# Is there any generalised “Lorentz“ transformation for a non-inertial moving frame?

I am looking for a formula to calculate the coordinate transformations—let’s say for an accelerated moving frame or a rotating frame. I would be very happy if you could quote a book or PDF that contains that formula.

• en.wikipedia.org/wiki/Rindler_coordinates – Charlie Jun 15 at 19:48
• Related: What is the proper way to explain the twin paradox?. John Rennie derives a formula for time dilation in non-inertial frames; it is not a transformation though. – Jonas Jun 15 at 20:26
• To clarify: What do you mean by frame? In this context, the word frame typically refers to a coord system that has somehow been associated with particular observer(s), but in general, coord systems have nothing to do with observers. Coord systems and observers are separate concepts, unfortunately conflated with each other in some introductions to special relativity. We can use one coord system (e.g., Minkowski) for all observers. The only reason to switch is for convenience. Are you asking which coord systems might be more convenient for scenarios involving accelerating/rotating observers? – Chiral Anomaly Jun 16 at 1:08
• @ChiralAnomaly This is the problem, I am trying to solve. We have 2 observers. There is a constant acceleration between both observers. Observer 1 sees a particle moving with a trajectory of x(t). Which trajectory will see observer 2? – Jorge Ramirez Jun 16 at 16:33

To transform between coordinate systems you need to be able to express the new coordinates $$(t', x', y', z')$$ in terms of the original coordinates $$(t, x, y, z)$$. There's no general formula for this, it entirely depends on how the coordinate systems are related: that is, in general $$t'$$ is a function of all of $$t, x, y, z$$ and it need not even be a linear function (e.g. when going from Cartesian coordinates to polar coordinates).
However, nobody stopped us from Lorentz transforming from a frame with velocity $$v$$ to a frame with velocity $$v+dv$$ - the change is too small, we can hop. But if you keep hopping, you can reach the accelerating frame. This is essentially what we do in a derivation of Rindler coordinates. However, even that involves successive Lorentz transformations between intermediate frames (separated differentially) to transform into the non-inertial one - there is no one big transform, to my knowledge.