# T_NAKÄ[uO

## ɁuVvʁFNEW SPACE-TIME METRICvǂށBiPj

<<   쐬 F 2016/10/25 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 ǂł܂B [VvʁFNEW SPACE-TIME METRICiPj]=================================================== In order to be consistent when working within the curved space-time domain, it is necessary to use Newtonfs law written in the curved space-time coordinates. Ȃ̈ŖȂ߂ɂ́AȂWŏꂽj[g̖@gp邱ƂKvłB It is necessary to use the proper time and the proper distance in the equation. ł͌ŗLԂƌŗLgƂKvłB It is strange that in Eq.27, which clearly leads to Schwarzschild solution of GTR, only the time was replaced by the proper time, but the coordinate distance was left alone. 炩ɈʑΘ_(GTR)̃VcVg𓱏o (27) ɂāAԂŗLԂƎւꂽAW͒PƂŎcꂽ̂͊łB This peculiarity will be corrected in this section. ُ̈ȓ_͂̏͂ŏCB The relationship between the coordinate and the proper distance is easily found and together with Newtonfs equation substituted into Eq.33. WƌŗL̊֌W͊ȒPɌAj[g̕Ƌ(33)ウB The goal is to obtain the new and correct metric for the space-time of the central gravitating mass. S[́ASɏd͌̐Vvʂ𓾂邱ƂłB It will be assumed that the proper distance (x) is a function of only the coordinate distance x for@the static case that is considered here. ŗL (x) ́Aōl@ÓIP[X̍W x ̊֐łƉ肳B Newtonfs equation written in terms of the proper distance and the proper time is as follows: (38) ŗLƌŗLԂŋLqj[g͈̕ȉ̒ʂłF @$\frac{d^{2}\rho (x)}{d\tau ^{2}}= -\frac{\kappa M_{s}}{\rho ^{2}(x)}\; \; \; \; \; \; (38)$ From Eq.26, the proper distance is defined by the following differential equation: (39) (26) AŗĹAȉ̔ɂĒF (39) @$d\rho (x)= e^{\frac{B(x)}{2}}dx\; \; \; \; \; \; (39)$ By carrying through the differentiation as indicated in Eq.38, the following relation is obtained: (40) (38) Ŏs邱ƂɂāAȉ̊֌W͓F @$\frac{d^{2}\rho (x)}{dx ^{2}}\left ( \frac{dx}{d\tau } \right )^{2}+\frac{d\rho(x) }{dx}\frac{d^{2}x}{d\tau ^{2}}= -\frac{\kappa M_{s}}{\rho ^{2}(x)}\; \; \; \; \; \; (40)$ In the next step, differentiating Eq.39 with respect to x, substituting the result into Eq.40, and using Eq.31,the result, after some rearrangement, becomes: (41) ̃XebvɂẮAx Ɋւ (39) A(40) ɌʂA(31) gāA炩̍ĐAʂ͎ɂȂF (41) @$\frac{dB}{dx}\: c^{2}(e^{B(x)}-1)+2e^{B(x)}\frac{d^{2}x}{d\tau ^{2}}= -\frac{2\kappa M_{s}}{\rho ^{2}(x)}\: e^{\frac{B(x)}{2}}\; \; \; \; \; \; (41)$ From Eq.31 also follows that: (42) (31) Ao @$2e^{B(x)}\frac{d^{2}x}{d\tau&space;^{2}}=c^{2}\frac{dB(x)}{dx}\;&space;\;&space;\;&space;\;&space;\;&space;\;&space;(42)$ Eliminating the second derivative of x from these two equations results in the following: (43) 2̕ x ̂QKƌʂ͎ɂȂF @$\frac{dB(\rho )}{d\rho}e^{B(\rho )}= -\frac{2\kappa M_{s}}{c^{2}\rho ^{2}}\; \; \; \; \; \; (43)$ This equation can now be easily integrated. ́̕AȒPɐϕ邱ƂłB Assuming again that at large distances from the origin the space-time is flat, the result for B, expressed as a function of the proper distance, using the Schwarzschild radius to simplify the notation, becomes: (44) where (45) Ăь_̑傫ꂽŎ󂪕łƉ肵āABiVcVgapČŗL̊֐Ƃĕ\ǰʂ́A\LPꎟƂȂF @$e^{B(\rho )}= 1+\frac{R_{s}}{\rho }\; \; \; \; \; \; (44)$ (45) @$R_{s}= \frac{2\kappa M_{s}}{c^{2} }\; \; \; \; \; \; (45)$ From this result the new metric line element for