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
+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} }},
-{{#tag:math|R_{\mu \nu} }} Ricci curvature tensor
+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.
-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.
+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):
-===Sign convention===
-The above form of the EFE is the standard established by [[Gravitation (book)|Misner, Thorne, and Wheeler]].{{sfnp|Misner|Thorne|Wheeler|1973|p=501ff}} The authors analyzed all conventions that exist and classified according to the following three signs (S1, S2, S3):
 :<math>
@@ Line 21: / Line 18: @@
 :<math>R_{\mu \nu}=[S2]\times [S3] \times {R^\alpha}_{\mu\alpha\nu} </math>
-With these definitions [[Gravitation (book)|Misner, Thorne, and Wheeler]] classify themselves as {{math|(+ + +)}}, whereas Weinberg (1972){{sfnp|Weinberg|1972}} and Peacock (1994){{sfnp|Peacock|1994}} are {{math|(+ − −)}}, Peebles (1980)<ref>{{cite book |last=Peebles |first=Phillip James Edwin |title=The Large-scale Structure of the Universe |location= |publisher=Princeton University Press |year=1980 |isbn=0-691-08239-1 }}</ref> and Efstathiou et al. (1990)<ref>{{cite journal |last=Efstathiou |first=G. |first2=W. J. |last2=Sutherland |first3=S. J. |last3=Maddox |title=The cosmological constant and cold dark matter |journal=[[Nature (journal)|Nature]] |volume=348 |issue=6303 |year=1990 |pages=705 |doi=10.1038/348705a0 }}</ref> are {{math|(− + +)}},  Rindler (1977){{citation needed|date=October 2014}}, Atwater (1974){{citation needed|date=October 2014}}, Collins Martin & Squires (1989)<ref>{{cite book |last=Collins |first=P. D. B. |last2=Martin |first2=A. D. |last3=Squires |first3=E. J. |year=1989 |title=Particle Physics and Cosmology |location=New York |publisher=Wiley |isbn=0-471-60088-1 }}</ref> are {{math|(− + −)}}.
+Taking the trace with respect to the metric of both sides of the EFE one gets
-Authors including Einstein have used a different sign in their definition for the Ricci tensor which results in the sign of the constant on the right side being negative
-:<math>R_{\mu \nu} - \tfrac{1}{2} R g_{\mu \nu} - \Lambda g_{\mu \nu} = -\frac{8 \pi G}{c^4} T_{\mu \nu}.</math>
-The sign of the (very small) cosmological term would change in both these versions, if the {{math|(+ − − −)}} metric [[sign convention]] is used rather than the MTW {{math|(− + + +)}} metric sign convention adopted here.
-===Equivalent formulations===
-Taking the [[Scalar curvature#Definition|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 {{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).