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Difference between revisions of "Einstein equations"

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''of the gravitational field''
 
''of the gravitational field''
  
 
Fundamental equations in the general theory of relativity. They connect the metric tensor of the space-time continuum, which describes the gravitational field, and the physical characteristics of different forms of matter, described by means of the energy-momentum tensor:
 
Fundamental equations in the general theory of relativity. They connect the metric tensor of the space-time continuum, which describes the gravitational field, and the physical characteristics of different forms of matter, described by means of the energy-momentum tensor:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035210/e0352101.png" /></td> </tr></table>
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$$R_{ik}-\frac12g_{ik}R=\frac{8\pi}{c^4}GT_{ik}.$$
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035210/e0352102.png" /> is the [[Ricci tensor|Ricci tensor]], which can be expressed in terms of the metric tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035210/e0352103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035210/e0352104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035210/e0352105.png" /> is the energy-momentum tensor, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035210/e0352106.png" /> is the speed of light in vacuum, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035210/e0352107.png" /> is the gravitational constant.
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Here $R_{ik}$ is the [[Ricci tensor|Ricci tensor]], which can be expressed in terms of the metric tensor $g_{ik}$, $R=R_i^i$, $T_{ik}$ is the energy-momentum tensor, $c$ is the speed of light in vacuum, and $G$ is the gravitational constant.
  
 
====References====
 
====References====

Latest revision as of 13:29, 15 August 2014

of the gravitational field

Fundamental equations in the general theory of relativity. They connect the metric tensor of the space-time continuum, which describes the gravitational field, and the physical characteristics of different forms of matter, described by means of the energy-momentum tensor:

$$R_{ik}-\frac12g_{ik}R=\frac{8\pi}{c^4}GT_{ik}.$$

Here $R_{ik}$ is the Ricci tensor, which can be expressed in terms of the metric tensor $g_{ik}$, $R=R_i^i$, $T_{ik}$ is the energy-momentum tensor, $c$ is the speed of light in vacuum, and $G$ is the gravitational constant.

References

[1] L.D. Landau, E.M. Lifshitz, "The classical theory of fields" , Addison-Wesley (1962) (Translated from Russian)


Comments

References

[a1] S. Weinberg, "Gravitation and cosmology" , Wiley (1972) pp. Chapt. 7
[a2] R.M. Wald, "General relativity" , Univ. Chicago Press (1984) pp. Chapt. 4
How to Cite This Entry:
Einstein equations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Einstein_equations&oldid=14498
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article