Lack of uniqueness of the metric in GR

[RockyMarciano]

That the metric tensor is not uniquely determined by the EFE and what this might entail has been a source of debate for about a century.
A way to view the problem is to decide what the manifold that has the property of diffeomorphism invariance and background independence exactly is in the theory.The options are a general pseudoriemannian differentiable manifold or a curved one.

 

[robphy]

The electromagnetic field is not uniquely determined by the Maxwell Equations. Initial conditions and boundary conditions are needed as well... as is the case for any differential equation.

 

[RockyMarciano]

The electromagnetic field is not uniquely determined by the Maxwell Equations. Initial conditions and boundary conditions are needed as well... as is the case for any differential equation.

Yes, the indeterminacy is well known to persist independently of the presence of initial conditions. Perhaps I should have specified more the context but I thought it unnecessary for an A-labelled question.
The basic difference with the EM case is that the metric tensor acts both as a dynamic field and as the background geometry, unlike the potentials in Maxwell's equations space.

 

[robphy]

Is there a specific question?
To invite folks to a conversation, it's good to provide context, details, and references.
References also help give a hint as to the level of background ("A"-level classification is helpful, as a first step.)

You might find interest in this chapter:
Geroch & Horowitz "Global Structure of Spacetimes"
https://books.google.com/books?id=pxA4AAAAIAAJ&pg=PA212&lpg=PA212&dq=geroch+horowitz

 

[martinbn]

That the metric tensor is not uniquely determined by the EFE and what this might entail has been a source of debate for about a century.
A way to view the problem is to decide what the manifold that has the property of diffeomorphism invariance and background independence exactly is in the theory.The options are a general pseudoriemannian differentiable manifold or a curved one.

This doesn't make sense.

 

[RockyMarciano]

This doesn't make sense.

which part?
What is your background in general relativity?

 

[RockyMarciano]

To be clearer, what should be the space in which the laws determined by the field equations of GR should be generally covariant, a general differentiable manifold M, or a curved pseudoriemannian manifold(a pair M,g with g a Lorentzian metric)?

 

[martinbn]

which part?

All of it. For example: you are contrasting general pseudo-riemannian manifolds and curved ones. But what do you mean? That these two types are distinct or that one could possible consider non metric connections, or something else? Unless you give more detail it is meaningless and no one can guess what you mean and comment.

What is your background in general relativity?

Self taught. What is yours?

 

[martinbn]

To be clearer, what should be the space in which the laws determined by the field equations of GR should be generally covariant, a general differentiable manifold M, or a curved pseudoriemannian manifold(a pair M,g with g a Lorentzian metric)?

This doesn't make things clearer to me. What do you mean by the field equations (I am guessing Einstein's field equations) if there is no metric? Also if you exclude compact manifolds, which are not physically interesting, every differentiable manifold admits a Lorentzian metric. The two classes are not really different.

 

[RockyMarciano]

What do you mean by the field equations (I am guessing Einstein's field equations) if there is no metric?

I'm trying to use the naive socratic questioning to construct a line of thought but I can see it is not working out.
So your answer is the second manifold. Everybody agrees?

Also if you exclude compact manifolds, which are not physically interesting, every differentiable manifold admits a Lorentzian metric. The two classes are not really different.

That every non-compact differentiable manifold admits a metric doesn't mean there is no valid distinction of category between smooth manifolds and (pseudo)riemannnan manifolds, i.e. not every smooth manifold is equipped with an inner product.
I should add that the differentiable manifold in the first case is meant to be equipped with a connection.

 

[RockyMarciano]

Is there a specific question?
To invite folks to a conversation, it's good to provide context, details, and references.
References also help give a hint as to the level of background ("A"-level classification is helpful, as a first step.)

You might find interest in this chapter:
Geroch & Horowitz "Global Structure of Spacetimes"
https://books.google.com/books?id=pxA4AAAAIAAJ&pg=PA212&lpg=PA212&dq=geroch+horowitz

I missed this, you are right.
I know the usual starting point in GR is a 4-manifold endowed with a Lorentzian metric, so my question seems strange, but I'm willing to include approaches to GR that have different starting points, like for instance Poincare gauge gravity(there is a recent insights article about it).

 

[PeterDonis]

I'm trying to use the naive socratic questioning to construct a line of thought but I can see it is not working out.

No, it isn't. Do you have a specific question about physics? If so, ask it. Otherwise we can just close this thread.

 

[RockyMarciano]

No, it isn't. Do you have a specific question about physics? If so, ask it. Otherwise we can just close this thread.

Fair enough. I'll make my question specific.
The invariance under general coordinate transformations(diffeomorphism invariance) is a key requirement for a theory to be considered physical. This is understood in practice as the statement that the field equations that hold on the manifold of the theory can be put in tensorial form.
If the manifold of the theory is a general differential manifold or isomorphic to it this is quite inmediate by definition.
In GR things are somewhat more complicated because the manifold on which the field equations must hold is not simply a differential manifold but a pair M,g with g a Lorentzian metric with a metric compatible connection(Levi-Civita connection). The presence of this Lorentzian metric and its vanishing covariant derivative seems to require that the diffeomorphisms of the manifold be also isometries, my question: is this so? is the diffeomorphism group of GR also an isometry group?

