Question on Spacetime dilation.
in [Relativity]
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From: RustyJames on 22 Dec 2008 16:32 On Dec 19, 6:02 am, Alen <al...(a)westserv.net.au> wrote: > On Dec 19, 9:12 am, "Ken S. Tucker" <dynam...(a)vianet.on.ca> wrote: > > > > > > > Hi XXein and guys.... > > ...> xxein: Welcome aboard to both you and Allen. I hope you enjoy the > > > ride. > > > I posted this to SPF... > > > To Theo and all theoreticians who embrace the signature (+---), > > I'm unable to understand it so far, but I'm trying! > > > On Dec 18, 12:16 pm, Theo Wollenleben <alpha0...(a)yahoo.de> wrote: > > > > Ken S. Tucker schrieb: > > > > On Dec 16, 2:50 pm, Theo Wollenleben <alpha0...(a)yahoo.de> wrote: > > > >> Ken S. Tucker schrieb: > > > >>> ds^2 =g_uv dx^u dx^v , {u,v=0,1,2,3}, > > > >>> = dx_u dx^u , > > > >>> = dx_0 dx^0 + dx_i dx^i , {i=1,2,3} , Eq.(1). > > > >>> dt' = ds = (0.6) dt, Eq.(2). > > > >>> What differential coefficients should be subbed > > > >>> into Eq.(1) to yield Eq.(2)? > > > >> g_uv = diag(1,-1,-1,-1) > > > >> dx_0 = dx^0, dx_i = -dx^i > > > >> ds^2 = (dx^0)^2 - (dx^i)^2 > > > >> ds = sqrt(1 - (dx^i/dx^0)^2)*dx^0 = sqrt(1-(v/c)^2)*cdt > > > > If "dr" is a spatial displacement *vector*, and > > > > e^i and e_i are the 3D spatial basis vectors, the > > > > text book suggests that, > > > > dr = e_i dx^i = e^i dx_i . > > > > and also > > > > dr^2 = dr.dr , (that's a dot/scalar product) > > > > = e_i.e_j dx^i dx^j = g_ij dx^i dx^j > > > > = e^i.e^j dx_i dx_j = g^ij dx_i dx_j > > > > does that seem reasonable to you Theo and all? > > > Where the scalar product is given by x.y=g(x,y). That is correct, though > > > "being spatial" in Minkowski space is not an invariant concept, of > > > course. If we choose another basis, the same vector could have a nonzero > > > time component. > > > Ok, I've studied metrics such as g_i0, however they are not popular, > > and I have been advised to understand the signature (1,-1,-1,-1) > > you (Theo) introduced above, that are conventionally used by most > > relativists and what they mean by that. > > > A minor problem I'm stuck on is this, if > > > dr = e_i dx^i = e^i dx_i , and dx^i = - dx_i then e_i = -e^i . > > > The tensor textbook requires that the Kronecker Delta " delta^u_v " > > has values of 1,0 such as > > > delta^u_v = {1,0}when {u=v , u=/=v}, defined by > > > delta^u_v = e^u.e_v , but > > > e^1.e_1 = -1 , e^0.e_0 = +1 , e^1.e_0 = 0 , etc. > > > gives 3 values for the Kronecker Delta. > > That's what I need to correct and better understand. > > Thanks, comments appreciated. > > Ken S. Tucker > > I would think that, if > > delta^u_v = e^u.e_v > > then e_v = 1/e^v > > so you will always get delta^u_v = +1 only, > or 0 otherwise? > > Alen- Hide quoted text - > > - Show quoted text - Whats the new variable associated with the increase of the velocity of the expansion of the dark matter witch has now been proven to be increasing faster with time.thats has to be considered if the medium in wich your clock is traveling through is dynamically changing with respect to time and velocity since the faster your velocity in the direction of the expantion would effect the distance traveled if the medium is changing dimensions outward in all directions.
