General Relativity Equation Formula Latex Code and Summary

General Relativity

Tags: #physics #relativity

Equation

$$\frac{\mathrm{d}^{2} x^{\alpha}}{\mathrm{d} s^{2}}+\Gamma^{\alpha}_{\beta\gamma}\frac{\mathrm{d} x^{\beta}}{\mathrm{d} s}\frac{\mathrm{d} x^{\gamma}}{\mathrm{d} s}=0$$

Latex Code

                                 \frac{\mathrm{d}^{2} x^{\alpha}}{\mathrm{d} s^{2}}+\Gamma^{\alpha}_{\beta\gamma}\frac{\mathrm{d} x^{\beta}}{\mathrm{d} s}\frac{\mathrm{d} x^{\gamma}}{\mathrm{d} s}=0

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Introduction

$\frac{\mathrm{d}^{2} x^{\alpha}}{\mathrm{d} s^{2}}+\Gamma^{\alpha}_{\beta\gamma}\frac{\mathrm{d} x^{\beta}}{\mathrm{d} s}\frac{\mathrm{d} x^{\gamma}}{\mathrm{d} s}=0$

Latex code for the principles of general relativity. I will briefly introduce the notations in this formulation.

The geodesic postulate: free falling particles move along geodesics of space-time with the proper time $\tau$ or arc length $s$ as parameter. For particles with zero rest mass (photons), the use of a free parameter is required because for them holds $\mathrm{d}s=0$ . From $\delta \int \mathrm{d}s = 0$ the equations of motion can be derived as : $\frac{\mathrm{d}^{2} x^{\alpha}}{\mathrm{d} s^{2}}+\Gamma^{\alpha}_{\beta\gamma}\frac{\mathrm{d} x^{\beta}}{\mathrm{d} s}\frac{\mathrm{d} x^{\gamma}}{\mathrm{d} s}=0$
The principle of equivalence: inertial mass ? gravitational mass->gravitation is equivalent with a curved space-time were particles move along geodesics.
By a proper choice of the coordinate system it is possible to make the metric locally flat in each point

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