Re: Are other things impossible too?
Here is a sci.physics.research post where the derivation of the Lorentz transformation is given, based only on the simplest possible assumptions of isotropy and homogeneity:
\n From: Danny Ross Lunsford (mail)\nSubject: Re: Unification of Electromagnetism and Gravity \n\n Newsgroups: sci.physics.research\nDate: 2001-12-22 09:38:07 PST \n\nBrian J Flanagan wrote:\n\n> "Danny Ross Lunsford" wrote:\n> \n>>Relativity theory is independent of the nature of particular forces.\n> \n> I'm not sure what you mean by "independent", here. Surely the special\n> theory has much to do with EM (in re: the constancy of c), just as the\n> general theory has much to do with gravitation?\n\nNot at all. In fact the framework of relativity is derives from a \ngroup-theoretic analysis based on very basic and simple assumptions about \nspace, time, and motion, taken from experience. The conclusion is that the \nallowed transformations depend on parameter with the dimensions of a \nvelocity that is either finite or not. In the real world, it turns out to \nbe finite. Of course, it is c. That light goes at this speed is incidental \nto the analysis, which would still be correct if light went at some other \nspeed. Light happens to go at c because the photon is massless.\n\nTo give this derivation, we assume\n\n1) The allowable transformations are linear between frames in uniform \nrelative motion, so that they depend on the relative (vectorial) velocity V\n\n2) That time can enter into the transformations in an essential way, that \nis we do not assume t' = t.\n\n3) That space and time are isotropic, that is, time flows evenly and there \nare no preferred directions.\n\nSpatial isotropy eliminates the directional character of V and so we can \nconcentrate on the simple case of one spatial dimension.\n\nTemporal isotropy implies that what is true of frame A in relation to frame \nB for V, is true of frame B in relation to frame A for -V.\n\nNow, writing\n\nx' = a(V) x + b(V) t\nt' = c(V) x + d(V) t\n\nx = a(-V) x' + b(-V) t'\nt = c(-V) x' + d(-V) t'\n\nso\n\na(V) a(-V) + b(-V) c(V) = 1\nd(V) d(-V) + b(V) c(-V) = 1\na(-V) b(V) + b(-V) d(V) = 0\na(V) c(-V) + c(V) d(-V) = 0\n\nand of course we can replace V -> -V and these still hold. So\n\na(V) b(-V) + b(V) d(-V) = 0\na(V) c(-V) + c(V) d(-V) = 0\n\nso b(V) = c(V).\n\nNow\n\na(V) a(-V) + b(V) b(-V) = 1\na(V) b(-V) + b(V) d(-V) = 0\n\nthus\n\nb(V) [ b(-V)^2 - a(-V)d(-V) ] = b(-V)\n\nand this also holds for V -> -V, which implies either\n\na(V) d(V) - b(V)^2 = 1, b(V) = -b(-V)\n\na(V) d(V) - b(V)^2 = -1, b(V) = b(-V)\n\nThe latter is ruled out by letting V=0 in which case\n\nx = x'\nt = t'\n\nWe can solve the equations now with\n\na(V) = d(V) = a(-V) = d(-V)\n\nand so\n\na(V)^2 - b(V)^2 = 1\n\nand \n\na(0) = 1, b(0) = 0\n\nThe origin x=0, which moves at speed V in the other frame, transforms as\n\nx' = b(V) t\nt' = a(V) t\n\nso\n\ndx'/dt' = V = b(V)/a(V)\n\nso\n\na(V) = 1 / sqrt(1 - V^2)\nb(V) = V / sqrt(1 - V^2)\n\nFinally we dimensionalize time vs. space and replace\n\nV -> V/C\nt -> Ct\n\nand write\n\nx' = 1/sqrt(1 - (V/C)^2) ( x + (V/C) Ct )\n\nCt' = 1/sqrt(1 - (V/C)^2) ( Ct + (V/C) x )\n\nIf we let C go to infinity,\n\nx' = x + Vt\nt' = t\n\nSo either C is finite, or not. Experience shows that it is finite.\n\nOf course, historically relativity emerged from the contradictions in \nelectron theory implied by the tacit, wrong assumptions about the nature of \nsimultaneity.\n\n-drl\n\n
Implicit in this is the assumption of homogeneity, which I should have stated. (All points are equivalent - doesn't matter if you're here or there.)
