# The hole argument and diffeomorphism invariance

1. May 19, 2006

### hellfire

The "hole argument" and diffeomorphism invariance

First, let me give a summary of my understanding of the "hole argument": Consider a space-time completely filled with matter with exception of a finite space-time volume that contains no matter (a hole). The hole is located between two spatial hypersurfaces. On the first spatial hypersuface the initial conditions for the hole are defined. One may be able to perform a active smooth transformation that leaves unchanged the metric and matter outside the hole, but that changes only the metric within the hole. These two different configurations differ only within the hole that has, however, same initial conditions. This means that both configurations must be the same physical solution in general relativity. Otherwise general relativity would not be deterministic, because matter and initial conditions should determine one unique solution to the equations of motion.

Now to my question. Consider the same situation, but instead of performing a smooth transformation, take a more general transformation. For example, a transformation that leaves unchanged the outside, but exchanges of two points (permutation) within the hole. The result and conclusions of the hole argument should be the same. However, general relativity is based "only" on the principle that two solutions are equivalent if they are related via smooth space-time transformations...?

2. May 24, 2006

### hellfire

May be I should not have started this thread in the general relativity subforum, because it may be more related to philosophy or even to other possible theories beyond general relativity. Anyway, I will make one more attempt to get some comments.

The usual conclusion to the hole argument is to regard the individuality of space-time points as devoid of physical meaning, if general covariance has to be valid.

John Stachel makes use of the terms quiddity and haecceity. The first one describes the universal qualities and the second one the individuality. He makes use of the analogy with quantum mechanical particles, that have quiddity but no haecceity. The permutation symmetry of systems of multiple particles leaves the equations of motion and the description of the system invariant.

Stachel postulates a principle of maximal permutability and argues that in discrete systems the symmetry transformations to be applied consist in permuting elements, but that in continuous systems the most general transformation that can be applied consists of diffeomorphisms.

However, it is clear that the conclusions of the hole argument are the same if one takes also more general transformations for space-time, like permuting two space-time points.

3. May 24, 2006

### pervect

Staff Emeritus
I hate to let a good thread like this go unanswered, when there are so many less-than-good threads out there.

Unfortunately, I don't have a lot to say on the topic.

4. Aug 14, 2008

### atyy

Re: The "hole argument" and diffeomorphism invariance

Perhaps the solution is that actually spacetime points do have absolute meaning. Spacetime points are absolute in the sense of the allowable coordinate systems defined on them. So if you exchange two points nonsmoothly in an absolute sense, then the allowable coordinates can never be smooth. So you will never be able to get a physically equivalent metric for that region of space.

5. Aug 14, 2008

### atyy

Re: The "hole argument" and diffeomorphism invariance

Also the ability to define coordinates is an absolute thing in the sense that it is physical. The ability to describe physical trajectories with a metric is another absolute thing. So GR has two absolute things - the ability to assign smooth coordinates, and the invariance of a particle's proper time under smooth coordinate changes. In thermodynamics, we can assign smooth coordinates like P,V,T, so we have the first absolute structure, but not the second absolute structure of a metric.

Last edited: Aug 14, 2008