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#### Welcome to Einstein Relatively Easy!

This web site is aimed at the general reader who is keen to discover Einstein's theories of special and general relativity, and who may also like to tackle the essential underlying mathematics.

Einstein's Relativity is too beautiful and too engaging to be restricted to the professionals!

Have fun!

*"I have no special talents*

* I am only passionately curious" *

Albert Einstein

## Einstein-Hilbert action

- Details
- Category: General Relativity

An alternative route to Einstein's equation is through the principle of least action, as we did previously to deduce the geodesic equation in curved spacetime in Geodesic equation from the principle of least action.

In this article, we will therfore go through the process of deriving the Einstein equations in vacuum and then in the presence of matter using **the variational approach**.

##### Action in vacuum

The derivation of the action from a set of equations of motion is very hard, not always possible, and there is no systematic way to do it. We therefore will begin by guessing the action and show that it gives the right answer.

So we will first seek an action S for gravitation that leads to the field equations of general relativity in the absence of matter and energy (in vacuum), that is, we will guess something like:

where L is a scalar Lagrange density and d^{4}V is the element of 4-volume. We thus need both a scalar and the 4-volume element.

## Palatini equation

- Details
- Category: Dictionary

This article looks at the process of deriving the so called** Palatini equation** and follows the demonstration found in D'Inverno *Introducing Einstein's relativity* , Chapter 11-1 (General Relativity from a variational principle, The Palatini equation).

Many tensor identities are derived most easily using the technique of geodesic coordinates in a Local Inertial Frame, where we choose an arbitrary point P at which the Christoffel symbols nullify, which in D'Inverno notation could be written as:

As we know from our article Riemann curvature tensor part III: Symmetries and independant components in this particular case, the Riemann tensor reduces to:

Looking now at a variation of the connection Γ^{a}_{bc} to a new connection Γ^{a}_{bc}(hat):

Then δΓ being the difference of two connections, is a tensor of type (1,2), and this variation results in a change in the Riemann tensor between two coordinate systems as:

since partial derivatives commute with variation and **is equivalent to covariant derivative in geodesic coordinates**.

## Newtonian limit

- Details
- Category: Dictionary

It should be clear that General Relativity describes gravitation in terms of curvature of spacetime and reduces to Special Theory of Relativity for Local Inertial Frame (LIF). However, it is important to explicitly check that **the description reduces to the Newtonian treatment** when we select the **correct boundary conditions**.

These conditions, referred to as **the Newtonian limit**, are applicable to physical systems exhibiting:

- - objects moving relatively
**slowly**(compared to the speed of light). - -
**weak**gravitional field. - - gravitational field does not change with time, ie it is
**static**.

Mathematically, this leads to the following approximations:

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