Simultaneity in Einstein's Train Thought Experiment

Note: This article summarises the ideas presented in: "Vlad, I. (2026) Time vs. Duration: A Reinterpretation of Special Relativity", a Vixra preprint:

Time vs. Duration: A Reinterpretation of Special Relativity

Introduction

The relativity of simultaneity is one of the central concepts of Einstein's theory of special relativity. It states that two events judged to occur simultaneously in one inertial reference frame are not necessarily simultaneous in another frame moving relative to the first. The principle follows directly from the Lorentz transformations and distinguishes Einstein's theory from the notion of absolute time in classical Newtonian mechanics.

Einstein illustrated this principle using a thought experiment involving a moving train and two lightning strikes. Since its publication in 1905, the example has become one of the best-known demonstrations of how different observers can assign different times to the same pair of events while remaining equally correct within their own inertial reference frames.

Einstein's original thought experiment

Einstein's thought experiment considers a train moving at a constant velocity relative to an embankment or railway platform.

Two lightning bolts strike opposite ends of the train. An observer standing on the platform is positioned midway between the two strike locations. Because light from both strikes reaches this observer simultaneously, the platform observer concludes that the lightning strikes occurred at the same time.

A second observer sits at the midpoint of the moving train. As the train moves towards one flash and away from the other, light from the forward strike reaches the train observer before light from the rear strike. The observer therefore concludes that the two lightning strikes were not simultaneous.

Special relativity explains that both conclusions are valid because each observer measures time within a different inertial reference frame. There is no preferred frame of reference, and simultaneity is therefore frame-dependent rather than absolute.

Alternative interpretation

Einstein's thought experiment can also be examined by explicitly comparing corresponding physical reference points on both the train and the track.

Rather than focusing solely on the arrival times of light signals, this approach considers the physical locations where the lightning strikes occur and whether those locations can be uniquely identified in both reference frames.

Apparent disagreements arise not from the relativity of simultaneity itself but from the selection and comparison of reference points. The observer on the train uses reference points only from the moving frame (the ends of the train and the lightning strikes), whereas the observer on the platform analyses reference points from both the inertial and moving frames. Consequently, the platform observer has access to a broader set of reference information when interpreting the events. According to this interpretation, when corresponding physical reference points are carefully identified and transformed between frames, observers can consistently identify the same physical events despite describing them with different coordinate systems.

If identical physical events are identified in both frames and appropriate Lorentz transformations are applied, the observers can compare equivalent measurements without ambiguity.

This interpretation does not reject the Lorentz transformations or the mathematical framework of special relativity. Instead, it proposes an alternative procedure for relating observations made in different frames.

Refined train scenario

The modified thought experiment considers a train moving at a constant velocity along a straight track.

Two lightning strikes occur simultaneously in the platform frame.

• a R(ed) lightning strike occurs on the left-hand side.

• a G(reen) lightning strike occurs on the right-hand side.

Each lightning bolt passes through a gap between two railway carriages before reaching the ground. As a result, every strike leaves a permanent mark both on the train and on the railway track.

The strike locations are labelled as follows:

• A′ on the train corresponds to A on the track.

• B′ on the train corresponds to B on the track.

The marks serve as physical reference points that may later be compared by observers in either frame (Figure 1).

Figure 1. A train is struck by two lightning bolts in the order RA’A–GB’B

Verification conditions:

The proposed interpretation defines three conditions for identifying simultaneous events.

• Common physical events

Each lightning strike must produce a mark on both the train and the railway track. If a strike leaves a mark on only one object, the two reference frames cannot be directly compared because the same physical event has not been recorded in both systems (non-simultaneity).

• Consistent correspondence

The physical correspondence between the marks must remain unchanged. The red strike must always correspond to points A and A′, while the green strike must always correspond to B and B′ (RA'A — GB'B). Changing these associations would mean that different physical events are being compared (non-simultaneity).

• Consistency of spatial measurements

The separation between the marks must remain consistent after transformation between reference frames.

In the platform frame, the length of a carriage at rest ($L_{0}$) appears contracted to $L'={\frac {L_{0}}{\gamma }}$. The observer compares the distance AB on the railway with the spatial projection of the segment A′B′ on the moving train. Therefore, $AB=A'B'=L'={\frac {L_{0}}{\gamma }}$

In the train frame, the observer compares the proper length of the train carriage with the Lorentz-contracted length of the corresponding track segment. $AB={\frac {A'B'}{\gamma }}={\frac {L_{0}}{\gamma }}$

These measurements should remain mutually consistent after applying the appropriate Lorentz transformation.

It is crucial to recognise that both observers agree on the length of segment AB, even though they observe it from different reference frames. This provides the first indication that simultaneity cannot depend on the reference frame.

Observer perspectives

Platform observer

The observer standing on the platform considers the railway to be stationary while the train moves with constant velocity.

The train therefore appears Lorentz-contracted.

Within this frame, the distance between the marks on the railway corresponds to the projected distance between the marks on the moving train.

Because both lightning strikes produce corresponding marks on the train and on the track, and because the order of the marks remains unchanged, the platform observer concludes that the strikes were simultaneous.

Train observer

The observer travelling on the train regards the train as stationary.

The train carriage therefore retains its proper length, while the railway appears Lorentz-contracted.

Although the train observer receives the light signals at different times, the physical marks remain permanently recorded on both the train and the track.

These physical records allow the observer to identify the same events as the platform observer after applying the appropriate Lorentz transformations between the two reference frames.

