# Would knowing only the, speed constant, and symmetries of spacetime in General Relativity be enough to derive General Relativity?

I read before that General Relativity can be derived from symmetries, but what I don't know is whether this means that symmetries alone can derive General Relativity or if this means that symmetries combined with additional information can be used to derive General Relativity.

For comparison if one said that special relativity can be derived from symmetries that could be interpreted to mean that special relativity can be derived from those symmetries combined with the speed constant. If however we knew nothing about a speed constant, length contraction, time dilation, or electromagnetism, and the only information we had related to special relativity, then we wouldn't have enough information to derive special relativity, as Euclidean Space, and Galilean Spacetime have all the symmetries that Spacetime in Special Relativity has.

Let's say that we didn't know about the existence of black holes, gravitational lensing, or gravitational interactions. The only information we had related to General Relativity were the symmetries of General Relativity, that there is a speed constant, and that spacetime is curved near a massive body. Would we have enough information to derive General Relativity in this case?

To be clear when I say the speed constant I'm referring to the speed that massless particle move at.

• What exactly do you mean by "derive general relativity". Do you mean deriving the Einstein field equations? Jul 30 '21 at 2:26
• @josephh Yes along with the other equations of General Relativity. Jul 30 '21 at 2:35

Symmetry alone is not enough to derive GR.

The action for GR is the Einstein-Hilbert action (I'll add the cosmological constant since we think it is there in Nature) $$$$S_{\rm EH} = \frac{1}{16\pi G}\int {\rm d}^4 x \sqrt{-g} \left[ R - 2\Lambda \right]$$$$ where I've set $$c=1$$ and $$G$$ is Newton's constant.

There are an infinite number of terms we could add with exactly the same symmetries but would lead to different field equations... $$$$S_{\rm other} = \int {\rm d}^4 x \sqrt{-g} \left[ c_1 R^2 + c_2 R^{\mu\nu}R_{\mu\nu} + c_3 R^{\mu\nu\rho\sigma} R_{\mu\nu\rho\sigma} + c_4 R^3 + ... \right]$$$$ However, GR is the unique self-consistent and stable interacting low energy Lorentz-invariant (two derivatives or less acting on the metric in the action) theory of a massless spin-2 particle [1]. So, while symmetry alone is not enough, there are a set of physical principles which do single it out as a special theory.

[1] There are many references, but a classic argument is in Feynman's lectures on gravitation, and other more complete arguments are given by Deser: https://arxiv.org/abs/gr-qc/0411023 and Boulware and Deser: https://doi.org/10.1016/0003-4916(75)90302-4

• For the $S_{other}$ I see that there's an $R^2$, an $R^3$, and some $R$ terms with symbols for index values instead of numbers. Is there a pattern for always getting the next $R$ term in the sequence? Jul 30 '21 at 12:34
• @AndersGustafson You can write any index contraction with an arbitrary number of $R_{\mu\nu\rho\sigma}$. Some more cubic terms would be $R_{\mu\nu}R^{\nu\rho}R^\mu_{\ \ \rho}$ and $R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} R$. I'm sure there's a systematic way to get all the possible contractions, but the point is more that there are an infinite number of possible terms that have the same symmetry (general covariance) as the Einstein-Hilbert action and only involve the curvature, so symmetry considerations alone are not enough to derive GR. Jul 30 '21 at 14:27
• Also you can write covariant derivatives of the curvature, like $\nabla_\mu \nabla_\nu R^{\mu\nu\rho\sigma}R_{\rho\sigma}$ or $R \square R$. Jul 30 '21 at 15:08
• You can also include contractions with the Levi-Civita tensor density $\epsilon_{\mu\nu\rho\sigma}$, or equivalently with the dual Riemann tensor $\tilde{R}_{\mu\nu\rho\sigma}$, and again covariant derivatives. This enables things like $\tilde{R}_{\mu\nu\rho\sigma}R^{\mu\nu} R^{\rho\sigma}$ and $\nabla_\mu \tilde{R}^{\mu\nu}_{\ \ \ \ \rho\sigma} \nabla_\nu R^{\rho\sigma}$ (the contractions just get crazier and crazier). Jul 30 '21 at 19:08
• Thanks for the information! Aug 1 '21 at 0:29

This answer only deals with the derivation of Einstein's field equation (EFE), since other things in GR are based on this equation. If the symmetry involved is the principle of general invariance (exact definition is given in Chapter 7 of Hans Ohanian's Gravitation and Spacetime, provided we knew the linear field equations for gravitation as well as assumed the field equation is second differential order and linear in the second derivative of the metric, then the EFE can be derived. For more details, have a read at Chapters 3 and 7 of the aforementioned book. Note that Ohanian derived the linearized gravitational field equation before deriving EFE, in contrast to many books that adopt the other way around.

Actually A. Einstein found the field equations (EFE) without knowing about black holes and gravitational lensing. Of course he knew that they were gravitational interactions. Without knowing about gravitational interactions one would have no theory at all (not even Newton's theory). The key point of Einstein's derivation of the field equations is the equivalence principle: the equivalence between gravitational mass and inertial mass. Probably one would not call that a symmetry.

The symmetry governing General Relativity is diffeomorphism invariance, which is a consequence of the relativity principle and the equivalence principle. But this would not uniquely define the gravitational theory. One would need another guidance: Actually the new gravitational theory should reproduce Newton's gravitational theory in the limit of weak fields and small velocities . Once applied this third principle one is directly led to the EFE (which might suffer from quantum corrections if one ever succeeds to quantize it consistently ...).

Finally one should not over-emphasize the role of symmetry in physics. One cannot derive all physical theories just from a symmetry. For instance the symmetry principle led to a strong interest in supersymmetry, but until now there is no experimental hint of supersymmetry found. Therefore symmetry should not be used as the only guiding principle because it would lead people to forget about other principles which might eventually play even a more important role.