Derivation of Relative Clock Rates (Traditional View)

Introduction

In an accelerated space trip where the traveler returns to an inertial frame at a younger age than the equivalent person who does not travel, the first question one might ask is "where and when did this differential aging occur". One way of answering this question is to compare clock rates at equivalent times. In this document I will derive these equivalent times from which the rate ratios can be obtained.

Equivalent Times

Lines of constant time are obtained from the Lorentz time transformation equation.

$c T_{i} = γ (c t - \frac{x v}{c}) + C_{i}$

(1.1)

where $γ = \sqrt{1 - v^{2} / c^{2}}$ and C_i is a constant
so that $c t_{i} = (v / c) x_{i}$ and therefore these lines have, in the (ct,x) spacetime coordinate system, slope v/c.

In order to get the times in the inertial frame we first have to scale that frame. For our acceleration program the total trip time is given by:

$T_{e n d} = 4 (c / a) \sinh \frac{a t_{a}}{c} + 2 t_{c} c o s h \frac{a t_{a}}{c}$

(1.2)

It is handy to have the value of the following integral

$θ (t) = \int_{0}^{t} a (t') d t'$

(1.3)

Then the value of v/c is

$v / c = \tanh θ$

(1.4)

and the value of the rocket displacement x as a function of time is

$x (t) = c \int_{0}^{t} t a n h θ (t') d t'$

(1.5)

Now we must project the line from x(t) back to x=0 with slope v/c. This results in

$x_{0} (t) = c t - \tanh θ (t) x (t)$

(1.6)

The value of T(t) is obtained from the value of x₀(t) by appropriate scaling of the left hand T axis:

$c T (t) = \frac{T_{e n d}}{4 t_{a} + 2 t_{c}} x_{0} (t)$

(1.7)

Relative Clock Rates

Having obtained cT(t) we can obtain the relative clock rates by the equation

$\frac{d T}{d t} (t) = \frac{T (t + δ t / 2) - T (t - δ t / 2)}{δ t}$

(1.8)