curvature of spacetime
in [Relativity]
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From: GSS on 1 Apr 2006 08:32 Tom Roberts wrote: > GSS wrote: >> Tom Roberts wrote: >>>Consider the case M=0, in which the Riemann curvature tensor of the >>>manifold is identically zero. The surfaces r=constant have nonzero >>>Gaussian curvature. >> How do you still consider the notion of *curvature of space* to be >> really valid? > Because geometrically it _is_ valid. That is, there are indeed > geometrical manifolds with nonzero curvature. <shrug> > > And you never responded to how a 2-d surface in a flat 4-d spacetime can > have nonzero curvature, and why that shows that the curvature of such > 2-d surfaces is useless in "describing" the geometry of the 4-d manifold.... If you say that the curvature of 2-d surfaces is useless in "describing" the geometry of the 4-d manifold....I am willing to agree with you. But I just wanted you people to help me visualize the intrinsic curvature of 3-d Schw. space. I was told that the Gaussian curvature of certain 2-d surfaces will represent the intrinsic curvature of 3-d Schw. space. When I wanted these 2-d surfaces to be identified, Jan PB had given some interesting suggestions. But now you say it is *useless*..... That means the notion of intrinsic curvature of space is either too complex that it cannot be visualized or it is just invalid. Mathematically it is good enough to state that in Riemannian geometry the Riemann tensor is non-zero. Where is the necessity of associating it with a cooked up fictitious term 'curvature of space'? Was it required to fool and mislead the 'layman'? >> The term *curvature* basically applies to the bending of curves and 2-d >> surfaces. > > Not in differential geometry or GR. The term "curvature" was borrowed by > analogy with 2-d surfaces, and has come to mean the Riemann curvature > tensor. That is, a manifold of _any_ dimension with nonzero Riemann > tensor is said to be curved. <shrug> Why *said* to be curved when it is actually not curved? >> It is the forced extension of this term used to describe the >> new notion of *curvature of space* in GR. > > It may be new to you, but it most definitely was not new in GR. Riemann, > Ricci, Levi-Civita, Christoffel, et al investigated this in the 1800s. > Einstein used _existing_ mathematics in his development of GR. <shrug> > > And it's the curvature of spaceTIME that is important in GR. I thought we had agreed last time that space-time is just a _model_!! And saying the associated Riemann tensor is zero or non-zero conveys the full mathematical sense of the situation. Appending the term *curvature* to it appears to be just superfluous and misleading. >> Inherently the new notion >> applies to the deformation of space induced by the change in metric >> (for M>0). > I repeat: there is no "change" in the metric. These are > _different_manifolds_. The Schw. manifold with M=0 is both geometrically > and topologically different from those with M>0. <shrug> > > You keep trying to discuss "change of a manifold" in ways that simply do > not make sense. Again let me remind you that we had agreed in a previous post that here we are dealing with a _model_ and not with actual 3-d space. It is a standard practice in the study and analysis of _models_ to *vary* certain crucial parameters in a controlled manner and to observe the overall effect of such variations on the _model_. Does it not make any sense to you? > Analogy: how would one "change" the Euclidean plane E^2 > to "become" the surface of a sphere S^2? -- the question > _does_not_make_sense_, and illustrates a very basic lack of > understanding of geometry. <shrug> I had explained it to you in detail in some previous post. You tend to forget! Let me repeat it here: ------------------------------------------------------------ I agree that the Euclidean and Riemannian geometries cannot be transformed into one another through admissible coordinate transformations without producing deformations in the associated space continuum. Let us therefore focus on this issue. When a surface is represented in the parametric form by 2-d surface coordinates, the intrinsic geometry of the surface is described by its 2-d metric tensor. The Riemann tensor composed from the 2-d metric components is non-zero for a curved surface and zero for a plane surface. Let us consider a large circular metal ring of radius R, filled inside with a plane thin film membrane (rubber membrane or soap film). The intrinsic geometry of any small region of this