Describing acceleration in special relativity is in principle straightforward, and for simple cases the resulting transformations are simple. Examples include circular motion and constant acceleration in the accelerating frame (the relativistic rocket). Anything more complicated is going to have to be done numerically, which is fine, but it's not immediately obvious to me how you'd go about this.
Let's call our frame $S$, and our metric is just the Minkowski metric. If we can write down an expression for the trajectory $x(t)$ in our coordinates $x$ and $t$ then everything is straightforward. But this isn't likely to be the case. It's more likely that the aceleration will be given in the accelerating object's frame $S'$ i.e. all we know is $a'(t')$.
So given that all we know is the form of $a'(t')$, how do we set about calculating the rocket's trajectory in our coordinates $S$? General principles will be fine as I'm sure I can work out the fine detail. It's just that I'm not sure where to start.
Assuming I'm not skirting too close to the homework event horizon, this might make a good blog type question. I've been thinking about writing an answer your own question post about acceleration in SR for some time.
Answer
This is an answer for motion in 1+1 dimensions. Let a dot stand for differentiation with respect to the rocket's proper time $t'$. The rocket's four-velocity is normalized, so
$$\dot{t}^2-\dot{x}^2=1\quad.\qquad (1)$$
Since the norm of the acceleration four-vector is invariant, we have
$$ \ddot{t}^2-\ddot{x}^2=-a'^2 \quad . \qquad (2)$$
Implicit differentiation of (1) gives
$$\ddot{t}=v\ddot{x} \quad ,$$
where $v=dx/dt$. If we substitute this into (2), we find
$$\ddot{x}=\gamma a'\quad.$$
Given $a'$ as a function of $t'$, this can be integrated numerically to find $x(t)$.
No comments:
Post a Comment