# Linearized gravity and local Lorentz symmetry

Action for linearized gravity is well-known, see for example David Tong: Lectures on General Relativity:

$$\mathbf{The\;Fierz-Pauli\;Action}$$

The linearised equations of motion can be derived from an action principle, the first written down by Fierz and Pauli,

$$$$\tag{5.8}S_{FP}\!=\!\frac{1}{8\pi G}\!\int \!d^{4}x \Big[\!-\!\frac{1}{4}\partial_{\rho}h_{\mu\nu}\partial^{\rho}h^{\mu\nu}\!+\!\frac{1}{2}\partial_{\rho}h_{\mu\nu}\partial^{\nu}h^{\rho\mu}\!+\!\frac{1}{4}\partial_{\mu}h\partial^{\mu}h\!-\!\frac{1}{2}\partial_{\nu}h^{\mu\nu}\partial_{\mu}h\Big]\!$$$$ This is the expansion of the Einstein-Hilbert action to quadratic order in $$h$$ (after some integration by parts). (At linear order, the expansion of the Lagrangian is equal to the linearised Ricci scalar $$(5.4)$$ which is a total derivative.)

$$-200-$$

This action is invariant under diffeomorphism transformation: $$\delta h_{\mu\nu} =\partial_\mu a_\nu + \partial_\nu a_\mu$$

Is this action invariant under local Lorentz transformations?

How construct local Lorentz invariant gravity action?

Since OP does not state explicitly the definition of $$h_{\mu\nu}$$ I will guess it is defined by $$g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}.$$ If OP corrects this then I will update this answer to reflect the change.

After expansion of the Einstein Hilbert action about $$g_{\mu\nu} = \eta_{\mu\nu}$$ all raising and lowering of indices is done by the flat space metric, $$\eta_{\mu\nu}$$. Moreover, it is clear from OP's action that the volume measure is $$d^{4}x = d^{4}x \sqrt{-\eta}$$ is the flat space measure. This leads me to argue that the action is invariant under transformations that preserve the Minkowski metric -- i.e. it is a (linearised) theory of a spin two field on a Minkowski space that has the reduced diff. invariance under transformations of $$h$$ mentioned by OP that survives from the originally diffeomorphism invariant theory. In terms of coordinate transformations, however, the best we could hope for is invariance under global Lorentz Transformations.

If OP has any corrections or comments let's discuss...

• Question about local Lorentz symmetry Commented Jan 18, 2020 at 20:12
• I argue not - only global Lorentz symmetry
– nox
Commented Jan 18, 2020 at 20:17
• I am looking for action with local Lorentz symmetry.. So sorry, you answer is not answer to my question:( Commented Jan 18, 2020 at 20:35
• Your question states "Is this action invariant under local Lorentz transformations?" My answer is "no" so I am answering at least half of your question.
– nox
Commented Jan 18, 2020 at 20:37
• One way in which you could move towards a theory with local Lorentz symmetry is by the vierbein formalism.
– nox
Commented Jan 18, 2020 at 20:42

Answer from talk On exotic six-dimensional supergravity theories:

No, the transformation should be complete.

"The Lorentz transformation is a linear transformation. It may include a rotation of space; a rotation-free Lorentz transformation is called a Lorentz boost. In Minkowski space, the mathematical model of spacetime in special relativity, the Lorentz transformations preserve the spacetime interval between any two events. This property is the defining property of a Lorentz transformation. They describe only the transformations in which the spacetime event at the origin is left fixed. They can be considered as a hyperbolic rotation of Minkowski space. The more general set of transformations that also includes translations is known as the Poincaré group. "

For more information the Wikipedia article on the Loerentz Transformation, which I quoted above, might prove to be helpful.

https://en.wikipedia.org/wiki/Lorentz_transformation

Gravity is a function of orientation in space, if you change space, you get a change of gravity.

• It is very strange comment. My question was about local Lorenz transformations, And gravity formulation in terms of tetrad, have such symmetry. My answer includes all information. Commented Jan 24, 2020 at 18:52
• Yes, I saw your answer. It was already posted. However you had the general question on Lorenz transformation which wasn't addressed. Regarding that is what I committed on. I hope that makes sense to you. Commented Jan 24, 2020 at 22:38