From: John Polasek on 24 Dec 2008 12:47 On Tue, 23 Dec 2008 20:04:28 -0800 (PST), "Ken S. Tucker" <dynamics(a)vianet.on.ca> wrote: >Hi John. > >On Dec 23, 1:33 pm, John Polasek <jpola...(a)cfl.rr.com> wrote: >> On Mon, 22 Dec 2008 11:51:38 -0500, John Polasek <jpola...(a)cfl.rr.com> >> wrote: >> >> >> >> >On Sun, 21 Dec 2008 22:05:16 -0800 (PST), "Ken S. Tucker" >> ><dynam...(a)vianet.on.ca> wrote: >> >> >>On Dec 21, 8:00 pm, John Polasek <jpola...(a)cfl.rr.com> wrote: >> >>> On Tue, 16 Dec 2008 11:57:32 -0800 (PST), "Ken S. Tucker" >> >> >>> <dynam...(a)vianet.on.ca> wrote: >> >>> >A clock (in K) moving at 0.8c (relative to K') is >> >>> >dilated 0.6 by t' = t*sqrt(1 - v^2/c^2), so that >> >>> >t'=(0.6)*t. >> >> >>> >In GR that is generalized to be, >> >> >>> >ds^2 =g_uv dx^u dx^v , {u,v=0,1,2,3}, >> >> >>> >and then by association equatable to >> >> >>> >= dx_u dx^u , >> >> >>> >= dx_0 dx^0 + dx_i dx^i , {i=1,2,3} , Eq.(1). >> >> >>> Firstly, you are incorporating .8c which is related to SR, and >> >>> admixing it into GR. >> >> >>"admixing", SR is a GR limit. >> >> >>> Secondly, if you retain GR's guv matrix, even with zero gravity, then >> >>> your first term should be negative, that is, -dx0dx0. >> >> >>WHY? We want logic, NOT idle pronouncements. >> >> >>> I think I'm right on this. >> >> >>Prove it. >> >g_00 = -1 + 2GM/rc^2 = -1 +eps, in the limit, -1 >> >g_11 = 1/(1 -eps), in the limit, 1. >> >dx_0 g00 dx^0 = -dx0^2 in the limit >> >>>Remember the signature -1 1 1 1. >> >> >>That's a conjecture I want to see proven. >> >>Regards >> >>Ken S. Tucker >> >John Polasek >> >> I just presented a proof; now it's up to you to disprove it, not just >> let it hang. Start with the original -c2t2 + x2 + y2 + z2 = s2. >> >> As far as deciding if -1111 or 1-1-1-1 is correct, neither one is. > >I happen to agree with you John, however >I have been asked to derive either of those >signatures from a set of basis vectors, they >are still in use by some. Sounds like homework. >> This is a construct of MTW's in 'Gravitation', as a means of getting >> rid of ict (see pg. 61 'Farewell to ict') in favor of x0 = ct. >> MTW have performed a mathematical travesty. You can't do that to space >> and time. > >I'm ok with x0=ct. > >> g00 does not pass the test x0g00 = x0', since it makes x0' = - x0, a >> 180 degree reversal. Here MTW invented the 'one-form' x0g00, which >> evidently is an inside out x0 (and is never heard of again). > >Wow, I'll check that MTW out! MTW stands for Misner, Thorne and Wheeler (1973) who produced a $72.50 1277 page paperback telephone book that, at first glance, certainly must contain anything anyone ever would need to know about Einstein's general relativity. (And at second glance, an appalling superfluity of really well-done diagrams that by all means contain far more than you really want to know). But it is a poor primer, and if you seek a concise derivation of bending of light in gravity you will be disappointed as it is either 'left as an exercise for the student' or you are dragged from one extrapolation to the next. Albert's little essays in the Dover book (The principle of relativity) are very worth while by comparison. >> g00 must be a matrix that specifically modifies the coordinates so >> x0' = x0g00 = x0, in the limit, leaving it unchanged and not turned >> around 180 degrees. > >I hesitate to call g00 a matrix. A square array is a matrix but it must meet additional requirements to qualify as a tensor. gij can't meet these; it must be rotatable so the gii can operate on any of x1 x2 x3 or x4. g00 can't. g00 cannot be rotated out of its position and can only operate on x0. > >> On the other hand if you stay with x0 = ict then g00 = 1 and the >> signature is 1111, and x0g00 = x0 . It finally has to be 1111, since xi*gij = xj where the gij are the partial coefficients gij = @xj/@xi from which x0g00 = x'0 (and to have to equate it to -x0 is an unreasonable inversion, since g00 = -1). > >I've derived 1111 with x0=ct together. > snip > >Ok, looking forward. I'm sure it will be done 'real soon now'. >> John Polasek > >Seasons Greetings, >Ken S. Tucker John Polasek
From: Ken S. Tucker on 24 Dec 2008 13:19
