How to describe arbitrary accelerations in special relativity

Question

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 1

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 $x(t)$.

Answer 2

This post is a continuation of Ben's answer. I will use $\alpha = a'$ to avoid notational clutter. As Ben showed, we can write for 1-dimensional motion $$ \ddot{x} = \gamma\alpha, $$ where the dots are derivatives wrt proper time. The problem is that $\gamma$ contains $v = dx/dt$, so it is a function of coordinate time $t$ instead of proper time. We can eliminate this though: since $$ \dot{x} = \gamma v,\qquad \dot{t} = \gamma,\qquad c^2\ddot{t} = v\ddot{x}, $$ we find $$ \begin{align} \ddot{x} &= \dot{t}\alpha, \tag{1}\\ c^2\ddot{t} &= \dot{x}\alpha. \tag{2} \end{align} $$ Differentiating equation (2) gives $$ \dddot{x} = \ddot{t}\alpha + \dot{t}\dot{\alpha} = \frac{\alpha^2}{c^2}\dot{x} + \frac{\dot{\alpha}}{\alpha}\ddot{x}, $$ which we can write as a 2nd-order differential equation $$ \ddot{u} - \frac{\dot{\alpha}}{\alpha}\dot{u} - \frac{\alpha^2}{c^2}u = 0, $$ with $u(t') = \dot{x}$. Further integration gives $x(t')$. Incidentally, if we differentiate eq. (2), we get the same equation: $$ \ddot{\gamma} - \frac{\dot{\alpha}}{\alpha}\dot{\gamma} - \frac{\alpha^2}{c^2}\gamma = 0, $$ with $\gamma(t') = \dot{t}$. This was expected, since $$ c^2\gamma^2 - u^2 = c^2\dot{t}^2 - \dot{x}^2 = c^2. $$ Integrating $\gamma(t')$ gives $t(t')$, from which we can derive $t'(t)$ and finally $x(t) = x(t'(t))$.

Answer 3

How to describe arbitrary accelerations in special relativity

In trying to address this (first) question, I recommend the following coordinate-free, and invariant (frame independent) way of describing the acceleration of a participant ("object", "rocket", ...) $A$:

Given the trajectory of $A$ (coordinate-free, and invariantly) as the (ordered) set $\{ O, ..., P, ... Q, ..., W, ... X \}$ of (generally) distinct participants who $A$ met (in passing) in the course of a particular trial,
where participant $O$ may also be called the "origin of $A$'s trajectory", in this trial, and participant $X$ the "destination",

and given the (coordinate-free, and invariant) interval ratios between pairs of the corresponding events in which $A$ had taken part in this trial,
i.e. the real number values of ratios

$$\frac{s^2[~\varepsilon_{A P}, \varepsilon_{A Q}~]}{s^2[~\varepsilon_{A O}, \varepsilon_{A X}~]}$$

for all pairs of participants (such as for instance $P$ and $Q$, and also including $O$ and $X$) whom $A$ had met in the course of the trial.

The magnitude $|~\mathbf a_A[~Q~]~|$ of $A$'s acceleration at the (event of) the meeting and passing of participant $Q$ can thereby be expressed as

