The discussion centers around the question of which tensor in the Einstein Field Equations is appropriate for describing the coordinates related to the curvature of spacetime, particularly in the context of graphing this curvature. Participants explore the implications of various tensors, including the metric, Ricci tensor, and Riemann tensor, in relation to visualizing spacetime curvature in a simplified manner.
Participants do not reach a consensus on which tensor is most appropriate for graphing the curvature of spacetime. There are competing views regarding the utility of the Ricci tensor versus the Riemann tensor, and the discussion remains unresolved.
Participants express uncertainty regarding the specific conditions under which different tensors may be applicable for visualizing spacetime curvature, particularly in simplified two-dimensional representations. The discussion highlights the dependence on the definitions and interpretations of the tensors involved.
Oh ok that makes sense that the metric would be able to describe all geometric properties. Thank you. But, what I'm still wondering is whether or not the Ricci tensor is the right one for graphing the curve. What I mean by "graphing the curvature of spacetime", is plotting points of stressed spacetime. You've read about spacetime being like a trampoline, where when you place a bowling ball on top of it, it bends. Similarly, I'm sure you've seen qualitative interpretations of bent spacetime, which seem to look like parabolas, (like this). To be specific, what I want to do is strip the time component of the curvature of space, and strip the z component of the curvature of space and have a 2 dimensional curve (much like a parabola). But which tensor will take me to the information necessary for those cartesian coordinates?cpsinkule said:I'm not sure what you mean by graphing the "curvature" of spacetime. However, every term in the field equations on the left hand side is the metric, or some form of derivative of the metric. In essence, every piece of geometric data is in the metric, but the Ricci tensor tells you directly about the curvature of the space in the sense that it is 0 if the space is "flat" (necessary but not sufficient, the Riemann tensor must also be 0). But you need the metric to calculate it, regardless.
jpescarcega said:In the Einstein Field Equations: Rμν - 1/2gμνR + Λgμν = 8πG/c^4 × Tμν, which tensor will describe the coordinates for the curvature of spacetime? The equations above describe the curvature of spacetime as it relates to mass and energy, but if I were to want to graph the curvature of spacetime, which tensor would I look to for the coordinates?