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## ɁuVvʂ̌ʁFCONSEQUENCES OF THE NEW METRICvǂށBiRj

<<   쐬 F 2016/11/09 00:01   >>

 "Remarks on the Equivalence of Inertial and Gravitational Masses and on the Accuracy of Einstein's Theory of Gravity." CONSEQUENCES OF THE NEW METRIC 𑱂܂B [Vvʂ̌ʁFCONSEQUENCES OF THE NEW METRICiRj]======================================= The last phenomenon that will be addressed in this article is the Mercuryfs perihelion advance. {eőΏŌ̌ۂ́A̋ߓ_ړłB The perihelion advance is one of the more sensitive tests for the metric, since this phenomenon is integrating. ߓ_ړƂۂ͂܂Ƃ߂Ă̂ŁA͌vʂ̂芴xeXĝЂƂłB The small single orbit perihelion advances accumulate over the long time periods and the resulting accumulation can thus be readily detected. Ȉ̋Oߓ_ړ͒Ԃ̒~ςAʂƂĂɌ邱ƂłB To calculate the perihelion advance from the metric, it is customary to use Lagrangian derived in Eq.54. vʂߓ_ړvZ邽߂ɁA (54) 瓱o郉OW֐ĝłB From the Lagrangian it is then simple to find EL equations of motion and their first integrals. OW֐A^̃IC[OWFƑϕ͊ȒPɋ܂B Since the motion of the planet is periodic it is no problem that the Lagrangian in Eq.54 is expressed in proper distances. f̉^Ił̂ŁA (54) ̃OW֐ŗLŕ\邱Ƃ͖ł͂ȂB The same result would be obtained if the proper distances were transformed to coordinate distances. ŗLWɕϊȂ΁Aʂ͓ł낤B The first integrals are therefore as follows: Pϕ͎̂悤ɂȂBF @$R^{2}(\rho )\frac{d\varphi }{d\tau }= \alpha \; \; \; \; \; \; \; (62)$ @$\frac{dt}{d\tau }= e^{B(\rho )} \; \; \; \; \; \; \; (63)$ @$\left ( \frac{d\rho }{d\tau } \right )^{2}=k+c^{2} e^{B(\rho )}-\frac{\alpha ^{2}}{R^{2}(\rho )} \; \; \; \; \; \; \; (64)$ where and k are the constants of integration (k = -c2). k ͐ϕ萔(k = -c2)B It is important to note that Eq.64 has a universal validity regardless of the coordinate system used. gWnɊ֌WȂA (64) ʓIȑÓ_ɒڂ邱ƂdvłB This follows directly from Eq.38 that also does not depend on the selected coordinates. ́A܂AI΂ꂽWɈˑȂ (38) AڃtH[B By using the customary substitution u = 1/ and by eliminating from Eq.62 and 64, the equations can be reduced to a single equation for u as function of ϕ as follows: (65) KIȑp u = 1/ pāA (62) Ǝ (64) ƂɂāAȉ̒ʂ ϕ ɑΉ u ̂߂Ɉ̕ɂȂF @$\left ( \frac{du }{d\varphi } \right )^{2}e^{2R_{s}u}=\frac{k}{\alpha^{2} }+\frac{c^{2}}{\alpha ^{2}} e^{R_{s}u}-u^{2} e^{R_{s}u} \; \; \; \; \; \; \; (65)$ Dividing Eq.65 by factor e2Rsu , expanding each exponential into the power series keeping only the first two terms, and differentiating the result with respect to ϕ, the equation becomes: (66) q e2Rsu Ŏ (65)āAeX̎w֐Ƒ񍀂ۂĂ鋉ɓWJāAϕ ɂƈȉ̂悤ɂȂBF @$\frac{d^{2}u }{d\varphi^{2}}+u= \frac{R_{s}c^{2}}{2\alpha ^{2}}+\frac{3}{2}R_{s}u^{2} \; \; \; \; \; \; \; (66)$ This is the standard form of equation describing the perihelion advance. ́Aߓ_ړĂ̕WłB The advance is calculated to be: (67) ړ́Aȉ̒ʂłƌvZF @$\Delta \varphi \approx \frac{3}{2}\pi R_{s}\left ( \frac{1}{R_{1}}+\frac{1}{R_{2}} \right ) \; \; \; \; \; \; \; (67)$@ where R1 and R2 are the perihelion and aphelion distances respectively. R1 R2 ꂼߓ_Ɖ_łB This formula gives the standard value commonly recognized today for the Mercuryfs perihelion advance due to gravitationally induced space-time curvature, which is equal to ϕ = 42.993h per century. ̎́Ad͂ɂ鎞pȂ琅̋ߓ_ړɍʂɔF߂li1Iɂ ϕ=42.993"j^B ===============================================================================================

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