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Minkowski Metric
"The Minkowski metric, unlike the Riemann
metric, by definition is NOT a generally covariant quantity. This is now conventional wisdom in the history and foundations of gravitational physics." wrote Z The above remark is wrong. inkowskiIn fact the Minkowski metric is a covariant second rank tensor under GCTs in globally flat spacetime and it is invariant under Lorentz boosts connecting geodesic observers. Even when you make the split in intrinsically curved space-time guv(curved) = (Minkowski)uv + huv(curved) each term on RHS is generally covariant separately. Consider 1+1 space-time for computational simplicity. The O(1,1) Lorentz boosts between geodesic inertial observers are xi' = Li'^ixi i = 0,1 x0' = #(x0 - @x1) x1' = #(-@x0 + x1) c = 1 # = (1 - @^2)^-1/2 In ordinary notation # = gamma, @ = v = relative speed between geodesic inertial observers L0'^0 = # L0'^1 = - #@ L1'^0 = - #@ = L0'^1 L1'^1 = # = L0'^0 (M)i'j' = L^ii'L^jj'(M)ij where "Minkowski" = M M00 = 1 M11 = -1 M01 = M10 = 0 M0'0' = L0'^iL0'^jMij = L0'^0L0'^0M00 + L0'^1L0'^1M11 = L0'^0L0'^0 - L0'^1L0'^1M11 = #^2 - #^2@^2 = +1 = M00 M0'0' = M00 M1'1' = L1'^iL1'^jMij = L1'^0 L1'^0M00 + L1'^1L1'^1M11 = L1'^0 L1'^0 - L1'^1L1'^1 = #^2@^2 - #^2 = -1 = M11 M1'1' = M11 M0'1' = L0'^iL1'^jMij = L0'^0L1'^0M00 + L0'^1L1'^1M11 = L0'^0L1'^0 - L0'^1L1'^1 = 0 = M1'0' = M10 = M01 Therefore the canonical form for the Minkowski metric is invariant under the linear Lorentz boosts connecting geodesic observers. Furthermore, the Minkowski metric (Minkowski)uv under GCT general nonlinear local coordinate transformations Xu'^u is a covariant 2nd rank symmetric tensor (Minkowski)u'v' = Xu'^u Xv'^v (Minkowski)uv |
#2
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Minkowski Metric
I wonder what it means?
covariant? Tensor? Metric? I never did know. I probably never will. Will it get me to alpha centuri or make me immortal? I don't think so. "Jack Sarfatti" wrote in message ... "The Minkowski metric, unlike the Riemann metric, by definition is NOT a generally covariant quantity. This is now conventional wisdom in the history and foundations of gravitational physics." wrote Z The above remark is wrong. inkowskiIn fact the Minkowski metric is a covariant second rank tensor under GCTs in globally flat spacetime and it is invariant under Lorentz boosts connecting geodesic observers. Even when you make the split in intrinsically curved space-time guv(curved) = (Minkowski)uv + huv(curved) each term on RHS is generally covariant separately. Consider 1+1 space-time for computational simplicity. The O(1,1) Lorentz boosts between geodesic inertial observers are xi' = Li'^ixi i = 0,1 x0' = #(x0 - @x1) x1' = #(-@x0 + x1) c = 1 # = (1 - @^2)^-1/2 In ordinary notation # = gamma, @ = v = relative speed between geodesic inertial observers L0'^0 = # L0'^1 = - #@ L1'^0 = - #@ = L0'^1 L1'^1 = # = L0'^0 (M)i'j' = L^ii'L^jj'(M)ij where "Minkowski" = M M00 = 1 M11 = -1 M01 = M10 = 0 M0'0' = L0'^iL0'^jMij = L0'^0L0'^0M00 + L0'^1L0'^1M11 = L0'^0L0'^0 - L0'^1L0'^1M11 = #^2 - #^2@^2 = +1 = M00 M0'0' = M00 M1'1' = L1'^iL1'^jMij = L1'^0 L1'^0M00 + L1'^1L1'^1M11 = L1'^0 L1'^0 - L1'^1L1'^1 = #^2@^2 - #^2 = -1 = M11 M1'1' = M11 M0'1' = L0'^iL1'^jMij = L0'^0L1'^0M00 + L0'^1L1'^1M11 = L0'^0L1'^0 - L0'^1L1'^1 = 0 = M1'0' = M10 = M01 Therefore the canonical form for the Minkowski metric is invariant under the linear Lorentz boosts connecting geodesic observers. Furthermore, the Minkowski metric (Minkowski)uv under GCT general nonlinear local coordinate transformations Xu'^u is a covariant 2nd rank symmetric tensor (Minkowski)u'v' = Xu'^u Xv'^v (Minkowski)uv |
#3
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Minkowski Metric
Chris wrote: I wonder what it means? covariant? Tensor? Metric? I never did know. I probably never will. They are mathematical concepts which are not of fundamental importance at all. They are but mere tools of confusion very active in the modern understanding of physics and mathematics. You aren't missing anything. Will it get me to alpha centuri or make me immortal? I don't think so. I used to think how far the frontiers of mathematics has gone. Now I realize it is the frontiers of confusion. Those concepts aren't important now and they never were - always remember Orwell's lesson - those who control the past determine the future. Those who control the present determine the past. |
#4
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Minkowski Metric
wrote in message
