Difference between revisions of "Web Element: Math"

Latest revision as of 13:57, 11 December 2019

Einstein field equations (EFE),

[math]R_{\mu \nu} - {1 \over 2}R \, g_{\mu \nu} + \Lambda g_{\mu \nu}= {8 \pi G \over c^4} T_{\mu \nu} [/math],

where [math]R_{\mu \nu} [/math] is the Ricci curvature tensor, [math]R[/math] is the scalar curvature, [math]g_{\mu \nu} [/math] is the metric tensor (general relativity), [math]Λ [/math] is the cosmological constant, [math]G[/math] is Newton's gravitational constant, [math]c[/math] is the speed of light in vacuum, and [math]T_{\mu \nu} [/math] is the stress–energy tensor.

The above form of the EFE is the standard established by Gravitation by Misner, Thorne, and Wheeler. The authors analyzed all conventions that exist and classified according to the following three signs (S1, S2, S3):

[math] \begin{align} g_{\mu \nu} & = [S1] \times \operatorname{diag}(-1,+1,+1,+1) \\[6pt] {R^\mu}_{\alpha \beta \gamma} & = [S2] \times \left(\Gamma^\mu_{\alpha \gamma,\beta}-\Gamma^\mu_{\alpha \beta,\gamma}+\Gamma^\mu_{\sigma \beta}\Gamma^\sigma_{\gamma \alpha}-\Gamma^\mu_{\sigma \gamma}\Gamma^\sigma_{\beta \alpha}\right) \\[6pt] G_{\mu \nu} & = [S3] \times \frac{8 \pi G}{c^4} T_{\mu \nu} \end{align} [/math]

The third sign above is related to the choice of convention for the Ricci tensor:

[math]R_{\mu \nu}=[S2]\times [S3] \times {R^\alpha}_{\mu\alpha\nu} [/math]

Taking the trace with respect to the metric of both sides of the EFE one gets

[math]R - \frac{D}{2} R + D \Lambda = \frac{8 \pi G}{c^4} T \,[/math]

where [math]D[/math] is the spacetime dimension. This expression can be rewritten as

[math]-R + \frac{D \Lambda}{\frac{D}{2} -1} = \frac{8 \pi G}{c^4} \frac{T}{\frac{D}{2}-1} \,.[/math]

If one adds [math]−\frac{1}{2} g_{\mu \nu} [/math] times this to the EFE, one gets the following equivalent "trace-reversed" form

[math]R_{\mu \nu} - \frac{ \Lambda g_{\mu \nu}}{\frac{D}{2}-1} = \frac{8 \pi G}{c^4} \left(T_{\mu \nu} - \frac{1}{D-2}Tg_{\mu \nu}\right) .\,[/math]

For example, in [math]D = 4 [/math] dimensions this reduces to

[math]R_{\mu \nu} - \Lambda g_{\mu \nu} = \frac{8 \pi G}{c^4} \left(T_{\mu \nu} - \tfrac{1}{2}T\,g_{\mu \nu}\right) .\,[/math]

Reversing the trace again would restore the original EFE. The trace-reversed form may be more convenient in some cases (for example, when one is interested in weak-field limit and can replace [math]g_{\mu \nu} [/math] in the expression on the right with the Minkowski metric without significant loss of accuracy).

@@ Line 1: / Line 1: @@
-Einstein field equations (EFE)
+Einstein field equations (EFE),
-{{#tag:math|R_{\mu \nu} - {1 \over 2}R \, g_{\mu \nu} + \Lambda g_{\mu \nu}= {8 \pi G \over c^4} T_{\mu \nu} }}
+{{#tag:math|R_{\mu \nu} - {1 \over 2}R \, g_{\mu \nu} + \Lambda g_{\mu \nu}= {8 \pi G \over c^4} T_{\mu \nu} }},
 where {{#tag:math|R_{\mu \nu} }} is the Ricci curvature tensor, {{#tag:math|R}} is the scalar curvature, {{#tag:math|g_{\mu \nu} }} is the metric tensor (general relativity), {{#tag:math|Λ }} is the cosmological constant, {{#tag:math|G}} is Newton's gravitational constant, {{#tag:math|c}} is the speed of light in vacuum, and {{#tag:math|T_{\mu \nu} }} is the stress–energy tensor.
@@ Line 21: / Line 21: @@
 :<math>R - \frac{D}{2} R + D \Lambda = \frac{8 \pi G}{c^4} T \,</math>
-where {{mvar|D}} is the spacetime dimension. This expression can be rewritten as
+where {{#tag:math|D}} is the spacetime dimension. This expression can be rewritten as
 :<math>-R + \frac{D \Lambda}{\frac{D}{2} -1} = \frac{8 \pi G}{c^4} \frac{T}{\frac{D}{2}-1} \,.</math>
-If one adds {{math|−{{sfrac|1|2}}''g{{sub|μν}}''}} times this to the EFE, one gets the following equivalent "trace-reversed" form
+If one adds {{#tag:math|−\frac{1}{2} g_{\mu \nu} }} times this to the EFE, one gets the following equivalent "trace-reversed" form
 :<math>R_{\mu \nu} - \frac{ \Lambda g_{\mu \nu}}{\frac{D}{2}-1} = \frac{8 \pi G}{c^4} \left(T_{\mu \nu} - \frac{1}{D-2}Tg_{\mu \nu}\right) .\,</math>
-For example, in {{math|''D'' {{=}} 4}} dimensions this reduces to
+For example, in {{#tag:math|D = 4 }} dimensions this reduces to
 :<math>R_{\mu \nu} - \Lambda g_{\mu \nu} = \frac{8 \pi G}{c^4} \left(T_{\mu \nu} - \tfrac{1}{2}T\,g_{\mu \nu}\right) .\,</math>
-Reversing the trace again would restore the original EFE. The trace-reversed form may be more convenient in some cases (for example, when one is interested in weak-field limit and can replace {{mvar|g{{sub|μν}}}} in the expression on the right with the [[Minkowski metric]] without significant loss of accuracy).
+Reversing the trace again would restore the original EFE. The trace-reversed form may be more convenient in some cases (for example, when one is interested in weak-field limit and can replace {{#tag:math|g_{\mu \nu} }} in the expression on the right with the Minkowski metric without significant loss of accuracy).