What is the most fundamental axiom of General Relativity?

[TrickyDicky]

I would like to get some foundational concepts straight about GR as it is currently understood (I guess it is not seen exactly the same it was almost a century ago even if the basic concepts remain, this I would like to elucidate here too).
For instance, I understand that a basic axiom of GR as it was presented by Einstein in 1915 is the existence of a curved Lorentzian 4-manifold that is taken as the starting point of the theory in which the EFE are applied. Is this generally agreed as a basic postulate of GR? It would seem Einstein's idea of gravity as curvature demands such a postulate.

My bringing this up arises from considering the always problematic concepts of background independence and gauge invariance in GR. According to this more modern view of GR usually put forward by QG theoriticians the metric of the manifold (as implied in the "curved Lorentzian manifold axiom above mentioned) seems to lose its axiomatic importance in favor of the connection and the existence of a smooth manifold as basic postulate, relegating metrics to particular solutions with particular physical (idealized or not) circumstances. So the doubt comes to mind what the most basic axiom of GR would be, the existence of a smooth manifold with a certain connection or the old gravity is curvature that demands the existence of a Lorentzian metric with curvature? Or both at the same time?

 

[WannabeNewton]

I don't quite understand your question there matey. If you're doing GR then the connection being used is the unique connection associated with the Lorentzian metric so why wouldn't the Lorentzian metric be presupposed (you can even start out assuming the linear connection is a priori unrelated to the Lorentzian metric but you will find in the end that the connection is necessarily compatible with the metric in GR if the metric is to solve Einstein's equations).

As a side note, the term "space-time" in general is simply a pair $(M,g_{ab})$ where $M$ is a smooth manifold and $g_{ab}$ is a Lorentzian metric*. There is no a priori requirement that $g_{ab}$ must solve the Einstein equations; it is only when we specialize to GR that such assumptions are made but until then we are just talking about the geometric structure of arbitrary Lorentzian 4-manifolds. Also note that there is no general existence theorem for pseudo-Riemannian metrics using paracompactness, which is something that does exist for Riemannian metrics. On the other hand, if a Lorentzian metric is endowed, it is easy to show that there will always exist a unique connection compatible with said metric.

*Some authors might define a space-time as a pair $(M,g_{ab})$ such that there exists a continuous (or smooth) time-like vector field on all of $M$ (such pairs are called time-orientable). This is to prevent pathological quotient manifolds where there exists a point and a lightcone in the tangent space at that point with no notion of future-directed and past-directed (think of the Mobius strip).

EDIT: Are you asking if a bundle theoretic approach to GR is more fundamental than Lorentzian metric approach? If so, read through this article: http://arxiv.org/pdf/1110.1018v1.pdf

 

[robphy]

It seems to me that your question is the following.
Although classical-GR appears to be successful with (M,g) is taken as the starting point,
the failure to find a quantum theory of gravity (QG) by "standard" quantization methods leads one to seek alternative foundational structures of "spacetime", which captures (M,g) in some classical limit.

Depending on one's approach to QG, different camps choose different starting points, emphasizing one aspect as fundamental and the others as derived [maybe only in special cases]. What structure one chooses is likely a function of what progress towards a theory that camp has been able to achieve. (Thinking out loud...) For example, can quantization methods be better applied to a connection (instead of the metric)? Other possible starting points... the full curvature tensor, the conformal tensor, the causal structure,... possibly fuzzed up by a discrete spacetime. (...and what of the foundations of Quantum Theory?)

A related question... staying classically... are there other sets of geometric structures [which may have better physical motivation] that can be used in place of (M,g)? Why the Lorentzian signature? Why 3+1? Why pseudo-Riemannian and not (say) Finslerian?

