated through transfers to local detectors. In any coordinate basis, \(T_{\mu\nu}=T(\partial_\mu,\partial_\nu)\); upper indices mean \(T^{\mu\nu}=g^{\mu\alpha}g^{\nu\beta}T_{\alpha\beta}\). The same raising rule applies to the hatted components using the inverse orthonormal metric.

To form a two-index equation with this source, contract the curvature components just calculated. Summing one output index with one input gives the Ricci tensor, and a further contraction gives the scalar curvature:

The source’s local conservation rule involves a derivative of its tensor. Extend the connection to tensor inputs using the product rule, then contract the derivative index with the first tensor index using the inverse metric:

Here \(\nabla_\alpha\) means \(\nabla_{\partial_\alpha}\). Every factor is evaluated at the same event. The resulting components define a covector field, the covariant divergence of \(T\).

We therefore seek a curvature expression with zero covariant divergence. The combination \(\mathcal G_{\mu\nu}:=R_{\mu\nu}-\tfrac12Rg_{\mu\nu}\), called the Einstein tensor, satisfies the geometric identity \(\nabla^\mu\mathcal G_{\mu\nu}=0\). It can match a locally conserved source.

The additional physical postulate is the Einstein field equation:

Newton’s constant \(G_{\mathrm N}>0\) has units \(\mathrm{m^3/(kg\,s^2)}\) and is calibrated by gravitational measurements with known source masses and distances. The coefficient matches the Newtonian source law in the weak, slow limit. Both sides then have units of inverse length squared. The geometric identity requires \(\nabla^\mu T_{\mu\nu}=0\), the local energy-momentum conservation equation.

This source equation does not follow from the equivalence principle or the clock experiment. Its predictions require experimental tests. We have neglected a possible cosmological term \(\Lambda g_{\mu\nu}\) on the left, where \(\Lambda\) has units of inverse length squared; its effects are negligible for the laboratory problems here. Solving for a metric also requires appropriate initial or boundary data. Even where \(T_{\mu\nu}=0\), the full curvature can be nonzero, as outside Earth or in a gravitational wave.

12. How are special and general relativity related?

Special relativity supplies the local clock, light, and inertial-motion rules. General relativity uses a metric to relate those measurements across a region and a field equation to predict that metric from sources and initial or boundary data. Where curvature is nonzero, one inertial coordinate system cannot describe the entire region.

The main question is answered within this model: one metric determines ideal-clock readings, and its connection determines the free paths of test objects and light. Curvature describes their tidal separation. Together with the source equation, these rules form general relativity. The rocket calculations explain the leading clock shift and light deflection; the weak, steady, slow-motion limit recovers the measured drop acceleration.

Take two long railcars moving along parallel tracks. Each railcar carries clocks fixed at several positions. We call each railcar together with its clocks a laboratory. Laboratory \(S\) calls its positions \(x\) and its clock readings \(t\). Laboratory \(S'\) uses \(x'\) and \(t'\).

Each observer records events locally, where they are, and may choose that location as the origin of their own coordinates. We later compare these records to determine how the observers’ descriptions relate. In this experiment, the clocks along each railcar share one coordinate system, with its origin at the railcar’s center; they occupy different positions within that system.

Place ourselves at the center of \(S\), where \(x=0\). We directly observe events here; observers or instruments elsewhere make their own local records. Equip each clock with a light detector and memory, forming a recording station. Each station stays at a fixed position \(x\) on \(S\) or \(x'\) on \(S'\).

When a station on \(S'\) passes a station on \(S\), a light pulse at their encounter triggers each station to record its own clock reading and position label. The timestamp records the pulse’s local arrival at the detector. We idealize their detectors as colocated at a single event \(P\) shared by both stations; the clock housings do not need to touch. The station on \(S\) stores \((t_P,x_P)\), and the station on \(S'\) stores \((t'_P,x'_P)\). These pairs are the event’s coordinates in the two laboratories.

We compare only events within the region covered by both recording networks. To record events beyond that region using the same procedure, we must extend the networks, calibrate the added stations, and synchronize their clocks.

The synchronized network is one way to assign coordinates to events. Alternatively, a single station at the origin could send a light pulse, have it reflected by a distant object, and record its return using the same clock. Assuming the vacuum light-speed rule introduced in Section 1, the object’s radar distance at the reflection event is

where \(t_{\mathrm{send}}\) and \(t_{\mathrm{return}}\) are the emission and return readings of that one clock. No distant clock needs to be synchronized. We use distributed stations here because they let us compare records made locally at the same event. See this radar construction for the alternative procedure.

Both the synchronized-network procedure and the radar alternative rely on a physical assumption: every inertial laboratory obtains the same constant round-trip vacuum light speed \(c\), in every direction throughout the modeled region. We keep this model within that region. Extending it to the whole universe would assume that flat spacetime applies everywhere; the general relativity post will explain why gravity can require a different description. The results we derive in this post depend on this light-speed assumption.

This model uses pointlike detectors with their response delays corrected. Real instruments require calibration, and any remaining timing error contributes measurement uncertainty.

We collect the records afterward. A stored timestamp describes the local event, not when its message reaches us at the origin. Transmission delays change when we obtain the records, not the timestamps already stored in them. The records can be sent to us; the clocks themselves need not be brought together.

Follow the center station of \(S'\) as it passes successive stations on \(S\). Each station on \(S\) stores its local \(t\) and \(x\) at the encounter. From these collected records, reconstruct \(x(t)\), the position of the center of \(S'\) at time \(t\) according to \(S\). The measurements fit

This is uniform relative motion. In \(S\) coordinates, \(S\) has velocity zero and \(S'\) has the measured velocity \(v\). The value of \(v\) is an input to the experiment, not a universal constant.

We ask:

What rule relates the pairs \((t,x)\) and \((t',x')\) recorded locally at the same event?

1. How does one laboratory synchronize its clocks?

The clocks on one railcar need a rule for assigning common time labels. Its center clock emits one light pulse when it reads \(t_{\mathrm{emit}}\). A clock at position \(x\) reflects the pulse immediately, and the pulse returns to the center clock when it reads \(t_{\mathrm{return}}\). Define

The first line synchronizes the remote clock: the remote clock is adjusted to read \(t_{\mathrm{reflect}}\) when it reflects the pulse. In the second, \(c\) is the measured round-trip speed of light.

We assume that each inertial laboratory obtains the same constant round-trip speed \(c\) for light in vacuum, using its own clocks and rulers, regardless of the apparatus’s orientation or the source’s motion. This is a physical input to the model, not something the following derivation proves. The synchronization convention then assigns that same speed to one-way propagation; it cannot force different laboratories to obtain the same round-trip result.

Establishing how accurately nature obeys this assumption requires actual experiments, with instrument uncertainties and environmental effects accounted for. Here we derive its consequences. The railcar thought experiment does not establish its validity everywhere in the universe, and the gravity sequel will retain the corresponding local principle rather than prove it without assumptions.

2. Which laboratories provide the simple comparison?

Release several small test carts inside one laboratory after shielding them as well as possible from springs, air resistance, electric fields, and other detectable interactions. Record each cart’s passages at the laboratory’s stations and reconstruct its position function \(x(t)\) from those local records. The observations may fit

where the constant

is the test cart’s measured velocity relative to the laboratory. A position function \(x(t)\) of this form is called a uniform trajectory.

A synchronized clock-and-position system in which sufficiently isolated carts obey this constant-velocity rule is called an inertial reference system. “Inertial” is not another word for stationary. Two such systems may move uniformly relative to one another.

This definition is operational but approximate. A laboratory is treated as inertial only over the region and duration in which free test carts show no detectable departure from the rule.

For the experiment in the opening, require both clock-and-position systems to be inertial. Choose the encounter of their center stations as the shared zero event:

The origin choice sets \(x(0)=0\). The measured rule from the opening therefore becomes

No new motion assumption has been added here; the origin choice only removes the constant \(x(0)\).

3. Why does this first rule fail for light?

The origin of \(S'\) has position function \(x(t)=vt\) according to \(S\). Now consider any event to which \(S\) assigns coordinates \((t,x)\). Subtracting the origin’s position \(vt\) from the event’s position \(x\) gives the first proposed spatial rule. This does not determine the time assigned by \(S'\). As a separate first hypothesis, suppose both laboratories assign the same time to every event. Consequently, if two events are simultaneous in \(S\), they are also simultaneous in \(S'\): \(\Delta t=0\) implies \(\Delta t'=0\).

For an event on the moving origin, its position coordinate is \(x=x(t)=vt\). Substitution gives

At the center-station encounter, a local detector triggers a lamp fixed to \(S\) to emit a light pulse toward increasing \(x\), with trigger delay neglected in our ideal model. Both laboratories assign the emission \((t,x)=(0,0)\) and \((t',x')=(0,0)\). Laboratory \(S\) assigns each later event on the pulse’s path coordinates satisfying

This rule predicts

Thus it predicts the light speed \(c-v\) in \(S'\). A pulse traveling in the opposite direction similarly acquires speed magnitude \(c+v\).

