Two seemingly unrelated arguments against superposition in GR

[bcrowell]

Here are two seemingly unrelated arguments to explain why GR is a nonlinear theory:

(1) By the equivalence principle, any form of mass-energy should cause gravitational fields. Since gravitational fields carry energy, they should cause gravitational fields.

(2) GR doesn't care what coordinates we use. Therefore there is no natural way to identify the points of one spacetime with the points of another. This means that we can't add two metrics in any natural way.

I think #1 is pretty standard. #2 is raised by more than one author in Callender 2001. It's an anthology about quantum gravity, but this particular argument seems to me to apply just as well to classical superposition as to quantum-mechanical superposition.

I haven't seen #2 applied to classical GR before. Am I correct to do so? Is either 1 or 2 more fundamental? They seem unrelated; is there any connection I'm missing?

Craig Callender and Nick Huggett, eds., Physics Meets Philosophy at the Planck Scale: Contemporary Theories in Quantum Gravity, 2001.

 

[TrickyDicky]

Wrt #2, if any physical theory can be formulated in a coordinate independent way, that would mean that all physical theories are non-linear.
Classical electrodynamics can be formulated in coordinate independent way, but is linear, so that seems a counterexample to #2.

The problems of #1 have been amply discussed in this forum.

 

[Bill_K]

(2) GR doesn't care what coordinates we use. [STRIKE]Therefore there is no natural way to identify the points of one spacetime with the points of another. This means that we can't add two metrics in any natural way.[/STRIKE]

Therefore there are an infinite number of ways to identify the points of one spacetime with another. This means that we can add two metrics in an infinite number of ways.

 

[TrickyDicky]

Therefore there are an infinite number of ways to identify the points of one spacetime with another. This means that we can add two metrics in an infinite number of ways.

so I guess you don't consider (2) a good way to tell whether a theory is linear, do you? IOW, you are not implying GR is linear, right?

 

[martinbn]

Wrt #2, if any physical theory can be formulated in a coordinate independent way, that would mean that all physical theories are non-linear.
Classical electrodynamics can be formulated in coordinate independent way, but is linear, so that seems a counterexample to #2.

But it is not defined in a background independent way.

 

[Bill_K]

You can always define a new spacetime by adding the g's, g_μν = g⁽¹⁾_μν + g⁽²⁾_μν. It doesn't have any physical meaning. More to the point, you can add sources: T_μν = T⁽¹⁾_μν + T⁽²⁾_μν, and presumably the metric resulting from this could be calculated. Both of these operations are always possible, and neither of them means the theory is linear.

The relationship between field and source is nonlocal, something like ◻g_μν = T_μν. In a linear theory, there's a linear response. Meaning that when you add the T's, there's a variable g describing the field that also adds. For a theory to be linear they must both happen together.

 

[martinbn]

You can always define a new spacetime by adding the g's, g_μν = g⁽¹⁾_μν + g⁽²⁾_μν. It doesn't have any physical meaning. More to the point, you can add sources: T_μν = T⁽¹⁾_μν + T⁽²⁾_μν, and presumably the metric resulting from this could be calculated. Both of these operations are always possible, and neither of them means the theory is linear.

You can add the metrics only if the underlining manifold is the same, but if they are different it is meaningless. How do you add the standard euclidean metric of the plane and the standard metric of the sphere?

 

[TrickyDicky]

But it is not defined in a background independent way.

Sure, but background independence is not mentioned in the OP.

 

[TrickyDicky]

You can add the metrics only if the underlining manifold is the same, but if they are different it is meaningless. How do you add the standard euclidean metric of the plane and the standard metric of the sphere?

By adding different manifold metrics, are you referring to adding different solutions of the EFE? In that case I agree, but I guess Bill is referring to superpositions within the same solution.

 

[martinbn]

By adding different manifold metrics, are you referring to adding different solutions of the EFE? In that case I agree, but I guess Bill is referring to superpositions within the same solution.

Yes, that's what I mean, metrics on different manifolds. If we have the same manifold of course we can add tensors (of the same type). Why do you think he means within the same solution? In the context of some quantum theory one would probably need superpositions of different solutions.

 

[martinbn]

Sure, but background independence is not mentioned in the OP.

Yes, my bad, but that's how I understood it.

 

[atyy]

The 2 arguments seem to see GR in different ways. #1 sees it as a field in flat spacetime, since in curved spacetime GR, the gravitational field does not have energy. #2 sees it as a curved spacetime, since in flat spacetime presumably one could identify points.

To the extent that one believes that the EP itself indicates its limitation as a local principle, then the EP would indicate curved spacetime, and that would connect arguments #1 and 2. (Perhaps #2 should really be that there is no global inertial frame in GR, since every theory even SR is generally covariant?)

Another thought is that Deser's derivation of GR uses both the EP (gravity couples to all "energy") and gauge invariance (which is analogous to general covariance): http://arxiv.org/abs/gr-qc/0411023, http://arxiv.org/abs/0910.2975 .

 

[TrickyDicky]

#1 sees it as a field in flat spacetime, since in curved spacetime GR, the gravitational field does not have energy.

There is only curved spacetime GR, so it is the only way to see it.

 

[bcrowell]

Therefore there are an infinite number of ways to identify the points of one spacetime with another. This means that we can add two metrics in an infinite number of ways.

Right, that's what I mean when I say, "This means that we can't add two metrics in any natural way."

The 2 arguments seem to see GR in different ways. #1 sees it as a field in flat spacetime, since in curved spacetime GR, the gravitational field does not have energy. #2 sees it as a curved spacetime, since in flat spacetime presumably one could identify points.

I would seem both 1 and 2 as describing the standard conceptual picture of GR, with curved spacetime.

Another thought is that Deser's derivation of GR uses both the EP (gravity couples to all "energy") and gauge invariance (which is analogous to general covariance): http://arxiv.org/abs/gr-qc/0411023, http://arxiv.org/abs/0910.2975 .

Interesting...this sparks a vague memory of something from the Feynman Lectures on Gravitation (not the undergrad Feynman lectures, a different book). I think he shows that the gravitational field has to be self-interacting.

 

[TrickyDicky]

Right, that's what I mean when I say, "This means that we can't add two metrics in any natural way."

But you were saying that shows nonlinearity. How does (2) serve as an argument to explain that GR is nonlinear? Take SR, it doesn't care either what coordinates we use, and it is linear.

 

[atyy]

Interesting...this sparks a vague memory of something from the Feynman Lectures on Gravitation (not the undergrad Feynman lectures, a different book). I think he shows that the gravitational field has to be self-interacting.

Yes, the Deser derivation is in the same spirit as Feynman's.

I think from the view of a field in flat spacetime, gauge invariance is not necessary, and the EP is all that is needed to argue for a nonlinear equation. The resaon for saying this is that Nordstrom's theory demonstrates a consistent relativistic theory of gravity that obeys the EP but uses a scalar field. If we require gauge invariance as an additional condition, then we do recover GR via arguments in the Feynman-Deser line.

From the point of view of curved spacetime, the EP would say every where is locally flat, but globally curved. If spacetime is globally curved, then there are no global inertial frames, and general covariance is the only covariance, and there are no canonical ways of adding curved objects. (Both Nordstrom's theory (spin 0) and GR (spin 2) have dual formulations as curved spacetime.)