# Is there anything more fundamental than spacetime

## Nature of gravity: gravitons, space-time curvature, or both?

Here's just a small note. It is possible to provide rigorous mathematical proof of the equivalence of these two images.

If you just start with the three (semi-experimental) facts: Lorentz invariance, 1 / r long distance tail of gravitational force and its one-way effect (attraction only), and the fact that the bending of light is almost independent of its frequency and polarization, then you will find that these facts are compatible (at a great distance limit) only with the massless helicity ± 2 particle exchange. Afterwards it was proven that special relativity theory and analytical properties of the scattering amplitude lead to the equivalence principle [1,2]. This theorem is a pure analog of the Gell-Mann-Low-Goldberger theorem for soft photons, which states that the power extension of the amplitude of photon scattering by a hadron (in terms of photon frequency) does not depend on the spin or internal structure of the Hadron (up to the second order). By taking into account the multigraviton scattering amplitudes, it can be demonstrated that all local corner points for soft gravitons correspond to the extent of the Einstein effect.

It means that the exchange of helicity ± 2 massless particles inevitably leads to classical general relativity (the opposite statement is trivial).

This program was initiated by Steven Weinberg [1,2] and ended by Deser and Boulware [3]. The full consideration can be found in their paper [3] entitled “ Classical general relativity derived from quantum gravity ". This paper is a real masterpiece of the clear physical explanation of this problem.

### References

[1] S. Weinberg, Photons and gravitons in the S-matrix theory: Derivation of charge retention and equality of gravitational and inertial mass , Phys. Rev. B135 (1964) 1049.

[2] S. Weinberg, Photons and Gravitons in Perturbation Theory: Derivation of Maxwell's and Einstein's Equations , Phys. Rev. B138 (1965) 988.

[3] DG Boulware, S. Deser, Classical general theory of relativity, derived from quantum gravity , Ann. Phys. 89 (1975) 193.

### Grisha Kirilin

@space_cadet try here tinyurl.com/25or7oz BTW, there is a small inaccuracy in my answer. It's about Gell-Mann-Low-Goldberger's theorem, more precisely about spin. The theorem claims that the first two terms (∝ω0 and ∝ω1) of expansion depend only on the total electric charge and the anomalous magnetic moment of the hadron. It is interesting that there is no correction like "anomalous moment" in quantum gravity.

### Grisha Kirilin

@space_cadet It's very simple - you can't just construct the operator like σμνF.μν since σμνR.μν = 0 for Ricci tensor. The rigorous theorem of soft gravitons was established by Gross and Jackiw in Phys. Rev. 166 (1968) 1287. As far as I can remember, they consider graviton scattering by a spin-0 particle, but the theorem applies to any spin for the reason given above.