Relating Einstein Tensor to Reality

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  • #1
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Hi,

So I have taken a math course dealing with the mathematics behind GR. In it I learnd how to calculate many of the objects that GR deals with such as curvature tensors, geodesics and so on. However, not much physical explination into how these object tie into gravity was discussed (due to time constraints) and the emphasis was in calculation rather than the theory behind.

So now i am trying to tie in the theory with what i can calculate using a metric.

I was wondering if i could get an explination on how to relate these curvature tenors back to physical space. For example, say I wanted to create a program that plots the amount of curvature around a spherical body using a colour scheme (darker = more curvature) what would I use to do this as far as the "tools" avaliable such as the tensors in the Einstein Tensor?

thanks
 

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  • #2
pervect
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MTW's gravitation has some good explanations of the physical significance of the curvature tensor. But I'll try to write a short post about it

Suppose you have Newtonian gravity, with some potential function V. Then you can think of the gravitational force as being [itex]\nabla_a V[/itex].

There is a second rank tensor, that's called the Newtonian Tidal tensor, giving by the covariant derivative of the above force, i.e. [itex]\nabla_b \nabla_a V[/itex]

Using different notation, there's a wiki article about the tidal tensor http://en.wikipedia.org/wiki/Tidal_tensor

The tidal tensor can be thought of as many ways, one way is that its components give the relative tidal acceleration as a function of distance.

The connection between the Riemann tensor R and the newtonian tidal tensor, which we will call E, is that

R_ptqt = E_pq

Furthermore, the Geodesic equation tells us that the tidal tensor is a map from the distance between geodesics that are initially parallel (and hence have zero relative velocity) to the acceleration between the worldlines of two particles following the geodesics.
 
  • #3
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Thanks this look very interesting.
 
  • #4
julian
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If you want a detailed derivation of the vacuum field equations (as outlined by pervect)...i.e physics...see Ray D'Inverno's book - pages 134 to 142.
 

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