What is the ontology of general relativity ?

[vanesch]

What is the "ontology" of general relativity ?

Hi,

First of all, though I know some GR, I'm far from an expert on it. I used to think of the "ontology" of GR as being a 4-dimensional manifold with a metric connection on it, but apparently that doesn't fly, because of the "hole argument":

http://plato.stanford.edu/entries/spacetime-holearg/

I used to think that general covariance simply meant that we could choose arbitrary coordinates on this manifold, but it seems that this is not sufficient.
So my question is: if the "ontology" of GR is NOT a 4-dim. manifold + metric connection, then what is this ontology ? What, according to (classical) GR, is "out there" ; how do relativists see their subject ?

Or can we still see the ontology of GR as being a 4-dim manifold + metric connection, and the different physical descriptions as "gauge-equivalent" ?

 

[selfAdjoint]

Equivalence classes of diffeomorphisms on the manifold. At least I think that's what Einstein finallly came to after years of musing on the hole argument.

 

[robphy]

Your question appears to be a topic of current interest among philosophers of science
http://alcor.concordia.ca/~scol/seminars/conference/
http://www.spacetimesociety.org/conferences/2006/
...not to mention researchers in quantum gravity.

 

[vanesch]

Equivalence classes of diffeomorphisms on the manifold. At least I think that's what Einstein finallly came to after years of musing on the hole argument.

I'm trying to imagine what that is. Ok, this question belongs probably more in the differential geometry section, but I'm having a hard time seeing how two diffeomorphic manifolds are actually different manifolds. Of course, if two manifolds have other structure (for instance, are the result of a certain construction), they can be "different" in a certain respect, and nevertheless have a diffeomorphism between them. But a smooth manifold, to me, IS already an equivalence class of atlasses. As such, a diffeomorphism just shuffles the atlasses around within the equivalence class, so two smooth manifolds which are diffeomorphic ARE the same manifold to me, no ?

 

[JesseM]

http://plato.stanford.edu/entries/spacetime-holearg/

Not being well-versed in GR or differential geometry, I was confused by this explanation, in particular the diagram here:

What if you decided that the event "E" in the diagram should represent a particular time-reading on the internal clock of a test particle--surely then there could be no ambiguity over whether this event coincides with a point along the worldline of a galaxy? If so, then in general, why can't you unambiguously decide whether two points in different coordinate systems represent the "same event" or not by imagining a test particle whose path travels through that point in one coordinate system, and seeing if the particle's worldline passes through the corresponding point (at the same time, according to its internal clock) in the second coordinate system? Or does the problem that the "hole argument" illustrates not have anything to do with ambiguity over whether two points in different coordinate systems represent the "same event" or not?

 

[Hurkyl]

If I've interpreted correctly what I've read, this was essentially a measurement problem for General Relativity, and was thought to be a fatal flaw until it was figured out how one might possibly be able to "internally" perform a measurement. (Which, I suppose, is an explanation for the popularity of measuring things as they traverse a loop)

 

[JesseM]

This page has a shorter summary of the hole argument...it's still beyond me, but it may help others understand the details better:

Many authors have written about hole argument, mostly from an historical or philosophical perspective.4 But none provide a simple concise account of the argument, its rebuttal, and its lessons. My purpose here is to provide such an account.

Other authors use a general spacetime in the hole argument. We use the Schwartzschild spacetime, whose simple geometry makes the argument more concrete and visualizable. This is sufficient to understand the lessons of the hole argument.

The summary seems to say that the problem arises when you try to imagine "events" in the spacetime manifold having distinct identities before you impose a gravitational field on the manifold...so maybe there is no problem with thinking of each event having a distinct identify after you put in the gravitational field, which is what was confusing me earlier.

The hole argument uses, in an essential way, the concept of a spacetime of events without a gravitational field. For the argument (tacitly) assumes that events have a physical identity independently of a metric: “Consider an event with radial coordinate $$r = r_0$$. The event is on a sphere of area $$4 \pi r_{0}^2$$ under G and on a sphere of area $$4 \pi f^2 ( r_0 )$$ under G'.” This is what allows
the argument to conclude that "G and G' are physically distinguishable", which
leads to "physically unsatisfactory" results.

On the other hand, we have seen that the general covariance of general relativity implies that G and G' are physically indistinguishable. To block the hole argument and retain general covariance, general relativity must forgo any concept of a spacetime without a gravitational field. Thus in general relativity space and time are inseparable from a gravitational field: no field, no spacetime. This justifies Einstein's words at the start of this note. It is remarkable that such a deep result can be obtained from such simple considerations.