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.
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3$\begingroup$ en.wikipedia.org/wiki/Rindler_coordinates $\endgroup$– CharlieJun 15, 2021 at 19:48
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$\begingroup$ 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. $\endgroup$– jng224Jun 15, 2021 at 20:26
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1$\begingroup$ 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? $\endgroup$– Chiral AnomalyJun 16, 2021 at 1:08
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$\begingroup$ @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? $\endgroup$– Jorge RamirezJun 16, 2021 at 16:33
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2 Answers
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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).
The Lorentz transformation is a particularly simple case (it corresponds to a kind of rotation in spacetime). But if either coordinate system is non-inertial then the transformations in general become more complicated, e.g. for rotating coordinate systems you're dealing with polar coordinates that vary with time. Convenient coordinate systems have been developed for a few of these special cases (Born coordinates for rotating frames, Rindler coordinates for linearly accelerating frames) but for the general case you have to do the hard work of explicitly defining coordinates for the non-inertial frame in terms of some other easier to handle coordinate system.
answered Jun 16, 2021 at 0:11
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There isn't one. Lorentz transformations deal with boosts and rotations of four-vectors. There is no acceleration parameter involved in defining the transformation. If such a transformation exists, it won't be called Lorentz.
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.
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$\begingroup$ In that case would it make sense to replace the ∆ with differentials, which would lead to dt‘=γ(t)(dt-v(t)/c^2*dx) and dx‘=γ(t)(dx-v(t)dt)?? $\endgroup$ Jun 15, 2021 at 23:32