T_NAKÄ[uO

Ɂuړ鎞vFMOVING CLOCKSvǂށBiRj

<<   쐬 F 2016/10/11 00:01   >>

 "Remarks on the Equivalence of Inertial and Gravitational Masses and on the Accuracy of Einstein's Theory of Gravity." MOVING CLOCKS ͒̂ł܂łB [ړ鎞vFMOVING CLOCKSiR|Pj]=========================================================== Up to this point, everything functions well without any problems, so it is possible to proceed with the second clock experiments. ̎_܂ł́AȂ2ڂ̎v𑱍s邱ƂłB The same clock is constructed but without charge Q and –Q, only the mutual gravitational force of the plates now facilitates the platefs attraction. A͓d Q –Q ͖A݂̏d͂m͂𑣐iB It is assumed that the gravitational field intensity, and from this the force of attraction between the plates is :(8)@where is the gravitational constant. d͏ꂪ肳A݊Ԃ͈̈͂ȉ̒ʂBAȂ͏d͒萔B @$F= \frac{4\pi \cdot \kappa \cdot m_{g}^{2}}{A}\; \; \; \; \; \; (8)$ Using this force, the time to collision becomes:(9) ̗͂gƁAՓ˂N܂ł̎Ԃ͈ȉ̒ʂB @$t_{c}^{2}= \frac{m_{i}\cdot a\cdot A}{4\pi \cdot \kappa \cdot m_{g}^{2}}\; \; \; \; \; \; (9)$ This clock rate is the rate seen by the co-moving observer. ̎v݂̍́AvƂƂɈړn̊ϑ҂ϑ̂łB The values of all the parameters in this formula are, of course, the rest reference frame values. 񂱂̎ɓoꂷp[^l͌ŗLnł̒lłB ================================================================================================== (8) o̍lʂ̏d͏ɂƂقړlłAQ{ɂȂĂ̂ɍLʂɈlɕzdׂ̍d̂Q̏ꍇƓł傤B {͔͗LȂ̂ŁAł̂HƋ^ȂłAPʖʐϓ̏d͎ mg/A ̈ f @$f=4\pi \kappa \left ( \frac{m_{g}}{A} \right )^{2}= \frac{4\pi \kappa m_{g}^{2}}{A^{2}}$ ƂȂ܂A F ͂ɖʐ A |̂Ȃ̂ @$F= \frac{4\pi \kappa m_{g}^{2}}{A^{2}}\cdot A= \frac{4\pi \kappa m_{g}^{2}}{A}$ (8) oĂ܂B (9) ͂Q̔݂ a/2 ړƏՓ˂邽߁AxƂ @$\frac{1}{2}a= \frac{1}{2}\alpha t^{2}\; \to \;t^{2}= \frac{a}{\alpha } \; ,\; F= m_{i}\alpha \to \frac{1}{\alpha }= \frac{m_{i}}{F}= \frac{m_{i}A}{4\pi \kappa m_{g}^{2}}$ ܂ @$t^{2} = \frac{m_{i}aA}{4\pi \kappa m_{g}^{2}}$ ƂȂ܂B ǂ̘_̐ɍ킹悤Ș_TŌtĂ̂ŁA玕؂ꂪłB [ړ鎞vFMOVING CLOCKSiR|Qj]=========================================================== The next step, however, presents a problem. ÃXebv́AB Unlike for the transformation of the electric field from the moving to the laboratory coordinate system, there are no linear explicit formulas for the forces dependent on velocity available from the Einsteinfs field equation of General Theory of Relativity (GTR). ĂnƎn̓d̍WϊƂ͈قȂAʑΘ_ (GTR) ɂACV^C瓱鑬xɈˑ̖͂mȐϊȂB However, several authors have performed linearization for a weak gravitational field and similar equations to Maxwellfs equations have been presented in many publications. Ǎ҂͎アd͏̂߂̐A}NXEF̕ւ̗ގ̘_ɎꂽB A relatively recent work on gravitomagnetic effects has been published by M.L. Ruggiero and A. Tartaglia. gravitomagneticʂ̔rIŋ߂̌́AM.L.bWF[A.