Feynman's bug (hot plate) in generality

[gerald V]

TL;DR

Can variable devices living on a flat manifold create the impression of intrinsic curvature in sufficient generality?

A textbook (in German language: Sexl-Urbantke, Gravitation und Kosmologie) says that intrinsic curvature of spacetime is a definition. The alternative is to postulate a flat spacetime together with measuring devices that do not have constant length. These definitions are equivalent up to aspects of topology.

I am aware of Feynmans bug (hot plate), but had thought that this only works for special configurations. I have difficulties to understand the general case.

I assume that the specific nature of time can be left aside and spacetime be regarded as 4-dimensional Euclidean. This means, it is sufficient to have a single rod, which however changes in length as a function of its position and orientation inside this flat space.


My questions:

Is this behaviour of the rod actually sufficient to create the impression of an intrinsically curved spacetime in all necessary generality?

Is there literature where the respective calculations are done?


Thank you very much in advance.

 

[pervect]

I'm familiar with Einstein writing about the hot plate, but not about Feynmann doing so.

My take on the problem is that locally, the "hot plate" analogy works fine, but on a global scale, I don't believe adding in the "heat" that distorts the length of "rulers" on the plate is able to change the global topology of the plate. I.e., for instance, it's connectdness or it's compactness, for instance.

I don't have a formal reference for this, it's just my intuition.

 

[PeterDonis]

on a global scale, I don't believe adding in the "heat" that distorts the length of "rulers" on the plate is able to change the global topology of the plate. I.e., for instance, it's connectdness or it's compactness, for instance.

That's correct. It also can't change the underlying global topological manifold--for example, it can't make a flat planar plate into a 2-sphere.

I don't have a formal reference for this, it's just my intuition.

It's obvious: distorting the lengths of rulers means changing the metric, but the metric is a tensor on an underlying topological manifold. It makes no sense to talk about changing the metric if the underlying topological manifold is not fixed; there is no way to even compare the "before" and "after" metrics if the "before" and "after" underlying topological manifolds are different.