# Gravitational Waves: Propagation Speed is Co-ordinate Dependent

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1 April 08 meetin of the American Physical Society, Columbus, Ohio. Poster presented on 4 th April, Gravitational Waves: Propaation Speed is Co-ordinate Dependent Stephen J. Crothers. Proof: Linearise Einstein s field equations by, h () where the h μν << and η μν represents the Galilean values (,-,-,-). The h μν and their derivatives are small, so all products of them are nelected. Contract the iemann-christoffel curvature tensor ρμνσ to obtain the icci tensor:,,,, () The Γ terms are products of the μν and their first derivatives, therefore nelected, so, which can be broken into two parts,,,,, (3),,,, (4) Choose co-ordinates x so that the second part of eq.(4) vanishes:, x x (5) 0,,, (6) Accordin to eq.(), and so on for hiher derivatives, hence,, h, h, x h x (5b) h h 0 h (6b),,, Contractin eq.(5b) yields the icci scalar: by Merin Black Holes, Hadronic Journal, n.3, Vol. 39, 06, pp.7-30,

2 April 08 meetin of the American Physical Society, Columbus, Ohio. Poster presented on 4 th April, x h x h x x (7) Einstein s field equations (without cosmoloical constant) are, T (8) In terms of h μν these become, usin eq.(5b) and eq.(7), x h x h x x T (8b) The d Almbertian operator is defined as, Hence, (9) x x (0) x y z c t c t where is the differential operator del (or nabla), defined as,,,. x y z Takin the dot product of del with itself ives the Laplacian operator, x y z. Settin x 0 =ct, x =x, x =y, x 3 =z, eq.(8b) can be written as, h h T () These are the linearised field equations. They are subject to the condition (6b), which can be condensed to the followin condition, h h 0 (6c) x Usin eq.(9), eq.(5b) can be written as, by Merin Black Holes, Hadronic Journal, n.3, Vol. 39, 06, pp.7-30,

3 April 08 meetin of the American Physical Society, Columbus, Ohio. Poster presented on 4 th April, For empty space this becomes, h μν = () h μν = 0 (3) which by eq.(0) describes a wave propaatin at the speed of liht in vacuo. But the speed of the waves is co-ordinate dependent, as the constraint at eq.(6) attests: different co-ordinates, different speeds. Einstein chose a set of co-ordinates that yields his presumed speed of propaation: an example of the loical fallacy of assumin as premise that which is to be demonstrated (petitio principii).. General elativity: Its Violation of the Usual Conservation Laws for a Closed System. We make a distinction hereafter between ravitational field and matter in this way, that we denote everythin but the ravitational field as matter. Our use of the word therefore includes not only matter in the ordinary sense, but the electromanetic field as well. Einstein, A., The Foundation of the General Theory of elativity, Annalen der Physik, 49, (96), 4 It must be remembered that besides the enery density of the matter there must also be iven an enery density of the ravitational field, so that there can be no talk of principles of conservation of enery and momentum of matter alone. Einstein, A., The Meanin of elativity, expanded Princeton Science Library Edition, (005) T enery-momentum of Einstein s matter alone t enery-momentum pseudotensor of Einstein s ravitational field alone, defined, t (.) The quantities t we call the enery components of the ravitational field. Einstein, A., The Foundation of the General Theory of elativity, Annalen der Physik, 49, (96), 5 But t is not a tensor and is co-ordinate dependent, contrary to the basic tenet of General elativity. It is to be noted that t is not a tensor. Einstein, A., the Foundation of the General Theory of elativity, Annalen der Physik, 49, (96), 5 Alleedly, t acts 'like a tensor', for linear transformations of co-ordinates under certain conditions. Einstein then takes an ordinary (not tensor) diverence, by Merin Black Holes, Hadronic Journal, n.3, Vol. 39, 06, pp.7-30, 3

4 April 08 meetin of the American Physical Society, Columbus, Ohio. Poster presented on 4 th April, and proclaims, t x 0 (.) This equation expresses the law of conservation of momentum and of enery for the ravitational field. Einstein, A., The Foundation of the General Theory of elativity, Annalen der Physik, 49, (96), 5 Einstein s total enery-momentum equation for his field and sources is, t T E (.3) This is not a tensor equation, so aain only an ordinary diverence, t T x 0 (.4) by which Einstein proclaimed: Thus it results from our field equations of ravitation that the laws of conservation of momentum and enery are satisfied. here, instead of the enery components t of the ravitational field, we have to introduce the totality of the enery components of matter and ravitational field. Einstein, A., The Foundation of the General Theory of elativity, Annalen der Physik, 49, (96), 7 The mathematical error is profound. Contract Einstein s pseudotensor: t t (.4) Thus the invariant t is a first-order intrinsic differential invariant, i.e. t depends solely upon the components of the metric tensor and their first derivatives. But the pure mathematicians proved in 900 that first-order intrinsic differential invariants do not exist (icci-curbastro, G., Levi-Civita, T., Méthodes de calcul différentiel absolu ET leurs applications, Matematische Annalen, B. 54, p.6, 900). Thus, by reductio ad absurdum, Einstein s pseudotensor is a meaninless concoction of mathematical symbols. It cannot therefore be assined to anythin in physics or to make any calculations. Nevertheless, Einstein and cosmoloists assin and calculate. Consider the followin two equivalent forms of Einstein s field equations, T T (.6) T (.7) by Merin Black Holes, Hadronic Journal, n.3, Vol. 39, 06, pp.7-30, 4

