Special Relativity (alternative Formulations) - Curvilinear Coordinates and Non-inertial Frames

Curvilinear Coordinates and Non-inertial Frames

Equivalent to the original ? Curvilinear is a generalization, but the original SR can be applied locally.

There can be misunderstandings over the sense in which SR can be applied to accelerating frames.

The confusion here results from trying to describe three different things with just two labels. The three things are:

• A description of physics without gravity using just "inertial frames", i.e. non-accelerating Cartesian coordinate systems. These coordinate systems are all related to each other by the linear Lorentz transformations. The physical laws may be described more simply in these frames than in the others. This is "special relativity" as usually understood.
• A description of physics without gravity using arbitrary curvilinear coordinates. This is non-gravitational physics plus general covariance. Here one sets the Riemann-Christoffel tensor to zero instead of using the Einstein field equations. This is the sense in which "special relativity" can handle accelerated frames.
• A description of physics including gravity governed by the Einstein field equations, i.e. full general relativity.

Special relativity cannot be used to describe a global frame for non-inertial i.e. accelerating frames. However general relativity implies that special relativity can be applied locally where the observer is confined to making local measurements. For example an analysis of Bremsstrahlung does not require general relativity, SR is sufficient. For examples see Can Special Relativity handle accelerations?, Differential aging from acceleration, an explicit formula and SR treatment of arbitrarily accelerated motion.

The key point is that you can use special relativity to describe all kinds of accelerated phenomena, and also to predict the measurements made by an accelerated observer who's confined to making measurements at one specific location only. If you try to build a complete frame for such an observer, one that is meant to cover all of spacetime, you'll run into difficulties (there'll be a horizon, for one).

The problem is that you cannot derive from the postulates of special relativity that an acceleration will not have a non-trivial effect. E.g. in case of the twin paradox, we know that you can compute the correct answer of the age difference of the twins simply by integrating the formula for time dilation along the trajectory of the travelling twin. This means that one assumes that at any instant, the twin on its trajectory can be replaced by an inertial observer that is moving at the same velocity of the twin. This gives the correct answer, as long as we are computing effects that are local to the travelling twin. The fact that the acceleration that distinguishes the local inertial rest frame of the twin and the true frame of the twin does not have any additional effect follows from general relativity (it has, of course, been verified experimentally).

In 1943, Moller obtained a transform between an inertial frame and a frame moving with constant acceleration, based on Einstein's vacuum eq and a certain postulated time-independent metric tensor, although this transform is of limited applicability as it does not reduce to the Lorentz transform when a=0.

Throughout the 20th century efforts were made in order to generalize the Lorentz transformations to a set of transformations linking inertial frames to non-inertial frames with uniform acceleration. So far, these efforts failed to produce satisfactory results that are both consistent with 4-dimensional symmetry and to reduce in the limit a=0 to the Lorentz transformations. Hsu and Hsu claim that they have finally come up with suitable transformations for constant linear acceleration (uniform acceleration). They call these transformations: Generalized Moller-Wu-Lee Transformations. They also say: "But such a generalization turns out not to be unique from a theoretical viewpoint and there are infinitely many generalizations. So far, no established theoretical principle leads to a simple and unique generalization."