# Lagrangian for the General Relativity

In summary, the conversation discusses the Lagrangian for the Gravitational field and how it can be used to derive the vacuum field equations. The question is then raised about the full Lagrangian including matter, which is explained to be the sum of the gravitational Lagrangian and the matter Lagrangian. An example is given with electromagnetic fields and the resulting field equations are shown. The conversation ends with a thank you for the helpful explanation.
I've always found that the lagrangian for the Gravitational field is that from the Einstein-Hilbert action:

$L=R$ (R is the Ricci scalar; I'm not including the factor of $\sqrt{-g}$)

but when variational principles are applied, we get the vacuum field equations (obviously). I'd like someone to tell me which would be the FULL lagrangian (with matter coupled) for getting the Einstein's field equations.

You simply take the gravitational Lagrangian density and add in the Lagrangian density of the matter fields: $L=L_G+L_M$.

For example, if you have electromagnetic fields, then the action is

$$S = \int \ast \mathcal{R} - \frac12 \int F \wedge \ast F$$

where $\ast$ is the Hodge dual and F is the electromagnetic 2-form (rescaled up to some factors of $2\pi$ which I don't remember...I use the above normalization in my research). Using the normalization I've given here and varying with respect to the inverse metric, you obtain

$$R_{\mu\nu} - \frac12 \mathcal{R} g_{\mu\nu} = \frac12 T_{\mu\nu}$$

where

$$T_{\mu\nu} = F_{\mu\rho} F_\nu{}^\rho - \frac14 g_{\mu\nu} F_{\rho\sigma} F^{\rho\sigma}$$

Wow, thankyou both, that was very useful! :)

## 1. What is the Lagrangian for General Relativity?

The Lagrangian for General Relativity is a mathematical expression that describes the dynamics of space and time in Einstein's theory of general relativity. It is a function of the metric tensor, which describes the curvature of space-time, and its derivatives.

## 2. How is the Lagrangian derived in General Relativity?

The Lagrangian for General Relativity is derived using a variational principle, also known as the principle of least action. This principle states that the path taken by a system between two points is the one that minimizes the action, which is the integral of the Lagrangian over time. This leads to the Einstein field equations, which describe the relationship between the curvature of spacetime and the matter and energy within it.

## 3. What role does the Lagrangian play in General Relativity?

The Lagrangian is an essential tool in the study of General Relativity. It allows us to understand the dynamics of space and time and make predictions about how matter and energy will interact with the curvature of spacetime. It also serves as the basis for the Einstein field equations, which are the cornerstone of General Relativity.

## 4. How does the Lagrangian differ from Newton's laws of motion?

The Lagrangian for General Relativity is a relativistic version of the Lagrangian used in classical mechanics, which is based on Newton's laws of motion. However, in General Relativity, the Lagrangian includes additional terms that account for the effects of gravity on the geometry of spacetime. These effects are not present in Newton's laws of motion, which only describe the dynamics of objects in flat, non-curving space.

## 5. Can the Lagrangian be used to solve problems in General Relativity?

Yes, the Lagrangian is a powerful tool for solving problems in General Relativity. By setting up and solving the equations of motion derived from the Lagrangian, we can make predictions about the behavior of matter and energy in curved spacetime. It also allows us to study the effects of different types of matter and energy on the curvature of spacetime, providing insights into the fundamental nature of the universe.

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