# General covariance

by zonde
Tags: covariance
 Share this thread:
Sci Advisor
P: 913
 Quote by PeterDonis I'm not sure that's true. Newton-Cartan theory has absolute time and an absolute slicing of the complete manifold into spacelike slices, so the spacetime isn't completely dynamic as it is in GR. (Btw, you said in an earlier post that N-C theory determines the "temporal metric" dynamically; I'm not sure that's true either. There is no gravitational time dilation in N-C theory.)
In NC one has a spatial and temporal metric, which are metric-compatibel defining a connection up to a two-form K. The temporal metric with lower indices is determined by its metric-compatibility. Inverses of these metrics are defined via projective relations, and the temporal metric with upper indices is not fixed by the metric compatibility conditions. One can then impose field equations as one likes in terms of the Riemann/Ricci tensor (the question if these equations can be derived via an action principle is a different matter), and the usual Newton-Cartan field equations are chosen such that all the dynamical metric components and components of the two-form K can be gathered into a Galilei-scalar, known as the Newton potential, and all the other metric components become constant. This last fact is the flat-space content of Newton-Cartan, and is an explicit choice; one could also choose other dynamics such that space is not flat, giving a Galilean theory of gravity with curved space (i.e. the transformations in the tangent space are the Galilei transformations).

Of course, because the field equations of NC just reproduce Newtonian gravity, there will be no time dilation. It also depends on what one calls "dynamics"; usually metric compatibility is not considered to be dynamics.

 I think the question here is, is GR the *only* possible theory that is BI in the way GR is? (I.e., with a *completely* dynamic spacetime metric.) I don't think anybody really knows the answer to that.
One could formulate Newton-Cartan theory without the flat space condition, giving an honest BI theory with metrics which are even after gauge-fixing dynamical. The question is if such a theory is always some limiting case of GR. One can also define stringy versions of Newton-Cartan, based on strings or even branes instead of point particles, see

http://arxiv.org/abs/1206.5176

These theories are not the usual Newtonian limits of GR, so in that sense GR (with the possible additional terms to the Einstein Hilbert action as Bien Niehoff mentions) doesn't seem to be the only BI theory.
 Sci Advisor P: 913 Anyway, I have the ambition to, once I thoroughly understand all this stuff, put it in some notes without all the usual explicit (often coordinate-free) mathematical mumbo-jumbo and vague terminology obscuring for me personally what's really going on. Somehow I still haven't found a nice and clear overview of the meaning of covariance, the meaning of and relation between active and passive coordinate transformations, etc. Even good books like Carroll couldn't really satisfy my needs. I sometimes have the feeling that a lot of physicists don't really care, and a lot of "philosophers of physics" make the discussion so obscure that it makes me wanting to run back to the "shut up and calculate"- mentality :D
Sci Advisor
PF Gold
P: 5,091
 Quote by haushofer Anyway, I have the ambition to, once I thoroughly understand all this stuff, put it in some notes without all the usual explicit (often coordinate-free) mathematical mumbo-jumbo and vague terminology obscuring for me personally what's really going on. Somehow I still haven't found a nice and clear overview of the meaning of covariance, the meaning of and relation between active and passive coordinate transformations, etc. Even good books like Carroll couldn't really satisfy my needs. I sometimes have the feeling that a lot of physicists don't really care, and a lot of "philosophers of physics" make the discussion so obscure that it makes me wanting to run back to the "shut up and calculate"- mentality :D
Did you try to read Ben Crowell's reference:

http://arxiv.org/abs/gr-qc/0603087

This makes a better attempt than most I've seen to try formalize what distinguishes GR from e.g. Newton-Cartan (for example). Unfortunately, its overall conclusion is that the matter is not yet resolved, after all these years; that ultimately, background independence, no prior geometry, no absolute structures, etc. is not yet subject to any rigorous, problem free definition.
Sci Advisor
P: 8,800
 Quote by haushofer But Newton-Cartan theory is in the same way "background independent": both the metrics (spatial and temporal) are dynamically determined by equations analogous to the Einstein equations of GR. The connection is not uniquely determined by both metrics; one obtains an extra vector field, but that does not change the matter. I could even apply the hole argument for Newton-Cartan theory in the same way as for General Relativity, because the EOM are gct-invariant. The solution should thus be found in the fact that GR is really a non-linear self-interacting theory of massless spin-2 particles which becomes clear after gauge-fixing, while for Newton-Cartan this cannot be said: after gauge-fixing one obtains a spin (spin is here wrt to the Galilei group!) 0 theory, which is static and non self-interacting. Does this make sense? And what does "background independence" really mean then?
Yes, I think the quantum spin-2 way is the best, since if one formulates GR as field on flat spacetime, then there is a flat spacetime which is clearly not background independent. Also, because of the comments in the paper PAllen mentions in #93.

What I'm not sure about is: does the field on flat spacetime contain cosmology? No need to include the "big bang singularity", but just the physically relevant bits that present observations constrain?
Sci Advisor
P: 913
 Quote by PAllen Did you try to read Ben Crowell's reference: http://arxiv.org/abs/gr-qc/0603087 This makes a better attempt than most I've seen to try formalize what distinguishes GR from e.g. Newton-Cartan (for example). Unfortunately, its overall conclusion is that the matter is not yet resolved, after all these years; that ultimately, background independence, no prior geometry, no absolute structures, etc. is not yet subject to any rigorous, problem free definition.
No, I still haven't due to other obligations, but I hope to read it carefully when I'm able to. It rather strikes me that all these terms are still not that well understood, and I like the fact that some people do make an effort to shed more light on it. The discussions here also help a lot, so thanks for that!

