What are the Einstein field equations

Definition/Summary

The Einstein Field Equations are a set of ten differential equations which express the general theory of relativity mathematically: they relate the geometry (the curvature) of spacetime to the energy/matter content of spacetime.

These ten differential equations may be written as a single second-order (two-index) symmetric tensor equation, relating the Ricci curvature tensor [itex]R_{\mu\nu}[/itex] to the stress-energy tensor [itex]T_{\mu\nu}[/itex].

Equations

Short version (using Einstein tensor [itex]G_{\mu\nu}[/itex]):

[tex]G_{\mu\nu}\ =\ \frac{8\pi G}{c^4}\,T_{\mu\nu}[/tex]

Using standard cosmological units with [itex]G\ =\ c\ =\ 1[/itex]:

[tex]G_{\mu\nu}\ =\ 8\pi\,T_{\mu\nu}[/tex]

Long version (using Ricci curvature tensor [itex]R_{\mu\nu}[/itex] and scalar curvature [itex]R\ =\ Tr(R_{\mu\nu})[/itex]):

[tex]R_{\mu\nu}\ -\ \frac{1}{2}\,R\,g_{\mu\nu}\ =\ 8\pi\,T_{\mu\nu}[/tex] or [tex]T_{\mu\nu}-\ \frac{1}{2}\,T\,g_{\mu\nu}\ =\ \frac{1}{8\pi}\,R_{\mu\nu}[/tex]

"Symmetric" decomposition, into scalar part:

[tex]R\ =\ -\,8\pi\,T[/tex]

and traceless symmetric tensor part:

[tex]R_{\mu\nu}\ -\ \frac{1}{4}\,R\,g_{\mu\nu}\ =\ 8\pi\left(T_{\mu\nu}\ -\ \frac{1}{4}\,T\,g_{\mu\nu}\right)[/tex]

Extended explanation

Cosmological units:

Cosmology is one of the few areas in which practitioners prefer not to use SI units.

Cosmological units are defined so that [itex]G\ =\ c\ =\ 1[/itex]


Structure of the EFE:

A second-order (two-index) tensor equation is the simplest possible equation which could describe the relationship between curvature and matter/energy.

The only two-index tensor describing matter and energy is the symmetric stress-energy tensor, [itex]T_{\mu\nu}[/itex].

The only two-index tensors describing the structure of space are the symmetric Ricci curvature tensor [itex]R_{\mu\nu}[/itex] and the symmetric metric tensor [itex]g_{\mu\nu}[/itex].

Also available, as scalar multipliers, are the traces [itex]R\ =\ Tr(R_{\mu\nu})[/itex] and [itex]T\ =\ Tr(T_{\mu\nu})[/itex]

The EFE is the only combination of these which, in the weak-field limit, gives the inverse-square law of Newtonian gravity.

A very small multiple of [itex]g_{\mu\nu}[/itex] may also be inserted into the EFE without noticeably affecting the weak-field limit: that multiple is the cosmological constant, [itex]\Lambda[/itex], whose value is estimated at less than [itex]10^{-35}\,s^{-2}[/itex]

Trace and traceless:

A symmetric tensor has one scalar invariant: the trace.

By comparison, an anti-symmetric tensor has two scalar invariants, usually written in the form [itex]E^2 - B^2[/itex] and [itex]\boldsymbol{E}\cdot\boldsymbol{B}[/itex]

A symmetric tensor equation can be split into two parts, a scalar trace equation, and a symmetric traceless tensor equation.

For the EFE, these show that (except for the factor [itex]8\pi[/itex]):
trace of Ricci curvature equals minus trace of stress-energy,
but traceless Ricci curvature equals traceless stress-energy:​

[tex]Tr(R_{\mu\nu})\ =\ R\ =\ -\,8\pi\,T\ =\ -\,8\pi\,Tr(T_{\mu\nu})[/tex]

[tex]Notr(R_{\mu\nu})\ =\ R_{\mu\nu}\ -\ \frac{1}{4}\,R\,g_{\mu\nu}\ =\ 8\pi\left(T_{\mu\nu}\ -\ \frac{1}{4}\,T\,g_{\mu\nu}\right)\ =\ 8\pi\,Notr(T_{\mu\nu})[/tex]

The notation "Notr" is not a standard notation.

[itex]Tr(A_{\mu\nu})[/itex] is defined as [itex]g^{\mu\nu}A_{\mu\nu}[/itex]

Note that (in a four-dimensional space) [itex]Tr(g_{\mu\nu})\ =\ \frac{1}{4}[/itex] and so [itex]Tr\left(A_{\mu\nu}\ -\ \frac{1}{4}\,g_{\mu\nu}\,Tr(A_{\mu\nu})\right)\ =\ (1\ -\ \frac{1}{4}\,g_{\mu\nu})Tr(A_{\mu\nu})\ =\ 0[/itex] and so it would be more convenient if trace were defined to be one-quarter of its standard definition.


The reason for the factor [itex]8\pi[/itex] is ultimately that [will someone please complete this paragraph? :wink:]

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