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.

 

[PAllen]

Getting back to the original question, rephrased as: can the intrinsic geometry of a topologically trivial piece of a general Lorentzian manifold be equivalently treated as flat spacetime with rulers that vary in length with position and orientation, similarly for clocks rate. I think the answer must be yes, because this reproduces what a metric tells you. I see this as exactly what the quoted author is claiming,

 

[Demystifier]

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.

Yes, but I think the right question is this: Can general relativity (GR) "change" topology? Or more precisely, is GR a theory of the spacetime topology?

I think it isn't. GR is a local theory, in that sense it is not much different from the theory of "hot plate". When you find an analytic expression for the solution of the Einstein equation, that expression dos not determine the global topology of the spacetime.

Let me illustrate this by a few examples.

Example 1: 1-dimensional metric
$$ds^2=dx^2$$
The topology of such space can be the topology of the real line $\mathbb{R}$, but it can also be topology of a circle $S^1$.

Example 2: 2-dimensional metric
$$ds^2=-dt^2+dx^2$$
The topology can be $\mathbb{R}^2$, but can also be a cylinder $\mathbb{R} \times S^1$, or a torus $S^1\times S^1$.

Example 3: 1-dimensional "Schwarzschild"
$$ds^2=\frac{dr^2}{1-\frac{2M}{r}}$$
Is there a region with $r<2M$? It can be, but doesn't need to. For example, you can define the tortoise coordinate
$$dr_* = \frac{dr}{\sqrt{1-\frac{2M}{r}}}$$
which is defined only for $r>2M$. The metric is
$$ds^2=dr_*^2$$
for $r_*\in (-\infty,\infty )$. With the tortoise coordinate, it is natural to think of $r<2M$ as "beyond infinity", namely non-existing.

Those examples are deliberately trivial, to make them easily understood. But the same type of reasoning can be applied to non-trivial and more physical 4-dimensional examples, like the 4-dimensional Schwarzschild. The discussions in the literature often create an impression that their global topology is unambiguous, but it is not. Determination of the global topology from the metric contains a lot of guess work based on physical intuition. In particular, the maximal analytic extension of the metric does not need to be the physical one.

 

[martinbn]

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)

Which page?