Any coordinate system in GR?
in [Math]
|
Prev: Prime numbers
Next: Am I a crank?
From: JanPB on 24 Aug 2006 19:13 Edward Green wrote: > [...] I believe I have found the answer to my own question (as > usual), and while I did not state the requirement 100% correctly, I was > on the right track. I happened upon the correct formulation in > Schwarzschild's 1916 paper on the field of a mass point: > > "The field equations ... have the fundamental property that > they preserve their form under the substitution of other > arbitrary variables in lieu of x1,x2, x3, x4, as long as the > determinant of the substitution is equal to 1." > > http://arxiv.org/PS_cache/physics/pdf/9905/9905030.pdf It's usually not a good idea to learn stuff from original papers. They were written for experts and reflect the then-current knowledge. For example, Schwarzschild didn't know at that time that the determinant restriction was unnecessary (the equations are invariant under _any_ change of variables). > If the old coordinates were orthogonal and the new coordinates remained > orthogonal (which requirement I did not state) and we kept the same > units, mine would be a sufficient condition to meet the above > requirement on the named determinate. Forget the determinants, it's complete mothballs. Read a modern textbook instead. > In general we have more latitude > than that, _but_ we are not free to chose the new coordinates > arbitrarily, even if they are invertible in a neighborhood. The unit of > _volume_ at least must be preserved. No, that's unecessary. -- Jan Bielawski
From: JanPB on 24 Aug 2006 19:15 I wrote: > > It's usually not a good idea to learn stuff from original papers. I meant classic, well-established stuff of course. -- Jan Bielawski
From: Koobee Wublee on 24 Aug 2006 20:25 Edward Green wrote: > Igor wrote: > > The metric will always compensate to make up for the difference. > > So you claim. I believe I have found the answer to my own question (as > usual), and while I did not state the requirement 100% correctly, I was > on the right track. I happened upon the correct formulation in > Schwarzschild's 1916 paper on the field of a mass point: > > "The field equations ... have the fundamental property that > they preserve their form under the substitution of other > arbitrary variables in lieu of x1,x2, x3, x4, as long as the > determinant of the substitution is equal to 1." > > http://arxiv.org/PS_cache/physics/pdf/9905/9905030.pdf This is an excellent article. It is very easy to read and understand. Thanks to Schwarzschild's ingenuity. > If the old coordinates were orthogonal and the new coordinates remained > orthogonal (which requirement I did not state) and we kept the same > units, mine would be a sufficient condition to meet the above > requirement on the named determinate. In general we have more latitude > than that, _but_ we are not free to chose the new coordinates > arbitrarily, even if they are invertible in a neighborhood. The unit of > _volume_ at least must be preserved. In the case of a one-dimensional > manifold this reduces to the requirement to keep the same units. Schwarzschild found a unique solution to the differential equations of Einstein Field Equations in free space. Hilbert found another one that he called it Schwarzschild Metric. Recently, Mr. Rahman also a contributor of this newsgroup presented another solution. The spacetime with this metric is ds^2 = c^2 dt^2 / (1 + K / r) - (1 + K / r) dr^2 - (r + K)^2 dO^2 The metric above indeed is another solution which anyone can easily verify because its simplicity. Notice Rahman's metric and Schwarzschild's original metric do not manifest black holes. However, since Schwarzschild Metric is much simpler than Schwarzschild's original solution, Schwarzschild Metric is embraced by the physics communities today. Mr. Bielawski and Igor have not understood Schwarzschild's original paper and choose to blindly reject Schwarzschild's original solution and others. As multiple solutions to the vacuum field equations are discovered, there are actually an infinite number of them. With infinite number of solutions, it is shaking the very foundation of GR and SR. The house of cards will soon inevitably collapse. However, refusing to give up GR and to comfort themselves in false sense of security, they choose to embrace Voodoo Mathematics. In doing so, they blindly claim all solutions are indeed the same regardless manifesting black holes, constant expanding universe, accelerated expanding universe. VOODOO MATHEMATICS REPRESENTS THE ACHIEVEMENT IN PHYSICS DURING THE LAST 100 YEARS. It is very sad that these clowns are regarded as experts in their field.
From: Edward Green on 24 Aug 2006 20:44 JanPB wrote: >. The unit of _volume_ at least must be preserved. > > No, that's unecessary. Well sir, I disremember what started me on this tack... maybe I was trying to figure out what was "physical" in GR. However, I arrived at the following waystation: Physically distinct (in a specifiable sense) 1-d manifolds with what look like identical representations of a metric. Is it a _problem_ if physically distinct situations look identical given only the variable labels and a representation of the metric? Is there some missing piece? Did I correctly transform the metrics given the manipulations of the manifold or coordinate system described? You would do me a kindness if you were to scan the second two paragraphs in the post you replied to, and indicate the faults in the argument. Oh... and learning from original papers... I'm not sure I completely agree with you. Sometimes you have a conceptual problem, and you find that the original thinkers were alive to the problem, although it is glossed over now. At least I think that's happened to me once, so I can freely generalize. Maybe even twice now. ;-)
From: JanPB on 24 Aug 2006 20:53
Koobee Wublee wrote: > > Schwarzschild found a unique solution to the differential equations of > Einstein Field Equations in free space. Hilbert found another one that > he called it Schwarzschild Metric. Recently, Mr. Rahman also a > contributor of this newsgroup presented another solution. The > spacetime with this metric is > > ds^2 = c^2 dt^2 / (1 + K / r) - (1 + K / r) dr^2 - (r + K)^2 dO^2 > > The metric above indeed is another solution which anyone can easily > verify because its simplicity. Notice Rahman's metric and > Schwarzschild's original metric do not manifest black holes. No, the metric above is equal to Schwarzschild's metric. The form above is obtained by a coordinate change from the original one, hence the metric remains the same (tensors do not change under coordinate changes). > However, since Schwarzschild Metric is much simpler than > Schwarzschild's original solution, Schwarzschild Metric is embraced by > the physics communities today. It's embraced because it's the same. > Mr. Bielawski and Igor have not > understood Schwarzschild's original paper and choose to blindly reject > Schwarzschild's original solution and others. There is nothing to reject. One can _prove_ Schwarzschild's metric is unique. Off the horizon it follows immediately from the particular form of the Einstein equation in the spherically symmetric case (which is what we have) and the uniqueness of the extension over the horizon is slightly more involved but it follows from a similar argument. > As multiple solutions to the vacuum field equations are discovered, > there are actually an infinite number of them. Yes, and 2+2=5. > With infinite number of > solutions, it is shaking the very foundation of GR and SR. Sure. My boots are all torn already. Give us a break. -- Jan Bielawski |