Einstein-Hilbert effect

The Einstein-Hilbert effect is a mathematical expression from the general theory of relativity , which was first given by David Hilbert . From this effect , the leave Einstein's field equations with the principle of least action derived. ${\displaystyle W_{\text{Gravitation}}[g]}$

Mathematically, the Einstein-Hilbert effect is formulated as follows:

${\displaystyle W_{\text{Gravitation}}[g]={\frac {c^{4}}{16\pi G_{\text{N}}}}\int {\sqrt {|\det {g}(x)|}}\,R(g(x))\,\mathrm {d} ^{4}x}$

It is

• ${\displaystyle c}$the speed of light
• ${\displaystyle G_{\text{N}}}$the Newtonian gravitational constant
• ${\displaystyle g}$the metric tensor
• ${\displaystyle R}$the curvature scalar .

The requirement that the variation in effect vanish for every variation in the metric provides the equations ${\displaystyle \delta W[g]}$ ${\displaystyle \delta g}$

${\displaystyle R_{mn}(x)-{\frac {R(x)}{2}}\,g_{mn}(x)=0\,,}$

where denotes the components of the Ricci tensor . ${\displaystyle R_{mn}(x)}$

These are the field equations in a vacuum in the absence of particles and fields and in the case of vanishing vacuum energy density . The right-hand side of the field equations, the components of the energy-momentum tensor , is obtained by varying the part of the effect that describes matter according to the metric. The pre-factor

${\displaystyle {\frac {c^{4}}{16\pi G_{\text{N}}}}}$

prior to the Einstein-Hilbert action determines the strength, with the energy and pulse the gravitational produce.

In order to get the cosmological constant in the field equations, the effect can be given a term ${\displaystyle \Lambda }$

${\displaystyle -{\frac {c^{4}}{16\pi G_{\text{N}}}}\int \!{\sqrt {|\det {g}(x)|}}\,2\,\Lambda \,\mathrm {d} ^{4}x}$

Add. Such a term can also be understood as a component of the energy-momentum tensor, which has the advantage that it provides a physical justification for the cosmological constant. Today (2016) there are a large number of models that try to explain a cosmological constant through the matter content of the universe with varying degrees of success.