Lagrangian of a graviton

Question

Recently in an interview for a phd program I was asked how would you write the lagrangian of a graviton. I answered that since graviton is a massless particle it's lagrangian should be similar to the one for photon with it's spin being two rather than one providing a few different terms. Was I right?

Answer 1

That's a correct though not complete answer. (Which may have been completely fine in the context of your PhD interview -- you are in a better place to judge that, so I am only focusing on the physics content of your question.)

The free (non-interacting) theory for a massless spin-2 particle on a Minkowski background can be written in the form (up to an overall normalization) \begin{equation} \mathcal{L} = \epsilon^{\mu\nu\rho\sigma}\epsilon^{\mu'\nu'\rho'}_{\ \ \ \ \ \ \ \ \ \ \ \sigma} \partial_\mu h_{\nu \nu'} \partial_{\mu'} h_{\rho\rho'} \end{equation} where $\epsilon^{\mu\nu\rho\sigma}$ is the totally antisymmetric Levi-Civita symbol and $h_{\mu\nu}$ is a rank two symmetric tensor representing the spin-2 field.

It has a gauge symmetry, corresponding to linearized diffeomorphisms (coordinate transformations), which generalizes the $U(1)$ gauge symmetry of electromagnetism in a way appropriate for a spin-2 field \begin{equation} h_{\mu\nu} \rightarrow h_{\mu\nu} + \partial_\mu \xi_\nu + \partial_\nu \xi_\mu \end{equation} where $\xi_\mu$ is the gauge parameter.

The non-linear completion of this linearized theory is GR, in the form of the Einstein-Hilbert action (in units with $c=1$) \begin{equation} S = \frac{1}{16\pi G_N} \int {\rm d}^4 x \sqrt{-g} R \end{equation} where the metric tensor $g_{\mu\nu}$ generalizes the spin 2 field used in the linear theory. To derive the linearized theory from GR, you would write the metric as a Minkowski background plus a small perturbation \begin{equation} g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu} \end{equation} then expand the Einstein-Hilbert action to quadratic order in $h_{\mu\nu}$. (Or, more efficiently, expand the equations of motion to linear order in $h_{\mu\nu}$, then do a general variation of an action with all possible contractions of two derivatives and two powers of $h$ and fix the coefficients so you get the same equations of motion).

You can also add several extra terms here:

• A cosmological constant, at the expense of losing Minkowski space as a background solution (which is fine -- that seems to be the case in our Universe).
• Matter fields which couple to the metric.
• And, from the modern effective field theory perspective, we also believe there are higher order terms that appear with more derivatives (things like $R^2$ or $\nabla R$), suppressed by powers of the Planck scale.