These employ a great deal of differential geometry, and quite a few things that I have not worked much with preveously, such as affine connections, the torsion tensor, parallel transport, affine geodetics, affine flatness, tangent / fibre bundle, sections, Lie derivative, covariant derivative, absolute derivative, Riemann tensor, Ricci tensor, Einstein tensor, Christoffel symbols, Levi-Civita connection, etc.
32.
In the case of a Lorentzian manifold, n = 4, the Einstein tensor G _ { ab } = R _ { ab }-1 / 2 \, g _ { ab } R has, by design, a trace which is just the negative of the Ricci scalar, and one may check that the traceless part of the Einstein tensor agrees with the traceless part of the Ricci tensor.
33.
In the case of a Lorentzian manifold, n = 4, the Einstein tensor G _ { ab } = R _ { ab }-1 / 2 \, g _ { ab } R has, by design, a trace which is just the negative of the Ricci scalar, and one may check that the traceless part of the Einstein tensor agrees with the traceless part of the Ricci tensor.
34.
Here, the Einstein tensor G _ { \ mu \ nu } describes the curvature of space-time, whilst the energy-momentum tensor T _ { \ mu \ nu } describes the local distribution of matter . ( \ kappa is a constant . ) The Einstein equations express " local " relationships between the quantities involved specifically, this is a system of coupled non-linear second order partial differential equations.
35.
The GR field equation ( ignoring some uninteresting constants ) looks like G + ?g = T, where G is the " Einstein tensor " describing spacetime curvature, ? is the " cosmological constant ", g is the " metric tensor " describing . . . well . . . spacetime curvature again, I suppose, and T is the " stress-energy tensor " describing everything else ( the other forces and the fermions ).
36.
Regardless of whether you use an inertial frame of reference or an accelerating frame of reference in this problem, the Riemann curvature tensor will be zero everywhere, so the Einstein tensor will be zero everywhere, and the Einstein field equations simplify down to 0 = 0 . ( The mass of the objects involved here has a negligible effect on the curvature, and so is taken to be zero . ) In other words, in this problem, even if you use an accelerating frame of reference, the Einstein field equations that describe gravity say nothing at all about the observed acceleration.
37.
Where G _ { \ mu \ nu } = R _ { \ mu \ nu }-{ R \ over 2 } g _ { \ mu \ nu } is the Einstein tensor, which combines the Ricci tensor, the scalar curvature and the metric tensor, \ Lambda is the cosmological constant, 0 T _ { \ mu \ nu } energy-momentum tensor of matter, \ pi is the irrational number originally introduced as the ratio of the circumference of a circle to its diameter, c is the speed of light, G Newton's gravitational constant.
