Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 141In local inertial coordinates, where the ball starts out at rest, we havev = (1, 0, 0, 0), so¨Vlim ∣ = −R 00 . (3.5)V −→0 V t=0In short, the Ricci tensor says how our ball of freely falling test particlesstarts changing in volume. The Ricci tensor only captures some of theinformation in the Riemann curvature tensor. The rest is captured bythe so–called the Weyl tensor (see e.g., [Penrose (1989); Penrose (1994);Penrose (1997)]), which says how any such ball starts changing in shape.The Weyl tensor describes tidal forces, gravitational waves and the like.Now, the Einstein equation in its usual form saysG αβ = T αβ .Here the right side is the stress-energy tensor, while the left side, the ‘Einsteintensor’, is just an abbreviation for a quantity constructed from theRicci tensor:G αβ = R αβ − 1 2 g αβR γ γ.Thus the Einstein equation really saysThis impliesbut g α α = 4, soR αβ − 1 2 g αβR γ γ = T αβ . (3.6)R α α − 1 2 gα αR γ γ = T α α ,−R α α = T α α .Substituting this into equation (3.6), we getR αβ = T αβ − 1 2 g αβT γ γ . (3.7)This is an equivalent version of the Einstein equation, but with the rolesof R and T switched [Baez (2001)]. This is a formula for the Ricci tensor,which has a simple geometrical meaning.Equation (3.7) will be true if any one component holds in all localinertial coordinate systems. This is a bit like the observation that all ofMaxwell’s equations are contained in Gauss’s law and and ∇ · B = 0.