curvature of spacetime
in [Relativity]
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From: Hexenmeister on 16 Mar 2006 12:30 Oh goody! Kook fight! Androcles. "Tom Roberts" <tjroberts(a)lucent.com> wrote in message news:7sgSf.47814$F_3.9199(a)newssvr29.news.prodigy.net... | George Hammond wrote: | > "Tom Roberts" <tjroberts(a)lucent.com> wrote in message | > news:CVCQf.51646$H71.7983(a)newssvr13.news.prodigy.com... | >> Here's a test of your understanding of that paper: Today | >> we commonly write the Einstein Field Equation (using | >> component notation and units comparable to Einstein's): | >> G^uv = 8 pi G T^uv | >> The name "Einstein Field Equation" comes from the fact | >> that Einstein first derived and presented it in this paper | >> -- in which equation of the paper is it presented? | >> | >> If you cannot answer this question, you don't understand | >> that paper. Or almost certainly, GR itself. | >> | > [Hammond] | > I notice Tucker never gave a direct answer to your | > question Tom,... but I will. I say the answer to your | > (Tom Roberts') specific question (above) is: | > "Equation 53, page 149" | > of Einstein's famous 1916 paper | > (ibid: Dover's "...Relativity") | | Yes. | | > ... and the reason why, is | > because G_uv (the Einstein | > tensor reduces to R_uv | > (the Ricci tensor) for | > sqrt(-g) =1 which is | > the case in eqn 53..... | | No. There is no situation in which G_uv "reduces" to R_uv -- G_uv is | defined: | G_uv = R_uv - 0.5 g_uv R | and your claim would imply either g_uv=0 or R=0, neither of which makes | any sense at all (R=0 applies in certain manifolds with unphysical | symmetries, but not in general). | | In fact, Einstein's Eq. 53 uses the alternate form of the Einstein field | equation: | R_uv = 8 pi G (T_uv - 0.5 g_uv T) | | | Exercise for the reader: show the relationship between the | LHS of Einstein's eq. 53 and R_uv. | | | > hence | > eqn 53 is actually the first | > appearance of "Einstein's EFE" | > in his famous 1916 paper! | | Yes. | | | Tom Roberts tjroberts(a)lucent.com
From: George Hammond on 16 Mar 2006 18:54 "Tom Roberts" <tjroberts(a)lucent.com> wrote in message news:7sgSf.47814$F_3.9199(a)newssvr29.news.prodigy.net... > > George Hammond wrote: >> >> "Tom Roberts" <tjroberts(a)lucent.com> wrote in message >> news:CVCQf.51646$H71.7983(a)newssvr13.news.prodigy.com... >> >>> [Roberts] >>> Here's a test of your understanding of that paper: Today >>> we commonly write the Einstein Field Equation (using >>> component notation and units comparable to Einstein's): >>> G^uv = 8 pi G T^uv >>> The name "Einstein Field Equation" comes from the fact >>> that Einstein first derived and presented it in this paper >>> -- in which equation of the paper is it presented? >>> >>> If you cannot answer this question, you don't understand >>> that paper. Or almost certainly, GR itself. >>> >> [Hammond] >> I notice Tucker never gave a direct answer to your >> question Tom,... but I will. I say the answer to your >> (Tom Roberts') specific question (above) is: >> "Equation 53, page 149" >> of Einstein's famous 1916 paper >> (ibid: Dover's "...Relativity") > > [Roberts] Yes. > [Hammond] Yea. >> >> [Hammond] >> ... and the reason why, is >> because G_uv (the Einstein >> tensor reduces to R_uv >> (the Ricci tensor) for >> sqrt(-g) =1 which is >> the case in eqn 53..... > > No. There is no situation in which G_uv "reduces" to R_uv -- G_uv is > defined: > G_uv = R_uv - 0.5 g_uv R > and your claim would imply either g_uv=0 or R=0, neither of which makes > any sense at all (R=0 applies in certain manifolds with unphysical > symmetries, but not in general). > [Hammond] R=0 holds for empty space in Gen. Relativity.... and we don't need "unphysical manifolds".... all we need is the absence of matter (i.e. flat space)! In that case Einstein frequently wrote the EFE as: R_uv = 0 rather than G__uv = 0 so in that sense "the EFE reduces from G_uv=0 to R_uv = 0", even though, as you correctly point out, G_uv does not reduce to R_uv in general... but only in empty space. ================================= But the point I was making is the point Einstein made in his paper (ibid) and that is that we must define g_uv such that det(g)=|g|=-1. i.e. sqrt(-g)=1. In that case, Einstein says "the tensor G_uv reduces to R_uv" (Einstein, p. 142, paragraph 1, line 10) Fact is, I am a bit confused on this point, perhaps you could clarify this point for me as I'm short on time and you probably already know the answer? For instance does sqrt(-g)=1 imply flat space? ================================== > In fact, Einstein's Eq. 53 uses the alternate form of the Einstein field > equation: > R_uv = 8 pi G (T_uv - 0.5 g_uv T) > > > Exercise for the reader: show the relationship between the > LHS of Einstein's eq. 53 and R_uv. > [Hammond] It is well known that there are two equivalent forms of the EFE: form 1) R_uv - 0.5 R = 8 pi G T_uv or form 2) R_uv = 8 pi G (T_uv - 0.5 g_uv T) as you point out Einstein's 1916 paper uses form 2) (eqn 53 p. 149)...... and he writes R_uv in terms of the Christoffel symbols in eqn 53. As for your "homework problem" of showing that his Christoffel expression on the LHS of eqn 53 is equal to R_uv..... Einstein has already stated that result two pages earlier (eqns 44 p.142... middle line). > >> hence >> eqn 53 is actually the first >> appearance of "Einstein's EFE" >> in his famous 1916 paper! > > Yes. > [Hammond] Again, yea. > > Tom Roberts tjroberts(a)lucent.com -- ======================================== SCIENTIFIC PROOF OF GOD WEBSITE http://geocities.com/scientific_proof_of_god mirror site: http://proof-of-god.freewebsitehosting.com ========================================