On Dec 24, 9:47 am, John Polasek <jpola...(a)cfl.rr.com> wrote: > On Tue, 23 Dec 2008 20:04:28 -0800 (PST), "Ken S. Tucker" > <dynam...(a)vianet.on.ca> wrote: > >Hi John. > > >On Dec 23, 1:33 pm, John Polasek <jpola...(a)cfl.rr.com> wrote: > >> On Mon, 22 Dec 2008 11:51:38 -0500, John Polasek <jpola...(a)cfl.rr.com> > >> wrote: > > >> >On Sun, 21 Dec 2008 22:05:16 -0800 (PST), "Ken S. Tucker" > >> ><dynam...(a)vianet.on.ca> wrote: > > >> >>On Dec 21, 8:00 pm, John Polasek <jpola...(a)cfl.rr.com> wrote: > >> >>> On Tue, 16 Dec 2008 11:57:32 -0800 (PST), "Ken S. Tucker" > > >> >>> <dynam...(a)vianet.on.ca> wrote: > >> >>> >A clock (in K) moving at 0.8c (relative to K') is > >> >>> >dilated 0.6 by t' = t*sqrt(1 - v^2/c^2), so that > >> >>> >t'=(0.6)*t. > > >> >>> >In GR that is generalized to be, > > >> >>> >ds^2 =g_uv dx^u dx^v , {u,v=0,1,2,3}, > > >> >>> >and then by association equatable to > > >> >>> >= dx_u dx^u , > > >> >>> >= dx_0 dx^0 + dx_i dx^i , {i=1,2,3} , Eq.(1). > > >> >>> Firstly, you are incorporating .8c which is related to SR, and > >> >>> admixing it into GR. > > >> >>"admixing", SR is a GR limit. > > >> >>> Secondly, if you retain GR's guv matrix, even with zero gravity, then > >> >>> your first term should be negative, that is, -dx0dx0. > > >> >>WHY? We want logic, NOT idle pronouncements. > > >> >>> I think I'm right on this. > > >> >>Prove it. > >> >g_00 = -1 + 2GM/rc^2 = -1 +eps, in the limit, -1 > >> >g_11 = 1/(1 -eps), in the limit, 1. > >> >dx_0 g00 dx^0 = -dx0^2 in the limit > >> >>>Remember the signature -1 1 1 1. > > >> >>That's a conjecture I want to see proven. > >> >>Regards > >> >>Ken S. Tucker > >> >John Polasek > > >> I just presented a proof; now it's up to you to disprove it, not just > >> let it hang. Start with the original -c2t2 + x2 + y2 + z2 = s2. > > >> As far as deciding if -1111 or 1-1-1-1 is correct, neither one is. > > >I happen to agree with you John, however > >I have been asked to derive either of those > >signatures from a set of basis vectors, they > >are still in use by some. > > Sounds like homework. Well I've used the (++++) metric since the 70's successfully, but I thought I'd review any new input on the old (+---) and (-+++) so far same old, same old. > >> This is a construct of MTW's in 'Gravitation', as a means of getting > >> rid of ict (see pg. 61 'Farewell to ict') in favor of x0 = ct. > >> MTW have performed a mathematical travesty. You can't do that to space > >> and time. > > >I'm ok with x0=ct. > > >> g00 does not pass the test x0g00 = x0', since it makes x0' = - x0, a > >> 180 degree reversal. Here MTW invented the 'one-form' x0g00, which > >> evidently is an inside out x0 (and is never heard of again). > > >Wow, I'll check that MTW out! > > MTW stands for Misner, Thorne and Wheeler (1973) who produced a $72.50 > 1277 page paperback telephone book that, at first glance, certainly > must contain anything anyone ever would need to know about Einstein's > general relativity. (And at second glance, an appalling superfluity of > really well-done diagrams that by all means contain far more than you > really want to know). > But it is a poor primer, and if you seek a concise derivation of > bending of light in gravity you will be disappointed as it is either > 'left as an exercise for the student' or you are dragged from one > extrapolation to the next. > Albert's little essays in the Dover book (The principle of relativity) > are very worth while by comparison. Agreed, I read MTW, it's a good introductory text. > >> g00 must be a matrix that specifically modifies the coordinates so > >> x0' = x0g00 = x0, in the limit, leaving it unchanged and not turned > >> around 180 degrees. > > >I hesitate to call g00 a matrix. > > A square array is a matrix but it must meet additional requirements to > qualify as a tensor. gij can't meet these; it must be rotatable so the > gii can operate on any of x1 x2 x3 or x4. g00 can't. g00 cannot be > rotated out of its position and can only operate on x0. > > >> On the other hand if you stay with x0 = ict then g00 = 1 and the > >> signature is 1111, and x0g00 = x0 . > > It finally has to be 1111, since xi*gij = xj where the gij are the > partial coefficients gij = @xj/@xi from which x0g00 = x'0 (and to have > to equate it to -x0 is an unreasonable inversion, since g00 = -1). Yeah, that sux. > >I've derived 1111 with x0=ct together. > > snip > > >Ok, looking forward. > > I'm sure it will be done 'real soon now'. We have a few briefs here, http://physics.trak4.com/ that I've been urged to publish, but I/we are working on a few new articles, that are hard to leave. Seasons Greetings, Ken S. Tucker > John Polasek |