# T_NAKÄ[uO

## ɁuVvʁFNEW SPACE-TIME METRICvǂށBiQj

<<   쐬 F 2016/10/27 00:01   >>

 "Remarks on the Equivalence of Inertial and Gravitational Masses and on the Accuracy of Einstein's Theory of Gravity." NEW SPACE-TIME METRIC 𑱂܂B [VvʁFNEW SPACE-TIME METRICiQj]================================================== It will be assumed that the metric coefficient standing by the coordinate distance r has generally the following form: (47) W r ̕t߂̌vʌW͈ʂɈȉ̌Ɖ肳F @$e^{B(r)}= 1+R_{s}f(r)\; \; \; \; \; \; \; (47)$ The static gravitational field energy stored in the space around the central gravitating body can be expressed, up to a multiplicative constant, as follows: (48)where the prime signifies the derivative d/dr. Sd͌̎̋ԂɕۑÓId͏GlM[͈ȉ̂悤ɕ\킷ƂłF (48) @$W\approx \int_{0}^{\infty }\frac{{f}'^{2}R^{2}(r)dr}{1+R_{s}f(r)}\; \; \; \; \; \; \; (48)$ AvC (') r ł̔ d/dr B@@ The metric coefficient introduced in the denominator of the fraction behind the integration sign is necessary, since the gravitational field intensity transforms as a covariant vector. d͏̋σxNgƂĕϊł邽߁AϕĽ̕̕ɂvʌWKvłB The Radius function will be found by finding the extremum of W. a֐́AW̋ɒl{oƂɂČ邾낤B This is accomplished by setting the variation of W to zero, W = 0. ́AW̕ϕ[iW = 0jɂ邱ƂɂĒBB The corresponding EL equation for this variational problem is: (49) ̕ϕ̂߂̑ΉIC[OW͈ȉ̒ʂłF @$\frac{d}{dr}\left ( \frac{2{f}'R^{2}}{1+R_{s}f} \right )= - \frac{{f}'^{2}R^{2}R_{s}}{(1+R_{s}f)^{2}}\; \; \; \; \; \; \; (49)$ Solution of Eq.49 is easily found to be: (50) (49) ̉ȉ̒ʂłƂɂ킩F @$R^{2}(r)= -\frac{\sqrt{1+R_{s}f(r)}}{{f}'(r)}\; \; \; \; \; \; \; (50)$ By expressing this result as function of the proper distance , Eq.50 simplifies to read: (51)where the dot represents the derivative d/d. ̌ʂŗLς̊֐Ƃĕ\ƂɂāA(50) ͎̂悤ɂPF @$R^{2}(\rho )=\frac{-1}{\dot{f}(\rho )}\; \; \; \; \; \; \; (51)$ Ńhbg(.)́A ł̔ d/d \킷B By inspection, it is clear that the Radius function derived from the metric line element given in Eq.46 is simply R() = . ƁA(46) ŗ^vʂ̐vfɗR锼a֐P $\110dpi R(\rho)=\rho$ ł邱Ƃ͖łB This is encouraging, since this is what would normally be expected. ꂪʏ\z邱Ƃł邽߁Aɂ͗ECÂB Unfortunately, when this metric is used to evaluate the Mercuryfs perihelion advance, the result is zero. cOȂƂɁǍvʂ𐅐̋ߓ_ړ]̂ɗpƂAʂ̓[łB From this unexpected result it was concluded that Newtonfs gravitational law have to be modified. ̗\zǑʂAj[g̏d͖@CȂ΂ȂȂƌ_ꂽB The formula in Eq.44 suggests that this may represent only the first two terms of a power series expansion of some more complex relation. (44) ̌́AꂪGȊ֌W̋̍ŏ̂Q\ȂƂB One such relation could, for example, be a geometric series as follows: (52) ̂悤Ȋ֌ẂAƂ΁Aȉ̒ʂ蓙䐔ł肦F (52) @$e^{B(\rho )}=\frac{1}{1-\frac{R_{s}}{\rho }}= 1+\frac{R_{s}}{\rho }+\left ( \frac{R_{s}}{\rho } \right )^{2}+\left ( \frac{R_{s}}{\rho } \right )^{3}+\cdots \; \; \; \; \; \; \; (52)$ This would bring us back to the existence of black holes and event horizons, the very thing this article is trying to show that do not exist. ́AubNz[Ɓi{e݂ȂƂƂĂjۂ̒n݂̑̋c_ɖ߂ƂɂȂB Nevertheless, it is interesting to calculate the Radius function for this metric. łǍvʂ̂߂̔a֐vZ邱Ƃ͋[ƂłB It is simply R( ) = ( | Rs ). ́AP$\110dpi R(\rho)=(\rho-R_{s})$łB This is an interesting result, actually very intuitive, which suggests that the classical black hole region needs to be excluded from the space when the gravitational space energy is calculated. ́i͔ɒϓIȁjʔʂłAd͋ԃGlM[vZƂAÓTIȃubNz[̈ԂՒfKv邱ƂB Unfortunately when this metric is used for the calculation of Mercuryfs perihelion advance,the result is twice the observed value. cOȂƂɁǍvʂ𐅐̋ߓ_ړ̌vZɎgƁAʂ͊ώ@ꂽl2{ɂȂB It is interesting to note that the classical result is obtained only when the Radius function, which does not correspond to the minimum of the static gravitational field energy, is used together with coordinate instead of proper distances. iÓId͏GlM[̍ŏƈvȂja֐ŗL̑ɍWƋɎg鎞AÓTIȌʂ_ɒӂ邱Ƃ́A[B This is clearly again a nonphysical result for the Einsteinfs theory of gravity. ́A܂炩ɃACV^C̏d͗_ɂƂĔ񕨗IȌʂłB =============================================================================================== ܂A(48) ̓o͗ǂ܂񂪁Aϕ֐ L Ƃ @$\frac{\partial L}{\partial {f}'}= \frac{2{f}'R^{2}}{1+R_{s}f}\; ,\; \frac{\partial L}{\partial f}=-\frac{{f}'^{2}R^{2}}{(1+R_{s}f)^{2}}$@ Ȃ̂ŁAIC[OWF @$\frac{d}{dr}\left ( \frac{\partial L}{\partial {f}'} \right )= \frac{\partial L}{\partial f}$ A(49) oĂ܂A̎̍ӂ̔{ƁAf" Ȃǂ̍oĂčӂƍȂȂ̂ŁAƍHv܂B @$g(r)\equiv R^{2}{f}'\; \to \; R^{2}= \frac{g}{{f}'}$@ ƂƁA(49) @$\frac{d}{dr}\left ( \frac{2g}{1+R_{s}f} \right )= - \frac{{f}'gR_{s}}{(1+R_{s}f)^{2}}\; \to \; 2\: \frac{{g}'(1+fR_{s})-{f}'gR_{s}}{(1+R_{s}f)^{2}}= - \frac{{f}'gR_{s}}{(1+R_{s}f)^{2}}$ @@$\to \;{g}'(1+fR_{s})= \frac{1}{2} {f}'gR_{s}\; \to \;\frac{{g}'}{g}= \frac{1}{2}\frac{{(1+fR_{s})}'}{(1+fR_{s})}\; \to \;\ln g= \frac{1}{2}\ln (1+fR_{s})$ ܂ @$g= \sqrt{1+fR_{s}}\; \to \;R^{2}= \frac{\sqrt{1+fR_{s}}}{{f}'}$ (50) oĂ܂B (39) @$\frac{d\rho }{dr}= e^{\frac{B(r)}{2}}$ (47) @$e^{\frac{B(r)}{2}}= \sqrt{1+R_{s}f}$ Ȃ̂ @${f}'(r)= \frac{df}{dr}= \frac{df}{d\rho }\frac{d\rho }{dr}= \dot{f}(\rho )e^{\frac{B(r)}{2}}= \dot{f}(\rho )\sqrt{1+R_{s}f}$ ƂȂA (50) ɑ (51) oĂ܂B āA (52) Ɖ肷 @$e^{B(\rho )}=\frac{1}{1-\frac{R_{s}}{\rho }}=\frac{\rho }{\rho -R_{s}}=\frac{\rho-R_{s}+R_{s} }{\rho -R_{s}}= 1+R_{s}\frac{1}{\rho -R_{s}}$ @@$\to f(\rho )= \frac{1}{\rho -R_{s}}\to \dot{f}(\rho )= -\frac{1}{(\rho -R_{s})^{2}}$ Ȃ̂ŁA (51) @$R^{2}(\rho )= (\rho -R_{s})^{2}\; \to \: R(\rho )= \rho -R_{s}$ ƂȂ܂B ͂̕ӂŁBB

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