From: Tom Roberts on 16 Mar 2006 23:39 George Hammond wrote: > "Tom Roberts" <tjroberts(a)lucent.com> wrote in message > news:7sgSf.47814$F_3.9199(a)newssvr29.news.prodigy.net... >> No. There is no situation in which G_uv "reduces" to R_uv -- G_uv is >> defined: >> G_uv = R_uv - 0.5 g_uv R >> and your claim would imply either g_uv=0 or R=0, neither of which makes >> any sense at all (R=0 applies in certain manifolds with unphysical >> symmetries, but not in general). >> > [Hammond] > R=0 holds for empty space in Gen. Relativity.... > and we don't need "unphysical manifolds".> I said "unphysical symmetries", and R=0 is certainly that. The universe we inhabit cannot be modeled well by any manifold with R=0, so in that sense this is indeed unphysical. >... all we need is the > absence of matter (i.e. flat space)! Absence of matter does NOT imply flat space (or rather, flat spaceTIME). E.g. the Schwarzschild manifold has no matter, yet is not flat. > In that case Einstein frequently > wrote the EFE as: > R_uv = 0 > rather than > G__uv = 0 Yes. Not because R=0 but rather because T^uv=0, which implies T=0. Remember there are two equivalent formulations of the Einstein field equation: R_uv - 0.5 g_uv R = 8 pi G T_uv R_uv = 8 pi G (T_uv - 0.5 g_uv T) The second form gives R_uv=0 in vacuum. > so in that sense "the EFE reduces from G_uv=0 to > R_uv = 0", even though, as you correctly point out, > G_uv does not reduce to R_uv in general... but only in empty > space. G_uv does not in general "reduce" to R_uv in empty space. It does so only in _flat_ spacetime [#], and 0=0 is not a very enlightening "reduction". [#] and certain other cases for which R=0 but R_uv!=0. > But the point I was making is the point Einstein made in his > paper (ibid) and that is that we must define g_uv such that > det(g)=|g|=-1. i.e. sqrt(-g)=1. In that case, Einstein says > "the tensor G_uv reduces to R_uv" (Einstein, > p. 142, paragraph 1, line 10) The condition sqrt(-g)=1 is not sufficient to imply R=0, and not sufficient to imply G_uv=R_uv. I suspect Einstein had more context.... > Fact is, I am a bit confused on this point, perhaps you > could clarify this point for me as I'm short on time and > you probably already know the answer? For instance > does sqrt(-g)=1 imply flat space? No. Nor does R=0. But R^i_jkl=0 does, because that is what we mean by "flat". Tom Roberts tjroberts(a)lucent.com
From: George Hammond on 17 Mar 2006 05:38 "Tom Roberts" <tjroberts(a)lucent.com> wrote in message news:74rSf.61048$dW3.43315(a)newssvr21.news.prodigy.com... .. > G_uv does not in general "reduce" to R_uv in empty space. It does so only > in _flat_ spacetime [#], and 0=0 is not a very enlightening "reduction". > [Hammond] Wrong. G_uv, R_uv, R, T_uv, and T are ALL zero in empty space even tho said empty space may not be flat! An example is the space outside the Earth above the atmosphere. G_uv, R_uv, R, T_uv and T are all zero, but spacetime is not flat at said location. In such a location the EFE can be written G_uv = 0 or R_uv = 0 Either equation will determine the Schwarzchild solution for instance. In that sense, it is said sometimes that G_uv reduces to R_uv in empty space.. P.A.M. Dirac put it this way in his celebrated monograph _General Theory of Relativity_ ---------------------------------------- Dirac says on page 43:: "The Einstein equations in the absence of matter are: (24.1) R_uv = 0 They lead to R=0; and hence (24.2) R_uv-1/2 g_uv R = 0 We may use either R_uv = 0 or R_uv-1/2 g_uv R=0 as ther basic equations for empty space. --------------------------------------- -- ======================================== SCIENTIFIC PROOF OF GOD WEBSITE http://geocities.com/scientific_proof_of_god mirror site: http://proof-of-god.freewebsitehosting.com ========================================
From: Tom Roberts on 17 Mar 2006 22:05
George Hammond wrote: > "Tom Roberts" <tjroberts(a)lucent.com> wrote in message > news:74rSf.61048$dW3.43315(a)newssvr21.news.prodigy.com... >> G_uv does not in general "reduce" to R_uv in empty space. It does so only >> in _flat_ spacetime [#], and 0=0 is not a very enlightening "reduction". > > Wrong. > G_uv, R_uv, R, T_uv, and T are ALL zero > in empty space even tho said empty space > may not be flat! OK. We are using different contexts. What I wrote above was in the context of abstract differential geometry, while yours is in the context of GR. I should not have switched contexts like that. Sorry. But in my defense I'll say that "G reduces to R" surely sounds like an attempt to make a general statement about differential geometry; in fact the EFE is required, and that limits it to the context of GR. Tom Roberts tjroberts(a)lucent.com |