That dependence conflicts with the equal round-trip speeds assumed in Section 1. Combining that physical input with our synchronization convention requires each laboratory to represent its light rays by

or, in its own coordinates,

This first rule cannot preserve both light-propagation rules. We therefore ask:

What coordinate rule maps every uniform trajectory to a uniform trajectory (because both \(S\) and \(S'\) were defined as inertial laboratories) and maps both light rays \(x=ct\) and \(x=-ct\) to light rays of the same speed in \(S'\)?

4. Construct the coordinate rule

Within our idealized region, we assume that no position along the tracks or clock reading is physically distinguished: repeating the experiment elsewhere or later under the same conditions gives the same coordinate rule. We therefore seek a linear rule, meaning that

where \(a\), \(b\), \(d\), and \(e\) are constant during the experiment. They may depend on the measured relative velocity \(v\), but not on the event’s coordinates \(x\) and \(t\).

Every event at the center of \(S'\) has \(x=vt\) and \(x'=0\). Substituting these values into \(x'=ax+bt\) gives

This must hold for every \(t\), so \(b=-av\). The coordinate rule is now

For the positive light ray \(x=ct\), the condition \(x'=ct'\) gives

For the negative light ray \(x=-ct\), the condition \(x'=-ct'\) gives

Adding these two equations yields

Subtracting the second equation from the first yields

The rule must therefore have the form

It remains to determine the coefficient \(a\). Experiments performed entirely inside either inertial laboratory are assumed to obey the same physical rules. This assumption is called the principle of relativity. The labels \(S\) and \(S'\) and the choice of positive direction are arbitrary. Exchanging the labels leaves \(a\) unchanged and replaces \(v\) with \(-v\). Apply these replacements to the spatial rule:

Substitute the expressions for \(x'\) and \(t'\):

Choose the positive root so that the rule approaches the identity when \(v=0\). Thus \(a=\gamma(v)\), where

This real factor is defined for \(\lvert v\rvert<c\). The resulting rule therefore treats \(c\) as a limiting relative speed for the inertial laboratories considered here.

For the chosen pair of laboratories, \(v\) is fixed, so \(\gamma(v)\) is constant across events. Write \(\gamma:=\gamma(v)\).

The required coordinate update is therefore

This is the Lorentz transformation in one spatial direction.

To state its input and output explicitly, define the dimensionless relative speed

Then the coordinate-pair transformation is the linear map

We use the ordered coordinates \((ct,x)\) so that both coordinates have units of length and every matrix entry is dimensionless. In these coordinates,

The matrix is a coordinate representation of the map; the two event-coordinate pairs are its input and output.

5. What does the rule predict for clocks?

Choose two locally recorded events in which the center station of \(S'\) passes two different stations on \(S\). Its clock is present at both events, so their position difference in \(S'\) is

The inverse time transformation is

Substituting \(\Delta x'=0\) gives

The one clock from \(S'\) supplies both readings used in \(\Delta t'\). Two different clocks from \(S\) supply the readings used in the larger interval \(\Delta t\). This prediction is called time dilation.

The reading accumulated by a clock along its own path is called its proper time, denoted \(\tau\). For this clock,

so

No clock is claimed to malfunction. The two quantities are produced by different measurement procedures: one clock accompanies both events, while a synchronized pair of clocks in the other laboratory records them at different positions.

6. What does the rule predict for simultaneous measurements?

Consider a rod at rest in \(S'\). Let its endpoint separation measured there be

for two endpoint events with \(\Delta t'=0\). This is the rod’s rest length.

To measure the moving rod’s length in \(S\), record both endpoint positions at the same \(S\) time:

The spatial transformation then gives

Therefore the length assigned by \(S\) is

This is length contraction along the direction of relative motion. It is a prediction about two simultaneous endpoint observations, not a claim that every observer selects the same pair of endpoint events.

Indeed, if two separated events are simultaneous in \(S\), then \(\Delta t=0\) and

Unless \(\Delta x=0\), laboratory \(S'\) does not call those events simultaneous. Relativity of simultaneity is not an additional postulate. It is required by the same coordinate rule that preserved the light observations.

7. How does the coordinate rule extend to three spatial dimensions?

To describe positions away from the \(x\)-axis, we now include \(y\) and \(z\) in the laboratories’ records.

Start with the \(x\)–\(y\) plane. Set \(t=t'=0\) when the two center stations meet at the same position; both laboratories label that position \(x=y=0\) and \(x'=y'=0\). Their axes remain aligned, and their relative motion is along \(x\). A general linear transformation of \((ct,x,y)\) has nine coefficients. The earlier longitudinal calculation fixes four of them, leaving

Here \(\beta=v/c\) as before; \(A,B,C,D,q\) are dimensionless unknown coefficients, constant for a fixed relative velocity \(v\).

First use records made at the centers of \(S\) and \(S'\) after they have passed each other, so \(t\ne0\).

For an event at \(S\)’s center, \(S\) records \(x=0\) and \(y=0\). The corresponding transverse coordinate in \(S'\) is \(y'=0\), since \(S\)’s center lies on the \(x'\)-axis. Substituting these coordinates into the third equation gives

Since \(ct\ne0\), this gives \(C=0\).

For an event at \(S'\)’s center, \(S\) records \(x=vt=\beta ct\) and \(y=0\). Laboratory \(S'\) assigns \(y'=0\) because the event occurs at its own center. Substituting these coordinates and using \(C=0\) gives

For \(v\ne0\), we have \(\beta ct\ne0\), so \(D=0\).

Thus \(y'=qy\). For \(v=0\), the aligned laboratories use the identity transformation.

Next assume that neither transverse direction is physically preferred. Reflect the setup across its longitudinal axis, replacing \(y\) and \(y'\) by \(-y\) and \(-y'\). The time and longitudinal-coordinate rules must stay unchanged, but the terms \(Ay\) and \(By\) reverse sign. Therefore \(A=B=0\).

Only the transverse scale remains. Write it as \(q(v)\) to show its possible dependence on relative velocity. As in Section 4, exchanging the laboratories must give the inverse rule:

Reversing the longitudinal direction replaces \(v\) by \(-v\) without changing the transverse coordinates. Assuming no preferred longitudinal direction therefore gives \(q(-v)=q(v)\). Combining these conditions yields

The positive \(y\) directions agree, so \(q(v)=1\) and \(y'=y\). The same argument applies to \(z\); reflection symmetry also rules out mixing between \(y\) and \(z\). Therefore

8. How does the clock result define a spacetime interval?

Section 5 already predicted the elapsed time on \(S'\)’s center clock. We now express that result using the three spatial coordinates introduced in Section 7. The labels \(S\) and \(S'\) can stand for any two inertial laboratories satisfying our setup.

How do \(S\) and \(S'\) express the elapsed time on the same clock using their own records?

Take the two encounters on \(S'\)’s center clock from Section 5. Its elapsed time is \(\Delta\tau\). Laboratory \(S\) records the time difference \(\Delta t>0\) and the center’s displacement \(\Delta x=v\Delta t\). Because the relative motion is along \(x\), \(\Delta y=\Delta z=0\). Consequently,

Substitute this into the clock result from Section 5:

We can replace \(S\) in this calculation with any other inertial laboratory that records the same two encounters. That laboratory uses its own time and position differences, and the clock’s velocity relative to it, in the same time-dilation rule. The result is again \(c^2\Delta\tau^2\): all these calculations predict the elapsed proper time on the same clock in \(S'\).

In particular, using the records of \(S\) and \(S'\) gives

The two laboratories assign different time and position differences, but this combination gives the same value \(c^2\Delta\tau^2\).

Using the convention with a negative time term, the spacetime interval is \(\Delta s^2:=-c^2\Delta t^2+\Delta x^2+\Delta y^2+\Delta z^2=-c^2\Delta\tau^2\).

What do the differentials act on?

We have written the clock rule using finite time and position differences. To understand its differential notation, follow \(S'\)’s center clock using the time and position coordinates recorded by \(S\).

Represent the modeled region as a smooth space of events \(M\). Laboratory \(S\) assigns coordinates through the map

Here \(p\) is an event, and \(t,x,y,z\) are the coordinate functions assigning its timestamp and position. Multiplying \(t(p)\) by \(c\) expresses all four components in units of length.

Write the center clock’s path as \(\zeta:I\to M\), where \(I\subseteq\mathbb R\) is the interval of \(S\)’s laboratory timestamps covered by the journey. The path parameter \(\lambda\in I\) is \(S\)’s recorded time, in seconds, at the clock’s position. Reserve \(t:M\to\mathbb R\) for the function assigning \(S\)’s timestamp to each spacetime event. Along the clock’s path,

At \(p=\zeta(\lambda)\), the path has a tangent vector

The space \(T_pM\), called the tangent space, consists of tangents to smooth paths through \(p\). This is the construction used for configuration space in Energy Beyond Particle Mechanics, applied here to paths of events. In \(S\)’s coordinates, our clock’s tangent has components

These components are coordinate rates per unit of \(S\)’s time. The capital \(V_p\) denotes the tangent; \(v\) remains the relative velocity of the laboratories.

