# Einstein's trains, revisited

Einstein's famous trains thought experiment (or "trains gedanken") was outlined in chapters 7 through 10 of his "Relativity" book, and has been reworked by multiple authors since, sometimes with the trains replaced by pairs of rocket-ships. The point of the thought-experiment is to illustrate how two different laboratories with different states of motion can reconcile the idea that the speed of light must be constant for each laboratory, throughout space.

With a relativistic acoustic metric, the scenario plays out slightly differently, with experimenters only required to find the speed of light to be constant in their own immediate vicinities.

## The experiment according to special relativity

Lightning strikes two bridges $A$ and $B$ that cross a railway line, and an observer standing mid-way between $A$ and $B$ sees both flashes at the same moment - for this observer, the flashes are simultaneous. At the same instant two trains are passing alongside the trackside observer, and observers on these two trains, at almost the same location, also see both flashes at the same moment. If we then consider a light-signal associated with the lightning-strike moving parallel to the track from $A$ to $B$, then if each observer believes that the signal progresses at a velocity $c$ relative to their own state of motion, if the two bridges are nominally one lightsecond apart, the trackside observer will expect the signal $A\rightarrow B$ to take one second, the observer in a train moving with the light will expect it to take less then one second, and an observer moving in the opposite direction to the light will expect it to take more than one second.

Einstein reconciles these different interpretations with the same observed behaviour by pointing out that the agreed events that all observers need to see can be assigned different nominal distance and time coordinates by the different observers, if they each believe that $c$is constant in their own frame.

#### relativity of simultaneity

• In train 1, the observer argues that since bridge $A$ is receding and bridge $B$ is approaching, when they see the two flashes happening at the same moment, the lightning strikes must have happened at an earlier time when $A$ was closer and $B$ was further away, so the light-distances travelled are not the same as the current distances of the bridges - for both flashes to be seen at the same time, $A$ must have been struck by lightning slightly later than $B$.
• In train 2, bridge $B$ is receding and $A$ is approaching, so for the two flashes to be seen at the same moment, the light from the approaching $A$ will be reasoned to have travelled a greater distance then that from the receding $B$, so it is $A$ must have been hit by lightning first.

#### relativity of distance

We can now consider the reflected flash from $A$, moving towards $B$.

• In train 1, since $A$ and $B$ are seen to be moving in the opposite direction to a flash, the events corresponding to the flash being emitted at $A$ and received at $B$ are considered closer together (because $B$ is still approaching while the light is in transit).
• In train 2, since $A$ and $B$ are seen to be moving in the same direction as the flash, $B$ gets a little further away while the signal moves $A \rightarrow B$, and therefore the locations for the start and end of the light-journey are further apart.

#### overall behaviour

Under special relativity, different observers assuming a fixed lightspeed in different frames and seeing the same distant events will assign different interpreted distances and times to those events. With the appropriate use of the Lorentz transform, this then allows each observer to insist that light is really travelling at $c$ with respect to themselves.

While the paragraphs above only tell us that a solution is conceivable, Einstein then goes on to derive the necessary relationships that have to hold if every observer believes that lightspeed is globally constant, and presents the Lorentz transform as a necessary component, required to make it all work.

## The experiment according to Advanced GR

If we use a relativistic acoustic metric, the apparent discrepancy that Einstein solves with a Lorentz transform doesn't arise.

Suppose that we stand on Bridge $B$, and watch the flash from lightning striking bridge $A$, with portions of the flash travelling through both trains. If we take our cue from real-world physics rather than from coordinate-system geometry, Fizeau's result, that a moving particulate medium drags light and creates a lightspeed anisotropy, means that the signals travelling through trains 1 and 2 should advance at physically different rates and reach $B$ at physically different times - we should see the flash transmitted through approaching train 1 before we see the corresponding flash transmitted through receding train 2.

#### local c-constancy in either case

If each train contained a mirror-equipped laboratory, and light was not dragged at all by the trains, each observer would still obtain the same experimental round-trip averaged value for lightspeed (from the same combination of opposing "fast" and "slow" signals inside their carriages). But if both trains completely dragged light, each local experiment would still be reporting a measurement of constant $c$ because lightspeed would physically be travelling at $c$ w.r.t. their equipment.

Measured local lightspeed constancy is expected regardless of whether bodies drag light or not, but the two approaches differ in what they predict an observer in one train seeing if they watch what's happening in the other train.

#### different implementations of the principle of relativity

Under special relativity, the wavefront moving from $A$ to $B$ moves through the region at exactly the same rate regardless of what the trains are doing, and each experimenter's local sense of c-constancy can be extrapolated into the surrounding region, including the interiors of any other moving laboratories.

In an acoustic system, things are different – an observer in one train who looks out of their window can see the light propagating though the other laboratory at a "wrong" rate - lightspeed is measurably anisotropic. However, although both observers agree that there is a relative offset in light-velocities between their two laboratories, they cannot agree whose fault this is – they are forced to agree to disagree. An occupant of train 1, considering themselves to be stationary, can argue that the light travelling through train 2' reaches its destination later because the light is retarded by the train's opposing dragging effects. An observer in train 2 may insist that their own train is stationary and therefore cannot be dragging light – but that the light passing through the structure of train 1 is arriving earlier than its correct time due to the forward-dragging effects of train 1's motion.

While both observers can compare notes and agree that there is a relative lightspeed anisoptropy – one-way light-signals raced along parallel paths through both laboratories arrive at their target at recognisably different times – there is no an obvious way for the two experimenters to agree how to assign blame.

## Notes

• If we have two spaceships passing each other at high speed in deep space, special relativity requires experimenters in each ship to be unable to identify their own supposed motion, and for signals moving though ${SHIPB}$ to be moving at $c_\text{SHIPA}$ and vice versa. In the acoustic system, light propagates at $c_A$ within system $A$ and at $c_B$ within system $B$, but the region between has a gravitoelectromagnetic distortion expressing the relative velocity of the two physical systems – a signal sent from $A$ to $B$ is emitted at $c_A$, but after riding the gradient arrives at $c_B$. The observerspace shift equivalence principle, which views velocity-shifts as gravitation-equivalent, has the side-effect that the terminal velocity for the distortion effect associated with the relative velocity of two bodies has exactly the same value as the relative velocity. This "GEM" mechanism for local lightspeed constancy is different to the mechanism used in special relativity.
• This leaves us with two different mechanisms for local lightspeed constancy. The "GEM" mechanism appears to be unavoidable in problems involving objects with significant gravitational fields, but seems to generate different relationships to those of SR. Since we want all objects to use the same equations of motion, this suggests that the GEM-compatible regulation mechanism may also have to apply at everyday and atomic scales, which would mean that experimental $c$-constancy in mechanics may have to be curvature-mediated.