# T_NAKÄ[uO

## ɁuVvʂ̌ʁFCONSEQUENCES OF THE NEW METRICvǂށBiQj

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 "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 METRICiQ-Pj]===================================== The speed of light is obtained from the new metric by setting ds = 0. Vvʂ ds = 0 ƒuƂŌB For the radial direction this (58) a͂̂ƂA @$\frac{dr}{dt}=c\cdot e^{-R_{s}/\rho (r)}\; \; \; \; \; \; \; (58)$ As the photon approaches the origin of coordinates its speed is gradually reduced until the photon finally stops. qW_ɐڋ߂čŌɎ~܂܂ŁȂx͏XɌĂB@ The oscillating electric and magnetic fields become DC fields. UĂd͒ɂȂB From this result it is also apparent that the particles created in the region within the Schwarzschild radius with very high velocity can escape. ̌ʂAɍŃVocVgä̗Őꂽq邱Ƃł邱Ƃ炩łB This result might represent a possible theoretical support for the observation of particle jets emanating from the centers of many galaxies. ̌ʂ́A̋͂̒S˂闱qWFbg̊ϑ̂߂̉\ȗ_IȃT|[g\ƂB =============================================================================================== (53) œaɌ肵 ds = 0 ƒu @$e^{-R_{s}/\rho }(cdt)^{2}-e^{R_{s}/\rho }dr^{2}= 0\; \to \; c^{2}e^{-2R_{s}/\rho }-\left ( \frac{dr}{dt} \right )^{2}=0$ @$\left ( \frac{dr}{dt} \right )^{2}=c^{2}e^{-2R_{s}/\rho }$ Ȃ̂ (58) oĂ܂B [Vvʂ̌ʁFCONSEQUENCES OF THE NEW METRICiQ-Qj]=================================== It is also interesting to calculate from the new metric the Newtonian acceleration expressed in coordinate distances. Wŕ\ꂽVvʂ̃j[gxvZ邱Ƃ[B This can be accomplished directly from Eq.38 when it is rewritten using the new metric. ͐VȌvʂgpďꂽƂA (38) 璼ڍsƂłB This becomes: (59) ͎̂ƂɂȂB @$\frac{d^{2}\rho }{d\tau ^{2}}= \frac{c^{2}}{2}\cdot \frac{d}{d\rho }e^{B(\rho )}\; \; \; \; \; \; \; \; (59)$ By replacing the proper distance and the proper time with the coordinate distance and the coordinate time the result becomes: (60) ŗLƌŗLԂWƍWɒuƁǍʂ͎̂ƂFi60j @$\frac{d^{2}r}{dt^{2}}= \frac{c^{2}{B}'(r)}{2}\left ( 3 e^{-3B(r)}-2 e^{-2B(r)}\right )\; \; \; \; \; \; \; \; (60)$ The second term in Eq.60 represents a repulsive force that dominates at small distances. (60)̑񍀂́AߋŎxzIȔ͂\ĂB By substituting for the metric coefficient into Eq.60 the result becomes: (61) (60) ɌvʌWu邱ƂɂāAʂ͎̂悤ɂȂB @$\frac{d^{2}r}{dt^{2}}= -\frac{c^{2}R_{s}}{2\rho ^{2}(r)}\cdot \frac{3e^{\frac{-R_{s}}{\rho (r)}}-2}{e^{\frac{3}{2}\frac{R_{s}}{\rho (r)}}}\; \; \; \; \; \; \; \; (61)$ As r approaches the origin of coordinates the repulsive acceleration approaches to zero. r W_ɋ߂ÂƔx̓[ɋ߂ÂB At large distances, r Rs , the expression reverts back to the standard Newtonian attractive acceleration. r Rs Ƃł́AWj[g͉xɖ߂B ============================================================================================== (38) @$\frac{d^{2}\rho }{d\tau ^{2}}=- \frac{\kappa M_{s}}{\rho ^{2}}$ (43) @$\frac{dB}{d\rho }\cdot e^{B}= -\frac{2\kappa M_{s}}{c^{2}\rho ^{2}}$ ̎ @$\frac{d}{d\rho } e^{B}= -\frac{2}{c^{2}}\cdot \frac{\kappa M_{s}}{\rho ^{2}}= -\frac{2}{c^{2}}\cdot\frac{d^{2}\rho }{d\tau ^{2}}$ ƕόł邽߁A (59) oĂ܂B (60) ̓ô͍Ƃ땪Ă܂BBŔǉƎv܂B (39) @$\frac{d\rho }{dr}= e^{\frac{B(r)}{2}}$ łA @$B(r)= \frac{R_{s}}{\rho (r)}$ Ȃ̂ @${B}'(r)= \frac{dB(r)}{dr }= \frac{dB}{d\rho }\frac{d\rho }{dx}= -\frac{R_{s}}{\rho ^{2}(r)}e^{\frac{B(r)}{2}}$ ƂȂ܂BāA (60) ̉Eӂ @$-\frac{c^{2}R_{s}}{2\rho ^{2}(r)}e^{\frac{B(r)}{2}}\left ( 3 e^{-3B(r)}-2 e^{-2B(r)}\right )= -\frac{c^{2}R_{s}}{2\rho ^{2}(r)}e^{-\frac{3B(r)}{2}}\left ( 3 e^{-B(r)}-2\right )$ @@$= -\frac{c^{2}R_{s}}{2\rho ^{2}(r)}\frac{3 e^{-B(r)}-2}{e^{-\frac{3B(r)}{2}}}$ ŁA (61) o܂B ͂̕ӂŁBB @

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