Spacetime Curvature: Which Tensor Gives Coordinates?

[jpescarcega]

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?

 

[cpsinkule]

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]

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.

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?

 

[Mentz114]

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?

Suppose you want to plot the field around a spherical source. The solution of the metric is known but the Ricci tensor is zero, so that is no use. I would say that you should plot the tidal effects which are got by projecting the Riemann tensor into the spacetime of a stationary observer or a freely falling observer. The result of this is three forces which act in the radial direction and in the two directions orthogonal to the radius.

According to Prof J Baez, this captures the meaning of the Einstein field equations. See http://math.ucr.edu/home/baez/einstein/