What is the physical meaning of metric compatibility and why is it important?

[TrickyDicky]

What exactly is the physical meaning of the fact that the covariant derivative of the metric tensor vanishes?

 

[LAHLH]

We choose this property of metric compatibility as it's known when we specify our connection. One could I believe choose different connections to Christoffell for which you wouldn't have metric compatibility, but then these are non physical. The fact that you geodesics are not just curves that parallel transport their tangent vectors wrt connection, but that they are also paths of extremal distance in spacetime is owing to the Christoffell connection being the connection we chose. Metric compat means lots of nice properties too, such as preservation of angle between two parallel transported vectors, or equivalently that geodesics can't change their character from null/timelike/spacelike.

I'm sure someone else will be able to give deeper and maybe more physical reasons still though...

 

[TrickyDicky]

We choose this property of metric compatibility as it's known when we specify our connection. One could I believe choose different connections to Christoffell for which you wouldn't have metric compatibility, but then these are non physical. The fact that you geodesics are not just curves that parallel transport their tangent vectors wrt connection, but that they are also paths of extremal distance in spacetime is owing to the Christoffell connection being the connection we chose. Metric compat means lots of nice properties too, such as preservation of angle between two parallel transported vectors, or equivalently that geodesics can't change their character from null/timelike/spacelike.

I'm sure someone else will be able to give deeper and maybe more physical reasons still though...

Thanks,LAHLH.
You are right that this property is fixed when we choose the connection so that the covariant derivative is coordinate transform invariant.
I'm more interested on any obvious physical consequences of this, i.e. wrt the gravitational field, or maybe is a statement about invariance of space translations and rotations and time translation, (conservation of the metric)...?

 

[bcrowell]

A covariant derivative is a derivative from which we've removed any contribution that is purely a result of the choice of coordinates. The first derivatives of the metric are the way they are purely as a result of the choice of coordinates.

 

[LAHLH]

I'm not sure how to answer your question in a more physical way, but I did stumble up what I thought sounded relevant:

The vanishing of the covariant derivative of the metric—the condition of
metric compatibility—is sometimes introduced perfunctorily in texts on general
relativity, but Schr¨odinger was right to call it “momentous” (Schr¨odinger 1985,
106). It means that the local Lorentz frames associated with a space-time point
p (those for which, at p, the metric tensor takes the form diag(1,−1,−1,−1)
and the first derivatives of all its components vanish) are also local inertial
frames (relative to which the components of the connection vanish at p).10 If
the laws of physics of the non-gravitational interactions are assumed to take
their standard special relativistic form at p relative to such local Lorentz charts
(the local validity of special relativity), then metric compatibility implies that
gravity is not a force in the traditional sense—an agency causing deviation from
natural motion—, in so far as the worldlines of freely falling bodies are geodesics
of the connection.
The full physical implications of the non-metric compatible connection in
Weyl’s theory remain obscure in the absence of a full-blown theory of matter.
Weyl’s hints at a solution to the Einstein objection seem to involve a violation of
minimal coupling, i.e. a violation of the prohibition of curvature coupling in the
non-gravitational equations, and hence of the local validity of special relativity.
But it seems that the familiar insight into the special nature of the gravitational
interaction provided by the strong equivalence principle—the encapsulation of
the considerations given in the previous paragraph—is lost in the Weyl theory.

 

[TrickyDicky]

A covariant derivative is a derivative from which we've removed any contribution that is purely a result of the choice of coordinates.

Right, that's what's been said.

The first derivatives of the metric are the way they are purely as a result of the choice of coordinates.

I'm not sure what you mean here, the second derivatives are the ones determined by the metric.

 

[bcrowell]

The first derivatives of the metric are the way they are purely as a result of the choice of coordinates.

I'm not sure what you mean here, the second derivatives are the ones determined by the metric.

I meant the non-covariant derivatives. A non-covariant first derivative like $\partial_a g_{bc}$ is what it is because of the choice of coordinates. It has no coordinate-independent meaning. Since a covariant derivative like $\nabla_a g_{bc}$ is supposed to exclude variation in g that occurs only because of the choice of coordinates, it has to be zero.

 

[LAHLH]

Since a covariant derivative like $\nabla_a g_{bc}$ is supposed to exclude variation in g that occurs only because of the choice of coordinates, it has to be zero.

I don't think this is true, there are connections that would be perfectly valid and serve to define covariant derivatives without the metric compat property.

You are right to say if we just used partial instead of covariant we get something non tensorial. This argument applies to any tensor not just the metric and if your argument was correct you could extend it to any tensor and say its covariant deriv had to be zero.

There are theories I think one due to Weyl to do with Electromagnetism that drops the met compat requirement, and on a related note I think the Palatini formulation of GR drops the torsion free requirement, I don't know enough about these things though.

 

[bcrowell]

I've never come across a clear explanation of what metric compatibility means (preferably both mathematically and physically), and why we care about it. LAHLH, would you care to take a shot at explaining it with crayons for those of us who are less mathematically sophisticated?

 

[atyy]

I think LAHLH said everything in his post #2. Intuitutively, there is no definition of parallel on a non-affine space. We are free to define parallel as we wish, and it turns out that there are many possible consistent definitions of parallel. In the presence of a metric, which we interpret as giving us the ability to define angle and distance, there is a natural definition of parallel, so we pick that definition above all the other possible definitions.

http://en.wikipedia.org/wiki/Levi-Civita_connection
http://en.wikipedia.org/wiki/Connection_(mathematics)

 

[George Jones]

I've never come across a clear explanation of what metric compatibility means (preferably both mathematically and physically), and why we care about it.

I think that the physical meaning of metric compatibility is given by the first paragraph of the passage that LAHLH quoted in post #5. See also section 13.3 of Misner, Thorne, and Wheeler.