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Tetrads in General Relativity

X. Curvature

Vacuum Solutions

   The Einstein field equations can have non-trivial solutions even in regions where there are no sources, i.e. in regions of spacetime that are devoid of matter and radiation. There the Einstein tensor must vanish:

10.27

\[ {\text{T(}}{{\mathbf{g}}_\mu }) = 0 \to {\text{G}}({{\mathbf{g}}_\mu })=0\]

This implies that the Ricci tensor vanishes: ${\text{Ric}}({{\mathbf{g}}_\mu }) = 0 \to R = 0$. But the relationship between the Ricci and Riemann is such that the vanishing of the Ricci not necessarily implies that the Riemann should be zero. If the Riemann tensor is not zero, then spacetime must be curved. Such solutions satisfying the Einstein equations (10.27) are called vacuum solutions. One of those is the Schwarzschild solution, which is the first, and arguably the most important, non-trivial vacuum solution of the Einstein field equations; see sections VIII,IX.

   In 1917 Einstein proposed to add to his tensor a new free constant $\Lambda $ according to

10.28

\[{G_{\mu \nu }} \to {G_{\mu \nu }} - \Lambda {g_{\mu \nu }}{\qquad}\operatorname{G} ({{\mathbf{g}}_\mu }) \to \operatorname{G} ({{\mathbf{g}}_\mu }) - \Lambda {{\mathbf{g}}_\mu }\]

thereby relaxing the requirement that ${G_{\mu \nu }}$ vanishes when spacetime is flat. The new tensor would still have all other properties of the original, but it would allow a static unchanging universe as one particular solution of his equations. Then, in 1929, Edward Hubble discovered that the universe was expanding, as Aleksandr Friedmann and Georges Lemaitre had predicted theoretically a few years earlier. Einstein subsequently retracted his proposal calling it “the biggest blunder of my life”.

   Still, to this day, the constant $\Lambda $, now called the cosmological constant, is a parameter in cosmological models to explain the actually observed accelerating expansion of the universe. It is also the simplest explanation for the existence of dark energy in the universe. The cosmological constant $\Lambda$ is a dimensionful parameter with units of inverse length squared. In the Einstein equation it has the same effect as an intrinsic energy density of the vacuum - the lowest energy state of the universe, This is now the common interpretation of the cosmological constant in the standard model of cosmology known as the ΛCDM model. [Wikipedia: Lambda-CDM model]