Special Relativity: What differential equation describes an accelerated object from a non-inertial reference frame?

Question

I am looking for a set of differential equations (to be solved numerically for an educational program) that would describe the position and apparent time of an accelerated clock relative to a non-inertial reference frame. For now, all movement is along a single spatial dimension. I am not interested in considering the time light takes to travel between observers; assume that they are clever enough to account for this delay. As inputs to this system of equations, I would have:

• An arbitrary, time-dependent acceleration a(t) as experienced by an observer in the non-inertial frame. In practice, this would be a simple list such as a(t=0) = 0; a(t=1) = 0.2; ...
• The initial position x(t=0), clock time t'(t=0), and velocity v(t=0) of the accelerated clock as seen in the non-inertial reference frame
• Another time-dependent acceleration a'(t') which describes the acceleration experienced by an observer traveling with the clock.

To make this (hopefully) more clear, I'll give an example: an observer A, initially at rest, sees a relativistic rocket pass by carrying observer B. When A's watch reads t=0, she notes that B's position is x(t=0), B's speed is v(t=0) and B's (prominently displayed) clock reads t'(t=0). A has a list of instructions a(t) which tell her to set her rocket engines to achieve a specific acceleration at specific times. B has a similar set of instructions a'(t'). Both A and B consult their own watches when determining when to change their acceleration.

I have seen this question: What is the displacement of an accelerated and relativistic object?, which tells me enough to model the situation for an inertial observer, but I have not figured out how to reapply this for a non-inertial observer. Similarly, I can compute the worldlines of non-accelerated objects from the perspective of a non-inertial observer, but it seems something sneaky happens when both the observer and observed are accelerating.

I have also tried computing both worldlines from an inertial reference frame and then using the Lorentz transformation [x' = g (x - vt); y' = g (t - vx/c^2)] to ask what B's worldline would look like from A's perspective, but this did not work (I can describe more about this, but for now it should suffice to say that the worldlines were not remotely correct).

Answer 1

I would approach this by specifying a third observer C who is in an inertial frame. It's relatively easy to calculate the time of an accelerated frame wrt the inertial frame, and vice verse. So if you take some proper time for A, you can convert this into the time for the inertial observer then convert it again into the proper time for observer B.

John Baez's article on the relativistic rocket gives the equations for converting from the accelerated time to observer C's time and vice versa. However note that these are the equations for constant acceleration. If you want to model time dependant accelerations then You'll need to delve into chapter 6 of Gravitation by Misner, Thorne and Wheeler.

Answer 2

There is no such thing as a non-inertial reference frame in SR at all, actually. Attempts to build such a frame (Rindler coordinates) meet some essentially GRstic issues, such as imcompleteness, event horizon, particle horizon, coordinate singularity, some peculiar (gravitationally-affected) kinematics and physics as seen by the observer.

However, non-inertial bodies and non-inertial observers can be described with no problem. For the observer, the picture s/he sees around herself/himself could be most easily calculated, if you assume the "instantly comoving" inertial frame, that is the frame moving at the same velocity as the observer at taken instant. (See also here.) This is exactly adequate, since the observer could switch off her/his jet for a tiny instant, and become inertial. Though you should be careful in which values to consider in such a frame. Some things are immediately observable, such as photons arrived from th space, and the apparent position and velocity of the other rocket as seen by these photons. They are physical and could be measured with instruments. And also there are unobservable things, like the "simultaneous position and velocity" of the distant rocket, which could not be seen and could only be calculated beforehand, based on the information the observer has an access to. If the information (for example, about future accelerations of the distant rocket) is unavailable, then such things could be calculated by no means.