For a smooth scalar function \(f:M\to\mathbb R\), its differential at \(p\) is a linear map

To evaluate it on any tangent at \(p\), differentiate \(f\) along a path with that tangent. For our clock, choosing the position function \(f=x\) gives

The time-coordinate differential acts on the spacetime tangent by differentiating \(t\circ\zeta\); the transverse coordinates stay zero:

A linear map from tangent vectors to real numbers is called a covector. The space of these maps is the dual space \(T_p^*M\), so \(dx_p\in T_p^*M\). The differential \(dx_p\) is the operation; \(v\) is its value on this particular clock’s tangent.

How do these differentials express the metric?

We now use the coordinate rates recorded by \(S\) to express how fast \(S'\)’s center clock ticks relative to \(S\)’s clocks. We write each squared rate as a product with two tangent inputs at the same event \(p\). For a second tangent \(W_p\in T_pM\), the product \(dx_p(V_p)\,dx_p(W_p)\) gives \(dx_p(V_p)^2\) when we use the clock’s tangent for both inputs, setting \(W_p=V_p\). This represents the clock rule through a map

The map will be linear in each input separately, which is the meaning of bilinear. The tensor product denotes these products of coordinate rates:

The interval’s coefficients then give the Minkowski metric

This expression defines \(\eta_p\) at each event using the coordinate differentials there. We write \(\eta\) for the metric whose value at event \(p\) is \(\eta_p\). Exchanging its two inputs leaves the result unchanged, so the map is symmetric.

Use our clock’s tangent for both inputs, substituting the coordinate rates above:

Define \(\tau:I\to\mathbb R\) so that \(\tau(\lambda)\) is the reading, in seconds, of \(S'\)’s center clock at the event \(\zeta(\lambda)\). To calculate its rate, choose a nonzero parameter increment \(\Delta\lambda\) with both \(\lambda\) and \(\lambda+\Delta\lambda\) in \(I\). Using \(t(\zeta(\lambda))=\lambda\) and Section 5’s clock rule gives

Divide the clock-reading difference by \(\Delta\lambda\) and take the limit. Then substitute the definition of \(\gamma(v)\) and the metric value \(\eta_p(V_p,V_p)=-c^2+v^2\):

This derivative measures the moving clock’s seconds per second recorded by \(S\). We now want to express the interval along this same path using either \(S\)’s time or the moving clock’s own time.

How can we express the interval using either clock’s time?

Changing the time parameter changes the path’s tangent and hence the scalar \(\eta_p(V_p,V_p)\). We will express the full bilinear rule on the parameter interval \(I\), so that we can write it in either time coordinate. Since \(\eta_p\) accepts spacetime tangents, we first need to send tangents of \(I\) to spacetime along the clock’s path. Fix a parameter value \(\lambda_0\) and let \(p=\zeta(\lambda_0)\). The interval’s tangent space is one-dimensional:

Here \(\partial/\partial\lambda\) is the tangent that differentiates functions of the parameter. Its dual, the cotangent space, consists of linear maps on these tangents:

The coordinate function on \(I\) is \(\operatorname{id}_I:I\to\mathbb R\), with \(\operatorname{id}_I(\lambda)=\lambda\). Apply our differential definition to this function to obtain the coordinate covector on the interval:

Thus \(dt_p\) acts on spacetime tangents in \(T_pM\), while \((d\lambda)_{\lambda_0}\) acts on parameter tangents in \(T_{\lambda_0}I\).

The derivative of the path transfers a parameter tangent to spacetime:

Use this map on both inputs before applying \(\eta_p\). For any \(A,B\in T_{\lambda_0}I\), define

This bilinear form on \(I\) is the pullback metric. To calculate its coefficient, write \(A=a\,\partial/\partial\lambda\) and \(B=b\,\partial/\partial\lambda\) at \(\lambda_0\). Linearity of \(d\zeta_{\lambda_0}\) and bilinearity of \(\eta_p\) give

These evaluations determine the bilinear form on every pair of inputs. We define the line element along the path, denoted \(ds^2\), by

The scalar \(\eta_p(V_p,V_p)\) is the coefficient of this tensor, with units \(\mathrm{m^2/s^2}\) for our time parameter.

The moving clock’s reading \(\tau\) is already a function on \(I\). Its differential is

Square \(d\tau/d\lambda=1/\gamma(v)\) at \(\lambda_0\) and use \(\eta_p(V_p,V_p)=-c^2+v^2\):

Substitute this coefficient into \((ds^2)_{\lambda_0}\), then use the expression for \(d\tau_{\lambda_0}\) above:

Write \(d\tau^2:=d\tau\otimes d\tau\). Then \(ds^2=-c^2d\tau^2\).

What if the clock’s velocity changes?

Now allow the clock to move independently of its railcar while both recording networks remain inertial.

How much time does the clock accumulate along its recorded path when its velocity changes?

Keep \(\lambda\) equal to \(S\)’s time along the new path. Write \(u_x(\lambda),u_y(\lambda),u_z(\lambda)\) for the clock’s velocity components in \(S\). At each \(p=\zeta(\lambda)\), with \(V_p=d\zeta/d\lambda\), the differentials give

The speed is \(u(\lambda)=\sqrt{u_x(\lambda)^2+u_y(\lambda)^2+u_z(\lambda)^2}\). Substituting these rates into the metric gives

We assume an ideal clock: during acceleration, its instantaneous ticking rate follows the constant-velocity rule at its current velocity, with no additional acceleration-dependent change. This extends the clock model beyond the uniform motion considered in Section 5. For a smooth path with \(u(\lambda)<c\), its rate is therefore

Integrate between parameter values \(\lambda_1\) and \(\lambda_2\), which are the starting and ending laboratory times, to predict the change on the moving clock:

The integral adds the clock increments along the full path. Applying the finite constant-velocity formula once to the journey’s endpoints generally gives a different result: different paths between those endpoints can accumulate different elapsed times.

9. How do coordinate transformations act on differentials?

Both laboratories assign coordinates to the same events in \(M\). To relate their coordinate differentials, use \(S\)’s chart \(\chi\) and define \(S'\)’s chart in the same coordinate order:

Here \(t',x',y',z':M\to\mathbb R\) assign \(S'\)’s timestamp in seconds and positions in metres. As coordinate charts, \(\chi\) and \(\chi'\) have smooth inverses on their coordinate images. The coordinate transition map converts \(S\)’s coordinates of an event into \(S'\)’s coordinates of that same event:

Write \(X=(X^0,X^1,X^2,X^3)\in\chi(M)\) for a coordinate tuple, with all four entries measured in metres. The Lorentz transformation from Sections 4 and 7 gives

Let \(F^\mu:\chi(M)\to\mathbb R\) denote component \(\mu\) of \(F\), with \(\mu=0,1,2,3\). Differentiate these component functions with respect to the input coordinates \(X^\nu\), keeping \(v\) constant. At \(X=\chi(p)\), the resulting Jacobian matrix \(J_p\in\mathbb R^{4\times4}\) is

The matrix \(J_p\) represents the derivative \(DF_{\chi(p)}:\mathbb R^4\to\mathbb R^4\). The chart derivatives \(d\chi_p,d\chi'_p:T_pM\to\mathbb R^4\) assign coordinate components to a spacetime tangent. Since \(\chi'=F\circ\chi\), the chain rule gives

Writing these linear maps in components gives the matrix multiplication

The first row, followed by division by \(c\), gives the time-coordinate differential:

10. What velocity does \(S'\) measure for a test cart?

Let a small test cart move along the common \(x\) direction. Write \(u\) and \(u'\) for its signed instantaneous velocities in \(S\) and \(S'\), measured in metres per second along their positive \(x\) axes. The laboratories retain their constant relative velocity \(v\).

What velocity \(u'\) does \(S'\) calculate for the same cart?

Write the cart’s smooth path as \(\rho:I\to M\), where \(I\) is its interval of laboratory timestamps in \(S\). Choose \(\lambda\) to equal that timestamp. At \(p=\rho(\lambda)\), define the cart’s tangent \(V_p\):

The coordinate differentials give the timestamp and position rates along this path:

Evaluate the first two rows of Section 9’s Jacobian relation on \(V_p\), dividing the time row by \(c\). Substituting these rates gives

The functions \(x'\circ\rho,t'\circ\rho:I\to\mathbb R\) give \(S'\)’s position and timestamp records along the cart’s path. Its measured velocity \(u'\) is the limiting ratio of their increments. Where \(dt'_p(V_p)\ne0\), divide both increments by \(\Delta\lambda\) and take the limit:

Substitute the evaluated rates and cancel \(\gamma(v)\). The nonzero-denominator condition is \(1-uv/c^2\ne0\):

This predicts \(u'\) at each event on the cart’s path from its instantaneous velocity \(u\) and the laboratory velocity \(v\) measured by \(S\). For a cart at rest in \(S\), substituting \(u=0\) gives \(u'=-v\). When both \(\lvert u\rvert\) and \(\lvert v\rvert\) are much smaller than \(c\), the denominator is nearly one and \(u'\approx u-v\).