Questions arise like: "how well does this choice of fundamental structure determine some other structure [in some appropriate limit]?"
(e.g. Google search on "determine the metric" )

I think it's fair to say that there is no definitive axiomatic structure until a successful theory of quantum gravity has been achieved.
Here are some possibly interesting references on foundational questions for GR/QG...

https://www.physicsforums.com/showthread.php?p=4377122#post4377122 [from a thread in May started by you]

https://www.physicsforums.com/showpost.php?p=878413&postcount=7 [old post listing numerous references]

http://arxiv.org/abs/gr-qc/0006061v3 [old (2001), but interesting review article on Quantum Gravity by Rovelli]

http://iopscience.iop.org/0034-4885/37/10/001 [old (1974), but interesting review article on Quantum Gravity by Ashtekar and Geroch]

 

[TrickyDicky]

I don't quite understand your question there matey.

I of course agree with all you say in your post. See below.

It seems to me that your question is the following.
Although classical-GR appears to be successful with (M,g) is taken as the starting point,
the failure to find a quantum theory of gravity (QG) by "standard" quantization methods leads one to seek alternative foundational structures of "spacetime", which captures (M,g) in some classical limit.

Depending on one's approach to QG, different camps choose different starting points, emphasizing one aspect as fundamental and the others as derived [maybe only in special cases]. What structure one chooses is likely a function of what progress towards a theory that camp has been able to achieve. (Thinking out loud...) For example, can quantization methods be better applied to a connection (instead of the metric)? Other possible starting points... the full curvature tensor, the conformal tensor, the causal structure,... possibly fuzzed up by a discrete spacetime. (...and what of the foundations of Quantum Theory?)

A related question... staying classically... are there other sets of geometric structures [which may have better physical motivation] that can be used in place of (M,g)? Why the Lorentzian signature? Why 3+1? Why pseudo-Riemannian and not (say) Finslerian?

Questions arise like: "how well does this choice of fundamental structure determine some other structure [in some appropriate limit]?"
(e.g. Google search on "determine the metric" )

I think it's fair to say that there is no definitive axiomatic structure until a successful theory of quantum gravity has been achieved.
Here are some possibly interesting references on foundational questions for GR/QG...

https://www.physicsforums.com/showthread.php?p=4377122#post4377122 [from a thread in May started by you]

https://www.physicsforums.com/showpost.php?p=878413&postcount=7 [old post listing numerous references]

http://arxiv.org/abs/gr-qc/0006061v3 [old (2001), but interesting review article on Quantum Gravity by Rovelli]

http://iopscience.iop.org/0034-4885/37/10/001 [old (1974), but interesting review article on Quantum Gravity by Ashtekar and Geroch]

Thanks, robphy, you caught my meaning quite well. I in fact took as departure pont this section of the wikipedia entry on LQG:
"General covariance and background independence
Main article: General covariance
Main article: background-independent
Main article: diffeomorphism

...

In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth. These are the defining symmetry transformations of General Relativity since the theory is formulated only in terms of a differentiable manifold.

In General Relativitiy, General covariance is intimately related to "diffeomorphism invariance". This symmetry is one of the defining features of the theory. However, it is a common misunderstanding that "diffeomorphism invariance" refers to the invariance of physical laws under arbitrary coordinate transformations; this is untrue and in fact every physical theory is invariant under coordinate transformations. Diffeomorphisms, as mathematicians define them, correspond to something much more radical; intuitively a way they can be envisaged is as simultaneously dragging all the physical fields (including the gravitational field) over the bare differentiable manifold while staying in the same coordinate system; diffeomorphisms are the true symmetry transformations of General Relativity, and come about from the assertion that the formulation of the theory is based on a bare differentiable manifold, but not on any prior geometry - the theory is background-independent (this is a profound shift, as all physical theories before general relativity had as part of their formulation a prior geometry). What is preserved under such transformations are the coincidences between the values the gravitational field take at such and such a `place' and the values the matter fields take there, from these relationships one can form a notion of matter being located with respect to the gravitational field, or vice versa. This is what Einstein discovered, physical entities are located with respect to one another only and not with respect to the spacetime manifold - as Carlo Rovelli puts it: "No more fields on spacetime: just fields on fields.".[4] This is the true meaning of the saying "The stage disappears and becomes one of the actors"; space-time as a `container' over which physics takes place has no objective physical meaning and instead the gravitational interaction is represented as just one of the fields forming the world. This is known as the relationalist interpretation of space-time. The realization by Einstein that General Relativity should be interpreted this way is the origin of his remark "Beyond my wildest expectations".