5 April 08 meetin of the American Physical Society, Columbus, Ohio. Poster presented on 4 th April, By eq.(.6), accordin to Einstein, if T 0 then 0. But by eq.(.7), if 0 then T 0. In other words, and T must vanish identically: 0 = 0. If there are no material sources then there is no ravitational field, and no universe. Consequently, Einstein s field equations must take the followin, G T 0 (.8) where G Comparin this to eq.(.3) the G constitute the enery-momentum components / of Einstein s ravitational field alone: after all, G describes Einstein s ravitational field. Equation (.8) also constitutes the total enery-momentum equation for Einstein s ravitational field and its material sources combined. space as opposed to what fills space, which is dependent on the coordinates, has no separate existence Einstein, A., elativity and the problem of space, Appendix 5 in the 5 th edition of elativity: the Special and the General Theory, Methuen, London, (954), pp I wish to show that space-time is not necessarily somethin to which one can ascribe a separate existence, independently of the actual objects of physical reality. Einstein, A., Preface to the 5 th edition of elativity: the Special and the General Theory, Methuen, London, (954) Unlike eq.(.3), eq.(.8) is a tensor equation. Its tensor diverence is zero and therefore constitutes the only conservation law for Einstein s ravitational field and its material sources combined. Since the total enery-momentum eq.(.8) is always zero, and the G / and the T must vanish identically because spacetime and matter have no separate existence in General elativity, ravitational enery cannot be localised, i.e. no possibility of ravitational waves. Since the total enery-momentum is always zero the usual conservation laws for enery and momentum of a closed system cannot be satisfied. General elativity is therefore in conflict with a vast array of experiments on a fundamental level. The so-called cosmoloical constant can be easily included as follows, G T 0 (.9) In this case the enery-momentum components of Einstein s ravitational field are G. When G or T is zero, all must vanish identically. by Merin Black Holes, Hadronic Journal, n.3, Vol. 39, 06, pp.7-30, 5

6 April 08 meetin of the American Physical Society, Columbus, Ohio. Poster presented on 4 th April, The mysterious dark enery is arbitrarily attributed to λ by cosmoloists. But accordin to Einstein, λ is not a material source for his ravitational field, instead vauely implicated in his ravitational field alone:... by introducin into Hamilton's principle, instead of the scalar of iemann's tensor, this scalar increased by a universal constant Einstein, A., Cosmoloical Considerations on the General Theory of elativity, Sitzunsberichte der Preussischen Akad. d. Wissenschaften, (97), 4 By eq.(.9) is part of the enery-momentum of the ravitational field, which however necessarily vanishes when T 0. Expand eq.(.9): T 0 (.9b) Einstein's scalar increased by a universal constant is the term ( λ)/. Aain, Einstein s field equations in the absence of matter, 0, have no physical meanin. Therefore the Schwarzschild solution also has no physical meanin. The 'cosmoloical constant' also falls afoul of de Sitter' empty universe, which possesses spacetime curvature (ravity) but contains no matter ( T 0 ), and is therefore physically meaninless. Consequently, the theories of black holes and ravitational waves are invalid ([] Crothers, S.J., On Corda's 'Clarification' of Schwarzschild's Solution, Hadronic Journal, Vol. 39, 06, [] Crothers, S. J., General elativity: In Acknowledement Of Professor Gerardus t Hooft, Nobel Laureate, 4 Auust, 04, 3. The Mathematical Theory of Black Holes: its Violation of the ules of Pure Mathematics and Loic Presumably the ravitational waves reported by LIGO-Viro are present inside some Bi Ban expandin universe as there has been no report that Bi Ban cosmoloy has been abandoned. The LIGO-Viro Collaborations have stated of first discovery : It matches the waveform predicted by eneral relativity for the inspiral and merer of a pair of black holes and the rindown of the resultin sinle black hole. Abbott, B.P. et al., Observation of Gravitational Waves from a Binary Black Hole Merer, PL 6, 060 (06) All black hole metrics are solutions to correspondin specific forms of Einstein's nonlinear field equations. Gravitational waves are however obtained from a linearisation of Einstein s field equations, combined with a selection of co-ordinates that produce the presumed propaation speed c. Because General elativity is a nonlinear theory, the Principle of Superposition does not hold. Let X be some black hole universe and Y by Merin Black Holes, Hadronic Journal, n.3, Vol. 39, 06, pp.7-30, 6

