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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

## 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 deerivatives 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:

## Variation of the metric determinant

- Details
- Category: General Relativity

This artilce looks at the process of deriving the variation of the metric determinant, which will be useful for deriving the Einstein equations from a variatioanl approach, in the next article Einstein-Hilbert action.

##### Matrix determinants and trace

Let us consider a matrix from a general form

Then **the trace of this matrix**, as for any square matrix, is **the sum of the elements on the main diagonal** (the diagonal from the upper left to the lower right), so

Trace(A) = tr(A)=a_{0} + a_{1}

so that

If we now consider** the exponential matrix e ^{A} **as:

then the **determinant of this matrix**, defined as the product of the elements on the main diagonal can be expressed as:

so that finally we can write

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