thin film can be represented by a 2-d flat metric with zero Riemann tensor. Let us now imagine that we exert a steady pressure over a small localized region of this film (say by impinging an air jet) in such a way that a small hemispherical bubble of radius r<<R is formed in this local region. The 2-d surface of this hemispherical bubble can be represented by a modified 2-d metric with non-zero Riemann tensor. **Obviously it is not difficult to visualize that the localized hemispherical bubble induced by a steady external pressure is actually a deformed (elongated/stretched) membrane with a curved surface in comparison to the undeformed plane membrane in the surrounding region.** By moving the impinging air jet sideways, location of the hemispherical bubble on the large plane membrane can be easily shifted. The state of deformation of the curved membrane in comparison to the plane membrane can be studied in detail by comparing the Riemannian metric of the curved surface with the Euclidean metric of the plane surface. It can be easily shown that all displacements produced on the curved surface of the membrane are continuous and finite. The essential point I am stressing here is that a plane membrane surface with Euclidean metric does get deformed into a curved surface with Riemannian metric under the influence of external pressure. It is precisely in the same way it has been postulated in GR that 'flat' space with Euclidean metric gets deformed to a 'curved' space with Riemannian metric under the influence of a steady state gravitational field. ------------------------------------------------------------ >> Effectively we have been discussing whether this notion of >> *curvature of space* does adequately describe the induced deformation >> of space or not. .... >> The discussions so far only show that it does not >> appear to be feasible to properly describe the induced deformation of >> space through the notion of *curvature of space* . > I have no idea what you are trying to say. The Einstein field equation > plus the usual notions of differential geometry do indeed adequately > describe the geometry of GR. <shrug> For detailed study of the Schw. space continuum we could simulate the points of the 3-d Schw space model on a computer and animate the gradual change in separation or displacement of neighborhood points by varying the value of M in the curvature tensor. I am sure this will immediately highlight the inconsistencies of the GR model. GSS
From: Tom Roberts on 4 Apr 2006 00:30 GSS wrote: > Tom Roberts wrote: >> And you never responded to how a 2-d surface in a flat 4-d spacetime can >> have nonzero curvature, and why that shows that the curvature of such >> 2-d surfaces is useless in "describing" the geometry of the 4-d manifold.... > > If you say that the curvature of 2-d surfaces is useless in > "describing" the geometry of the 4-d manifold....I am willing to agree > with you. But I just wanted you people to help me visualize the > intrinsic curvature of 3-d Schw. space. I was told that the Gaussian > curvature of certain 2-d surfaces will represent the intrinsic > curvature of 3-d Schw. space. When I wanted these 2-d surfaces to be > identified, Jan PB had given some interesting suggestions. But now you > say it is *useless*..... _SOME_ 2-d surfaces can be useful in describing the geometry of 4-d spacetime, in particular those spanned by a 2-d vector space of geodesics. But you were discussing 2-d surfaces defined by coordinates, and _those_ are useless because coordinates are completely arbitrary, and introducing that arbitrariness destroys their usefulness. > That means the notion of intrinsic curvature of space is either too > complex that it cannot be visualized or it is just invalid. No. But in many cases using a ball of dust particles is a better visualization tool than 2-d surfaces. > Mathematically it is good enough to state that in Riemannian geometry > the Riemann tensor is non-zero. Where is the necessity of associating > it with a cooked up fictitious term 'curvature of space'? Mathematicians and physicists are human. We share the common desire to communicate with each other easily, accurately, and concisely -- that's why technical vocabularies were invented. > Was it > required to fool and mislead the 'layman'? Your problem, not mine (ours). But this technical vocabulary is not secret or unfathomable, it just takes _STUDY_. <shrug> >>> The term *curvature* basically applies to the bending of curves and 2-d >>> surfaces. >> Not in differential geometry