# Is GR a 2nd order approximation in g?

• nickyrtr
In summary: Bianchi identity.In summary, atyy's summary of the content of the conversation is that the Einstein Equation can be rewritten as a second order approximation in g with higher order terms that don't really lead anywhere. Einstein himself argued that the only term he could "add by hand", without creating some sort of conflict with the postulates, was the term containing the cosmological constant. Higher order gravity theories have been pursued for a long time, and Stelle suggests that with the added R2 terms, gravity can be renormalized as a quantum field theory. However, higher order terms in R are forbidden in 4...dimensional general relativity due to the Bianchi identity.
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?

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.

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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:

$g_{\mu\pi}g^{\pi\sigma}g_{\sigma\rho}g^{\rho\tau} = \delta_{\mu}^{\sigma}g_{\sigma\rho}g^{\rho\tau} = \delta_{\mu}^{\sigma}\delta_{\sigma}^{\tau} = \delta_{\mu}^{\tau}$

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.

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

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

note: This paper claims that Lagrangian terms higher than linear order in R are forbidden in 4 dimensions.

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## 1. What is GR?

GR stands for General Relativity, which is a theory of gravitation developed by Albert Einstein in the early 20th century. It describes the gravitational force as a curvature of spacetime caused by the presence of mass and energy.

## 2. What does it mean for GR to be a 2nd order approximation in g?

This means that the theory of General Relativity is a second-order theory, meaning that it takes into account the first and second derivatives of the gravitational field with respect to the coordinates of spacetime. The "g" in this context refers to the gravitational field.

## 3. Why is it important to know if GR is a 2nd order approximation in g?

Understanding if GR is a 2nd order approximation in g is important for accurately predicting and understanding the behavior of gravitational systems. It allows us to make more precise calculations and predictions about phenomena such as the motion of planets and the bending of light by massive objects.

## 4. How is GR a 2nd order approximation in g related to other theories of gravity?

GR is the most accurate theory of gravity that we have currently, and it is considered a second-order theory because it takes into account the effects of curvature on the gravitational field. Other theories, such as Newton's theory of gravity, are considered first-order theories as they only take into account the first derivative of the gravitational field.

## 5. Are there any limitations to GR as a 2nd order approximation in g?

While GR is a highly accurate theory of gravity, it is not a complete theory and has limitations. For example, it does not account for the effects of quantum mechanics and cannot explain the behavior of extremely small particles. Additionally, it does not fully explain the observed acceleration of the expansion of the universe, leading scientists to search for a more complete theory of gravity.

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