Notes on Relativity and Cosmology - Physics Department, UCSB

5.1. THE UNIFORMLY ACCELERATING WORLDLINE 125
frame in which it is computed. The boost parameter of the reference frame itself
just cancels out.
What about measuring time? Well, as we have discussed, the ‘natural’ measure
of time along a worldline is the proper time. The proper time is again
independent of any choice of reference frame.
OK, so how does this relate to our above discussion? Let’s again think about
computing the proper acceleration α(E) at some event E using the momentarily
co-moving inertial frame. We have
α(E) = dv E
dt E
. (5.2)
What we want to do is to write dv E and dt E in terms of the boost parameter
(θ) and the proper time (τ).
Let’s start with the time part. Recall that the proper time τ along the rocket’s
worldline is just the time that is measured by a clock on the rocket. Thus,
the question is just “How would a small time interval dτ measured by this
clock (at event E) compare to the corresponding time interval dt E measured
in the momentarily co-moving inertial frame?” But we are interested only in
the infinitesimal time around event E where there is negligible relative velocity
between these two clocks. Clocks with no relative velocity measure time intervals
in exactly the same way. So, we have dt E = dτ.
Now let’s work in the boost parameter, using dθ to replace the dv E in equation
(5.1). Recall that the boost parameter θ is just a function of the velocity v/c =
tanhθ. So, let’s try to compute dv E /dt E using the chain rule. You can use the
definition of tanh to check that
Thus, we have
dv
dθ =
dv
dτ = dv dθ
dθ dτ =
c
cosh 2 θ .
c
cosh 2 θ
dθ
dτ . (5.3)
Finally, note that at event E, the boost parameter θ of the rocket relative to
the momentarily co-moving inertial frame is zero. So, if we want dvE
dτ
we should
substitute θ = 0 into the above equation 1 :
In other words,
α = dv E
dτ =
c
dθ
cosh 2 ∣
θ θ=0 dτ = cdθ dτ . (5.4)
dθ
dτ
= α/c. (5.5)
1 Note that this means that our chain rule calculation (5.3) has in fact shown that, for small
velocities, we have approximately θ ≈ v/c. This may make you feel even better about using
boost parameters as a measure of velocity.