Mentz114 said:
OK. My point is that frames are irrelevant in the sense that we need only find a frame independent analysis. The problem can be expressed in a frame independent way if can find a Lorentz scalar that tells us if the ships are getting further apart. There is a tensor, ##\theta_{ab}=\partial_a u_b## which has a contraction ##\theta={\theta^a}_a## which tells us if the rockets move apart or get closer. For the case in question it is ##\gamma^3 \partial_t\beta=\gamma^3\ a##. The ships are moving apart.
I'm sorry, did you already define \theta? The way you've defined it seems to treat u_b as a vector field, but how is that defined?
I tried to come up with a covariant formulation of the problem, and it seemed sort of complicated. Here are my thoughts on this:
Suppose you just want to characterize how stretched a string is, in a covariant way. Here's my approach: We label a point along the string connecting the rockets with a number \lambda, which is the distance along the string from one end to that point. Then we could define \mathcal{P}(\lambda, \tau) to be the event that the point labeled \lambda passes through at proper time \tau.
In terms of \mathcal{P}(\lambda, \tau) we can define two partial derivatives:
D^\mu = \frac{\partial}{\partial \lambda} \mathcal{P}
U^\mu = \frac{\partial}{\partial \tau} \mathcal{P}
U^\mu is just the 4-velocity of the point on the string labeled \lambda
So if we start at \lambda, \tau and consider a nearby piece of the string at \lambda + \delta \lambda, \tau + \delta \tau, the separation will be, to first order:
S^\mu = D^\mu \delta \lambda + U^\mu \delta \tau
Now, here's the tricky part, it seems to me. If we want to know the
spatial separation between two nearby pieces of string, that means that we have to choose \delta \tau so that S^\mu is purely spatial, in the local comoving rest frame of the string. That means that
U^\mu S_\mu = 0
because the 4-velocity is purely temporal, in that comoving rest frame. So that implies that:
U^\mu D_\mu \delta \lambda + U^\mu U_\mu \delta \tau = 0
Since U^\mu U_\mu = 1 for any object (in units with c=1), it follows that:
\delta \tau = - \delta \lambda \ U^\mu D_\mu
This is the value of \delta \tau that makes \mathcal{P}(\lambda, \tau) and \mathcal{P}(\lambda + \delta \lambda, \tau + \delta \tau) simultaneous, in the comoving rest frame of the string at label \lambda. So for this value of \delta \tau, the separation between the nearby pieces of string is given by:
S^\mu = \delta \lambda (D^\mu - (U \cdot D) U^\mu)
If there were no stretching or compression, then the separation between the two points would be of magnitude \delta \lambda. So a measure of the stretching or compression is the factor:
Q = - \frac{S_\mu S^\mu}{\delta \lambda^2} (the minus sign is because the square of a spatial vector is negative, in my convention)
If there is no stretching or compression, then Q = 1. If Q > 1, that means the string is being stretched. If Q < 1, that means the string is being compressed.
So writing it out,
Q = - (D^\mu D_\mu - (D^\mu U_\mu)^2) (where I again used U^\mu U_\mu = 1)
Just as a check, if every part of the string is at rest in Rindler coordinates, then that means
x(\lambda, \tau) = \lambda cosh(\frac{\tau}{\lambda})
t(\lambda, \tau) = \lambda sinh(\frac{\tau}{\lambda})
(The usual Rindler coordinates use X, T instead of \lambda, \tau. The relationship is just \lambda = X, \tau = \lambda gT)
then Q = 1 (I'm skipping the proof). So if the string is at rest in Rindler coordinates, then it is unstretched.
On the other hand, if every part of the string accelerates together at the rate, then:
x(\lambda, \tau) = \lambda + f(\tau)
t(\lambda, \tau) = g(\tau) (independent of \lambda.
Then
Q = 1 + U^2
where U = \frac{dx}{d\tau}
So Q > 1. So if the points on the string are undergoing simultaneous acceleration, then the string stretches.
