# Connection between pseudometric and Einstein elevator

I have a hard time understanding GR. I understand a lot (from a math point) about (pseudo)Riemannian manifolds, and I also learned about Einstein's elevator thought experiment. So let me elaborate:

• From a physics point of view, you can take the elevator and derive that light has to bend, also that there has to be gravitational time delay. So far so good.

• Then almost all the literature I saw, turns to the next chapters, and assumes from the previous discussion that it is clear the (pseudo)Riemannian metric is all what matters now.

For me, there is a bit gap, a 'how'/'why'/'what' in between. I do not see how the metric tensor relates to accelerated reference frames. I feel like I am missing something very obvious, but what is it? Can somebody elaborate?

lalala asked: I do not see how the metric tensor relates to accelerated reference frames.

The purpose of the elevator experiment is to showcase that a uniform gravitational field can be switched off merely by a change of reference frame. Such observers would be in freefall and they would experience them to be moving in uniform and straight line motion in space (i.e. locally inertial although for this case it holds globally). Equivalence principle states that by no experiment can these observers determine the existence of a gravitational force. So one needs to develope a formalism where the prediction of the equivalence principle be reproduced while also resulting in consistent prediction for the effect of gravity for observers that aren't freely falling.

For example, if we consider a particle in orbit around earth, then it's going in circular motion. However, this does not appear to be a straight line motion in space. Turns out that promoting space to spacetime and absorbing the influence of gravity into a pseudo-Riemannian metric tensor can make this motion into a straight line motion in "spacetime". In mathematical terms this is stated as $$\nabla_X X =0$$ where $$X$$ is the tangent vector along the geodesic of the particle in spacetime and $$\nabla$$ is the covariant derivative (which is itself derived from the metric).

Secondly, the elevator thought experiment does not cause the metric to become pseudo-Reimannian. Instead the latter is required for dynamics in spacetime to be predictive. As a pseudo-Reimannian metric sets the light cone structure, therefore it determines what can be causally influenced by a given event and vise versa.

In more detail, the purpose of the metric is to induce an inner product on spacetime. This allows us to measure length of worldlines. The metric additionally sets the light cone structure which allows us to distinguish timelike and spacelike paths. In short, the metric determines the geometrical nature of spacetime.

• The "absorption of the influence of gravity into the metric" can be seen for example by considering the slow velocity weak and static almost flat limit of the geodesic equation and then comparing it to newton's modified 2nd law under accelerated coordinate transformations. Accepting then the (strong) equivalence principle is then accepting that that correspondence is generally valid Commented Nov 23, 2023 at 12:03

I do not see how the metric tensor relates to accelerated reference frames.

The relationship is:

1. The metric tensor for an arbitrailty accelerating reference frame in flat spacetime has (pseudo)-Riemannian form, and furthermore various physical observations of what takes place in such a frame (especially inertial forces) are just like commonly observed physical consequences of gravitation.
2. Generalising now, we can consider the set of all manifolds in which the metric tensor has everywhere a (pseudo)-Riemannian form. This is not a generalization logically proved by or derived from point (1), but its possible interest does get some motivation from (1).
3. We can assert immediately that the same facts about physical phenomona we found in (1) will now be found in the more general scenario of (2) if we limit the discussion to small regions of spacetime. Of course we will also get a whole lot of new observations about larger regions of spacetime.

Finally, I think it is worth adding that it is NOT possible to "switch gravity off" by adopting an accelerating reference frame. What gets "switched off" is just the acceleration of a particle in free-fall relative to the chosen frame, and nearby particles have similar (but not identical) free-fall motions. But the tidal effects of gravity do not go away and, in particular, the curvature tensor is just that---a tensor---so it cannot be made to vanish by adopting any special frame (if it is non-zero).

• Thanks for the comment regarding switching off gravity locally. I have modified my slopy answer to switching off a uniform gravitational field to reflect this now.
– S.G
Commented Nov 23, 2023 at 20:07