# The Einstein Express

In order to explain the effects of high speed movement on observations of length and time many textbooks on special relativity propose a train moving at a constant speed slightly less than the speed of light.  A passenger on the train makes some measurements of an event. The event is something happening such as measuring the time taken by the passenger to pass between two markers on the railway embankment (a term used by Einstein on page 25 of his book Relativity, the Special and General Theory, Methuen, fifteenth edition 1952).  An observer at rest on the embankment measures the time interval taken by the event. Do their results agree? Who is correct, the passenger or the observer on the embankment? Hop aboard for our ride on the Einstein express!

## The Relativity of Simultaneity

Einstein considered two points A and B on the embankment. The midpoint of AB is M. The train is moving in the direction from A towards B. The observer standing on the embankment at M measures simultaneous lightning strikes at A and B. This means that the observer receives the light from the strikes at the same instant. When the lightning strikes occur, according to the observer on the embankment, the person in the train is alongside M at a position M'. The light from the strikes travels towards in all directions and as the person in the train is moving away from the light coming from A and towards the light coming from B, the person in the train receives the strike at B before the strike from ATo quote Einstein, "events which are simultaneous with reference to the embankment are not simultaneous with respect to the train, and vice versa". He called this principle the relativity of simultaneity.

## The View from the Embankment

A large number of school textbooks incorrectly describe the visual appearance of the high speed train as it passes in front of the observer on the embankment. These textbooks correctly state that the high speed has no affect on the vertical height of the train but incorrectly state that the train appears contracted in length due to the Lorentz contraction. The correct description is that the light from the train is seen to be redshifted in colour and the train looks to be rotated in its own plane away from the observer and from side-on appears the same overall length as when seen at rest. These effects are due to the different travel times of the light arriving at the observer's eye at each instant. For example, the light coming from the far end of the train when it is further away will arrive at the observer at the same instant as light coming from the near end of the train when it is closer. The combination of the different travel times of the light rays and the length contraction effect due to special relativity results in a visual rotation of the train. A clear description of these effects is given by A P French on page 150 of his book Special Relativity, Van Nostrand Reinhold, 1984. French considers a rectangular board moving at a very high speed and viewed from a direction perpendicular to its velocity in its own plane. Using simple geometry, he shows that the observer sees a foreshortened view of the back end of the train and a length contracted view of the side of the train. The combination of these two effects is to produce a rotated view of the train with the overall sideview length of the train as seen by the observer the same as when the train is at rest. Another description of the combined effects of rotation and contraction is given on page 96 of the first edition of Spacetime Physics by  Edwin F Taylor and John Archibald Wheeler, W H Freeman and Company 1963.

## The View from the Train

The person on the train considers buildings on the embankment to be moving towards them at a very high speed. They see the approaching buildings to be blue shifted in colour and they appear to curve inwards so that their vertical height is reduced. The visual distortion is greatest at large distances from the train. The front edge of the buildings, normally not visible at very low speeds, now appears and becomes more visible as the speed increases. Buildings alongside the train appear to be redshifted and rotated towards the person on the train with their trailing edge being rotated away from the train. These buildings remain at the same vertical height and their overall side-on view length being the same as when seen from rest due to the combination of the effects of Lorentz contraction and rotation due to the different travel times of light rays coming from different points on the embankment arriving at the person's eye at the same instant. Buildings on the embankment behind the leaving train appear to be distorted in height and red shirted in colour. The person on the train sees the train as being identical in colour, shape and size as when it is seen at rest relative to the embankment.

## Which View of the World Is Correct?

Both views are correct. Einstein stated that all inertial reference frames have equal footing in terms of describing the world using the laws of physics. Due to the constancy of the speed of light, the effects of time dilation, length contraction and the travel times of light rays reaching the passenger's and observer's eyes the outside world appears distorted due to the very high relative speeds.

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