Nonlinearity of GR equations

[hilbert2]

Most of the fundamental equations of nature happen to be *linear* partial differential equations... This includes Maxwell equations and the Dirac equation (don't know about QCD field eqs).

On the other hand, the gravitational field equation of general relativity is a nonlinear PDE. As far as I know, the nonlinear behavior is significant only in extreme situations, like near a black hole. In most situation one can approximate with a linearized field equation.

Is there any easily explained reason why gravitational field should differ from other fields in this respect? Does the difficulty of quantizing gravity have anything to do with this nonlinearity? Can gravitational field exhibit chaotic behavior like many other systems governed by nonlinear PDE:s (turbulence in Navier-Stokes flow, for example).

 

[atyy]

The Yang-Mills equations for QCD are also nonlinear, but QCD has a UV complete quantization.

Formally, the nonlinearity of the gravitational field equations are enforced by requiring covariant energy conservation. Heuristically, the gravitational field is a "source" for itself. However, because the gravitational field does not have a local energy tensor, this concept shows up in the nonlinearity of the vacuum equations.

I think the "mixmaster" solutions are chaotic.

 

[Bill_K]

Most of the fundamental equations of nature happen to be *linear* partial differential equations... This includes Maxwell equations and the Dirac equation (don't know about QCD field eqs).

This is only true for free fields. As soon as you include any interaction the equation(s) become nonlinear.

On the other hand, the gravitational field equation of general relativity is a nonlinear PDE. As far as I know, the nonlinear behavior is significant only in extreme situations, like near a black hole. In most situation one can approximate with a linearized field equation.

GR requires the source be conserved. Linearized GR cannot affect the sources. As soon as you consider the action of the field on its sources, nonlinear GR becomes necessary. Such as in cosmology (and I don't mean just the big bang.)

 

[hilbert2]

Thanks for the responses. So, if I have understood correctly, the gravitational field has an associated energy density and therefore it itself acts as a source of gravity. So this is what makes the equation nonlinear.

 

[atyy]

Thanks for the responses. So, if I have understood correctly, the gravitational field has an associated energy density and therefore it itself acts as a source of gravity. So this is what makes the equation nonlinear.

No, I meant that usually we consider that source of the gravitational field to be the stress-energy tensor of matter. Since gravity does not have a stress-energy tensor, in order for it to act similarly to a "source" for itself, its equations have to nonlinear, ie. in the absence of matter, there are vacuum solutions to the equations with curved spacetime.

 

[WannabeNewton]

As far as I know, there is no generally accepted notion of the local energy density of the gravitational field; what's usually mentioned is the Landau-Lifgarbagez pseudotensor: http://en.wikipedia.org/wiki/Stress–energy–momentum_pseudotensor

 

[hilbert2]

I think I'll have to read my GR textbook (Ohanian&Ruffini) a bit more before I understand this better, but thanks anyway...

As a related question, are the QCD field eqs nonlinear because gluons interact with each other?

 

[WannabeNewton]

I can't recall of Ohanion goes into the problems involved with the local energy density of the gravitational field and the relation to non-linearity but Wald certainly goes into this, as well as d'inverno, and MTW goes a great deal into this. I can't answer your related question (my knowledge of QFT is cute at best) so sorry about that

 

[atyy]

As a related question, are the QCD field eqs nonlinear because gluons interact with each other?

Yes. Actually, Carroll draw the analogy between graviton-graviton scattering and QCD in http://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll4.html (see the bottom of the section).

Just a note regarding quantization of gravity. Gravity can be quantized, and it works, but only far below the Planck energy. In this respect, quantum GR is like quantum electrodynamics in that both theories fail at high energies. The energies at which quantum GR and QED fail are much higher than any experiments we do on earth, so it doesn't matter. The quantum corrections to classical gravity are believed to be very small at low energies, which explains why we can use classical GR. (I believe the graviton-graviton scattering part of the theory is unreliable, which is why Carroll mentions he shouldn't really draw them, but am not sure about this point.) This treatment of quantum GR is called the effective field theory treatment of quantum gravity