Natural Science Forum / Physics / Research / July 2008

The equation you seek is easy to derive. It is a part of my standard
reply for frequently asked questions regarding a constantly
accelerating frame of reference.

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. 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 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 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 =96L] 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) =961]

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 |v/c| < 1.

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.

Shubee
http://www.everythingimportant.org/relativity/special.pdf