For a light pulse travelling along positive \(x\), substitute \(u=c\):

as required by the light-propagation assumption.

11. How do momentum and energy fit this geometry?

We have answered the coordinate-comparison question without a force law. In Rediscovering Mechanics and Energy from Questions, integrating force over time gave a momentum difference; integrating force along a displacement gave an energy difference.

Can the spacetime geometry suggest a momentum that lets us repeat those endpoint calculations?

What geometric object could represent momentum?

Keep the cart’s path \(\rho:I\to M\) and laboratory-time parameter \(\lambda\) from Section 10. Model the cart as a particle with fixed rest mass \(m>0\), measured in kilograms by the low-speed inertial calibration of the first post and assumed identical in both laboratories. Allow its signed velocity \(u:I\to(-c,c)\) in \(S\) to vary. For this cart, let \(\tau:I\to\mathbb R\) record an ideal clock carried along its path, in seconds.

To describe motion per unit of the cart’s own elapsed time, divide its laboratory-time tangent by the clock rate from Section 8. At \(p=\rho(\lambda)\), write \(u=u(\lambda)\) and define the four-velocity \(U_p\) and a mass-weighted candidate \(P_p\):

All four components of \(P_p\) have momentum units, \(\mathrm{kg\,m/s}\). Its spatial component suggests the function \(p_x:(-c,c)\to\mathbb R\),

Because \(P_p\) is a tangent vector, Section 9’s chain rule gives its components in the other chart:

This is the transformation law for a contravariant vector, or tensor of type \((1,0)\). The vector \(P_p\) is the same object in \(T_pM\); the two charts assign different component columns to it. The chain rule gives this Jacobian law for any smooth change of coordinates, including the Lorentz transformation used here.

The metric \(\eta_p\) is a tensor of type \((0,2)\): it takes two tangent vectors and returns a scalar. Evaluating the same metric on the same vectors gives the same number in either chart. For example,

Which endpoint quantities follow from a force law?

Geometry alone does not establish that \(P_p\) describes momentum exchanged in interactions. We adopt conservation of total four-momentum as an additional physical law: summing the particles’ \(P_p\) components in a single inertial chart gives the same totals before and after an isolated interaction.

Let \(F_x:I\to\mathbb R\) be the signed force in \(S\), in newtons, supplied by an independently calibrated interaction law. We also adopt the relativistic force equation. Integrating it from laboratory time \(\lambda_1\) to \(\lambda_2\), with \(u_i:=u(\lambda_i)\), gives

This is the first post’s impulse calculation with the new momentum function. To calculate the work \(W\) in joules, use \(d(x\circ\rho)/d\lambda=u(\lambda)\), substitute the force equation, and apply the chain rule:

To find an endpoint expression, differentiate \(\gamma(u)=(1-u^2/c^2)^{-1/2}\):

Substituting into the work integral gives

Define the cart’s energy in \(S\) by \(E:(-c,c)\to\mathbb R\), measured in joules:

The time component of \(P_p\) is therefore \(E/c\). Write \(E'\) and \(p'_x\) for the cart’s energy and momentum in \(S'\). The first two rows of the Lorentz Jacobian give

Energy and momentum components depend on the laboratory. Conservation of total four-momentum includes conservation of the sum of these energies in each inertial laboratory.

The work integral at fixed mass determines energy differences; identifying \(E/c\) with the time component of the conserved four-vector fixes the reference used here. It assigns rest energy \(E(0)=mc^2\). In reactions that change the particles’ rest masses, these rest terms enter the same conserved energy sum. The kinetic energy \(K:(-c,c)\to\mathbb R\), measured in joules relative to rest, is

These relations have experimental support: Bertozzi’s electron measurements tested the speed–kinetic-energy relation, and Rainville and colleagues compared atomic mass differences with emitted gamma-ray energies to test the rest-energy relation.

12. Answer to the main question

Under the stated assumptions, the Lorentz transformation relates the two inertial laboratories’ coordinates for the same event:

Time dilation, length contraction, and relativity of simultaneity follow from this rule. For a uniformly moving clock, the spacetime interval recovers the same elapsed time from either laboratory’s records. The metric lets us apply the clock rule along a path with changing velocity.

With the additional momentum and force laws, we recovered momentum and energy as endpoint quantities for impulse and work, extending the first post’s construction to relativistic speeds.

This is the flat-spacetime description of special relativity, used here within the modeled region. To account for gravity beyond a region where curvature can be neglected, we face a further question:

What rule predicts clocks, light, and freely falling objects when gravitational effects vary across a region?

From Free Fall to Curved Spacetime begins with that question.

Only after the real two-dimensional construction is complete will we ask why the same state is usually written with complex numbers.

1. The prediction problem

Send particles one at a time into a two-path interferometer. A first beam splitter gives each particle two possible paths, \(A\) and \(B\). The paths later meet at a second beam splitter, followed by detectors \(D_0\) and \(D_1\).

Our question is:

Given what happens along each path, what fraction of the particles will arrive at each detector?

The adjustable rectangle in path \(B\) does not move either mirror. It represents a phase-shifting element designed to leave the outgoing beam on the same geometric route. For light, a transparent material makes propagation through a segment behave as though the path were longer: a segment of geometric length \(\ell\) and refractive index \(n\) contributes the effective optical length \(n\ell\). Changing the material or the thickness traversed changes that effective length without redirecting the beam away from \(M_B\).

Let the effective path lengths be \(L_A\) and \(L_B\). First make them equal, so the apparatus can be adjusted until every particle is detected at \(D_0\). Then use the element to change the effective length of path \(B\) while leaving its geometric route and path \(A\) fixed. The independently controlled effective path-length difference is

For each chosen \(\Delta L\), send many particles through the apparatus. The detection frequency at a detector means the number of clicks there divided by the total number of particles sent. These measured frequencies estimate the detection probabilities. As \(\Delta L\) is varied, they change continuously and eventually repeat:

Individual particles produce individual detector clicks. The repeating pattern appears only after many trials.

The symbol \(\lambda\) denotes the smallest positive change in \(\Delta L\) after which the detector statistics repeat. There is no recursive definition: the experimenter sets and measures \(\Delta L\), then uses the resulting detector frequencies to measure \(\lambda\) once. After that calibration, the measured \(\lambda\) predicts the probabilities for new choices of \(\Delta L\).

Let \(\theta_A\) and \(\theta_B\) denote the phases accumulated along the two paths. Only their difference affects the detector probabilities. Because path \(A\) remains fixed while the adjuster changes path \(B\), define

Throughout the rest of this post, \(\theta\) always means this relative phase. At this stage it means “position within the repeating interference pattern.”

More generally, changes in travel time or potential along a path can move the same interference pattern. Two successive changes shift the pattern by the sum of their individual amounts, so their phase changes add:

This is why phase is said to accumulate along a path: consecutive portions of the journey contribute consecutive shifts within the observed cycle.

If both paths are changed by the same amount, \(\theta_A\) and \(\theta_B\) increase equally, so their difference \(\theta\) and the detector statistics do not change.

Operationally, phase belongs to a path’s contribution to the later interference. Its meaning is revealed only when that contribution is compared with another path.

We compare four apparatus configurations:

Let \(C\) denote the chosen configuration:

In configuration \(C_p\), let the recorded path be the random variable

The first beam splitter is balanced, so the measured path frequencies are

The separate-input configurations calibrate the second beam splitter:

2. Why are these probabilities not enough?

Suppose that, in configuration \(C_0\), every trial possesses one exclusive path value \(X\in\Omega\) and the single-path calibrations can be transferred unchanged into that configuration. The assumptions are

Because \(X=A\) and \(X=B\) are mutually exclusive and exhaustive within this proposed model, the law of total probability gives

But the measured probability in the same configuration varies with \(\theta\):

For example,

The conjunction of the transfer assumptions has failed: probabilities measured after changing the apparatus to \(C_A\), \(C_B\), or \(C_p\) cannot be inserted as though they were conditional probabilities inside \(C_0\).

3. Can one state and recombination rule predict \(C_p\) and \(C_0\)?

The subscript in \(F_0\) refers to detector \(D_0\).

