What is the name of this formula?
$$ G_{\mu\nu} = 8 \pi T_{\mu\nu} $$
2 Answers
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That is the Einstein equation for general relativity:
$$ G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu} $$
Specifically it is the form of the equation in geometrical units where speed of light and the gravitational constant are both equal to one, and the cosmological constant is zero.
answered Jun 22, 2019 at 9:36
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$\begingroup$ How is this formula derived ? $$ G_{\mu\nu} = 8 \pi T_{\mu\nu} $$ $\endgroup$– user222524Commented Jun 22, 2019 at 9:55
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2$\begingroup$ @Cujo they're the field equations of relativity, they can't really be derived since they define the theory. Would you for example say that you can derive Maxwell's equations? $\endgroup$ Commented Jun 22, 2019 at 10:33
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$\begingroup$ @Cujo You will find a huge amount of material on this equation on the internet, including many videos and papers describing their development and use. The mathematics is quite involved (Tensor calculus). $\endgroup$ Commented Jun 22, 2019 at 11:46
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1$\begingroup$ I might express the open words as "That is a special case of the Einstein [...] with the cosmological constant set to zero". $\endgroup$ Commented Jun 22, 2019 at 15:13
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$\begingroup$ the Einstein equations can be derived from thermodynamic considerations arxiv.org/pdf/gr-qc/9504004.pdf $\endgroup$– d_bCommented Jun 22, 2019 at 19:32
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It's called Einstein Field Equation for gravitational fields.
The LHS term $G_{\mu v}$ is the Einstein Tensor
$$G_{\mu v}=R_{\mu v}+\frac{1}{2}g_{\mu v}R$$
which gives information about how the geometry of spacetime is altered by the presence of e.g. matter, pressure, energy and momentum in the universe, which are described mathematically by $T_{\mu v}$ - the Energy-Momentum Tensor on the RHS.
To be able to derive this, you need to have a rigorous mathematical background in
-Partial differential Equations
-Linear Algebra
-Tensor calculus/differential geometry.
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$\begingroup$ Dose it apply to black holes ? $$ G_{\mu\nu} = 8 \pi T_{\mu\nu} $$ $\endgroup$– user222524Commented Jun 22, 2019 at 13:07
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1$\begingroup$ Yes, closely related. The existence of black holes was predicted by one of the solutions of this equation (the Schwarzschild solution). $\endgroup$ Commented Jun 22, 2019 at 15:54