In LQG this aspect of General Relativity is taken seriously..."End Quote wikipediaSo if the formulation of the theory is based only on the differentiable manifold if we are to keep the key principle of general covariance and no prior geometry, I was wondering if this view about the basic postulate of GR is more fudamental than the one that stresses gravity as curvature that needs more structure as a basic postulate since curvature implies a metric, that is a Pseudoriemannian manifold, not just a differentiable manifold.

 

[WannabeNewton]

Oh well in that case I would have to agree with you. The notion of background independence is in and of itself more fundamental than coming down to Earth and talking about the physical gravitational field itself.

 

[TrickyDicky]

Oh well in that case I would have to agree with you. The notion of background independence is in and of itself more fundamental than coming down to Earth and talking about the physical gravitational field itself.

Hi WN. I didn't take sides yet so I'm not sure we agree or exactly about what. It was a question. I certainly consider the gravitational field pretty fundamental in a theory of gravitation.

 

[WannabeNewton]

Hi WN. I didn't take sides yet so I'm not sure we agree or exactly about what. It was a question. I certainly consider the gravitational field pretty fundamental in a theory of gravitation.

Sure but the point is that since the gravitational field takes the role of the geometry and in GR there is no a priori fixed background geometry assumed as we must actively solve the field equations for the metric tensor i.e. they will have their own dynamical evolution on a smooth manifold, the background independence is in and of itself more primitive I would say. So while the physical gravitational field and its effects can be associated with the dynamical nature of the metric tensor, the idea that there exists field equations for the geometry in the first place i.e. that it is never fixed beforehand is quite a more profound statement, mainly in the context of higher theories.

Nevertheless, I'm having trouble seeing why background independence would make the levi-civita connection more fundamental than the metric tensor, and I specifically mention the levi-civita connection since this is the one we choose in GR. The idea of an arbitrary linear connection as a differential operator on smooth sections of vector bundles which is in general independent of the dynamical metric tensor is certainly more primitive in the sense of background independence but if you fix the levi-civita connection then surely it depends on the dynamical geometry.

 

[TrickyDicky]

Sure but the point is that since the gravitational field takes the role of the geometry and in GR there is no a priori fixed background geometry assumed as we must actively solve the field equations for the metric tensor i.e. they will have their own dynamical evolution on a smooth manifold, the background independence is in and of itself more primitive I would say. So while the physical gravitational field and its effects can be associated with the dynamical nature of the metric tensor, the idea that there exists field equations for the geometry in the first place i.e. that it is never fixed beforehand is quite a more profound statement, mainly in the context of higher theories.

Nevertheless, I'm having trouble seeing why background independence would make the levi-civita connection more fundamental than the metric tensor, and I specifically mention the levi-civita connection since this is the one we choose in GR. The idea of an arbitrary linear connection as a differential operator on smooth sections of vector bundles which is in general independent of the dynamical metric tensor is certainly more primitive in the sense of background independence but if you fix the levi-civita connection then surely it depends on the dynamical geometry.