We want a \(\theta\)-dependent path state \(s_\theta:\Omega\to\mathbb R\) and a fixed detector rule \(F_0:\mathbb R^2\to\mathbb R_{\ge0}\) satisfying

In this notation, the rejected classical model used

First test real path values and a squared-linear rule, so that the two terms can cancel while the result remains nonnegative:

The state \(s_\theta\) above is inferred from the path frequencies measured in \(C_p\). Independently, the configurations \(C_A\) and \(C_B\) calibrate how \(F_0\) responds when only one path enters the second beam splitter. If \(X\) is extended here to label that input path, then passing \((1,0)\) means setting

while passing \((0,1)\) means setting

These are single-path calibration inputs, not possible values of the two-path state \(s_\theta\). They determine the rule \(F_0\) independently:

The calibrated rule is then applied to the \(C_p\)-derived input \(\bigl(s_\theta(A),s_\theta(B)\bigr)\) to predict \(C_0\).

Expanding the definition of \(F_0\) gives

Because \(s_\theta(A),s_\theta(B)\in\{-1/\sqrt2,+1/\sqrt2\}\),

But

so no such real-valued family \(s_\theta\) and fixed rule \(F_0\) can satisfy both requirements.

The missing information is a continuously variable relation between the two path contributions. Replace the real-valued path state \(s_\theta\) by

Choose path \(A\) as the phase reference, so \(\theta_A=0\) and \(\theta_B=\theta\). Define \(\boldsymbol\psi_\theta\) by

Their squared lengths reproduce the path probabilities:

Their angle retains the additional quantity measured by the interference experiment:

Correspondingly, replace \(F_0:\mathbb R^2\to\mathbb R_{\ge0}\) by

where

The two calibration probabilities are \(1/2\) for every \(\theta\), so this is one fixed function. Applied to the new path state,

This already answers the original question about \(D_0\). The state \(\boldsymbol\psi_\theta\) retains the path probabilities and relative phase; the rule \(\mathcal F_0\) uses that state to produce \(P(D_0\mid C=C_0,\theta)\). Nothing else is required if \(D_0\) is the only output we want to predict.

This motivates the definition:

A path amplitude is a two-dimensional real vector whose squared length gives the probability of that path in configuration \(C_p\) and whose direction retains the phase needed to predict later interference.

3.1. How does a basis describe the path state?

The successful path states form

With pointwise vector addition and scalar multiplication, \((\mathbb R^2)^\Omega\) is the path-state space. Its elements are complete path states; a basis is only a particular set of elements used to expand them.

Because \(\Omega=\{A,B\}\), the coordinate map

is a linear isomorphism:

Thus a path state may be represented as a block vector with two \(\mathbb R^2\) entries. For example,

For each \(X\in\Omega\), define the scalar-valued path selector

Thus \(e_A\) selects path \(A\) and \(e_B\) selects path \(B\). Define the standard basis vectors of each path’s amplitude plane by

Their combination is the function

Therefore, by the probability rule for amplitudes,

is a basis of \((\mathbb R^2)^\Omega\). The previously defined state has the explicit expansion

More generally, for \(\boldsymbol u,\boldsymbol v\in\mathbb R^2\),

Thus the earlier detector rule becomes

In \(C_A\) only the path-\(A\) component is present; in \(C_B\) only the path-\(B\) component is present. Use the concrete unit amplitude

Then

The definition of \(\mathcal F_0\) already assumes that a common rotation of the amplitude plane changes no probability. For every \(R\in\mathbb R^{2\times2}\) satisfying \(R^{\mathsf T}R=I\) and \(\det R=1\),

Therefore every unit direction gives the same single-path result. In particular, because \(\lVert\boldsymbol r_2\rVert^2=1\),

Thus testing \(\boldsymbol r_2\) would add no information. Without this rotational-symmetry assumption, the two basis directions and their linear combinations would have to be calibrated separately.

The first line evaluates \(\mathcal F_0\) on the ordered path-amplitude pair \((\boldsymbol r_1,\boldsymbol 0)\) used in \(C_A\); the second uses \((\boldsymbol 0,\boldsymbol r_1)\) in \(C_B\). Applying the inverse coordinate map defines the corresponding calibration path states \(\boldsymbol\psi_{C_A}\) and \(\boldsymbol\psi_{C_B}\):

Their squared component lengths express the single-path inputs:

These calibration states are distinct from the two-path state \(\boldsymbol\psi_\theta\), whose component lengths come from \(C_p\) and which is used to predict the detector probabilities in \(C_0\).

3.2. Can one rule predict both detector outputs?

The apparatus also contains \(D_1\). The rule \(\mathcal F_0\) returns only the probability at \(D_0\) and discards the resulting output amplitude. We could define a separate rule for \(D_1\), but then the two rules would not express that both outputs come from the same second beam splitter.

We therefore generalize \(\mathcal F_0\) to one map that returns an amplitude for every detector. Squared lengths of those amplitudes will give both detector probabilities, and the output amplitudes remain available if another component is placed after the beam splitter.

Define the detector-label set

In \(C_A\), only the amplitude from path \(A\) enters the second beam splitter. In \(C_B\), only the amplitude from path \(B\) enters it. We now want those two measurements to predict what happens in \(C_0\), where both amplitudes enter together. To combine the two single-path responses, assume that the beam splitter acts linearly.

Therefore define the beam-splitter map and its output by

Define the detector-coordinate map

For each detector label \(D\in\Delta\) and path label \(X\in\Omega\), let \((M_{\mathrm{BS}})_{D,X}\) denote the coefficient mapping the amplitude from path \(X\) to detector \(D\).

The same two normalized inputs now calibrate the response at both detectors. In detector coordinates,

For every \((D,X)\in\Delta\times\Omega\),

Therefore

For each detector output, choose the axes of its amplitude plane so that the coefficient from input \(A\) is positive. At \(\theta=0\),

Therefore linearity gives, for each \(D\in\Delta\),

The experiment gives

By the probability rule for amplitudes,

Every coefficient has magnitude \(1/\sqrt2\). The first equality therefore requires the two coefficients at \(D_0\) to have the same sign; the second requires those at \(D_1\) to have opposite signs. Hence

Therefore

This matrix was not chosen independently of the experiment: the single-input calibrations fixed the magnitudes \(1/\sqrt2\), and the constructive and destructive outputs at \(\theta=0\) fixed the relative signs.

Applying \(M_{\mathrm{BS}}\) gives

For \(D_0\), the new map reproduces the earlier rule exactly:

The detector probabilities are the squared lengths of these output amplitudes:

Thus the same state predicts both the path probabilities and the interference probabilities. The state alone does not specify a probability until the experimental configuration specifies which output amplitudes are to be calculated.

4. Why use complex numbers?

The real two-dimensional amplitudes already reproduce the experiment. Complex numbers give the same vector addition, length, and rotation with less notation.

Identify

Then

The path state is the function

or equivalently the vector

The vector space of two-path states is

The beam-splitter map has the same matrix in complex notation:

so

The detector probabilities are

5. Where does the group structure enter?

If the adjuster adds a further relative phase \(\delta\), it changes the state by

Successive adjustments satisfy

The unit complex numbers form the group

In the equivalent real description, the rotation generator is

and

Under the identification with complex numbers, multiplication by \(J\) becomes multiplication by \(i\). The imaginary unit is therefore the concise representation of the real rotation structure required to retain continuously varying phase.

Multiplying the entire state by the same phase does not affect any probability:

Here \(j\in\{0,1\}\). The angle \(\chi\) is called a global phase; only relative phases affect the detector statistics.

The construction above derives a state that predicts both path and detector measurements and a beam-splitter map calibrated from the experiment. The next question—how such a state changes with time—is developed in From Quantum State to the Schrödinger Equation.

Given the state now, what rule predicts the state later?

1. What experiment produces a time-update problem?

The previous post defined

For the ordered path labels \((A,B)\), define

After choosing path \(A\) as the phase reference, its two-path state satisfies

The parameter \(\theta\) labels relative phase, not time. To create a time-update experiment, drive the phase adjuster at a constant measured angular frequency \(\omega\). Choose \(t=0\) when \(\theta=0\) and choose the sign convention

Repeated trials then give

The state at time \(t\) is

so after an additional interval \(\tau\),

Using the same linearity postulate as for the beam splitter, extend the observed component update to an arbitrary input state:

The original experiment has now supplied one example of a time-update map. The general problem is to determine which properties any isolated-system update

must have.

1.1. What must the update preserve?

Define the inner product and norm on \(\mathcal H\) by

The squared norm is total probability. An isolated-system update must therefore satisfy

for every \(\psi\in\mathcal H\). Amplitudes also obey linear superposition:

If the physical rule does not change with the clock’s starting time, successive intervals compose as

and

Finally, assume that \(U(\tau)\) changes differentiably with \(\tau\). These assumptions state the prediction problem precisely; they do not yet specify a particular system’s dynamics.

2. What generates a continuous probability-preserving update?

Define the instantaneous generator

Differentiability then gives the short-interval expansion

Here \(o(\tau)\) is an operator remainder for which every matrix entry of \(o(\tau)/\tau\) tends to zero as \(\tau\to0\).

