# What is the effective Lagrangian in General Relativity?

• Replusz
In summary, the conversation discusses the concept of effective Lagrangian in relation to free particles traveling along geodesics in general relativity. There are two ways to obtain the connection coefficients for this Lagrangian: through a formula or by variation of action. The conversation also mentions the use of two different actions, one with a square root and one without, which both lead to the same equations of motion.
Replusz
Homework Statement
Connection coefficients
Relevant Equations
part c) of the problems

Attempt: I don't know what they mean by effective Lagrangian.
I am aware there is something called 'the lagrangian' that goes as L=g_ab * dx^a/dk * dx^b/dk, but i don't see how this gives me any of the chrostoffel symbols...

cheers

Well, there are two ways of getting the connection coefficients.
One way is getting them via the formula where they are defined as Christoffel symbols of the second kind:
$$\Gamma^\mu_{\nu\rho} = \frac{1}{2}g^{\mu\sigma}(\partial_\nu g_{\sigma\rho} + \partial_\rho g_{\nu\sigma} - \partial_\sigma g_{\nu\rho})$$
The second way is by variation of action:
$$I = \frac{1}{2}\int g_{\mu\nu}\dot{x^\mu}\dot{x^\nu}d\lambda$$
The Lagrangian in this action is so called geodesic Lagrangian, and by variating the action ##\delta I = 0##, you find geodesic equations from which you can read connection coefficients. You variate with respect to every component ##x^\mu##. Since in general relativity, free particles travel along geodesics, this is what is meant by effective Lagrangian for free particles(I think, at least, I always called it geodesic Lagrangian, but there should be no other). Of course, ##\dot{x^\mu} \equiv \frac{dx^\mu}{d\lambda}## where ##\lambda## is parameter of the geodesic.
Sometimes the action is also defined as:
$$I = \int \sqrt{g_{\mu\nu}\dot{x^\mu}\dot{x^\nu}}d\lambda$$
but these two actions have same equations of motion. The second action is precisely the line element form, but it's pretty much the same, and I like using the one without the square root.

DEvens

## 1. What is General Relativity?

General Relativity is a theory of gravity developed by Albert Einstein in the early 20th century. It explains how massive objects in the universe interact with each other and how they affect the fabric of space-time.

## 2. How does General Relativity differ from Newton's theory of gravity?

Unlike Newton's theory, which states that gravity is a force acting between objects, General Relativity explains gravity as the curvature of space-time caused by the presence of massive objects. It also accounts for the effects of acceleration and the speed of light.

## 3. Can you provide a simple example to illustrate General Relativity?

Imagine a large rubber sheet representing the fabric of space-time. If you place a heavy object, like a bowling ball, on the sheet, it will cause a dip in the sheet's surface. This represents how massive objects create a curvature in space-time. Other smaller objects, like marbles, will roll towards the dip created by the bowling ball, just like how planets orbit around a star in space.

## 4. How has General Relativity been proven to be correct?

General Relativity has been tested and confirmed by numerous experiments and observations, such as the bending of light around massive objects and the precise predictions for the orbit of Mercury. The recent detection of gravitational waves also provides strong evidence for the theory.

## 5. Is General Relativity easy to understand?

General Relativity can be a complex and abstract concept, but there are many resources available to help understand it better. With some effort and dedication, it is possible to grasp the basic principles and implications of the theory.

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