# T_NAKÄ[uO

## ɁuVvʁFNEW SPACE-TIME METRICvǂށBiRj

<<   쐬 F 2016/10/31 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 ̍ŌƂȂ܂B [VvʁFNEW SPACE-TIME METRICiRj]================================================ Finally, as will be shown in the next section, the correct result for the Mercuryfs perihelion advance is obtained when the metric coefficient has additional higher order terms in the metric element expansion starting with 1/ 2( Rs/ )2 . ŌɁÃZNVŎ悤ɁA 1/2(Rs/)2 n܂鋉ŁA荂̌vʌWƂA̋ߓ_ړ߂̐ʂ͓B Newtonfs gravitational law needs therefore a modification to agree with observations. āAj[g̏d͖@́Aώ@ʂƍv邽߂ɁACKvƂB To add more expansion terms to the metric coefficient is possible and finally arrive at a simple analytic formula for B(), B() = Rs/. XȂ鋉vʌWɉ邱ƂłAŏIIɁABiρj̊ȒPȉ͌ B() = Rs/ ɂǂ蒅B Such a metric predicts the same perihelion advance as the metric with only the second order term within the accuracy of current observations. ̂悤Ȍvʂ́Å݂ώ@̐m͈͓̔āA2œߓ_ړ\B Therefore, it will be assumed that the extrapolated expression correctly describes the space-time of the studied problem. āAO}ꂽ\Ȗ̎𐳂LqƉ肳B The metric line element and the corresponding Lagrangian for the space-time in the vicinity of a gravitating non-rotating body, which agree with observations, are therefore as follows: (53) (54) āAvʂ̐vfƁAiϑɍvj]ĂȂ̂̋ߖTɑΉ郉OW֐͈ȉ̒ʂłF (53) (54) @$ds^{2}= e^{\frac{-R_{s}}{\rho }}(cdt)^{2}-e^{\frac{R_{s}}{\rho }}dr^{2}-\rho ^{2}e^{\frac{-R_{s}}{\rho }}(d\vartheta ^{2}+\sin ^{2}\vartheta \cdot d\varphi ^{2})\; \; \; \; \; \; (53)$ @$L= e^{\frac{-R_{s}}{\rho }}\left ( \frac{cdt}{d\tau } \right )^{2}-e^{\frac{R_{s}}{\rho }}\left ( \frac{dr}{d\tau } \right )^{2}-\rho ^{2}e^{\frac{-R_{s}}{\rho }}\left \{\left ( \frac{d\vartheta }{d\tau } \right ) ^{2}+\sin ^{2}\vartheta \left ( \frac{ d\varphi}{d\tau } \right ) ^{2} \right \}\; \; \; \; \; \; (54)$ It is clear that this metric does not have any pathology at the Schwarzschild radius, does not have any censored singularities with event horizons and that it covers the whole space-time region. ̌vʂ̓VcVgaŕaIȗlAۂ̒nƓِȂƂ͖łAS̎n𕢂B }QDzʂ1.4{̐ɂW̊֐ƂĂ̌ŗL̐}BŗLƍW̖Ș́AVcVgä̗ɋNB It is also clear that the new metric approaches the Schwarzschild metric for large distances, so it can be expected that the observational confirmations of GTR, including the light deflection by Sun and Shapiro delay, will also apply here. Vvʂł̓VcVgvʂɋ߂ÂƂ͖Ȃ̂ŁAzɂ̘pȂVsx܂ވʑΘ_(GTR)̊m؂ɓĂ͂܂Ɗ҂B This can be made more obvious by expressing the proper distance as a function of the coordinate distance. ́AW̊֐ƂČŗL\ƂɂĂ薾炩ɂ邱ƂłB Unfortunately, an approximate analytic formula for large distances is possible only when r is expressed as a function of . cOȂƂɁAr ς̊֐Ƃĕ\鎞Ał͓̉Iߎ\łB This becomes:(55)where is the Euler constant = .5772156649c. ͎ɂȂAŃ ̓IC[萔 = .5772156649cłB @$r(\rho )= \rho \cdot e^{\frac{-R_{s}}{2\rho }}\left \{ 1-\frac{R_{s}}{2}\ln \left ( \frac{2\rho }{R_{s}} \right ) \right \}+\frac{R_{s}}{2}\gamma \; \; \; \; \; \; (55)$ For small distances, both the constant and logarithmic terms are omitted. ߋł́A萔Ƒΐ֐̍͏ȗB The graph of as function of r can be calculated numerically and the result is shown in Fig.2. r ̊֐ɂƂẴς̃Ot͐lIɌvZ邱ƂłAʂ͐}2ɎB The derivation of the light deflection formula and Shapiro delay will be left for future publications. ̘pȎƃVsx̓o͏̘_̂߂ɎcĂB The results of light deflection and Shapiro delay, to the first order of Rs/r, are identical with the corresponding GTR formulas. ̘pȂƃVsx̌ʂ́ARs/r̂Pߎł́AΉʑΘ_(GTR)@ƓłB ================================================================================================ ͓e𗝉ĂȂׂAIȎt܂B ȂĂȂāAxǂł݂Kv悤łB

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