In special relativity the coordinates are those of a rectilinear inertial frame, so the Christoffel symbols term vanishes, but sometimes when authors use curved coordinates in order to describe an accelerated frame, the frame of reference isn't inertial, they will still describe the physics as special relativistic because the metric is just a frame transformation of the Minkowski space metric.
42.
:Could you explain ( just for my peace of mind ) how your mathematics, and the " Gruber and Price " examples, square with " " This is because gravitational time dilation is manifested in accelerated frames of reference " " in the article Gravitational time dilation, and how they square with the Equivalence principle ? ( I think I need to do a lot more reading on this before I understand it ! )"
43.
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.
44.
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.
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