 

[atyy]

Hawkiing and Ellis, p64: Two models (M,g) and (M',g') are taken to be equivalent if they are isometric, that is if there is a diffeomorphism $\theta$: M→M' which carries the metric g into the metric g', ie. $\theta_{*}$g = g'. Strictly speaking then, the model for spacetime is not just one pair (M,g), but the whole equivalence class of pairs (M',g') which are equivalent to (M,g).

 

[vanhees71]

This is also a kind of gauge invariance. We've had a very nice thread about the topic and also a recent Insights article about GR as a theory gauging the Lorentz (or more generally Poincare) symmetry, which is another point of view of its foundations, which helps a lot to its understanding in terms of the modern paradigm of gauge invariance as a model-building scheme for the fundamental particles and interactions:

https://www.physicsforums.com/insights/general-relativity-gauge-theory/

 

[RockyMarciano]

This is also a kind of gauge invariance. We've had a very nice thread about the topic and also a recent Insights article about GR as a theory gauging the Lorentz (or more generally Poincare) symmetry, which is another point of view of its foundations, which helps a lot to its understanding in terms of the modern paradigm of gauge invariance as a model-building scheme for the fundamental particles and interactions:

https://www.physicsforums.com/insights/general-relativity-gauge-theory/

Indeed, this is the insights article I referred to above.
But this alternative point of view spontaneously breaks the symmetry of general coordinate transformations(as explained in the article and also in https://en.wikipedia.org/wiki/Higgs_field_(classical) ), leaving only local diffeomorphism invarriance. So I guess it can't be exactly equivalent to the usual one as stated in the Hawking quote by atyy above.
This is evident when checking that the alternative take on GR is only valid for the vacuum equations with unique solution. While the standard GR admits matter fields described with a nonvanishing energy-momentum tensor.

 

[RockyMarciano]

Hawkiing and Ellis, p64: Two models (M,g) and (M',g') are taken to be equivalent if they are isometric, that is if there is a diffeomorphism $\theta$: M→M' which carries the metric g into the metric g', ie. $\theta_{*}$g = g'. Strictly speaking then, the model for spacetime is not just one pair (M,g), but the whole equivalence class of pairs (M',g') which are equivalent to (M,g).

So this quote appears to answer my question in the affirmative. But it would be interesting to know if this equivalence class of pairs (M',g') is meant to refer to a particular solution g in which case it has no bearing on the diffeomorphism invariance of the field equations, that cannot be referred to a unique g unless they had a unique solution, which we know cannot be the case for the EFE since there are infinite different energy-momentum distributions.

 

[vanhees71]

Of course, you can as well couple matter fields in the gauge-theoretical approach. The only difference with Einsteinian GR is that if you do this with the Dirac field, you get a spacetime with torsion. For classical electrodynamics, where you have only classical charge-current distributions and the electromagnetic Abelian gauge field no torsion occurs. So there is no problem with general relativistic "macroscopic physics". Whether or not the occurrence of torsion when minimally coupling to fermions, I can't say.

 

[RockyMarciano]

Of course, you can as well couple matter fields in the gauge-theoretical approach. The only difference with Einsteinian GR is that if you do this with the Dirac field, you get a spacetime with torsion. For classical electrodynamics, where you have only classical charge-current distributions and the electromagnetic Abelian gauge field no torsion occurs. So there is no problem with general relativistic "macroscopic physics". Whether or not the occurrence of torsion when minimally coupling to fermions, I can't say.

Yes, with torsion you get another theory, Einstein-Cartan theory that is quite different mathematically, and I would disagree that physically they are "macroscopically" similar, i.e. there are no singularities(therefore no BHs, no Bing-Bang) and no propagation of torsion outside matter fields(therefore no gravitational waves), so it is a case where certain apparently small details lead to a big difference. In any case I'm only considering general relativity here.

 

[vanhees71]

I'm not familiar with Einstein-Cartan theory, but I cannot believe that it is so different for what's observed. In the macroscopic classical realm, including astrophysics and cosmology the only forces relevant are gravity and electromagnetism. The matter is macroscopic matter with the quantum effects pretty much irrelevant, i.e., you don't have spinors around to describe what's going on, and that's why there's no torsion of spacetime visible in the macroscopic realm. So I'd say, there's very little difference between standard GR and the gauge theory described in @haushofer 's Insight article as far as the yet observable applications of GR in astrophysics and cosmology are concerned.

Maybe the experts can shed more light on this question.

 

[RockyMarciano]

But the gauge theory described by haushofer has torsion set to zero and therefore is just GR in vacuum, it is not the Einstein-Cartan theory where you have to have torsion and nonsymmetric Ricci and Energy-momentum tensors, etc.

 

[RockyMarciano]

Continuing with my #17, logic dictates that the metric g that the quote atyy wrote in #14 refers to is of course some unspecified g (it just must obviously be a Lorentzian metric because the equations must hold on a Lorentzian manifold).

But then there's something worth clarifying, if diffeomorphisms (i.e. general coordinate transformations) in the manifold described in the quote are also isometries, then invariance under general coordinate transformations for the equations implies invariance under unspecified metrics g? How can the equations lead to a determined specific metric field then?