Is GR a 2nd order approximation in g?

[nickyrtr]

While studying the Einstein Equation, I noticed something curious, at least to me with little experience in General Relativity. Start with the usual formulation of the equation:

$$R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R + g_{\mu\nu}\Lambda = \frac{8{\pi}G}{c^2}T_{\mu\nu}$$

Then, apply the definition of R, the scalar curvature:

$$R = g^{ij}R_{ij}$$

and one can rewrite the Einstein Equation as follows:

$$R_{\mu\nu} + g_{\mu\nu}\Lambda - \frac{1}{2}g_{\mu\nu}g^{ij}R_{ij} = \frac{8{\pi}G}{c^2}T_{\mu\nu}$$

Intuitively, this looks a lot like some sort of second order approximation in g. One can imagine there might be higher order terms, insignificant except in the most extreme environments. Something like this:

$$R_{\mu\nu} + g_{\mu\nu}\Lambda - \frac{1}{2}g_{\mu\nu}g^{ij}R_{ij} + \frac{1}{3!}g_{\mu\pi}g^{\pi\sigma}g_{\sigma\nu}\Lambda_3 - \frac{1}{4!}g_{\mu\pi}g^{\pi\sigma}g_{\sigma\rho}g^{\rho\tau}R_{\tau\nu} + ... = \frac{8{\pi}G}{c^2}T_{\mu\nu}$$

The above is completely arbitrary and made-up, just trying to paint an intuitive picture of the idea. Is the possibility something we can rule out on mathematical grounds? Has this kind of extension to GR been explored?

 

[Naty1]

Don't know excactly what you did (my inability, not yours) but it's has always seemed likely that if quantum mechanics/gravity and GR don't agree in some situations, and neither produces finite results, then likely BOTH are currently approximations of a more general and precise (unified) theory. Perhaps akin to different string theories being different viewpoints of M theory. (all of which are only partially developed.)

I do know there ARE other formulations of GR, even Einstein had developed several, and only after arriving at his intuitive "equivalence principle" could he eliminate those which did not comport with uniform acceleration.

 

[xepma]

Your higher order terms don't really lead to higher orders in g... Namely, the metric with upper indices is the inverse of the metric with lower indices. So, for instance:

$g_{\mu\pi}g^{\pi\sigma}g_{\sigma\rho}g^{\rho\tau} =<br /> \delta_{\mu}^{\sigma}g_{\sigma\rho}g^{\rho\tau} =<br /> \delta_{\mu}^{\sigma}\delta_{\sigma}^{\tau} =<br /> \delta_{\mu}^{\tau}$

But I can understand your point about adding higher order terms, in the sense that other contributions might be added to the equations of motion. However, I think Einstein himself argued somehow that the only term he could "add by hand", without creating some sort of conflict with the postulates, was in fact the term containing the cosmological constant.

On the other hand, for instance in superstring theory the Einstein equations of motion roll out as approximate equations of motion and that indeed higher order corrections to these equations are present.

 

[atyy]

Possibly: "It can be shown (Lovelock 1972) that a linear combination of Gab and gab is the most general two-index symmetric tensor that is divergence-free and can be constructed locally from the metric and its derivatives up to second order." Ludvigsen, p100, http://books.google.com.sg/books?id=hQdh3SVgZ8MC

Also: Higher order gravity theories and their black hole solutions, Charmousis, http://arxiv.org/abs/0805.0568

 

[nickyrtr]

Thank you, atyy, for that reference. Indeed it appears that higher order gravity theories have been pursued for a long time. Here is another such publication:

http://prola.aps.org/abstract/PRD/v16/i4/p953_1"

From what I can tell, Stelle adds terms to the Einstein Equation that are quadratic in R, the scalar curvature, not g. This make sense to me now since, as another poster pointed out, higher powers of g do not really lead anywhere. So let me rephrase the original question as "Is GR a linear approximation in spacetime curvature?"

Interestingly, Stelle claims that with the added R² terms, gravity can be renormalized as a quantum field theory (though problems still remain). He also suggests that, in his view, quantum gravity may ultimately require non-perturbative techniques for detailed calculations. Perhaps "lattice quantum gravity" codes will one day occupy the world's largest supercomputers.

Here are some other related publications I stumbled upon, for those interested in the history of this topic:

http://prola.aps.org/abstract/PRD/v50/i6/p3874_1"
preprint: http://arxiv.org/abs/gr-qc/9405057"

http://prola.aps.org/abstract/PRD/v50/i8/p5039_1"
preprint: http://arxiv.org/abs/gr-qc?papernum=9312008"
note: the term "nonlinear gravity" here refers to a Lagrangian nonlinear in R, i.e. having terms higher than linear order.

http://adsabs.harvard.edu/abs/1985NuPhB.248..392H"

http://iopscience.iop.org/0264-9381/9/4/006"

http://www.sbfisica.org.br/bjp/download/v18/v18a44.pdf"

http://www.springerlink.com/content/c4147451w66g4611/"
note: This paper claims that Lagrangian terms higher than linear order in R are forbidden in 4 dimensions.