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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 clock remains at a fixed position \(x\) on \(S\). Likewise, each clock on \(S'\) remains at a fixed position \(x'\). This allows a local recording device to store the pair \((t,x)\) or \((t',x')\) as soon as an event occurs there, without waiting for information to travel to another part of the laboratory.

Attach a contact switch and memory to each clock. When a clock on \(S'\) passes a clock on \(S\), their switches touch at one point and trigger both memories. This local contact is event \(P\). The clock on \(S\) stores its reading \(t_P\) and its fixed position \(x_P\). The clock on \(S'\) stores \(t'_P\) and \(x'_P\). The pairs \((t_P,x_P)\) and \((t'_P,x'_P)\) are the event’s coordinates in the two laboratories.

We idealize the switches as pointlike and subtract their measured internal delays. The records are collected afterward; no signal must travel across either laboratory when \(P\) occurs.

Focus on the center clock of \(S'\). Each time it contacts a clock on \(S\), that clock stores \(t\) and \(x\). Use these pairs to define \(x(t)\), the position of the center of \(S'\) at time \(t\) according to \(S\). The measurements fit

\[x(t)=x(0)+vt.\]

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 \((t',x')\) stored at the same local contact event?

From Light Signals to Spacetime constructed the rule relating observations made by inertial laboratories. We now begin with a laboratory near Earth and ask:

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.