Curvature and the Einstein field equation

Use the parallel transport rule from the previous topic to carry a vector around a small closed loop. Its leading change depends on the loop's oriented area and the starting vector. The Riemann tensor records this local curvature information. One unchanged vector or one unchanged loop does not prove that every curvature component vanishes.

With the cosmological constant set to zero, the field equation is Gμν=(8πG/c⁴)Tμν. Here Gμν=Rμν−(1/2)Rgμν is the Einstein tensor, built from curvature. The stress-energy tensor Tμν describes energy density, momentum, pressure and stresses. The symbol G without indices is Newton's gravitational constant.

Loop rotation is evidence of curvature when the vector has been parallel-transported. Rotating an arrow by hand is not the same test. For a sufficiently small loop, the leading mismatch scales with its area. To recover local curvature information, compare mismatches for chosen vectors and oriented loop areas.

Here Φ is gravitational potential, meaning potential energy per unit mass; U=mΦ. Newton's Poisson equation ∇²Φ=4πGρ relates spatial variation of Φ to mass density ρ. In Cartesian coordinates, ∇²Φ sums its second derivatives in the three spatial directions. Matching the weak, nearly static, slow-motion limit fixes the coupling to 8πG/c⁴ in our convention. A weak field alone is not enough. Near a compact object or at relativistic speeds, use the relativistic equations instead.

Stress-energy constrains geometry, but specifying it does not by itself select one metric. Initial or boundary data are also needed. Different vacuum gravitational-wave data can give different geometries even with the same local stress-energy. Matter moving through the resulting geometry must also obey its equations of motion.

In vacuum, Tμν=0 and, with zero cosmological constant, Rμν=0. This does not require the full Riemann tensor to vanish. The empty region outside a spherical mass can have tidal curvature. Contraction pairs one upper and one lower tensor index and sums over that pair. Ricci curvature is such a contraction of Riemann curvature. Setting that sum to zero does not erase every Riemann component.