For any linear operator \(L:\mathcal H\to\mathcal H\), its adjoint \(L^\dagger\) is defined by

for all \(\phi,\psi\in\mathcal H\). By the polarization identity, a linear norm-preserving map also preserves the inner product:

Using the definition of the adjoint,

for all \(\phi,\psi\in\mathcal H\), so

Substitute the short-interval expansion:

Equality with \(I\) requires

Thus the time generator is anti-Hermitian. The experiment, however, labels its oscillations by the real number \(\omega\). To represent such real frequencies by a Hermitian operator, define

Then

so \(G\) is Hermitian and

From \(\psi(t+\tau)=U(\tau)\psi(t)\),

Therefore

For an operator \(L:\mathcal H\to\mathcal H\), define its matrix in the ordered coordinates \((A,B)\) by

For the driven two-path example,

which produces

The abstract derivation has therefore recovered the update already measured in the motivating experiment.

3. Why identify the generator with energy?

Norm preservation tells us that \(G\) is Hermitian, but it does not identify the physical quantity represented by its eigenvalues. That identification comes from experiment.

For a state with a single measured oscillation frequency \(\omega\), experiments give

where the universal constant \(\hbar>0\) converts angular frequency into energy. Define

Because \(G^\dagger=G\),

If \(\psi_E\in\mathcal H\) satisfies

then

The measured energies are therefore the eigenvalues of \(\widehat H\). Multiplying the generator equation by \(\hbar\) gives

This is the abstract Schrödinger equation. The operator \(\widehat H\) is the Hamiltonian, or energy operator.

For an energy eigenstate,

and therefore

4. Why can the imaginary unit not simply be omitted?

Suppose the same Hermitian energy operator instead generated

For \(\widehat H\psi_E=E\psi_E\), the solution would be

so

Except when \(E=0\), total probability would change with time. The factor \(-i\) instead turns the Hermitian energy operator into the anti-Hermitian generator required by norm preservation:

The imaginary unit changes real exponential growth into phase rotation:

5. How does the spatial Schrödinger equation arise?

For a particle whose possible positions form \(\mathbb R^3\), define the position-state space

At time \(t\), the state is the wavefunction

Let the result of a position measurement be the random vector \(\mathbf X\in\mathbb R^3\). For every measurable region \(R\subseteq\mathbb R^3\), the position probability rule is

A normalized state satisfies

To apply the abstract time-update equation, we must determine the spatial action of \(\widehat H\).

5.1. What operator represents momentum?

Matter-wave experiments relate measured momentum \(\mathbf p\) to measured wave vector \(\mathbf k\):

Write \(\mathbf x=(x_1,x_2,x_3)\) and define

An idealized plane wave is

Its spatial derivative satisfies

Therefore the linear operator

has the experimentally required value on every plane wave:

5.2. What operator represents kinetic and potential energy?

For a nonrelativistic free particle of measured mass \(m>0\), experiment gives

Using the momentum operator,

If the particle’s potential-energy function is

define its action on a wavefunction by

The nonrelativistic energy operator is

Substitution into the abstract equation gives

Every term now has a defined role:

6. Do we take the real part at the end?

No. A quantum state’s relative phase affects later probabilities through interference.

For example,

Both states give path probabilities \(1/2\) and \(1/2\), but their relative phases differ. The beam-splitter map therefore gives different detector probabilities. Replacing \(\psi_2\) by its real part would discard information required for that prediction.

Real measurement probabilities instead arise from the squared-amplitude rule. If a selected measurement outcome is represented by a normalized \(\phi\in\mathcal H\) and the system state is the normalized \(\psi\in\mathcal H\), then

For a Hermitian observable \(\widehat A:\mathcal H\to\mathcal H\), its expected value in state \(\psi\) is the real scalar

7. Could the same theory use only real numbers?

Yes. Replace each complex amplitude by its two real components and represent multiplication by \(i\) with

Here \(I_2\) is the \(2\times2\) identity matrix.

This gives an equivalent real description with twice as many scalar components and a distinguished linear operation \(J\). The rotation structure remains; only its notation changes.

Therefore the strongest conclusion is:

Continuous, reversible, probability-preserving quantum evolution requires a phase-rotation structure. Complex numbers provide its economical representation.

The symbol \(i\) is not removed at the end of a quantum calculation. It represents the rotation that converts energy into norm-preserving time evolution rather than exponential growth.

Are kinetic and potential energy the only possible kinds of energy, or are they merely the parts that happened to be useful for that problem?

We will follow the same method as before: begin with a question, use the measured update rules to answer it, and introduce a new energy term only when the calculation requires one.

1. What did energy accomplish for one cart?

For the cart and spring, the original question was to predict quantities such as the cart’s maximum speed without calculating every intermediate position. The update rules led to two endpoint quantities:

The spring’s work increased one by exactly the amount that it decreased the other:

Therefore,

remained constant, allowing us to compare the initial and final states directly.

That was the role of energy: it compressed many moment-by-moment updates into a relation between states. Calling the two terms kinetic and potential energy did not establish that nature contains exactly two fundamental substances. It identified two quantities that were convenient—and, given the measured dynamics, mathematically effective—for answering the question.

Does the same construction appear without a moving mass?

Consider an ideal electrical circuit containing an inductor and a capacitor. Let \(Q(t)\) be the capacitor’s charge and let

be the current. The circuit’s update rule is

where \(L\) is the inductance and \(C\) is the capacitance.

Ask the same kind of question as before:

Can we construct an endpoint quantity that remains constant and lets us relate the circuit’s present and future states?

The update rules give

Therefore the circuit has the conserved quantity

It has exactly the same mathematical pattern as the cart-spring energy:

The circuit contains no cart whose position is \(Q\) and no mechanical mass equal to \(L\). The analogy shows that the two-part structure can arise from the form of the update rules, while the physical interpretation of its terms changes with the system.

The new problem is to determine what happens when the state of a system requires more than one changing value.

2. What is a degree of freedom?

A degree of freedom is one independent value needed to specify a configuration.

For a particle, the configuration at time \(t\) is specified by finitely many coordinates, such as \(x(t)\). For a field, the configuration at time \(t\) is an entire function \(\phi(\mathbf{x},t)\). Thus a field has effectively one degree of freedom at every point in space.

This definition raises a concrete question:

If different degrees of freedom affect one another, what endpoint quantity lets us predict their possible later states without solving every intermediate update?

We can answer it first for only two degrees of freedom.

3. Why does a coupling term appear?

Consider two equal carts, each with mass \(m\), connected by a spring with stiffness \(k\). Let \(x_1(t)\) and \(x_2(t)\) be their displacements from equilibrium. The spring’s change in extension is then \(x_1-x_2\). Suppose experiments give the update rules

We ask:

Given the present positions and velocities, can we construct a state quantity that remains fixed and therefore constrains the carts’ later velocities?

We do not need a new method. From the one-cart problem, the change in kinetic energy equals the work done. For both carts together, define

The total work done by the connecting spring from time \(t_1\) to \(t_2\) is the sum of the work done on the two carts:

where

Because \(\dot r=\dot x_1-\dot x_2\), the integral becomes

Thus the spring’s work depends only on the initial and final separations. Following the same endpoint construction as for the one-cart spring, define

The work relation is therefore

But the work-energy relation also gives

Combining them yields

The coupling term is potential energy: it is the stored energy of the connecting spring. The word “coupling” does not name a third kind of energy. It says that this potential depends on a relationship between two degrees of freedom, \(x_1-x_2\), rather than on either position alone. The term arose because the measured update of \(x_1\) depended on \(x_2\) and vice versa.

4. Why does a field have spatial-gradient energy?

Begin with the spring force already developed in Section 3. Fix two identical springs to opposite walls and attach both to one mass. Let each wall be a horizontal distance \(a\) from the mass’s equilibrium position, and let each spring’s natural length be \(\ell_0<a\).

At \(y=0\), each spring pulls with magnitude

The left and right forces point in opposite directions, so they cancel. Give this already-familiar spring force a shorter name:

This is tension. It is not a new kind of force; it is the pulling force already carried by each stretched spring.

Now move the mass vertically by \(y\). Each spring has length

and pulls with magnitude

Here \(F_{\mathrm{s}}\) is the magnitude of the entire spring force, not its horizontal component. Put the displaced mass at \((0,y)\) and the two anchors at \((-a,0)\) and \((a,0)\). The vectors from the mass toward the anchors are

Both have length \(\ell(y)\), so their unit directions are \(\mathbf r_{\mathrm{L}}/\ell(y)\) and \(\mathbf r_{\mathrm{R}}/\ell(y)\). Because each spring pulls along its own length, the two force vectors are

Adding them gives

Thus the horizontal components cancel, while the vertical components add:

The magnitude law \(F_{\mathrm{s}}=k(\ell-\ell_0)\) and the fact that a spring pulls along its length belong to the measured spring model. Obtaining the horizontal and vertical components from that force is geometry.

