Accelerated Reference Frames in Special Relativity

Eugene Shubert

The equations for a coordinate system undergoing constant proper acceleration are:

x=x'cosh(ct'/x')
t=(x'/c)sinh(ct'/x') for all x'>0.

Permit me to illustrate their meaning. Think of an accelerating rocket. c is the speed of light. x' is a general position location in the accelerating rocket. (Call x' and t' rocket coordinates if you like). Believe it or not, in relativity, if a long rocket accelerates without any part of the ship being compressed or stretched during its motion, then an astronaut at the bottom of the rocket will feel a greater acceleration than an astronaut at the tip of the rocket. i.e., g=(c^2)/x'.

Here is how to use the equations:

Suppose a rocket ship, which looks like a long rod, begins to accelerate at t=t'=0. Suppose that the bottom of the rocket ship undergoes a constant proper acceleration g. For that case, assign the bottom of the ship the fixed point x' =(c^2)/g. (x=x' at t=0). Suppose the ship has a rest length of L. Then the tip of the ship has a proper acceleration of g' = (c^2)/(x'+L). Every point on the accelerated ship is carrying a clock. If some event happens on the ship, it will be at some x' with the clock at that point reading time t'. According to the coordinates of the stationary frame, the event will happen at point x at time t. All clocks are synchronized to read zero time at the instant acceleration begins.

Again:

x=x'cosh(ct'/x')
t=(x'/c)sinh(ct'/x') for all x'>0.

Eugene Shubert · Posted: Mon Jan 10, 2005 11:37 pm Post subject: Rigid Motion

Keeping your shape in every instantaneously-comoving inertial frame

Let x' be a fixed point on a resting rocket and L the proper length of the rocket. The equation of motion for the point x' at all subsequent times can be written as x'=f(x,t). I'm interested in the simplest case of one spatial dimension. What equation in special relativity ensures that there is zero stress at all points 0 < x' < L for the rocket during acceleration?

One point of an accelerating rocket determines the acceleration of every other point.
Long ago I thought I derived the equation describing the requirement that there is no compression or stretching of an accelerating rocket. The equation is

This equation is certainly satisfied for inertial motion and in the case of constant proper acceleration. Is it true for arbitrary acceleration?

Eugene Shubert · Posted: Sun Jan 16, 2005 7:11 pm Post subject: For the Accelerated Observer

Generalized Lorentz Transformation for an Accelerated Frame of Reference

This paper by Robert A. Nelson purports to derive an exact, explicit coordinate transformation between an inertial frame of reference and a frame of reference having an arbitrary time-dependent, nongravitational acceleration and an arbitrary time-dependent angular velocity.

The sophisticated formalism of this article is beyond my mathematical training. I would like to study a simplified, non-rotating, 1+1-dimensional version of the transformation. Can it be written out for someone at the undergraduate level? What is the exact, explicit coordinate transformation between an inertial frame of reference and an arbitrarily accelerated frame of reference for one spatial dimension?