7 April 08 meetin of the American Physical Society, Columbus, Ohio. Poster presented on 4 th April, be some Bi Ban universe. Then the linear combination (i.e. superposition) X + Y is not a universe. Indeed, X and Y pertain to completely different sets of Einstein field equations havin absolutely nothin to do with one another. Similarly if W and Z are both black hole universes, be they the same or not, and if U and V are Bi Ban universes, be they the same or not. A black hole universe cannot co-exist with any other black hole universe or with any Bi Ban universe. Black Hole Universes Bi Ban Universes () Have no k-curvature. () Have a k-curvature. () Are spatially infinite. () Are either spatially finite (k = ) or spatially infinite (k = or k = 0). (3) Are eternal. (3) Not eternal (~3.8 billion years old) (4) Contain only one mass. (4) Contain arbitrarily many masses. (5) Are not expandin (not non-static). (5) Are expandin (non-static). (6) Are asymptotically flat. (6) Are not asymptotically anythin. (7) Cannot be superposed with anythin. (7) Cannot be superposed with anythin. Since a black hole universe is a solution to a specific set of Einstein's nonlinear field equations it is not possible to extract from it any ravitational waves produced from linearised field equations. No ravitational waves can in fact be extracted from Einstein's nonlinear field equations. Superposin solutions obtained from the nonlinear system with those from the linearised system violates the mathematical structure of General elativity. Accordinly, ravitational waves cannot exist in any black hole universe. Neither can they exist in any Bi Ban universe because all Bi Ban models are in fact sinle mass universes by mathematical construction (uniform macroscopic density and pressure). Nonetheless the LIGO-Viro Collaborations superpose everythin. In can be shown that the mathematical theory of black holes latently requires that the absolute value of a real number must take on values less than zero, which is impossible ([] Crothers, S.J., The Painlevé-Gullstrand Extension - A Black Hole Fallacy, American Journal of Modern Physics, 5, Issue -, 06, pp.33-39, [] Crothers, S. J., On the Generation of Equivalent Black Hole Metrics: A eview, American Journal of Space Science, v.3, i., pp.8-44, July 6, 05, 4. The Secret Methods of LIGO For their discovery the LIGO-Viro teams numerically manufactured 50,000 model waveforms ( templates ) from meaninless equations. A 'eneric' noise was initially reported by LIGO, after which LIGO s computers extracted GW5094 from the LIGO manufactured meaninless database as a curve of best fit to the noise: by Merin Black Holes, Hadronic Journal, n.3, Vol. 39, 06, pp.7-30, 7

8 April 08 meetin of the American Physical Society, Columbus, Ohio. Poster presented on 4 th April, The initial detection was made by low-latency searches for eneric ravitationalwave transients [4] and was reported within three minutes of data acquisition [43]. Subsequently, matched-filter analyses that use relativistic models of compact binary waveforms [44] recovered GW5094 as the most sinificant event from each detector for the observations reported here. Abbott, B.P. et al., Observation of Gravitational Waves from a Binary Black Hole Merer, PL 6, 060 (06) With such powerful computin resources and so many derees of freedom it is possible to best fit just about any LIGO instability with an element of its numerically manufactured database of curves. The LIGO-Viro Collaborations have manaed to best fit a numerically manufactured curve for and to entities that do not exist. This amplifies the futility of applyin numerical and perturbation methods to enerate desired outcomes. There are no known Einstein field equations for two or more masses and hence no known solutions thereto. There is no existence theorem by which it can even be asserted that Einstein's field equations contain latent capability for describin confiurations of two or more masses. General elativity cannot account for the simple experimental fact that two fixed suspended masses approach one another upon release. For precisely these reasons all Bi Ban models make the universe a sinle mass: an ideal indivisible fluid of uniform macroscopic density and pressure that permeates the entire universe. ([] Crothers, S. J., Flaws in Black Hole Theory and General elativity, Proceedins of the XXIX th International Workshop on Hih Enery Physics, Protvino, ussia, 6-8 June 03, [] Crothers, S. J. On the Invalidity of the Hawkin-Penrose Sinularity Theorems and Acceleration of the Universe with Neative Cosmoloical Constant, Global Journal of Science Frontier esearch Physics and Space Science, Volume 3 Issue 4, Version.0, 03, [3] Crothers, S. J., On the Generation of Equivalent Black Hole Metrics: A eview, American Journal of Space Science, v.3, i., pp.8-44, 05, [4] Crothers, S.J., LIGO at the University of Queensland, by Merin Black Holes, Hadronic Journal, n.3, Vol. 39, 06, pp.7-30, 8