This is my standard reply for frequently asked questions regarding a constantly accelerating frame of reference.

The basic 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. This force is given by the equation g(x')=-(c^2)/x'.

Many interesting problems can be solved in special relativity by first learning how to interpret and use these three elementary equations.

Suppose a rocket ship, which looks like a long rod, begins to accelerate at t=t'=0. Let's say that the ship has a rest length of L. (x=x' at t=0). Suppose that the tip of the rocket ship undergoes a constant proper acceleration g. For that case, assign the tip of the ship the fixed point x'_b =(c^2)/g. Then the bottom of the ship undergoes a proper acceleration of g' =(c^2)/(x'_a) where x'_a = x'_b - 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.

From these equations, I shall derive the exact redshift/distance equation for light that is moving from the bottom of the rocket to the furthermost tip of the rocket. I propose the following strategy.

One helpful insight in physics is the observation that if we had a constant gravitational field, then the equation for the Doppler shift would be trivial. I reason as follows:

Let f be the initial frequency of a photon at some initial point in a constant gravitational field. Suppose that the photon is moving radially. Since the force felt by the photon is constant along its path, the Doppler shift would depend only on the distance traveled by the photon through the constant accelerative force. In this instance, the Doppler shift equation for a photon traveling a distance x would be

f' = fH(x).

If the photon would move an additional distance y, then

f'' = f'H(y).

Consequently, f''=fH(x+y) = fH(x)H(y) and we are left with the simple functional equation

H(x+y) = H(x)H(y)

The only continuous solution for this equation is of the form

H(x) =exp (alpha x) where alpha is some constant.

Comparing this equation with Einstein’s approximation leads us to the conclusion that alpha = -g/c^2.

Now for the more general problem of a photon moving radially in a varying gravitational field where the accelerative force -g is a known function of x, simply recall baby calculus and divide the photon path into infinitely many infinitesimal layers, at least in the limit, where the gravitational force is approximately constant at each layer. The infinite product

f|(final) =f [exp g(x'1)dx'/c^2] [exp g(x'2)dx'/c^2] ... [exp g(x'n)dx'/c^2] would convert to the integral

f|(final) =f exp[ integral g(x')dx'/c^2 ]

where g(x') = -c^2/x'

The limits of integration would be from x'= x'_a to x'= x'_b.

Remembering to divide g(x') by c^2, the exponential of the integral that is to be evaluated between the limits of [c^2/g -L] to [c^2/g] is easily computed to be 1-gL/c^2.

So f' = f (1-gL/c^2).

It is easy to check that this reasoning is correct. I shall now derive the equation in a completely different way and without exploiting Einstein's approximation. Consider the instantaneously co-moving inertial frame of reference at the moment that a photon at the base of the rocket (x'= x'_a) is emitted toward the tip (x'= x'_b). I will first determine the time it takes for the photon to arrive, in inertial coordinates, given that the front of the rocket (x'= x'_b) is moving at a constant proper acceleration g. I shall then figure out the final velocity from that.

From the perspective of the initial co-moving inertial reference frame

ct= L + c^2/g[sqrt(1+(gt/c)^2) -1]

or equivalently,

1-gL/c^2 = sqrt(1+(gt/c)^2) - gt/c

At this point, I insert the well-known formula:

gt/c = sinh gt'/c where t' is the elapsed proper time for any clock at x'_b.

Thus 1-gL/c^2 = exp (-gt'/c)

Now recall that v/c = tanh (gt'/c) is the very well-known formula for the final velocity of a clock undergoing constant proper acceleration g and recall that

arc tanh x = 1/2 ln[(1+x)/(1-x)] is always applicable for |x| < 1. Let x=v/c.

Consequently,

1-gL/c^2 = exp (-gt'/c) = sqrt [(1-v/c)/(1+v/c)], which again is the correct factor in the Doppler shift equation.

Thus f' = f (1-gL/c^2).

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?

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?