One of the reasons why I got so interested in this whole notion of "background independence" was because I heard the claim from some string theory critics that "any good theory of quantum gravity should be BI" and "string theory is not BI". But the meaning of this becomes, after these discussions, a bit blurry to say the least. (The primary reason was actually that for my master thesis I had to read Wald's article on "black hole entropy is Noether charge". It then occured to me I'd never really understood this whole business.)
Sci Advisor
P: 1,690
 Quote by atyy Yes, I think the quantum spin-2 way is the best, since if one formulates GR as field on flat spacetime, then there is a flat spacetime which is clearly not background independent.
Again, it depends on definitions. If you are going by the definitions that Ben Niehoff for instance is using above, then the above *is* background independant.

Note the difference between the two following definitions...

1) Background dependance is tantamount to using the background field method for gravity, and ONLY for gravity (eg the metric tensor is split into a classical but arbitrary fixed background metric + a small perturbation). The approximation is valid up to some cutoff, whereupon the backreaction of the pertubation on the background can no longer be ignored.

2) Background independance is like asking whether the metric field is dynamical or not in the Lagrangian of the theory. In the sense that if you look at the variation in the action and consider (d/d&G), then you look for something that vanishes. So for instance, coupling a topological field theory to a theory with curvature invariants is clearly background independant in this definition. The terms with curvature invariants, owing to their general covariance, will integrate out any metric dependance, and terms that are topological have no metric dependancy at all. Contrast that with something like a Maxwell term, which when acted with the operator, will instantly pull out the nondynamical and absolute fixed structure.

Both definitions (as well as anyone that you can think off) are not going to generalize universally, or serve as a theory 'filter'. The first problem is that the word 'background' is often generalized in the literature to mean something more than just a classical solution of Einstein's equations. Second, its a little bit unclear what physical principle you are trying to capture that is so damn important, considering that even classical GR can be written in ways that make it look background dependant. (Consider writing GR like field theorists for the first case and consider the pure connection formalism for the 2nd)
 Sci Advisor P: 913 I finally read the paper by Giulini, and it is nice. He defines NC gravity being not background independent because of the appearence of "absolute structures": most of the metric components of NC gravity only have 1 solution (modulo gct's), whereas all the other components are gathered into the Newton potential. I think I found some nice insights in the paper :) It's also nice to compare the general-covariantization of the Poisson equation (giulini does it for the Schrodinger equation, but the difference is only a time derivative) with the formulation of Newton-Cartan. The latter can be seen as a much less trivial general-covariantization of Newton.
Sci Advisor
P: 2,215
 Quote by TrickyDicky I think it is actually a misnomer and somewhat ill-defined but anyway I always understood it as something like the difference between the EM theory of Maxwell and the GRT of Einstein, in the sense that the former equations refer to fields that act in " a background space" and so are "background dependent" while the latter equations refer to a field that "is" the spacetime in itself and therefore "background independent". I say it is a misnomer and superfluous term because at least since Riemann we know manifolds don't need to be embedded in any "background space" to be defined.
The way that I heard it described once is this:

You can formulate just about any theory of physics in a generally covariant form, which really means writing it in terms of geometric objects that can be defined independently of a choice of coordinates: Scalar fields, vector fields, tensor fields.

The theory is "background free" if there are no nondynamic scalar, vector or tensor fields. By "nondynamic", I mean a field that appears in the equations of physics (when written in generally covariant form) but which is not itself governed by the physics.

For example, in Newtonian physics, universal time is a scalar field that is nondynamic. In Special Relativity, the metric tensor is a tensor field that is nondynamic. General Relativity has no nondynamic fields.
Sci Advisor
P: 913
 Quote by stevendaryl The way that I heard it described once is this: The theory is "background free" if there are no nondynamic scalar, vector or tensor fields. By "nondynamic", I mean a field that appears in the equations of physics (when written in generally covariant form) but which is not itself governed by the physics. For example, in Newtonian physics, universal time is a scalar field that is nondynamic. In Special Relativity, the metric tensor is a tensor field that is nondynamic. General Relativity has no nondynamic fields.
I'd say that is rather vague. E.g., in Newton-Cartan theory the Ricci tensor is determined by the matter density, and this Ricci tensor is defined by the connection, which on its turn depends on the temporal metric!

What is true, is that up to gct's this temporal metric (with lower indices) only has one solution. That's a hint that it is non-dynamical, and that the theory has been "Stückelberged".

All the metric components of NC-theory turn out to be non-dynamical this way, except for a combination of components which form the Newton potential. This potential does have more solutions up to gct's, and as such is the only dynamical field in the theory.

In your definition I can always postulate EOM for the non-dynamical field. The simplest example for this was already given; if one has a Minkowksi metric and partial derivatives in a theory, just general-covariantize this and impose as extra EOM that the Riemann tensor for this metric vanishes.

 Related Discussions Special & General Relativity 1 Beyond the Standard Model 6 Special & General Relativity 5