# Is the spacetime for gravity described with gravitons flat?

Gravity has two equivalent descriptions. One is general relativity, the other is the mechanism by the exchange of gravitons. Is in the latter the spacetime flat?

• Gravitons are described using a background spacetime, which is not necessarily flat. – Slereah Jun 12 '16 at 13:59
• Afaik in any theory about gravitons it is a requirement that it should reproduce GR in big distances. – peterh - Reinstate Monica Jun 12 '16 at 14:02
• I don´t believe gravitons exist, but if they exist they couple to mass, not to spacetime. How does that make spacetime curve? – descheleschilder Jun 12 '16 at 14:07
• Probably you should learn the actual theory before believing or not in its existence, don't you think? – OON Jun 12 '16 at 14:19
• You´re absolutely right, but I don´t see how the coupling of gravitions to mass and themselves produces a curved spacetime. And although I know about math, that doesn´t show me the mechanism. – descheleschilder Jun 12 '16 at 15:14

The action for classical gravitation is the Hilbert-Palatini action $$S = \int d^4x\sqrt{-g}R$$ for $R$ the Ricci scalar. The Goto or Polyakov action is of the form $$S = -\frac{1}{4\pi\alpha'}\int d\tau d\sigma g_{\mu\nu}\left(\frac{\partial X^\mu}{\partial\tau}\frac{\partial X^\nu}{\partial\tau}~-~\frac{\partial X^\mu}{\partial\sigma}\frac{\partial X^\nu}{\partial\sigma}\right).$$ The question is then whether we can derive the gravitational acton from the string action. One approach is to consider the string expanded around a small variation in the spacetime variables $$X^\mu = X_0^\mu + \nabla_\alpha X^\mu\delta x^\alpha + \nabla_\alpha\nabla_\beta X^\mu\delta x^\alpha\delta x^\beta$$ The derivation is a bit involved, but a sketch can be given. The terms linear in $\delta x$ are connection coefficients that drop out and the second order terms give curvatures. This term is $O(\alpha')$ and thus the string tension, and all QM, disappears. The Hilbert-Palintini action can in fact be derived, and in this form it pertains to tree level graviton interactions.
Feynman pointed out that tree level graviton contributions are no different than classical gravitation. This first set of expansions simply involves the dynamics of the string on the target spacetime. With this expansion we can however continue onwards to get an action of the form $$S = \int d^4x\sqrt{-g}(R + \alpha'O(R^2) + \alpha'^2(R^4) + \dots).$$ These higher order terms are now corrections to the classical spacetime. This is where the quantum mechanics of the graviton in a perturbative sense enters into the physics. This is a bit of a low energy (low being removed from the Planck energy or Hagedorn energy/temperature) result.