relativity paradox in exponentially accelerating train

kochanskij

Using Einstein's train thought experiment, suppose a train is accellerating exponetially so that the engine remains forever ahead of a beam of light without it ever reaching the speed of light.

V = c[ 1 - e^(-t)] for the train

The light would approach the engine but never quite reach it. So it would pass the second car in a finite time. If you are sitting in the second car, you could wait til the light beam passes you outside, then get up and walk to the front of the engine. Your speed would be very slow relative to the train and fast but less than the speed of light relative to the ground.
But in doing this walk, you will have caught up to and passed a beam of light. This should be impossible!

So what prevents this from happening? Is it impossible to walk to the front of the train? If so, why? It is a short finite distance and the acceleration of the train is never infinite.

Jeff
kochanskij@yahoo.com

JesseM

So all parts of the train have the same velocity as a function of time as seen in some inertial frame, meaning the train maintains a constant length in this inertial frame? If so, then if you look at the train's length in the instantaneous inertial rest frame of an observer who's accelerating with the train from one moment to the next, the length is continually expanding--see the Bell spaceship paradox. So my guess is that if you construct a non-inertial frame where a point on the second car is at rest (with the time coordinate in the non-inertial frame matching up with the proper time of a clock at that point, and simultaneity and distance in the non-inertial frame agreeing with the instantaneous inertial rest frame of that point at any given moment), then whatever the coordinate speed of the light beam as it passes that point as measured in the non-inertial frame, any observer who sets off from that point at a slower coordinate speed will never reach the front because the front is continually expanding away.

By the way, if someone wants to do a detailed mathematical analysis, the problem may be easier if instead of "exponential" acceleration you assume every part of the train is accelerating with the same constant proper acceleration (constant G-force as measured by anyone on the train, and constant coordinate acceleration in the instantaneous inertial rest frame of any point on the train at each moment). An observer with constant proper acceleration will still have a Rindler horizon where light emitted from points at or beyond that horizon will never catch up with them. And this observer's velocity as a function of time as seen in the inertial frame where they had a velocity of 0 at t=0 is given on the relativistic rocket page as v = at / sqrt(1 + (at/c)²), where a is the proper acceleration.