or GR. The term "curvature" was borrowed by >> analogy with 2-d surfaces, and has come to mean the Riemann curvature >> tensor. That is, a manifold of _any_ dimension with nonzero Riemann >> tensor is said to be curved. <shrug> > > Why *said* to be curved when it is actually not curved? <sigh> The nuances of English. I was discussing the usage of words and not the concepts they represent. A manifold of _any_ dimension with nonzero Riemann tensor is indeed curved. That's what we _mean_ by that phrase. >> And it's the curvature of spaceTIME that is important in GR. > > I thought we had agreed last time that space-time is just a _model_!! Certainly. And in that model the curvature of space is almost useless, but the curvature of spaceTIME is very important. <shrug> > And saying the associated Riemann tensor is zero or non-zero conveys > the full mathematical sense of the situation. Appending the term > *curvature* to it appears to be just superfluous and misleading. It is the way people in the field communicate with each other. <shrug> >> You keep trying to discuss "change of a manifold" in ways that simply do >> not make sense. > > Again let me remind you that we had agreed in a previous post that here > we are dealing with a _model_ and not with actual 3-d space. It is a > standard practice in the study and analysis of _models_ to *vary* > certain crucial parameters in a controlled manner and to observe the > overall effect of such variations on the _model_. Does it not make any > sense to you? That makes sense for continuous parameters that are changed in a continuous manner. Example: In discussing the viscosity of water it makes sense to change the temperature by a small amount, _except_ across its boiling or freezing point. It does _not_ make sense for incommensurate concepts. In classical mechanics you can usually add any object to a given problem without changing the geometry at all; in GR that simply is not true. Minkowski spacetime and Schwarzschild spacetime are incommensurate, and it is not possible to "change" one into the other -- they are models of _completely_ different physical situations. IOW: the parameter M of Schw. manifolds is _not_ continuous in the limit M->0 -- infinitesimal is just plain _different_ from exactly 0. >> Analogy: how would one "change" the Euclidean plane E^2 >> to "become" the surface of a sphere S^2? -- the question >> _does_not_make_sense_, and illustrates a very basic lack of >> understanding of geometry. <shrug> > > I had explained it to you in detail in some previous post. [...] Your "explanation" completely misses the point: the topology is different, and the geometry must be consistent with the topology, and is likewise different. You avoided that by changing the subject from E^2 and S^2 to D^2 and S^2-P (a 2-d disk and the surface of a sphere minus one point). These latter have the same topology, but you didn't even realize the difference. This subject is subtle, and you won't understand it until and unless you sit down and _study_ it. <shrug> > For detailed study of the Schw. space continuum we could simulate the > points of the 3-d Schw space model on a computer and animate the > gradual change in separation or displacement of neighborhood points by > varying the value of M in the curvature tensor. I am sure this will > immediately highlight the inconsistencies of the GR model. Clearly you have never done anything of the sort. In particular, these manifolds are _continuous_ and your "simulation" would be completely unable to represent that important property. <shrug> Tom Roberts tjroberts(a)lucent.com
From: Hexenmeister on 4 Apr 2006 07:55 "Tom Roberts" <tjroberts(a)lucent.com> wrote in message news:ZDmYf.51582$2O6.5573(a)newssvr12.news.prodigy.com... | GSS wrote: | > Tom Roberts wrote: | >> And you never responded to how a 2-d surface in a flat 4-d spacetime can | >> have nonzero curvature, and why that shows that the curvature of such | >> 2-d surfaces is useless in "describing" the geometry of the 4-d manifold.... | > | > If you say that the curvature of 2-d surfaces is useless in | > "describing" the geometry of the 4-d manifold....I am willing to agree | > with you. But I just wanted you people to help me visualize the | > intrinsic curvature of 3-d Schw. space. I was told that the Gaussian | > curvature of certain 2-d surfaces will represent the intrinsic | > curvature of 3-d Schw. space. When I wanted these 2-d surfaces to be | > identified, Jan PB had given some interesting suggestions. But now you | > say it is *useless*..... | | _SOME_ 2-d surfaces can be useful in describing the geometry of 4-d | spacetime, in particular those spanned by a 2-d vector space of | geodesics. But you were discussing 2-d surfaces defined by coordinates, | and _those_ are useless because coordinates are completely arbitrary, | and introducing that arbitrariness destroys their usefulness. | "Real" has nothing to do with it. -- Tom Roberts | > That means the notion of intrinsic curvature of space is either too | > complex that it cannot be visualized or it is just invalid. | | No. But in many cases using a ball of dust particles is a better | visualization tool than 2-d surfaces. "Real" has nothing to do with it. -- Tom Roberts | | | > Mathematically it is good enough to state that in Riemannian geometry | > the Riemann tensor is non-zero. Where is the necessity of associating | > it with a cooked up fictitious term 'curvature of space'? | | Mathematicians and physicists are human. We share the common desire to | communicate with each other easily, accurately, and concisely -- that's | why technical vocabularies were invented. "Real" has nothing to do with it. -- Tom Roberts | | > Was it | > required to fool and mislead the 'layman'? | | Your problem, not mine (ours). "Real" has nothing to do with it. -- Tom Roberts | | But this technical vocabulary is not secret or unfathomable, it just | takes _STUDY_. <shrug> "Real" has nothing to do with it. -- Tom Roberts | | | >>> The term *curvature* basically applies to the bending of curves and 2-d | >>> surfaces. | >> Not in differential geometry or GR. The term "curvature" was borrowed by | >> analogy with 2-d surfaces, and has come to mean the Riemann curvature | >> tensor. That is, a manifold of _any_ dimension with nonzero Riemann | >> tensor is said to be curved. <shrug> | > | > Why *said* to be curved when it is actually not curved? | | <sigh> The nuances of English. I was discussing the usage of words and | not the concepts they represent. A manifold of _any_ dimension with | nonzero Riemann tensor is indeed curved. That's what we _mean_ by that | phrase. Humpty Roberts in Wonderland:- `I don't know what you mean by "observations",' Alice said. Humpty Roberts smiled contemptuously. `Of course you don't -- till I tell you. I meant "there's a nice knock-down argument for you!"' <shrug> `But "observations" doesn't mean "a nice knock-down argument",' Alice objected. `When I use a word,' Humpty Roberts said, in rather a scornful tone, <shrug>, `it means just what I choose it to mean -- neither more nor less.' <shrug> `The question is,' said Alice, `whether you can make words mean so many different things.' `The question is,' said Humpty Roberts, `which is to be master -- that's all.' <shrug> Alice was too much puzzled to say anything; so after a minute Humpty Roberts began again. `They've a temper, some of them -- particularly verbs: they're the proudest -- adjectives you can do anything with, but not verbs -- however, I can manage the whole lot of them! Impenetrability! That's what I say!' <shrug> With thanks to Lewis Carroll. Professor Androcles. | | >> And it's the curvature of spaceTIME that is important in GR. | > | > I thought we had agreed last time that space-time is just a _model_!! | | Certainly. And in that model the curvature of space is almost useless, | but the curvature of spaceTIME is very important. <shrug> | | | > And saying the associated Riemann tensor is zero or non-zero conveys | > the full mathematical sense of the situation. Appending the term | > *curvature* to it appears to be just superfluous and misleading. | | It is the way people in the field communicate with each other. <shrug> | | | >> You keep trying to discuss "change of a manifold" in ways that simply do | >> not make sense. | > | > Again let me remind you that we had agreed in a previous post that here | > we are dealing with a _model_ and not with actual 3-d space. It is a | > standard practice in the study and analysis of _models_ to *vary* | > certain crucial parameters in a controlled manner and to observe the | > overall effect of such variations on the _model_. Does it not make any | > sense to you? | | That makes sense for continuous parameters that are changed in a | continuous manner. | | Example: In discussing the viscosity of water it makes | sense to change the temperature by a small amount, _except_ | across its boiling or freezing point. | | It does _not_ make sense for incommensurate concepts. In classical | mechanics you can usually add any object to a given problem