oups.com... Chris wrote: I wonder what it means? covariant? Tensor? Metric? I never did know. I probably never will. [Schoenfeld] They are mathematical concepts which are not of fundamental importance at all. They are but mere tools of confusion very active in the modern understanding of physics and mathematics. You aren't missing anything. I used to think how far the frontiers of mathematics has gone. Now I realize it is the frontiers of confusion. Those concepts aren't important now and they never were. [hanson] VERY WELL SAID, Schoenfeld. Kudos! [Schoenfeld] - always remember Orwell's lesson - those who control the past determine the future. Those who control the present determine the past. [hanson] True, as long as Orwell meant "the peasantry" by "those" The peasants are perennial like the grass, feeding and controlling the intelligentsia, which goes with and is blown away by the winds of time... perennially so too. ahahahaha... ahahaha |
#6
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Minkowski Metric
ahahahaha.... "Mike" elastic at yahoo.greek wrote in message
ups.com... wrote: I used to think how far the frontiers of mathematics has gone. Now I realize it is the frontiers of confusion. Those concepts aren't important now and they never were - always remember Orwell's lesson - those who control the past determine the future. Those who control the present determine the past. [Mike on one of his elastic binges burbs:] and those who control the future determine the present. Just to make it complete..... hahahahahahaha.....where is Hanson.....hahahahahaha..... maybe he finalkly got an Abo boyfriend...hahahahahahaha hahahahahaha....Abos put a curse on Auusies. No rain for the next 200 years...hahahahahaha [hanson] ahahahaha... Mikey, you should not be boozing it up so early in the morning. What "Auusies" and "200 years" and "Abo boy friends"? ... ahahahaha... Did your jealousy in your ethanolic Greek style fantasy yearnings overwhelm you?... ahahaha... Thanks for the laughs.... ahahahahanson [Mike to Schoenfeld] By the way, did you get around these laws of motion? hahahahahahaha http://groups.google.gr/group/sci.ph...c2bb07e?hl=en& Mike |
#7
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Minkowski Metric
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#8
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Minkowski Metric
"Chris" wrote in message o.uk... I wonder what it means? covariant? Tensor? Metric? I never did know. I probably never will. Will it get me to alpha centuri or make me immortal? I don't think so. To familiarize yourself with these items you should take a quick look at what tensors are. Then you'll be on your way to understanding the terms you seek. See my web page at http://www.geocities.com/physics_wor...tro_tensor.htm If anything is unclear then please let me know. Good luck Pete |
#9
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Minkowski Metric
Pmb wrote: "Chris" wrote in message o.uk... I wonder what it means? covariant? Tensor? Metric? I never did know. I probably never will. Will it get me to alpha centuri or make me immortal? I don't think so. To familiarize yourself with these items you should take a quick look at what tensors are. Then you'll be on your way to understanding the terms you seek. See my web page at http://www.geocities.com/physics_wor...tro_tensor.htm If anything is unclear then please let me know. Good luck Pete Back in 1983, the International Standards of Units agreed to set, Length (n meters) = c Time (second), with "n" a fixed scalar. Did you (Pete and anyone else) agree with that decision or have any input into that decision? Ken |
#10
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Minkowski Metric
JanPB wrote: wrote: Chris wrote: I wonder what it means? covariant? Tensor? Metric? I never did know. I probably never will. They are mathematical concepts which are not of fundamental importance at all. They are but mere tools of confusion very active in the modern understanding of physics and mathematics. You aren't missing anything. Why don't you just say what you really think: "I'm going to declare unimportant anything I don't understand". The problem is I do understand them but the understanding of them did not come "naturally". I realize now it is because they are unnatural concepts. Matrices / Quaternions / Spinors / Tensors are either lacking structure one fundamental aspect or are unnecessarily complicated in another. They describe "snippets" of useful structure in some places but carry a whole bunch of useless baggage which always needs to be dealt with. Geometric Algebra describe all of that and much more in the simplest way, but no simpler (to paraphrase Albert). |
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