# T_NAKÄ[uO

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 "Remarks on the Equivalence of Inertial and Gravitational Masses and on the Accuracy of Einstein's Theory of Gravity." MERCURYfS PERIHELION ADVANCE ɓ܂B [̋ߓ_ړFMERCURYfS PERIHELION ADVANCEiPj]============================================== It is well known that the perihelion advance is a curved space-time phenomenon and the domain of GTR. ߓ_ړȂ̌ۂłʑΘ_(GTR)̗̈ł邱Ƃ͂悭mĂB The metric for the curved space-time, however, has not been introduced yet. AȂ̌vʂ́A܂ȂB The thought experiments were performed only within the bounds of STR. vĺAꑊΘ_(STR)͈͓̔łsꂽB Nevertheless, it is interesting to see what effect the new mass equivalence principle has on this phenomenon. łAVʓ̌ۂɂǂȉeyڂ邱Ƃ͋[B Considering the Sun with a large mass Ms being placed in the origin of the XYZ coordinate system, neglecting the mass of Mercury relative to the mass of Sun, and choosing the orbital plane the XY plane, it is possible to write the following two component equations that follow directly from Newtonfs second and gravitational laws: (17) (18) 傫Ȏ Ms zXYZWň_ɂƍl邪Azʂɔׂď̎ʂ𖳎AOʂXYʂƂƁAj[g̑Q@Əd͖@璼ڂɎ̂Q̕ƂoB @@$\frac { d }{ dt } \left[ m_{ i }\frac { d }{ dt } \left( r\cos { \varphi } \right) \right] =-\kappa \frac { m_{ g }M_{ s } }{ r^{ 2 } } \cos { \varphi } \;\;\;\;\;\;(17)$ @@$\frac { d }{ dt } \left[ m_{ i }\frac { d }{ dt } \left( r\sin { \varphi } \right) \right] =-\kappa \frac { m_{ g }M_{ s } }{ r^{ 2 } } \sin { \varphi } \;\;\;\;\;\;(18)$ By multiplying Eq.17 with sinϕ and Eq.18 with cosϕ, subtracting the results, and after some simple algebraic manipulations, it is possible to derive the first integral of motion, corresponding to the conservation of angular momentum, in the form: (19) where it was, of course, considered that mi and mg both depend on velocity. (17) cos |A(18) sin |A炩̒Pȑ㐔A^̑ϕ߂邱ƂłAIɂ͊p^ʕۑƈvB @@$m_{i}r^{2}\frac{d\varphi}{dt }= \alpha \; \; \; \; \; \; (19)$@@ A mi mg ƂƂ͗ƂɈˑB The introduced constant is the constant of integration representing the angular momentum. ꂽ萔́Ap^ʂӖϕ萔łB Similarly, after some algebra, it is possible to derive the following equation for r: (20) lɁA኱̑㐔Ar ̂߂Ɉȉ̕oƂł: (20) @@$\frac{1}{m_{i}}\: \frac{dm_{i}}{dt}\: \frac{dr}{dt}+\frac{d^{2}r}{dt^{2}} -r\left ( \frac{d\varphi }{dt} \right )^{2}= -\kappa \:\frac{m_{g}M_{s}}{m_{i}r^{2}} \; \; \; \; \; \; (20)$@ This equation can be simplified, as is typically done, by substituting u for 1/r and by using Eq.19. ɂ (19) g 1/r u ウ邱ƂɂĂ͒̕P邱Ƃł The result becomes: (21) ʂ͎̂悤ɂȂB @@$\frac{d^{2}u}{d\varphi ^{2}}+u= \kappa \: \frac{m_{g}m_{i}M_{s}}{\alpha ^{2}}\ \; \; \; \; \; \; (21)$@ The solution of this equation leads to a classical result, describing elliptical orbits of Newtonian mechanics,when all the masses are considered constant. ׂĂ̎ʂ萔ƍlƁẢ́̕Aj[g͊w̑ȉ~OAÓTIȌʂɎB ============================================================================================== (17) ̍ӁF @$\frac { d }{ dt } \left[ m_{ i }\frac { d (r\cos{\varphi}) }{ dt } \right] = \frac{dm_{i}}{dt}\frac{d(r\cos{\varphi})}{dt}+m_{i}\frac{d^{2}(r\cos{\varphi})}{dt^{2}}$ (18) ̍ӁF @$\frac { d }{ dt } \left[ m_{ i }\frac { d (r\sin{\varphi}) }{ dt } \right] = \frac{dm_{i}}{dt}\frac{d(r\sin{\varphi})}{dt}+m_{i}\frac{d^{2}(r\sin{\varphi})}{dt^{2}}$ łɊeX(17) cos |A(18) sin |ĈƉEӂ͓̂ŁAR[ɂȂł傤B āA̗vfvZĂ܂B @@$\frac { d (r\sin{\varphi}) }{ dt }= \sin{\varphi}\frac{dr}{dt}+r\cos \varphi \frac{d\varphi }{dt}\; ,\; \frac { d (r\cos{\varphi}) }{ dt }= \cos{\varphi}\frac{dr}{dt}-r\sin \varphi \frac{d\varphi }{dt}$ @@$\frac { d^{2} (r\sin{\varphi}) }{ dt^{2} }= \sin{\varphi}\frac{d^{2}r}{dt^{2}}+2r\cos \varphi\frac{dr}{dt} \frac{d\varphi }{dt}-r\sin \varphi \left ( \frac{d\varphi }{dt} \right )^{2}+r\cos \varphi \frac{d^{2}\varphi }{dt^{2}}$ @@$\frac { d^{2} (r\cos{\varphi}) }{ dt^{2} }= \cos{\varphi}\frac{d^{2}r}{dt^{2}}-2r\sin \varphi\frac{dr}{dt} \frac{d\varphi }{dt}-r\cos \varphi \left ( \frac{d\varphi }{dt} \right )^{2}-r\sin \varphi \frac{d^{2}\varphi }{dt^{2}}$ A @@$\cos \varphi \frac { d (r\sin{\varphi}) }{ dt }-\sin \varphi \frac { d (r\cos{\varphi}) }{ dt }= r \frac{d\varphi }{dt}$ @@$\cos \varphi \frac { d^{2} (r\sin{\varphi}) }{ dt^{2} }-\sin \varphi \frac { d^{2} (r\cos{\varphi}) }{ dt^{2} }=2 r \frac{dr }{dt}\frac{d\varphi }{dt}+r\frac{d^{2}\varphi }{dt^{2}}$ Ȃ̂ŁA(18)~sin-(17)~cos vZƁA @@$\frac{dm_{i}}{dt} r \frac{d\varphi }{dt}+m_{i}\left ( 2 r \frac{dr }{dt}\frac{d\varphi }{dt}+r\frac{d^{2}\varphi }{dt^{2}} \right )= 0$ ɂȂ܂BőŜ r |Đ @@$\left ( \frac{dm_{i}}{dt }\right ) r^{2} \frac{d\varphi }{dt}+m_{i}\left ( 2 r \frac{dr }{dt}\right )\frac{d\varphi }{dt}+m_{i}r^{2}\left ( \frac{d^{2}\varphi }{dt^{2}} \right ) = 0$ Ȃ̂ŁAӂ mir2(d/dt) Ƃς t Ŕ̂łAŏIIɂ @@$\frac{d}{dt}\left ( m_{i} r^{2} \frac{d\varphi }{dt} \right )= 0\; \to \; m_{i} r^{2} \frac{d\varphi }{dt}= \alpha$ ƂȂ܂B A @@$\sin \varphi \frac { d (r\sin{\varphi}) }{ dt }+\cos \varphi \frac { d (r\cos{\varphi}) }{ dt }= \frac{dr }{dt}$ @@$\sin \varphi \frac { d^{2} (r\sin{\varphi}) }{ dt^{2} }+\cos \varphi \frac { d ^{2}(r\cos{\varphi}) }{ dt ^{2}}= \frac{d^{2}r}{dt^{2}}-r\left ( \frac{d\varphi }{dt} \right )^{2}$ (18)~cos-(17)~sin vZƁA @@$\frac{dm_{i}}{dt}\frac{dr}{dt}+m_{i} \left \{ \frac{d^{2}r}{dt^{2}}-r\left ( \frac{d\varphi }{dt} \right )^{2} \right \}=-\kappa \frac { m_{ g }M_{ g } }{ r^{ 2 } }$ Ȃ̂ŁAŜ mi Ŋ @@$\frac{1}{m_{i}}\frac{dm_{i}}{dt}\frac{dr}{dt}+ \frac{d^{2}r}{dt^{2}}-r\left ( \frac{d\varphi }{dt} \right )^{2} =-\kappa \frac { m_{ g }M_{ g } }{m_{i} r^{ 2 } }$ (20) oĂ܂B āA(20) mi ŜɊ| @@$\frac{dm_{i}}{dt}\frac{dr}{dt}+ m_{i}\frac{d^{2}r}{dt^{2}}-m_{i}r\left ( \frac{d\varphi }{dt} \right )^{2} =-\kappa \frac { m_{ g }M_{ g } }{ r^{ 2 } }\; \; \; \; \; \; \; (A)$ ܂A(19) @@$\frac{d\varphi }{dt}= \frac{\alpha }{m_{i}r^{2}}\; \; \; \; (B)\; ,\; \frac{dt}{d\varphi }= \frac{m_{i}r^{2} }{\alpha}\; \; \; \; (C)$ łA$u\equiv r^{-1}$ Ƃ @@$\frac{du}{d\varphi }= \frac{dr^{-1}}{d\varphi }=\frac{dr^{-1}}{dt }\frac{dt }{d\varphi }=-\frac{1}{r^{2}}\frac{dr}{dt }\frac{dt }{d\varphi }=-\frac{1}{r^{2}}\frac{m_{i}r^{2}}{\alpha }\frac{dr}{dt }=-\frac{m_{i}}{\alpha }\frac{dr}{dt }$ ƂȂ܂Bꂩ @$\frac{d^{2}u}{d\varphi^{2} }= -\frac{1}{\alpha }\left \{\frac{dm_{i}}{d\varphi } \frac{dr}{dt } +m_{i}\frac{d}{d\varphi }\left ( \frac{dr}{dt } \right )\right \} = -\frac{1}{\alpha }\left \{\frac{dm_{i}}{dt }\frac{dt}{d\varphi } \frac{dr}{dt } +m_{i}\frac{dt}{d\varphi }\frac{d}{dt }\left ( \frac{dr}{dt } \right )\right \}$ @@$= -\frac{1}{\alpha }\frac{dt}{d\varphi } \left ( \frac{dm_{i}}{dt } \frac{dr}{dt } +m_{i}\frac{d^{2}r}{dt^{2} } \right )= -\frac{m_{i}r^{2}}{\alpha^{2} }\left ( \frac{dm_{i}}{dt } \frac{dr}{dt } +m_{i}\frac{d^{2}r}{dt^{2} } \right )$ łB܂@@@@@ @$u= \frac{1}{r}= -\frac{m_{i}r^{2}}{\alpha^{2} }\left (- \frac{\alpha ^{2}}{m_{i}r^{3}} \right )= -\frac{m_{i}r^{2}}{\alpha^{2} }\left \{ -m_{i}r\left ( \frac{\alpha }{m_{i}r^{2}} \right )^{2} \right \}= -\frac{m_{i}r^{2}}{\alpha^{2} }\left \{ -m_{i}r\left ( \frac{d\varphi }{dt} \right )^{2} \right \}$ @$\frac{d^{2}u}{d\varphi^{2} }+u= -\frac{m_{i}r^{2}}{\alpha^{2} }\left \{ \frac{dm_{i}}{dt } \frac{dr}{dt } +m_{i}\frac{d^{2}r}{dt^{2} }-m_{i}r\left ( \frac{d\varphi }{dt} \right )^{2} \right \}$ ƂȂ܂BŉEӂ { } ̒ (A) ̍ӂƓȂ̂ŁA (A) ̉EӂŒu @$\frac{d^{2}u}{d\varphi^{2} }+u= -\frac{m_{i}r^{2}}{\alpha^{2} }\left ( -\kappa \frac{m_{g}M_{s}}{r^{2}} \right )= \kappa \frac{m_{i}m_{g}M_{s}}{\alpha^{2} }$ ƂȂ (21) o܂B [̋ߓ_ړFMERCURYfS PERIHELION ADVANCEiQj]============================================= There have been attempts in the past to calculate the Mercuryfs perihelion advance based strictly on STR assuming identical relativistic mass corrections for both the inertial and gravitational masses. ߋɁAʂƏd͎ʂƂɓ̑Θ_Iʂł̕␳sꑊΘ_(STR)ɊÂ̋ߓ_ړɌvZ鎎݂B However, the calculations yielded only a fraction of the observed value. ǍvZ͊ϑl킸ȕ^łB This clearly indicates a problem but further attempts to resolve this discrepancy were unfortunately abandoned in favor of the GTR solution. ͖𖾂炩ɎA̐HႢXȂ鎎݂́AcOȂƂɁAʑΘ_̉̕I邱ƂŒfOꂽB STR should provide either a correct result or a zero result. ꑊΘ_(STR)́Aʂ[񋟂Ȃ΂ȂȂB A partial agreement is not reasonable or acceptable. Iȍӂ͍IłȂAełȂ̂łB However, when the new assumption about the velocity dependence of the gravitational mass is used, the perihelion advance is zero. Ad͎ʂ̑xˑ̐V肪̗pƁAߓ_ړ̓[łB The right hand side of Eq.21 becomes a constant again as in the Newtonian case. j[gEP[Xł́A(21) ̉Eӂ͍Ăђ萔ɂȂB This is a very encouraging result clearly indicating that the assumption about the gravitational mass dependency on velocity is correct. ́Ad͎ʂ̑xˑɂẲ肪Ƃ𖾂炩ɎĂɗL͂ȌʂB Since no curved space-time was yet assumed the advance of the perihelion must be correctly calculated as zero. ܂ŋȂ肵Ȃ̂ŁAߓ_ړ͐[ƌvZȂ΂ȂȂB This result now becomes consistent with STR. ̌ʂ́AݓꑊΘ_(STR)ƈvĂB The curved space-time concept is addressed in the next section. Ȃ̊TÓA͂ŏqׂB =============================================================================================== @$m_{g}\cdot m_{i}= \left \{m(rst)_{g} \sqrt{1-\frac{v^{2}}{c^{2}}} \right \}\left \{m(rst)_{i} \frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}} \right \}= m(rst)_{g}m(rst)_{i}$ Ȃ̂ŁA(21) ̉Eӂ͒萔ɂȂł傤Bāuߓ_ړɂĂ炢iQjvlƋߓ_ړ͔Ȃ̂m܂ˁB ̏͂̌_肭oȂ̂łA^Ȏł̑ȉ~OꑊΘ_ōlł͋ߓ_ړ͏oĂȂƌĂ悤ɕ܂B

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