26
3 The Motion of Particles
Then from (3.23) above we get an elegant differential relation between rapidity and
the proper acceleration.
dθ
dτ
=
a
c
.
(3.25)
That is, the derivative of rapidity with respect to rocket proper time is the proper
acceleration divided by c. Accordingly if we are given the acceleration as a function
of proper time we may suppose that (3.25) has been solved to give the rapidity as a
function of the proper time (see Exercise 3.3 for the case of constant acceleration).
Now we may easily integrate to get the spacetime trajectory of the rocket. From
the definition of θ and the fundamental Lorentz relation dt/dτ = γ we can write β
as
β =
dx
d(ct)
=
1
c
dx
dτ
dτ
dt
=
1
cγ
dx
dτ
.
(3.26)
We thereby get a differential relation between dx and dτ
dx = βγ cdτ = (tanh θ cosh θ )cdτ = sinh θ cdτ.
(3.27)
Similarly we can get a differential relation for cdτ
cdt = c
dt
dτ
dτ = cγ dτ = cosh θ cdτ.
(3.28)
Using (3.27) and (3.28) we integrate to obtain the trajectory in terms of the rapidity,
which we may take to be a known function of proper time
ct =
τ
0
cosh θ dτ, x = c
τ
0
sinh θ dτ.
(3.29)
This is a complete solution to the general problem in one dimension; with the proper
acceleration given as a function of proper time equation (3.25) gives θ (τ ) and (3.29)
gives the trajectory. You should work out the special case of constant acceleration
as requested in Exercise 3.3; the result is a hyperbolic trajectory.