The discussion revolves around the units of the metric tensor in the context of general relativity, particularly in relation to the cosmological constant and the momentum-energy tensor. Participants explore the implications of these units and their roles in the equations governing spacetime geometry.
Participants express differing views on whether the metric tensor has units, with some asserting it has none while others raise questions about the implications of unit discrepancies in related tensors. The discussion remains unresolved regarding the units of the metric tensor.
There are unresolved questions about the assumptions underlying the units of the metric tensor and the momentum-energy tensor, as well as the implications of these units in the context of general relativity.
This discussion may be of interest to those studying general relativity, particularly in understanding the mathematical and conceptual foundations of the metric tensor and its applications in physics.
Antonio Lao said:I'm still not clear why the righthand side of Einstein's field equations, the momentum-energy tensor has components of different units (units in pressures, and units in densities).
The METRIC is the star of general relativity. It describes everything about the geometry of spacetime, since it let's us measure angles and distances. Einstein's equation describes how the flow of energy and momentum through spacetime affects the metric. What it affects is something about the metric called the "curvature". The biggest job in learning general relativity is learning to understand curvature!
Mathematically, the metric g is a tensor of rank (0,2). It eats two tangent vectors v,w and spits out a number g(v,w), which we think of as the "dot product" or "inner product" of the vectors v and w. This let's us compute the length of any tangent vector, or the angle between two tangent vectors. Since we are talking about spacetime, the metric need not satisfy g(v,v) > 0 for all nonzero v. A vector v is SPACELIKE if g(v,v) > 0, TIMELIKE if g(v,v) < 0, and LIGHTLIKE if g(v,v) = 0.
The STRESS-ENERGY TENSOR. The stress-energy is what appears on the right side of Einstein's equation. It is a tensor of rank (0,2), and it defined as follows: given any two tangent vectors u and v at a point p, the number T(u,v) says how much momentum in the u direction is flowing through the point p in the v direction. Writing it out in terms of components in any coordinates, we have
T(u,v) = T_ab u^a v^b