We are basically on the same page but I'm not sure background independence make the Levi-Civita connection more fundamental than the metric, either one implies the other. Rather it seems to favor the arbitrary linear connection of the smooth manifold view in the "no prior geometry/no absolute geometrical objects" previous step, that would only be fixed when some preferred coordinates/frames that break the general covariant symmetry are chosen(i.e. Schwarzschild, ...) for the case at hand/specific solution of the EFE.
It is important to make a distinction between two situations: all the vacuum solutions (Rab=0) where one usually introduces only some geometrical assumption like spherical symmetry or the like to get to some specific solution. One can easily change those geometrical conditions without much troble since we have agreed our theory is generally covariant only up to difeomorphisms which means we have background independence in the sense of no absolute geometry.
And the other situation is the one where we use the EFE including the stress-energy tensor, like for instance when we obtain the Friedmann solution, in thsi case we have to add as boundary condition a specific stress-energy tensor with a specific property like being a perfect fluid. It is somewhat harder IMO to justify physically that GR has "no prior stress tensor" in the same way it has "no prior geometry".

 

[atyy]

Some people use the Palatini action instead of the Hilbert action in order to vary the connection.

 

[TrickyDicky]

Of course here is where the energy conditions in GR enter as explained in their WP article: " the Einstein field equation is not very choosy about what kinds of states of matter or nongravitational fields are admissible in a spacetime model. This is both a strength, since a good general theory of gravitation should be maximally independent of any assumptions concerning nongravitational physics, and a weakness, because without some further criterion, the Einstein field equation admits putative solutions with properties most physicists regard as unphysical, i.e. too weird to resemble anything in the real universe even approximately. The energy conditions represent such criteria."

I would say it is not only a weakness but a serious problem of the theory as the SET represents matter (and radiation and all the nongravitational interactions) and while in general physicists have no problem admitting that abstract geometries have no absolute existence so background independence is seen even as something desirable in a physical theory, in the case of matter and physical interactions, not considering them as something defined and unique should be hard to swallow, and fixing it with arbitrary energy conditions appears not to be very satisfactory.

 

[atyy]

Within classical GR, the EP or "minimal coupling" is usually regarded as as the other fundamental principle of GR (in addition to background independence of the spacetime metric). If one starts from an action containing the Einstein-Hilbert term and the generally covariant form of the matter actions of special relativity, we then get the EFE and the correct form of the stress-energy tensor, and additional "equations of state" such as Maxwell's equations that describe the matter. In old books, this form of the EP is stated as "comma goes to semicolon". The EP in GR is the same as the "gauge principle" in field theory. Both govern the coupling of "forces" and "charges", and are more accurately called minimal coupling prescriptions.

The story has been argued to be cleaner in quantum GR, treating GR as the theory of quantum spin 2. Weinberg has argued that quantum spin 2 implies the EP. (http://arxiv.org/abs/1007.0435 see section 2.2, especially section 2.2.2.)

 

[TrickyDicky]

Within classical GR, the EP or "minimal coupling" is usually regarded as as the other fundamental principle of GR (in addition to background independence of the spacetime metric). If one starts from an action containing the Einstein-Hilbert term and the generally covariant form of the matter actions of special relativity, we then get the EFE and the correct form of the stress-energy tensor, and additional "equations of state" such as Maxwell's equations that describe the matter. In old books, this form of the EP is stated as "comma goes to semicolon". The EP in GR is the same as the "gauge principle" in field theory. Both govern the coupling of "forces" and "charges", and are more accurately called minimal coupling prescriptions.

The story has been argued to be cleaner in quantum GR, treating GR as the theory of quantum spin 2. Weinberg has argued that quantum spin 2 implies the EP. (http://arxiv.org/abs/1007.0435 see section 2.2, especially section 2.2.2.)

Atyy this is fine and it is taken into account in what I have posted but it is a little orthogonal to what I'm saying about the problems of the SET in GR that demand energy conditions to be "put by hand".
Minimal coupling is actually a heuristic mathematical procedure for going from SR to GR in the simplest form and that was implicitly used for many decades before wording it as a principle , and once it was it has been given several theoretical justifications from the EP and Hilbert variations as you mention to the general covariance itself etc...

 

[WannabeNewton]

I'm not fully understanding your juxtaposition of the nature of the SET in GR and background independence of the geometry. Could you restate it in some way? Perhaps elucidate a bit further? I'm just having trouble interpreting your words. Cheers.