Einstein field equations

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Representation of the curvature given by Einstein's field equation on the plane of the ecliptic of a spherical star: This equation relates the presence of matter to the curvature acquired by space-time.

In physics, Einstein's field equations, Einstein equations or Einstein-Hilbert equations (known as EFE , for Einstein field equations) are a set of ten equations from Albert Einstein's theory of general relativity that describe the fundamental interaction of gravitation as a result of spacetime being being curved by matter and energy.

First published by Einstein in 1915 as a tensor equation, the EFE equations equate the curvature of local spacetime (expressed by the Einstein tensor) with the local energy and momentum within that spacetime (expressed by the stress-energy tensor).

Einstein's field equations relate the presence of matter to the curvature of space-time. More precisely, the greater the concentration of matter, represented by the energy-momentum tensor, the greater the components of the Ricci curvature tensor.

In the classical non-relativistic limit, that is, at small velocities compared to light and relatively weak gravitational fields, Einstein's field equations reduce to Poisson's equation for the gravitational field, which is equivalent to the Newton's law of gravitation.

Mathematical form of Einstein's field equations

In Einstein's field equations, gravity is given in terms of a metric tensor, a quantity that describes the geometric properties of four-dimensional space-time and from which curvature can be calculated. In the same equation, matter is described by its stress-energy tensor, a quantity that contains the density and pressure of matter. These tensioners are 4 X 4 symmetric tensioners, so they have ten independent components. Given the freedom of choice of the four space-time coordinates, the independent equations reduce to six. The coupling force between matter and gravity is determined by the universal gravitational constant.

For each point in spacetime, Einstein's field equation describes how spacetime is curved by matter and is in the form of a local equality between a curvature tensor for the point and a tensor describing the distribution of matter around the point:

${displaystyle {text{G}}_{mu nu }={8pi {text{G}} over {text{c}}^{4}}T_{mu nu }}$
Symbol	Name
${displaystyle G_{mu nu }}$	Einstein curvature tensor, formed from second derivatives of the metric tensor $g_{munu}$
${displaystyle T_{mu nu }}$	Tensor moment-energy
$c$	Speed of light
$G$	Constant of universal gravitation

This equation holds for every point in space-time.

The Einstein tensor of curvature can be written as:

${displaystyle {text{G}}_{mu nu }=R_{mu nu }-{1 over 2}Rg_{mu nu }+Lambda g_{mu nu }}$


Symbol	Name
${displaystyle R_{mu nu }}$	Ricci curvature voltage
$R$	Ricci curvature scaler
$Lambda$	Cosmetic Constant

The field equation can therefore also be given as follows:

${displaystyle R_{mu nu }-{1 over 2}Rg_{mu nu }+Lambda g_{mu nu }={8pi {text{G}} over {text{c}}^{4}}T_{mu nu }}$

where ${displaystyle g_{mu nu },}$ is a symmetrical tensor 4 x 4, so it has ten independent components. Given the freedom of choice of the four coordinates of space-time, the independent equations are reduced by number to six. These equations are the basis for the mathematical formulation of general relativity. Note that considering the contraction on the two indices of the last relationship is that the curvature scale is related to the trace of the tensor energy impulse and the cosmological constant by:

${displaystyle R-2R+4Lambda ={8pi G over c^{4}}T,}$

This relationship allows us to write the field equations equivalently as:

${displaystyle R_{mu nu }-g_{mu nu }Lambda ={8pi G over c^{4}}left(T_{mu nu }-{1 over 2}T,g_{mu nu }right)}$

Geometric interpretation of Einstein's equation

Einstein's equation implies that, for each observer, the curvature scale $kappa ,$ of space is proportional to apparent density $rho ,$ :

${displaystyle kappa ={16pi G over c^{2}}rho }$
Symbol	Name	Value	Unit
$c$	Speed of light	299792458	m/s
$G$	Constant of universal gravitation	6.6741E^-11	N m² / kg²

In accordance with the geometric meaning of scalar curvature, this equality states that in a sphere of mass M and constant density, the radial excess (the difference between the real radius and the radius that would correspond in Euclidean geometry to a sphere of equal area) is equal to

${displaystyle Delta r={GM over {3c^{2}}}}$

For example, in the case of the Earth the radial excess is 1.5 mm and in the case of the Sun it is about 495 m.

It is amazing that this equation, which introduces minimal corrections in the formulas of Euclidean geometry, includes almost all the known equations of macroscopic physics. Indeed, when the speed of light c tends to infinity, Newton's law of universal gravitation, Poisson's Equation and, therefore, the attractive character of the forces are derived from it. gravitational forces, the equations of fluid mechanics (continuity equation and Euler's equations), the laws of conservation of mass and momentum, the Euclidean character of space, etc.

Equally all relativistic conservation laws are derived, and that the existence of gravitational and mass fields is only possible when space has more than two dimensions. Furthermore, if space is assumed to have four dimensions (the three we see daily plus a tiny extra circular dimension, roughly the size of the so-called Planck length, 10^-33 cm) from the equation From Einstein the classical theory of electromagnetism is deduced: Maxwell's equations and, therefore, Coulomb's law, the Conservation of electric charge and Lorentz's law.

Classic limit

Main article: Classic Limit#Classic Limit of General Relativity

In the classic boundary the only non-nul component of the Ricci tensor is the temporary component ${displaystyle scriptstyle R_{00}}$ . To obtain the classic limit it must be assumed that the gravitational potential is very small in relation to the square of the speed of light and then the limit of equations should be taken when the speed of light tends to infinity. By doing these manipulations it is obtained that Einstein's field equations for the gravitational field are reduced to Poisson's differential equation for gravitational potential. Assuming that for weak gravitational fields the time-space metric can be written as a disruption of Minkowski metric:

${displaystyle g_{alpha beta }(mathbf {x})=eta _{alpha beta }+{frac {h_{alpha beta }(mathbf {x})}{c^{2}}},qquad h_{00}approx -2phi _{g}}$

The time component of the Ricci tensor turns out to be:

${displaystyle R_{00}approx -{frac {1}{2}}sum _{i}{frac {partial ^{2}h_{00}}{partial (x^{i})^{2}}}={frac {4pi G}{c^{2}}}(rho c^{2})Rightarrow nabla ^{2}phi _{g}=4pi Grho }$

The last expression is precisely the Poisson equation, which is the classical expression that relates the gravitational potential to the density of matter.

Solutions of Einstein's field equation

A solution of Einstein's field equation is some appropriate metric for the given distribution of mass and pressure of matter. Some solutions for a given physical situation are as follows.

Symmetric and static spherical mass distribution

The vacuum solution around a symmetrical, static spherical mass distribution is Schwarzschild metric and Kruskal-Szekeres metric. It applies to a star and leads to the prediction of a horizon of events beyond which it cannot be observed. Predicts the possible existence of a given mass black hole $M$ of which no energy can be extracted, in the classic sense of the term (i.e., not mechanical-quantum).

See also: Radio de Schwarzschild and Schwarzschild black hole.

Rotating Axisymmetric Mass

The solution for the empty space around a rotationally axisymmetric mass distribution is the Kerr metric. It is applied to a rotating star and leads to the prediction of the possible existence of a rotating black hole of given mass M and angular momentum J, from which rotational energy can be extracted.

Isotropic and homogeneous (or uniform) universe

The solution for an isotropic and homogeneous universe, filled with constant density and negligible pressure, is the Robertson-Walker metric. It applies to the universe as a whole and leads to various models of its evolution that predict an expanding universe.

See also: Friedman-Lemaître-Robertson-Walker and Acceleration of the expansion of the Universe.

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