without | changing the geometry at all; in GR that simply is not true. | | Minkowski spacetime and Schwarzschild spacetime are incommensurate, and | it is not possible to "change" one into the other -- they are models of | _completely_ different physical situations. IOW: the parameter M of | Schw. manifolds is _not_ continuous in the limit M->0 -- infinitesimal | is just plain _different_ from exactly 0. | | | >> Analogy: how would one "change" the Euclidean plane E^2 | >> to "become" the surface of a sphere S^2? -- the question | >> _does_not_make_sense_, and illustrates a very basic lack of | >> understanding of geometry. <shrug> | > | > I had explained it to you in detail in some previous post. [...] | | Your "explanation" completely misses the point: the topology is | different, and the geometry must be consistent with the topology, and is | likewise different. You avoided that by changing the subject from E^2 | and S^2 to D^2 and S^2-P (a 2-d disk and the surface of a sphere minus | one point). These latter have the same topology, but you didn't even | realize the difference. | | This subject is subtle, and you won't understand it until and unless you | sit down and _study_ it. <shrug> | | | > For detailed study of the Schw. space continuum we could simulate the | > points of the 3-d Schw space model on a computer and animate the | > gradual change in separation or displacement of neighborhood points by | > varying the value of M in the curvature tensor. I am sure this will | > immediately highlight the inconsistencies of the GR model. | | Clearly you have never done anything of the sort. In particular, these | manifolds are _continuous_ and your "simulation" would be completely | unable to represent that important property. <shrug> | | | Tom Roberts tjroberts(a)lucent.com
From: JanPB on 6 Apr 2006 16:29 I wrote: > > Now look inside r<2M. Not only isn't this portion spacelike anymore, it > is also funnel-shaped which means it doesn't match the "Einstein-Rosen" > portion outside the horizon without a crease This analogy is a bit stretched here since the funnel is a Lorentz (indefinite metric) surface, so it cannot be embedded in any Euclidean space. The real problem is Schwarzschild's t puts a Riemannian surface on the outside right next to a Lorentz surface inside. -- Jan Bielawski
From: GSS on 7 Apr 2006 14:25
Tom Roberts wrote: > GSS wrote: ...... >> That means the notion of intrinsic curvature of space is either too >> complex that it cannot be visualized or it is just invalid. > No. But in many cases using a ball of dust particles is a better > visualization tool than 2-d surfaces. Noted. >> Mathematically it is good enough to state that in Riemannian geometry >> the Riemann tensor is non-zero. Where is the necessity of associating >> it with a cooked up fictitious term 'curvature of space'? > Mathematicians and physicists are human. We share the common desire to > communicate with each other easily, accurately, and concisely -- that's > why technical vocabularies were invented. There is nothing *technical* in the fictitious term *curvature of space*. ..... >> And saying the associated Riemann tensor is zero or non-zero conveys >> the full mathematical sense of the situation. Appending the term >> *curvature* to it appears to be just superfluous and misleading. > It is the way people in the field communicate with each other. <shrug> >>> You keep trying to discuss "change of a manifold" in ways that simply do >>> not make sense. >> Again let me remind you that we had agreed in a previous post that here >> we are dealing with a _model_ and not with actual 3-d space. It is a >> standard practice in the study and analysis of _models_ to *vary* >> certain crucial parameters in a controlled manner and to observe the >> overall effect of such variations on the _model_. Does it not make any >> sense to you? > That makes sense for continuous parameters that are changed in a > continuous manner. > > It does _not_ make sense for incommensurate concepts. In classical > mechanics you can usually add any object to a given problem without > changing the geometry at all; in GR that simply is not true. Or that 'GR simply is not true'?? > Minkowski spacetime and Schwarzschild spacetime are incommensurate, and > it is not possible to "change" one into the other -- they are models of > _completely_ different physical situations. IOW: the parameter M of > Schw. manifolds is _not_ continuous in the limit M->0 -- infinitesimal > is just plain _different_ from exactly 0. ....... Let us