curvature of spacetime
in [Relativity]
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From: JanPB on 25 Mar 2006 21:57 Nick wrote: > Electrons and protons are electric and magnetic. How does the > electromagnetic force hold electrons to the surface of an atom in > precise shells? Quantum mechanics. -- Jan Bielawski
From: Nick on 25 Mar 2006 22:01 You don't say much do you?
From: GSS on 26 Mar 2006 11:48 Tom Roberts wrote: > GSS wrote: > > Tom Roberts wrote: > >> This makes no sense, as the original Euclidean space is incommensurate > >> with the Schw. spacetime > > How come your response appears to be quite > > passimistic in comparison with that of Jan PB and Hammond? > Perhaps because I am more familiar with the geometry involved. Perhaps you *are* more familiar with geometry but not with physical reality. > > Kindly clarify whether in your opinion the 'foliated' 3-d space > > obtained from Schw. spacetime actually represents the physical 3-d > > space or not. > There _is_ no "physical 3-d space". And hence everything that exists in the "physical 3-d space" (including you and me) must be an illusion!! Is it ??? > In GR there is only spaceTIME. Space-time is not any physical entity that exists. It is a sum total of three different descriptions - PAST history, PRESENT existence and FUTURE that is yet to exist. In GR spacetime is just an_arbitrary_human construct, a model constructed in an attempt to give an alternative mathematical representation to the phenomenon of gravitation. I hope you do understand this fact. > You can choose to foliate spacetime into time and 3-space, but > there is nothing "physical" about that, as it is completely arbitrary. No > physical phenomena can depend upon an _arbitrary_ human choice. Do you admit that the actual physical phenomenon of gravitation cannot depend upon an _arbitrary_ human construct of spacetime Riemannian geometry? > > After all I am just trying to visualize how various plane surfaces in > > the original Euclidean space get curved in the 'foliated' 3-d Schw > > space. > There is no "original Euclidean space", either. Some time your pronouncements are quite axiomatic!! Kindly look at your very first sentence above. > You can consider either Euclidean space, > or Schwarzschild spacetime. But these are _different_ manifolds, > and they are _incommensurate_ as they have different topologies. ....... > Surfaces in one manifold cannot be related like that to surfaces > in another incommensurate manifold. <shrug> OK let us consider only the foliated 3-d Schw space. Happy? As shown earlier, there are three non-zero distinct components of Riemann curvature tensor for this space as given below: R_1212 = -(GM/c^2 r).(1/(1 - 2GM/c^2 r)) R_1313 = -(GM.Sin^2 (q)/c^2 r).(1/(1 - 2GM/c^2 r)) R_2323 = 2GM r.Sin^2 (q)/c^2 Kindly identify three 2-d surfaces in this space, the Gaussian curvature of which is represented by the above mentioned three components of the curvature tensor. GSS
From: Tom Roberts on 26 Mar 2006 20:05 GSS wrote: > Tom Roberts wrote: >> There _is_ no "physical 3-d space". > > And hence everything that exists in the "physical 3-d space" (including > you and me) must be an illusion!! Is it ??? Yes, it is an _approximation_ -- something our brains apparently cooked up because it is good enough for ordinary life. I don't think you can divorce your "existence in 3-d space" from time. >> In GR there is only spaceTIME. > > Space-time is not any physical entity that exists. Yes, of course. It is a _model_. <shrug> >> You can choose to foliate spacetime into time and 3-space, but >> there is nothing "physical" about that, as it is completely arbitrary. No >> physical phenomena can depend upon an _arbitrary_ human choice. > > Do you admit that the actual physical phenomenon of gravitation cannot > depend upon an _arbitrary_ human construct of spacetime Riemannian > geometry? Geometry is a _model_, not a "cause". >>> After all I am just trying to visualize how various plane surfaces in >>> the original Euclidean space get curved in the 'foliated' 3-d Schw >>> space. >> There is no "original Euclidean space", either. > > Some time your pronouncements are quite axiomatic!! Yes. Statements about mathematics are often like that -- they are English translations of _theorems_. Tom Roberts tjroberts(a)lucent.com
From: JanPB on 27 Mar 2006 02:13
GSS wrote: > > As shown earlier, there are three non-zero distinct components of > Riemann curvature tensor for this space as given below: > > R_1212 = -(GM/c^2 r).(1/(1 - 2GM/c^2 r)) > R_1313 = -(GM.Sin^2 (q)/c^2 r).(1/(1 - 2GM/c^2 r)) > R_2323 = 2GM r.Sin^2 (q)/c^2 > > Kindly identify three 2-d surfaces in this space, the Gaussian > curvature of which is represented by the above mentioned three > components of the curvature tensor. These values depend strongly on certain irrelevancies of the coordinate system. To obtain values reflecting the geometry (as opposed to geometry *together with* coordinate choice irrelevancies) it's much better to use either the corresponding sectional curvatures or components with respect to a Cartan moving frame since they are equal to Gaussian curvatures of something that's actually in there. Either way, you get: S_12 = -M/r^3 S_13 = -M/r^3 S_23 = 2M/r^3 The first two are curvatures of what looks like regular 2D planes through the origin (minus the origin). Because of the odd metric they are not really flat. The third one is the curvature of the surface made (at a fixed point) of geodesics starting (from that point) in directions which are linear combinations of d/dtheta and d/dphi (at that point). Because of the symmetry, it is sufficient to calculate just one of such geodesics and then make the surface of revolution out of it (spinning it around the radius). In fact, one could probably in this case reverse the procedure and compute the geodesics *from* S_23 using the surface of revolution curvature formula - although this may not be at all simpler than using the geodesic equations straight. I'm not sure if this way of looking at this space is as useful as, say, studying its geodesics. -- Jan Bielawski |