Virtual black hole: Difference between revisions

where <math>G_{\mu\nu} = R_{\mu\nu} - {R \over 2} g_{\mu\nu}</math> is the [[Einstein tensor]], which combines the [[Ricci tensor]], the [[scalar curvature]] and the [[metric tensor]]; <math>\Lambda</math> is the [[cosmological constant]]; а <math>T_{\mu\nu}</math> is the energy-momentum tensor of matter; <math>\pi</math> is the mathematical constant [[pi]]; <math>c</math> is the [[speed of light]]; and <math>G</math> is Newton's [[gravitational constant]].

 

 

For any tensor field <math>N_{\mu\nu...}</math>, we may call <math>N_{\mu\nu...}\sqrt{-g}</math> a tensor density, where <math>g</math> is the [[determinant]] of the [[metric tensor]] <math>g_{\mu\nu}</math>. The integral <math>\int N_{\mu\nu...}\sqrt{-g}\,d^4x</math> is a tensor if the domain of integration is small. It is not a tensor if the domain of integration is not small, because it then consists of a sum of tensors located at different points and it does not transform in any simple way under a transformation of coordinates.<ref>[https://vk.com/doc264717166_454951866 P.A.M. Dirac(1975), General Theory of Relativity, Wiley Interscience], p.37</ref> Here we consider only small domains. This is also true for the integration over the three-dimensional [[hypersurface]] <math>S^{\nu}</math>.