What is the name of this formula?

$$ G_{\mu\nu} = 8 \pi T_{\mu\nu} $$


2 Answers 2


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.

  • $\begingroup$ How is this formula derived ? $$ G_{\mu\nu} = 8 \pi T_{\mu\nu} $$ $\endgroup$
    – user222524
    Commented Jun 22, 2019 at 9:55
  • 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? $\endgroup$
    – jacob1729
    Commented Jun 22, 2019 at 10:33
  • $\begingroup$ @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
  • 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
  • $\begingroup$ the Einstein equations can be derived from thermodynamic considerations arxiv.org/pdf/gr-qc/9504004.pdf $\endgroup$
    – d_b
    Commented Jun 22, 2019 at 19:32

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.

  • $\begingroup$ Dose it apply to black holes ? $$ G_{\mu\nu} = 8 \pi T_{\mu\nu} $$ $\endgroup$
    – user222524
    Commented Jun 22, 2019 at 13:07
  • 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

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