the spherically symmetric and static gravitational field should be: (46)@where a Radius function R() has been introduced into the metric in accordance with the theorem derived in reference. ̌ʂAΏ̂̐ÓId͏̐Vvʂ̐vf́Aȉ̒ʂłȂ΂ȂȂF @$ds^{2}= \left ( 1+\frac{R_{s}}{\rho (r)} \right )^{-1}c^{2}dt^{2}-\left ( 1+\frac{R_{s}}{\rho (r)} \right )dr^{2}-R^{2}(\rho )(d\vartheta ^{2}+\sin ^{2}\vartheta d\varphi ^{2})\; \; \; \; \; \; \; \; (46)$ ŁAå֐ R() ͊藝ɏ]ČvʂɓꂽB@ Unfortunately, Newtonfs law will not help to determine the Radius function in the metric line element and some other physical reasoning will have to be used to find it. cOȂƂɁAj[g̖@͌vʂ̐vfŔå֐肷̂ɂȂ炸A̕IȐpȂ΂ȂȂB It will be considered that the Radius function, or more precisely the metric coefficient standing by the angular coordinates, can be determined by minimizing the energy of the gravitational field in the space around the gravitating body. d͌̎̋Ԃ̏d͏̃GlM[ŏɂ邱ƂɂAå֐A܂͊pWɂĎ萳mȌvʌW肳ɈႢȂƍl@B This assumption is in a sharp contrast with the Einsteinfs approach, where the Riemannian curvature instead of the field energy is minimized. ̉́ÃGlM[̑Ƀ[}ȗŏACV^C̃Av[Ƃ͉sΏƂłB =============================================================================================== ܂A(26) @$-c^{2}d\tau ^{2}= e^{B(x)}dx^{2}-e^{A(x)}dt^{2}$ Ȃ̂ŁAEӑ񍀂[Ƃ @$d\rho(x) ^{2}= e^{B(x)}dx^{2}\; \; \to \; \; d\rho(x) = e^{B(x)/2}dx$ ƁA(39) oĂ܂B܂ @$\frac{d\rho }{d\tau }= \frac{d\rho }{dx}\frac{dx}{d\tau } \to \frac{d^{2}\rho }{d\tau ^{2}}=\frac{d}{d\tau } \left ( \frac{d\rho }{d\tau } \right )= \frac{d\rho }{dx}\left ( \frac{dx}{d\tau } \right )^{2}+ \frac{d\rho }{dx}\frac{d^{2}x}{d\tau^{2} }$ Ȃ̂ŁA (38) ̍ӂɑƁA(40) ƂȂ܂B ܂A(39) @$\frac{d\rho }{dx}= e^{B/2}\; \to \; \frac{d^{2}\rho }{dx^{2}}= \frac{1}{2}\frac{dB}{dx} e^{B/2}$ Ȃ̂ŁA (31) (40) ̍ӂɑ @$\frac{1}{2}\frac{dB}{dx} e^{B/2}e^{-B}c^{2}(e^{B}-1)+e^{B/2}\frac{d^{2}x}{d\tau ^{2}}= -\frac{\kappa M_{s}}{\rho ^{2}}$ Ȃ̂ŁA @$\frac{1}{2}\frac{dB}{dx} e^{-B/2}c^{2}(e^{B}-1)+e^{B/2}\frac{d^{2}x}{d\tau ^{2}}= -\frac{\kappa M_{s}}{\rho ^{2}}$ ŁAŜ 2eB/2 | (41) oĂ܂B ܂ (31) @$\left ( \frac{dx}{d\tau } \right )^{2}= e^{-B}c^{2}(e^{B}-1) \to \left ( \frac{dx}{d\tau } \right )^{2}= c^{2}(1-e^{-B})$ Ə̂ŁAӂтŔ邱Ƃl܂B܂Aӂ @$\frac{d}{d\tau }\left ( \frac{dx}{d\tau } \right )^{2}= 2 \frac{dx}{d\tau }\frac{d^{2}x}{d\tau ^{2}}$ łAEӂ @$-c^{2}\frac{d(e^{-B})}{d\tau }= c^{2}e^{-B}\frac{dB}{dx}\frac{dx}{d\tau }$ Ȃ̂ŁAŜ dx/d Ŋ (42) oĂ܂B@ (42) (41) ̍ӂɑĐ @$\frac{1}{ e^{\frac{B(x)}{2}} }\frac{dB(x)}{dx}\:e^{B(x)}= -\frac{2\kappa M_{s}}{ c^{2}\rho ^{2}(x)}$ ƂȂ܂A(39) l (43) ƂȂ܂B (43) (45) ӎ @$\frac{dB(\rho )}{d\rho}e^{B(\rho )}= -\frac{R_{s}}{\rho ^{2}}\; \to \; \frac{d e^{B(\rho )}}{d\rho}= -\frac{R_{s}}{\rho ^{2}}$ Ȃ̂ŁAϕ @$e^{B(\rho )}= \frac{R_{s}}{\rho}+K$ łA @$\lim_{\rho \to \infty} e^{B(\rho )}= 1$ Ȃ̂ŁAK = 1 ƂȂ @$e^{B(\rho )}= 1+\frac{R_{s}}{\rho}\; \; \to \; \; e^{A(\rho )}= e^{-B(\rho )}= \left (1+\frac{R_{s}}{\rho} \right )^{-1}$ ƂȂł傤BꂩAvʂ @@$ds^{2}= \left ( 1+\frac{R_{s}}{\rho (r)} \right )^{-1}c^{2}dt^{2}-\left ( 1+\frac{R_{s}}{\rho (r)} \right )dr^{2}-R^{2}(\rho )(d\vartheta ^{2}+\sin ^{2}\vartheta d\varphi ^{2})$ ƂȂ܂B ͂̕ӂŁBB

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