curvature of spacetime
in [Relativity]
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From: JanPB on 27 Mar 2006 15:28 I keep writing: > > S_12 = -M/r^3 > S_13 = -M/r^3 > S_23 = 2M/r^ I use the indices purely as convenient labels here - sectional curvature is not a tensor and S_12, etc. are not its "components" wrt to a basis. (It's not a tensor because it's not a function of vectors but a function of 2D tangent planes.) -- Jan Bielawski
From: GSS on 28 Mar 2006 09:21 Tom Roberts wrote: > 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. _approximation_ of what? Of a space-time *model*? In my opinion, we can not afford to undermine the capability or potential of our brains. > >> In GR there is only spaceTIME. > > Space-time is not any physical entity that exists. > Yes, of course. It is a _model_. <shrug> At long last we agree here. Thanks. > >> 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". Good. Let us now examine whether this geometrical _model_ of gravitation (in GR) is truly valid on all counts or whether there is some inconsistency in this _model_. > >>> 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. To aid the visualization of 2-d surface curvature induced by the Riemann curvature tensor, we could simulate the points of the 3-d Schw space model on a computer and animate the gradual curving of typical surfaces by varying the value of M in the curvature tensor. (I understand even the collision of galaxies has been animated on computer.) I hope you will have no objection for such an exhaustive study of this _model_. GSS
From: GSS on 28 Mar 2006 11:35 JanPB wrote: > 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. Let us agree to use only that spherical polar coordinate system which has been adopted in the Schw. solution for the static spherically symmetric gravitation problem. Kindly point out the *irrelevancies* of this coordinate system which hinder the identification of 2-d surfaces in the Schw space the curvature of which is induced by the components of the curvature tensor. I hope you can mentally visualize the surfaces under discussion. If so the same can then be simulated on computer if required. > 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. Do you mean the planes passing through the polar axis theta=0? > 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). Kindly confirm that you can visualize this surface and can it be simulated on computer? Can you visualize how the shape of this surface will change when we vary the value of M in the curvature components from zero to M1 say? > 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. In my opinion the study of geodesics does not help us in detailed examination of the nature of deformation of space induced by the curvature tensor or the associated metric. GSS
From: JanPB on 29 Mar 2006 00:11 GSS wrote: > JanPB wrote: > > 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. > > Let us agree to use only that spherical polar coordinate system which > has been adopted in the Schw. solution for the static spherically > symmetric gravitation problem. Kindly point out the *irrelevancies* of > this coordinate system which hinder the identification of 2-d surfaces > in the Schw space the curvature of which is induced by the components > of the curvature tensor. I hope you can mentally visualize the surfaces > under discussion. If so the same can then be simulated on computer if > required. The difference between R_1212 and R_1313 is one example of coordinate-induced irrelevancy. The situation is spherically symmetric so any coordinate-independent quantification of something actually *there* should give the same result both in the (d/dr, d/dtheta) and (d/dr, d/dphi) directions. Another example of a coordinate phenomenon is that the length of d/dtheta is not equal to the length of d/dphi (r and r sin(theta), resp.) - yet by symmetry there is no difference between these two directions. > > 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. > > Do you mean the planes passing through the polar axis theta=0? Any "plane" through the origin. The S_12 and S_13 at any given point (r0, theta0, phi0) refer to Gaussian curvatures at that point of the planes spanned by (d/dr, d/dtheta) and (d/dr, d/dphi), resp. > > 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). > > Kindly confirm that you can visualize this surface and can it be > simulated on computer? You have to solve some differential equations to get the exact geodesics but you can imagine at every point (r0, theta0, phi0) a snippet of surface of positive curvature tangent to the (d/dtheta, d/dphi)-plane at that point and curving toward the origin. The exact equation of the geodesic is quite messy and even when you reduce the order from 2 to 1 by using the constants of the motion, the resulting equations are still nasty. If gamma(t) is a geodesic then: <dgamma/dt, dgamma/dt> = 1 (*) and since d/dphi induces an isometry (rotation around theta=0): <dgamma/dt, d/dphi> = const. (**) Given some reasonable initial conditions, like: gamma(t) = (r0, pi/2, 0) and dgamma/dt(0) = d/dphi ....then (*) and (**) become (assuming I didn't make a mistake): phi-dot = r0^2/r^2 (r-dot)^2/(1 - 2M/r) + r0^4/r^2 = 1 ....where "dot" is the time derivative. Ugh. Using the other approach (reverse engineering geodesics from the given curvature of the surface of revolution) is not as easy as I thought - the ambient space is not Euclidean so the formulas are even more complicated. > Can you visualize how the shape of this surface will change when we > vary the value of M in the curvature components from zero to M1 say? Well, it'll be uncurling and approaching the plane spanned by (d/dtheta, d/dphi) as the surface moves away from the origin. Another way to explore the geometry would be to calculate Gaussian curvatures of "planes" not passing through the origin. They are probably going to look like "funnels" of finite depth that are less and less funnel-like and more and more planar-like as their distance from the origin increases. -- Jan Bielawski
From: Tom Roberts on 29 Mar 2006 09:19
GSS wrote: > Tom Roberts wrote: >> GSS wrote: >>> Tom Roberts wrote: >>>> There _is_ no "physical 3-d space". >> 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. > > _approximation_ of what? Of a space-time *model*? Yes, of course. > In my opinion, we can not afford to undermine the capability or > potential of our brains. Of course not. Nor should we expect that phenomena on the scales which our brains can directly appreciate are the only phenomena there are. Specifically: the very small (<<1 mm), the very large (>>1 km), and the very fast (v ~ c) are not within our brains' evolutionary history, and are not handled well by brains that have not studied such phenomena very carefully. <shrug> Tom Roberts tjroberts(a)lucent.com |