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

Question

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.

Answer 1

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.

Answer 2

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.