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Ahmed Samra
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When does the arc of the space curvature is large and when is it small?
Would you expect it to be in reverse?Ahmed Samra said:I read it in the Internet. Anyway, large masses lead to a large space curvature while small masses lead to a small space curvature right?
It does matter if you are 1m away from a mass or 1000km.Ahmed Samra said:What do you mean by at the same distance?
You could, for example, compute the Riemann curvature tensor ##R^{a}_{bcd}## in the coordinate basis using the formulas here:http://en.wikipedia.org/wiki/Riemann_curvature_tensor#Coordinate_expression and then form a curvature scalar such as ##R^{abcd}R_{abcd}##. If you do this for the Schwarzschild metric, you will get ##R^{abcd}R_{abcd} = \frac{48G^{2}M^{2}}{c^{4}r^{6}}##.Ahmed Samra said:Is there a formula to know how large or small is the space-time curvature? And what is the formula?
When does the arc of the space curvature is large and when is it...
What notion of "space curvature" are you using that allows you to make such claims? How are you even defining "space curvature"?Naty1 said:(SPACE curvature is not coordinate-free; a change of coordinates makes space flat
WannabeNewton said:If "space curvature" is being used to refer to the induced curvature on a one-parameter family of space-like hypersurfaces that foliate the spacetime, then the very notion of "space curvature" is utterly meaningless unless such a foliation exists.
IF such a foliation exists, there will be a unit normal field ##n^{a}## to this family
and there will be a spatial metric ##h_{ab}## induced on each hypersurface given by ##h_{ab} = g_{ab} + n_{a}n_{b}##.
Each hypersurface has an extrinsic curvature given by ##K_{ab} = h_{a}{}{}^{c}\nabla_{c}n_{b}##.
It isn't confusing at allpervect said:Such a foilation exists in this case, and I would call that normal vector field [itex]\hat{t}[/itex], I hope the notation isn't confusing.
Yes indeed that would be the spatial metric on a single hypersurface of ##t = const.##.pervect said:That would be, in this example
[tex]
h_{ab} = \left( 1 + \frac{m}{2r} \right)^4 \left(dx^2 + dy^2 + dz^2 \right) [/tex]
correct?
It is raised by ##g^{ab}## so the first one would the one to use.pervect said:is
[tex]h_{a}{}^{c} = g^{bc} h_{ab}[/tex]
?
or is it
[tex]h_{a}{}^{c} = h^{bc} h_{ab}[/tex]
??
I'm not exactly sure what you mean here but in ##K_{\mu\nu} = h_{\mu}{}{}^{\alpha}\nabla_{\alpha}n_{\nu}## if we fix a particular value of ##t## then we would get the extrinsic curvature of a single space-like hypersurface as represented in this coordinate system, but until then it is giving us a way of assigning the extrinsic curvature to any member of the family as ##t## varies. Note however that even though in this particular case we are evaluating all these things in a particular coordinate system, the original relations given were not coordinate dependent.pervect said:I think I'm beginning to loose it here, ## \nabla_{c}n_{b}## looks like it should be 4-d, though, and ## K_{ab} ## should be 3-d...
wannabe says:...nowhere has a precise definition of the term “gravitational field” been given --- nor will one be given. Many different mathematical entities are associated with gravitation; the metric, the Riemann curvature tensor, the curvature scalar … Each of these plays an important role in gravitation theory, and none is so much more central than the others that it deserves the name “gravitational field.”
I never thought such a statement was anything but simple...I simply mean, for example, if the ds interval of GR is invariant, how could the space component [by itself] not be?? Or another way I think about it, if there is observer dependent space contraction, how could practically any curvature escape that??Quote by Naty1
(SPACE curvature is not coordinate-free; a change of coordinates makes space flat
What notion of "space curvature" are you using that allows you to make such claims? How are you even defining "space curvature"?
The problem is that the notion of "space curvature" is ill-defined as mentioned above. If you have some reference as to some standard definition for "space curvature" for arbitrary space-times then I would much appreciate it but if you mean the curvature of members of a space-like foliation of space-time then that has been discussed above and only applies to space-times where such a foliation exists.Naty1 said:I never thought such a statement was anything but simple...
WannabeNewton said:if you mean the curvature of members of a space-like foliation of space-time then that has been discussed above and only applies to space-times where such a foliation exists.
WannabeNewton said:Indeed, so as Peter explained and gave examples of, "space curvature" can have so many meanings that it's hard to make general statements by just saying "space curvature".
By the way, just to clarify, I did state that ##K_{ab}## was the extrinsic curvature in post #15 when I first mentioned it.PeterDonis said:Yes, and just to throw yet another distinction into the foodmixer , I believe the quantity ##K_{ab}## that you mentioned, from Wald, is related to the extrinsic curvature--
WannabeNewton said:Pervect that is a nice decomposition indeed! Do you know of any book(s) that go into more detail on it's geometrical interpretations for the most physically relevant space-times?
Thank you very much for looking that up pervect. I'll take a look at it as soon as I can. Cheers!pervect said:There's a short bit in MTW - it doesn't give the components names, but it breaks them down in the same way. See exercise 14.14 on pg 360. The 3x3 submatrixes are called E,F, and H
The Arc of Space Curvature refers to the bending or warping of space caused by the presence of massive objects, such as stars and galaxies. This phenomenon is described by Einstein's theory of general relativity.
The Arc of Space Curvature can have a significant impact on the motion and behavior of large objects, such as planets and stars. It can cause the objects to orbit around each other, as seen in the case of the Earth orbiting around the Sun.
Yes, the Arc of Space Curvature can be observed in everyday life through the phenomenon of gravitational lensing. This occurs when the curvature of space caused by a massive object, such as a galaxy, bends the path of light from a distant object, making it appear distorted or magnified.
The Arc of Space Curvature plays a crucial role in shaping the structure and evolution of the universe. It influences the distribution of matter and energy, leading to the formation of galaxies, clusters, and superclusters.
No, the Arc of Space Curvature can vary in different regions of the universe. It is influenced by the distribution of matter and energy, and it can change over time due to the expansion of the universe.