For \(\lvert y\rvert\ll a\), we have \(F_{\mathrm{s}}(y)\approx T\) and \(\ell(y)\approx a\), so

Thus tension is useful because the geometry converts two familiar spring forces into a restoring force for transverse motion.

How can the disturbance travel?

One mass can oscillate, but there is nowhere for a disturbance to travel. Now place finitely many masses in a row and join adjacent masses by spring segments that are already stretched and carry the same force \(T\). Hold the two ends fixed. Let \(\phi_i(t)\) be the vertical displacement of mass \(i\), and let neighboring equilibrium positions be separated by \(a\).

For small displacements, the right segment contributes the vertical force

while the left segment contributes

Adding them gives the update equation

This answers the propagation question: changing \(\phi_i\) changes the forces on its neighbors, which then change their neighbors.

The segment joining sites \(i\) and \(i+1\) has length

For \(\lvert\phi_{i+1}-\phi_i\rvert\ll a\), its increase in length is approximately

Increasing a segment already carrying force \(T\) by this amount requires the work

The finite chain therefore has the approximate energy

Why introduce a field?

A guitar string is a continuous elastic object rather than a literal collection of carts. The finite chain tells us what information matters, but using one coordinate for every microscopic piece would be impractical. At scales much larger than the atomic spacing, define

This is a field: at each time, its state is an entire function of position.

Divide the string into finite segments of length \(\Delta x\) and set \(m=\mu\Delta x\). The finite-chain energy becomes

As the largest segment length tends to zero, the finite differences approach the spatial derivative and this Riemann sum approaches

The first term is kinetic energy distributed along the string. The second is potential energy: a bent string is longer than a straight one, so tension stores additional energy. Its update equation is the wave equation

The spatial-gradient term is therefore not a third fundamental kind of energy. It is the potential energy required by a real question about a continuous object: predicting how a plucked string moves.

Does a real system ever need a local term?

The guitar string does not need one. Its tension responds to differences between nearby displacements. To motivate a local term, we need a different physical question:

Can each degree of freedom be pulled toward equilibrium even when it agrees with all its neighbors?

Consider a row of identical pendulums whose neighboring bobs are connected by light springs:

For pendulum \(i\), gravity gives the exact potential energy

This is local because it depends on \(\theta_i\) alone. For small angles,

The spring between neighboring bobs responds to their relative horizontal displacement. For small angles that displacement is approximately \(L(\theta_i-\theta_j)\), so its potential has the form

where \(\kappa\) includes the spring stiffness and the factor of \(L^2\). The approximate total energy is therefore

Let \(U_{\mathrm{total}}\) denote both potential-energy sums. Only two neighbor terms contain \(\theta_i\), so the update equation follows directly:

If all pendulums have the same angle \(\theta_i=\delta\), every neighbor difference is zero:

Gravity still contributes

so every pendulum is pulled back toward the vertical. This is the concrete purpose of the local term.

For a closely spaced system described by a continuous field, these same dependencies produce the general form

The guitar string has \(V(\phi)=0\). The coupled-pendulum system has a nonzero local gravitational potential. The displayed formula has three terms, but still only the broad classical split

The gradient and local terms are two contributions to potential energy, distinguished by whether they depend on relationships between nearby values or on one value itself. They are not new fundamental categories of energy.

5. Can a change of variables alter the decomposition?

Return to the two coupled carts. Their state can be described by \(x_1\) and \(x_2\), but it can also be described by their center position and separation:

With total mass \(M=2m\) and reduced mass \(m_r=m/2\), the same energy becomes

Nothing physical changed. We only changed the values used to describe the state. Yet two individual-cart kinetic terms became a center-motion term and a relative-motion term, while the coupling became an ordinary potential depending on \(r\).

This gives another question:

If the displayed pieces change when the descriptive variables change, how can we state the theory without committing to any particular coordinates?

We now keep the structure discovered in the examples while removing the coordinate-dependent notation.

The space of configurations

Let \(Q\) be the set of every possible configuration of the system, called its configuration space. A point \(q\in Q\) is one complete configuration.

Coordinates are merely numbers chosen to label points of \(Q\). The physical configuration does not depend on which labels we use.

Describing change without coordinates

At a configuration \(q\), the tangent space \(T_qQ\) is the vector space of all possible infinitesimal directions through \(q\). Equivalently, its vectors describe the possible first-order changes of curves passing through \(q\).

A trajectory is one such curve, so its velocity is one such change:

For one cart, this is the familiar velocity \(\dot x\). For a field, it is the entire time-change function \(\partial\phi(\mathbf{x},t)/\partial t\).

Momentum and force answer a different kind of question: what scalar result do we obtain when they act on a possible change? Objects that assign a number linearly to each tangent vector form the cotangent space \(T_q^*Q\). Thus

Pairing force with a displacement gives infinitesimal work; pairing it with velocity gives power. This is why force naturally belongs to the dual, or cotangent, space rather than the tangent space containing velocities.

Generalizing mass

In one dimension, mass converted velocity into momentum:

and measured kinetic energy:

For a general configuration space, the corresponding object is a mass metric \(g\). At each configuration, it takes two tangent vectors and returns a scalar:

It is a symmetric, bilinear inner product. Thus it compares two possible instantaneous changes; it does not itself subtract them. For example, the squared separation between two velocity vectors is \(g_q(v-w,v-w)\).

This is the same mathematical kind of object as a Riemannian metric, but it need not be the first fundamental form. The first fundamental form is specifically the metric a surface inherits from an ambient spatial metric. Here \(g\) instead weights motion by mass, so it measures kinetic-energy cost rather than ordinary spatial distance.

Fixing one input to be the velocity turns the metric into the momentum covector, while using the velocity in both inputs gives kinetic energy:

The dot in \(g_q(v,\mathord{\cdot})\) leaves one input open: supplying any possible change produces the number assigned to it by the momentum \(p\).

For one particle in Euclidean space, if \(h\) is the ordinary spatial metric, then

so \(g(v,v)=m\lVert v\rVert^2\). Mass does not alter spatial distance here; it weights the spatial metric for the purpose of describing motion.

The distinction becomes useful for several particles. A tangent vector now contains all their velocities, \(v=(v_1,\ldots,v_N)\), and

Thus one geometric object records how every particle’s possible motion is weighted by its mass. In generalized coordinates, its components form the mass matrix.

Generalizing the interaction

A potential assigns a scalar to every configuration:

Its differential \(dV_q\) records how that scalar changes in every possible direction from \(q\). The conservative force is

For the two carts, \(V\) can contain \(\tfrac12k(x_1-x_2)^2\). For a field, it can include both local terms and the integral of the spatial-gradient term. What looked like several potential pieces in coordinates is one function on the full configuration space.

The coordinate-free conservative theory

Within the class of conservative systems considered here, the general model is therefore specified by

These three objects answer three questions:

The coordinate-free equation of motion is

The left side is acceleration defined using the inertial geometry \(g\), including the corrections needed when coordinates curve or vary from place to place. The right side is the direction in which \(V\) decreases most rapidly, measured using the same geometry. For one Cartesian coordinate, this reduces to

The complete state \((q,v)\) is a point of the tangent bundle \(TQ\). The corresponding energy is therefore the scalar function

defined by

When \(g\) and \(V\) have no explicit time dependence, this energy remains constant along solutions of the equation of motion.

Coordinates turn these statements into formulas containing masses, matrices, gradients, and individually named terms. Those formulas may rearrange when coordinates change. The coordinate-free data \((Q,g,V)\) and the total energy assigned to a physical state do not.

6. Why combine systems?

The motivating problem

Consider two carts moving without friction. While they do not interact, knowing the state of cart 1 is enough to predict cart 1, and likewise for cart 2. Each cart’s kinetic energy remains constant.

Now join the carts with a spring. The question changes:

If cart 1 starts moving and cart 2 starts at rest, what quantity remains fixed while the spring transfers motion between them?

Neither cart can now be predicted from its own state alone. Its next update also depends on the position of the other cart. We therefore enlarge the system boundary to contain both carts and the spring.

What does combining them mean mathematically?

If the individual configuration spaces are

then the combined configuration space is their product:

A configuration is \(q=(x_1,x_2)\), and a complete state is

Equivalently,

The product is necessary because answering the new question requires both positions and both velocities.

What quantity remains fixed?

Let the spring have stiffness \(k\) and natural length \(\ell_0\). Its extension is

The energy function on the combined state space is

with

The first two terms are contributions to \(K\). The last is an interaction contribution to \(V\): it belongs to the relation between the carts’ positions, so it cannot be found in either cart’s state considered alone. It is not a third broad kind of energy; the total still has the form \(E=K+V\).

The equations of motion are

The motion \(q(t)\) is a curve in \(Q\), and its position and velocity form the state curve \(z(t)=(q(t),\dot q(t))\) in \(TQ\). Consequently, \(E_{\mathrm{total}}(z(t))\) is an ordinary scalar function of time. Its time derivative means differentiation along this curve; coordinate-free,

where \(dE_{\mathrm{total},z(t)}\in T_{z(t)}^*(TQ)\) acts on \(\dot z(t)\in T_{z(t)}(TQ)\).