agree to vary the parameter M strictly within a *continuous* range. Therefore let M vary from M1 to Ms where M1 is just one kilogram and Ms is equal to one solar mass. >> For detailed study of the Schw. space continuum we could simulate the >> points of the 3-d Schw space model on a computer and animate the >> gradual change in separation or displacement of neighborhood points by >> varying the value of M in the curvature tensor. I am sure this will >> immediately highlight the inconsistencies of the GR model. > Clearly you have never done anything of the sort. Let us do it now!! The first step before actual simulation of the phenomenon on computer is to mentally visualize it. We begin with this step. Let us assume that a spherically symmetric body of mass M and radius R_0, is located at the origin O of a polar coordinate system. Due to the 'static' gravitational field in its vicinity (i.e. r > R_0 > 0), the (modified) metric coefficients g_ij are given by the Schwarzschild solution as: g_11 = 1/(1 - 2GM/c^2 r) ; g_22 = r^2 ; g_33 = r^2.Sin^2 (theta) .... (1) Thus the (modified) radial metric coefficient g_11 at any particular space point P(r,theta,phi) is always greater than unity for M>0. The arc element or the (modified) separation distance ds between two neighboring space point positions P and Q in this region will be given by: (ds)^2 = g_11 (dr)^2 + g_22 (d,theta)^2 + g_33 (d,phi)^2 = (1/(1 - 2GM/c^2 r)).(dr)^2 + r^2.(d,theta)^2 + r^2.Sin^2 (theta).(d,phi)^2 .... (2) Let us assume that R_0 is some finite radius representing a physical boundary of a body of mass M and that R_0 remains fixed throughout this analysis. Further, let us assume that the gravitational field of the body of mass M practically vanishes at infinitely large radial distance, say R_max so that for r>R_max g_11 reduces to unity. R_max may be of the order of say one light year and we can assume it to be a fixed constant value for the present analysis. Now let us imagine a spherical surface at r=R_n where R_n is some constant value. For different discrete values of R_n we can visualize a set of concentric spherical surfaces. Let the discrete values of R_n be such that: R_1 = R_0 + dr R_2 = R_1 + dr R_3 = R_2 + dr R_4 = R_3 + dr ............. R_n = R_n-1 + dr R_n+1 = R_n + dr ..... (3) ............ Further to simulate this set of concentric spherical surfaces on computer, let us imagine that the surfaces represented by odd values of n in r=R_n are depicted with red colored pixels and those with even values of n are depicted with blue colored pixels. Let us work out the *radial* separation distance ds between any two neighboring red and blue surfaces depicted by say r=R_n+1 and r=R_n. Since in the radial direction d,phi=0 and d,theta=0 from equation (2) above we get, (ds_n)^2 = (1/(1 - 2GM/c^2 r)).(dr)^2 and for (2GM/c^2 r) << 1 ds_n = (1 - 2GM/c^2 r))^(-1/2).dr = dr + (GM/c^2).(dr/r) = dr + (GM/c^2).(dr/R_n) ......(4) Let us take the radial increment dr to be equal to one unit. Then, ds_n = 1 + (G/c^2). M/R_n ...... (5) Or ds_1 = 1 + (G/c^2). M/R_1 ds_2 = 1 + (G/c^2). M/R_2 ds_3 = 1 + (G/c^2). M/R_3 ds_4 = 1 + (G/c^2). M/R_4 ds_5 = 1 + (G/c^2). M/R_5 .......... ds_n = 1 + (G/c^2). M/R_n Now making use of the above relations, we can easily visualize that (a) For any value of M, ds_1>ds_2>ds_3>ds_4 ... >ds_n>ds_n+1 (b) That is the inner pairs of red and blue surfaces get separated much more than the outer pairs of red and blue surfaces. (c) As we vary the value of M in equation (5), say from nearly zero to Ms the separation distance ds between any adjacent pair of red and blue surfaces keep getting *increased* proportionately. (d) If we repeatedly keep varying the value of M from zero to Ms and back to zero, we can well imagine that the set of spherical surfaces represented by r=R_n will constitute an animated *breathing* sphere. Isn't it fantastic? Next let us work out the *total* radial separation distance S_n of a spherical surface represented by r=R_n from the innermost spherical surface represented by r=R0. That is, S_n = ds_1 + ds_2 + ds_3 + ds_4 + .... + ds_n = n + (G/c^2). M.(1/R_1 + 1/R_2 + 1/R_3 + .... + 1/R_n) If we extend this process to compute the radial separation distance S_max of a spherical surface represented by r=R_max from the innermost spherical surface represented by r=R0, it will really tend to infinity!!! After this critical examination of the geometrical _model_ of gravitation propounded in the *General Theory of Relativity*, would you still like to believe in its validity??? GSS |