The differing arrival times of light are therefore regarded as affecting observation rather than the physical identification of the events themselves.

Illustrative examples

The alternative interpretation is illustrated using a series of hypothetical examples intended to demonstrate how different physical arrangements affect the comparison of events between the train and platform reference frames.

Example 1: Strike recorded only on the train

In the first example (Figure 2), the green lightning strike reaches points B′ and B, while the red lightning strike reaches only point A′ on the train. Because the train has moved, the red strike passes through the side of a carriage instead of through the gap between carriages and therefore leaves no corresponding mark on the railway.

Figure 2. A train is struck by two lightning bolts in the order RA’–GB’B.

Under the proposed verification criteria:

• only one lightning strike produces corresponding marks on both the train and the track;
• the sequence of recorded marks is incomplete;
• no complete comparison between the two reference frames is possible.

These conditions prevent the events from being classified as simultaneous because one of the required physical correspondences is missing.

Example 2: Different physical locations

In the second example (Figure 3), the train changes position between the two lightning strikes.

Figure 3. A train is struck by two lightning bolts in the order RB’A–GA’B.

The first strike (Red) leaves marks at B′ on the train and A on the track. After the train moves, the second strike (Green) leaves marks at A′ on the train and B on the track.

Although both lightning strikes leave marks on both objects, the train and track marks no longer correspond to the same physical locations.

The sequence of reference points therefore differs from that required by the proposed verification conditions (RB’A–GA’B instead of RA'A—GB'B).

The observers are no longer comparing identical physical events, and the strikes are consequently regarded as non-simultaneous.

Example 3: Changing separation between reference points

The third example (Figure 4) considers a situation in which the lightning strikes occur at different positions along the train as it moves.

Figure 4. A train is struck by two lightning bolts in the order RA’A–GB’B (non-simultaneous)

The green strike occurs first and leaves corresponding marks at B′ on the train and B on the track.

Later, after the train has advanced, the red strike leaves corresponding marks at A′ on the train and A on the track.

As a result, the distance $AB\not ={\frac {A'B'}{\gamma }}$ , measured in the train frame and $AB\not =A'B'$ in the platform frame.

The changing spatial relationship between the recorded marks indicates that the two events do not satisfy the required verification conditions for simultaneity.

The role of reference points

A central feature of the proposed interpretation is the distinction between observations and physical reference points.

In Einstein's original thought experiment, the emphasis is placed on the arrival of light signals at observers located in different inertial frames. The differing arrival times are interpreted through the Lorentz transformations and lead naturally to the conclusion that simultaneity depends on the observer's frame of reference.

The alternative interpretation instead emphasises the permanent physical evidence produced by the lightning strikes.

Because every strike leaves a corresponding mark on both the train and the railway, the marks provide identifiable reference points that remain available after the events have occurred.

Comparing these corresponding reference points enables observers in different frames to determine whether they are analysing the same physical events before applying coordinate transformations.

The procedure is compared to converting between units of measurement. Just as a length measured in centimetres must be converted before being compared with one measured in inches, measurements obtained in different reference frames must first be transformed using the Lorentz equations before direct comparison.

Within this interpretation, the Lorentz transformation remains the mathematical tool that relates the two frames. The emphasis is placed on ensuring that equivalent physical reference points are identified before the transformation is applied.

Comparison with Einstein's interpretation

The alternative interpretation differs from Einstein's original presentation primarily in the aspects of the thought experiment that it emphasises.

Einstein's analysis begins with the finite speed of light and demonstrates that observers moving relative to one another receive light signals in different ways. This leads to the conclusion that simultaneity is not absolute but depends upon the observer's inertial frame.

This reinterpretation does not dispute the finite speed of light or the Lorentz transformations. Instead, it argues that the identification of corresponding physical reference points should play a more explicit role when comparing observations between frames. A delay in receiving the information does not alter the occurrence of the event.

According to this view, disagreement between observers may arise not because simultaneity is frame-dependent but because different physical locations are inadvertently treated as equivalent (both observers must compare reference points from both frames).

The proposal therefore places greater emphasis on verifying that identical physical events are being compared before conclusions about simultaneity are drawn.

Discussion

The interpretation represents an alternative conceptual framework for analysing Einstein's train thought experiment.

Its principal contribution is the introduction of explicit verification criteria based on permanent physical markers created by the lightning strikes. These criteria are intended to ensure that observers compare corresponding physical events rather than relying solely on the timing of received light signals. This approach provides a clearer distinction between the observation of an event and the event itself.

This reinterpretation is therefore best regarded as an alternative conceptual analysis of Einstein's thought experiment rather than a replacement for the theory of special relativity itself.

Conclusion

• Rather than focusing primarily on the arrival times of light signals, this reinterpretation it emphasises the identification of corresponding physical reference points left by the lightning strikes. According to this interpretation, careful matching of these reference points, together with the application of Lorentz transformations, enables observers in different frames to compare the same physical events consistently.

• We should not conflate simultaneity with observing (simultaneity plane with observational slices through spacetime).

• While observational slices through spacetime are frame-dependent, simultaneity is determined by the temporal coordinate (t), which maintains its value across different frames.

• When two events occur simultaneously, they possess their own temporal reference frame—a simultaneity frame—which is preserved when transferred across different reference frames.

• When reference points are chosen carefully and proper transformations are applied, all observers should agree on whether events are simultaneous. Apparent paradoxes arise not from the physics itself, but from inconsistent application of reference points or failure to apply necessary corrections when transferring measurements between frames.