$$ \begin{array}{ll} |~\mathbf a_A[~Q~]~| := \frac{c}{\sqrt{\stackrel{~}{|~s^2[~\varepsilon_{A O}, \varepsilon_{A X}~]~|}}} \times {\text{Limit}}_{ \large{\left\{\frac{s^2[~\varepsilon_{A P}, \varepsilon_{A W}~]}{s^2[~\varepsilon_{A O}, \varepsilon_{A X}~]} \rightarrow 0 \right\} }} &~ \cr \scriptsize{ \left[ ~~ \sqrt{ \stackrel{~}{\frac{s^2[~\varepsilon_{A O}, \varepsilon_{A X}~]}{s^2[~\varepsilon_{A P}, \varepsilon_{A W}~]} }} \times \sqrt{ \eqalign{ \stackrel{~}{ \left( \frac{s^2[~\varepsilon_{A P}, \varepsilon_{A W}~]}{s^2[~\varepsilon_{A P}, \varepsilon_{A Q}~]} \right)}~ \left( \frac{s^2[~\varepsilon_{A P}, \varepsilon_{A W}~]}{s^2[~\varepsilon_{A Q}, \varepsilon_{A W}~]} \right)~ + \left( \frac{s^2[~\varepsilon_{A P}, \varepsilon_{A Q}~]}{s^2[~\varepsilon_{A Q}, \varepsilon_{A W}~]} \right)~ + \left( \frac{s^2[~\varepsilon_{A Q}, \varepsilon_{A W}~]}{s^2[~\varepsilon_{A P}, \varepsilon_{A Q}~]} \right)~ \\ - 2 \left( \frac{s^2[~\varepsilon_{A P}, \varepsilon_{A W}~]}{s^2[~\varepsilon_{A P}, \varepsilon_{A Q}~]} \right)~ - 2 \left( \frac{s^2[~\varepsilon_{A P}, \varepsilon_{A W}~]}{s^2[~\varepsilon_{A Q}, \varepsilon_{A W}~]} \right)~ - 2 } } ~~~ \right] }; \end{array} $$

and the average direction of $A$'s acceleration throughout a trial from event $\varepsilon_{A O}$ until event $\varepsilon_{A X}$ would be expressed in terms of families of suitable participants; namely as "towards" any one participant $B$ (if there exists one), for each pair of events in which $B$ took part as well, such as $\varepsilon_{A B P} \equiv \varepsilon_{A P}$ and $\varepsilon_{A B W} \equiv \varepsilon_{A W}$, for which

$$ \sqrt{ \frac{s^2[~\varepsilon_{A B P}, \varepsilon_{B Y}~]}{s^2[~\varepsilon_{A B P}, \varepsilon_{A B W}~]} } + \sqrt{ \frac{s^2[~\varepsilon_{B Y}, \varepsilon_{A B W}~]}{s^2[~\varepsilon_{A B P}, \varepsilon_{A B W}~]} } = 1, $$

and the instantaneous direction of $A$'s acceleration at the (event of) the meeting and passing of participant $Q$ would be expressed by the partial ordering of such families, wrt. a partial ordering of trials which all include event $\varepsilon_{A Q}$.

Now, the inverse problem may be addressed as well:

[...] that the ac[c]eleration will be given [...]

i.e. given the real number values (normalized acceleration magnitudes)

$$ |~\mathbf a_A[~Q~]~| \frac{\sqrt{\stackrel{~}{|~s^2[~\varepsilon_{A O}, \varepsilon_{A X}~]~|}}}{c} $$

for each coincidence event $\varepsilon_{A Q}$ of $A$ having met and passed a participant $Q$ within a trial from having left the origin (participant $O$) until having reached the destination (participant $X$) to derive the possible values of ratios $\frac{s^2[~\varepsilon_{A O}, \varepsilon_{A Q}~]}{s^2[~\varepsilon_{A O}, \varepsilon_{A X}~]}$, $\frac{s^2[~\varepsilon_{A Q}, \varepsilon_{A X}~]}{s^2[~\varepsilon_{A O}, \varepsilon_{A X}~]}$, $\frac{s^2[~\varepsilon_{A P}, \varepsilon_{A Q}~]}{s^2[~\varepsilon_{A O}, \varepsilon_{A X}~]}$, etc.;
foremost "as such" ("intrinsically"),
or also subject to additional "external" constraints, referring to additional participants, conditions related to the direction(s) of acceleration and/or conditions on "the geometry" expressed by the sought values of interval ratios.

Unfortunately, at the moment I don't know specific and effective mathematical methods to tackle such problems in general; but having stated the problem in principle one may at least consider the "brute force method": to take stock of all imaginable assignments $$s^2 : \mathcal S \times \mathcal S \rightarrow \mathbb R,$$ where $\mathcal S$ denotes the set of all coincidence events of the participants to be considered,
and to "just check which of those fit" the given acceleration values and constraints.

Some might even consider sprinkling distinct coordinate tuples on the distinct participants (and/or on the distinct events) being considered ...