Along a motion of the combined system, the three contributions generally do not remain constant separately:

In general, none of these three derivatives is zero. But together,

Thus the individual cart energies and the spring’s interaction energy can change, but the energy of the complete isolated system remains fixed. We combine systems because the smaller system boundary no longer contains everything needed either to predict the update or to obtain a conserved energy.

7. Answer to the main question

Within the conservative models described by \((Q,g,V)\), total energy has exactly two components:

Here \(K\) measures change of configuration and \(V\) depends on configuration. Terms such as spring, gravitational, local, and spatial-gradient energy are not additional categories alongside kinetic and potential energy; they are different contributions to \(V\). Likewise, a system may contain several contributions to \(K\).

The model may still split those two components into useful subterms:

The number of these subterms depends on the system boundary, variables, interactions, and grouping used in the model. That secondary decomposition is coordinate- and model-dependent; the broad kinetic–potential decomposition is fixed for the conservative theory considered here.

Energy remains useful for the same reason it was useful in the first cart problem: it lets us constrain a final state from an initial state without reconstructing every intermediate update.

The central example is a cart attached to a spring. From it, we will motivate position, velocity, acceleration, inertial mass, force, momentum, impulse, kinetic energy, and potential energy.

1. The prediction problem

Consider a cart on a nearly frictionless track, attached to a spring. Pull the cart 10 centimeters to the right and release it.

Our first question is concrete:

When will the cart first reach the spring’s relaxed position?

We want a rule that predicts the answer for different carts, springs, and release positions without repeating every possible experiment.

2. Position: what outcome are we predicting?

Choose the spring’s relaxed position as the origin:

Record the cart’s position at a sequence of times. The desired arrival time is the first time for which \(x(t)=0\).

Position is necessary, but it is not enough to predict what happens next. The cart can pass through the same position while traveling either left or right.

3. Velocity: which way is the state developing?

To distinguish those cases, measure the rate at which position changes:

Over a short interval \(\Delta t\), this gives the approximate update:

But velocity also changes while the spring acts. Knowing \(x\) and \(v\) now does not yet tell us the next velocity.

4. Acceleration: how is velocity changing?

Measure the rate of change of velocity:

Then, over a short interval,

The prediction problem has now become more precise:

Given the present spring extension and the particular cart, what acceleration should we expect?

Acceleration is still only a measured description. We have not yet explained how the spring state and the cart jointly determine it.

5. Solve the problem

Measure the cart’s acceleration at several extensions:

The measurements suggest the empirical rule

Equivalently, write the measured coefficient as \(\omega^2\):

Here \(\omega\) is not a new explanation or a universal constant. It is merely a compact way of reporting what this cart-spring pair did in the experiment.

If we release the cart from rest at \(x_0\), the measured rule predicts

Therefore the cart first reaches the relaxed position when

so for this experiment

The original prediction problem is now solved. We did not need mass or force. Those ideas become relevant only when we ask a new question:

Can one rule predict the motion after we replace the cart or the spring?

6. Inertial mass: comparing different carts

Keep the spring fixed, but repeat the acceleration measurements using another cart:

Each cart has its own predictive rule: \(a_A=-4x\) and \(a_B=-2x\). More importantly, their acceleration ratio remains the same at every tested extension:

This raises a natural question:

Can the stable difference be assigned to the carts, leaving a separate description of the spring?

Define relative inertial mass by the inverse acceleration ratio when the carts are tested under the same interaction:

If A accelerates twice as much as B, B is assigned twice the inertial mass. Assigning one reference cart a unit mass fixes the overall scale.

This definition is useful only if the same mass ratios work with different springs, collisions, and other controlled interactions. That cross-experiment consistency is an empirical discovery, not a consequence of the notation.

7. Force: the part that belongs to the interaction

Once masses have been calibrated, inspect the measurements again. At any fixed extension, they reveal

The common value depends on the spring’s extension but not on which test cart we attached. This is exactly the object we were looking for: a description of the interaction that can be reused when the cart changes. We give it the name force:

This definition was not needed to solve the one-cart problem. It earns its role only because experiments reveal that \(ma\) separates the spring from the cart. If no cart-independent common value existed, this definition would not provide the desired generalization.

For the measured spring, that common value varies linearly with extension, so we report its behavior as

The constant \(k\) characterizes the spring in our chosen mass units. The minus sign records that the acceleration points toward the relaxed position. Combining the cart-independent spring rule with the cart’s inertial mass gives

Now the solution works for any calibrated cart attached to that spring:

and hence

Force is therefore the reusable interaction-side factor that emerged from trying to generalize an already solved experiment.

8. Momentum and impulse: the accumulated effect

Consider one cart whose initial velocity \(v_1\) is known. Suppose its acceleration during the interval from \(t_1\) to \(t_2\) has been measured or modeled. Ask:

Can we predict its final velocity \(v_2\) without already knowing it?

Because acceleration is the rate of change of velocity,

Integrating the known acceleration gives the answer:

or

That completely solves the one-cart problem. Momentum and impulse are not needed yet.

Now replace the cart-specific acceleration description with the reusable interaction description discovered in the previous section:

For constant mass, the prediction becomes

The same time integral appears whenever we ask for the accumulated effect of a force history, so give it a name—impulse:

This integral calculates the total effect of the changing force over the interval—the area under its force-versus-time graph. Once \(J\) is known, dividing it by the cart’s mass gives the total velocity change.

The final-velocity prediction is now

Multiplying by \(m\) gives

This repeated endpoint quantity motivates defining momentum:

The prediction then takes the compact update form

Momentum is a property of the cart’s state, not an object-independent quantity. Impulse summarizes the force accumulated during a particular interaction, and momentum records how that interaction changed the cart.

9. The question that motivates energy

Return first to the single cart-spring pair from Section 5. We know its measured acceleration rule \(a(x)=-\omega^2x\) and release it from rest at \(x_0>0\). Ask:

What will the cart’s maximum speed be, and can we find it without calculating its position at every intermediate instant?

While the cart moves from \(x_0\) toward the center, its acceleration and velocity point in the same direction, so its speed increases until it reaches \(x=0\). We therefore want its speed at the center.

The acceleration is given as a function of position rather than time. The chain rule connects it to the change of velocity with position:

Integrating from an initial state \((x_1,v_1)\) to a final state \((x_2,v_2)\) gives

For our cart, \(v_1=0\), \(x_1=x_0\), and \(x_2=0\). Therefore,

so

That solves the one-cart problem without force, work, or energy.

Now replace the cart-specific acceleration rule with the reusable interaction description from Section 7:

Multiplying the general one-cart result by the constant mass gives

The integral calculates the accumulated effect of the force as the cart moves through space—the signed area under the force-versus-position graph. Give this quantity the name work:

The velocity-side integral can be split explicitly at the reference velocity \(0\). Using \(u\) as the integration variable,

The two terms have the same form; only the endpoint velocity changes. This motivates assigning that quantity to each state and naming it kinetic energy:

The prediction then takes the compact update form

Kinetic energy belongs to the cart’s state. Work summarizes how a particular force changes that state while the cart moves through a displacement. The factor \(\tfrac12\) is not chosen arbitrarily; it arises from integrating \(v\,dv\).

10. Potential energy: recording reversible storage

The work formula answers a question about one particular trip from \(x_1\) to \(x_2\). But calculating a new integral for every pair of endpoints repeats the same work. This suggests a new question:

Can we assign one number to each spring configuration so that the work between any two configurations is obtained by subtracting their numbers?

For the spring, the force law is

Split its work integral at the relaxed position \(x=0\):

Each term now depends on only one endpoint. Define the potential energy of a configuration, relative to the relaxed spring, by

The minus sign makes work done by the spring equal a loss of potential energy. Indeed, the split integral becomes

For \(F(u)=-ku\), the assigned state quantity is

The zero at \(x=0\) is a convenient reference; adding the same constant to every value of \(U\) would not change any prediction.

Section 9 showed that the spring’s work changes the cart’s kinetic energy:

The endpoint construction above shows that the same work is the negative change of potential energy:

Therefore,

This motivates defining the total energy of the isolated cart-spring system:

Unlike the separate values of \(K\) and \(U\), this sum remains constant during the motion.

For a cart released from rest at \(x_0\), the center speed follows immediately:

Therefore,

Energy provides a compressed route from the initial state to constraints on the final state. It can answer some questions without reconstructing every intermediate moment.

The construction works when the interaction’s work depends only on its endpoints. If different paths between the same endpoints produce different work, no single-valued potential energy can summarize the interaction in this way. When it does work, the general relation is

Different interactions give different potential functions, such as

The particle-mechanics construction is now complete. It leaves a new question:

Are kinetic and potential energy the only possible kinds of energy, or are they merely the two parts that appear in this particular description?

Energy Beyond Particle Mechanics starts from that question.