^[^OAɂĔ\ꂽB By adapting their equations to the static case that is being studied here, the force of attraction for the moving plates can be expressed as(10)@where $\vec{E}_{g}$ and $\vec{B}_{g}$ are the gravito-static and the gravito-magnetic field intensities defined by the following@differential equations(11):representing the mass density and the mass current density respectively. ނ̕ŌĂP[XɓKƁAm߂͎̈͂ŕ\킳: @$\vec{F}= \frac{m_{g}}{m_{i}}\left ( \vec{E}_{g}+\frac{1}{c}\vec{v} \times \vec{B}_{g}\right )\; \; \; \; \; \; (10)$@ ɁAłgravito-d $\vec{E}_{g}$ gravito-ꋭ $\vec{B}_{g}$ ́Aʖx $\rho _{g}$ Ǝʓdx $\vec{j} _{g}$ \ȉ̔ɂĒ: @$\vec{\nabla}\cdot \vec{E}_{g}= -4\pi \cdot \kappa \cdot \rho _{g}\; \; \; \; \; \; (11)$ @$\vec{\nabla}\times \frac{1}{2}\vec{B}_{g}= -\frac{1}{c}\cdot 4\pi \cdot \kappa \cdot \vec{j} _{g}\; \; \; \; \; \; (12)$ By using the Gauss and Stokes formulas, as in the previous case of the electric and magnetic fields, these equations can be easily solved for the mass configuration that is being investigated here and the formula for the time to collision derived. KEXƃXg[NX̒藝pāAdꂨ̃P[X̂悤ɁA̎ʂ̍\̂߂͂̕ŌĂP[XɂĊȒPɉƂłAՓˎԂ̎͂ɗRB This becomes (13) ͎̂悤ɂȂB @$t_{c}^{2}= \frac{\frac{m(rst)_{i}}{\sqrt{1-v^{2}/c^{2}}}\cdot a\cdot A(rst)\sqrt{1-v^{2}/c^{2}}}{4\pi \cdot \kappa \cdot m(rst)_{g}^{2}\cdot \frac{(1-2\cdot v^{2}/c^{2})}{(1-v^{2}/c^{2})}}\; \; \; \; \; \; (13)$ It should be noted that Eq.10 was derived only for small velocities and weak gravitational field. (10)ᑬxƎアd͏œoꂽ_ɗӂKvB These conditions are satisfied in this experiment. ̎ł́ȀB The velocity in the direction of plates' attraction and the plates' mass can be assumed arbitrarily small. Ԃ͕̈̑xƔ̎ʂ͏Ɖ肵B No restriction is necessary for the velocity in the Z direction. Z̑x̂߂ɂ͋K͕KvłȂB From the result derived in Eq.13, it can be concluded that GTR for the weak gravitational fields and small velocities is approximately consistent with the Lorentz coordinate transformation, but Eq.13 is not a good fit for larger velocities. (13) œo錋ʂAアd͏ƒᑬxł̈ʑΘ_ (GTR)[cϊƂقڈvĂ邱Ƃ邪A(13) ͂荂xɑ΂Ă͍v̂ł͂ȂB The well-know factor of two from the left hand side of Eq.12 now appears in one of the velocity factor brackets of Eq.13 and spoils the LC. (12) ӂ 2 Ƃǂm炽q (13) ̊ʂ̒̑xqɌA[cqƂɂȂB @ This is a seldom-discussed problem among several pointed out by Logunov. ́ALogunovOmtɂĎwEꂽA܂c_ĂȂłB For this reason a different approach to the theory of static gravitational field will be followed in this article. ̗RŁAÓId͏̗_ւ̈قȂAv[{eŒǏ]B In the following derivations, it will be assumed that the inertial and the gravitational masses depend on velocity differently according to the following equations: (14)(15) and no velocity dependent gravito-magnetic forces are present. ȉ̓oɂ Aʂŏd͎ʂȉ̕ɂĊeXɈˑƉ肳AxˑȂ gravito-݂͂͑ȂB @$m_{g}= m(rst)_{g}\sqrt{1-\frac{v^{2}}{c^{2}}}\; \; \; \; \; \; (14)$ @$m_{i}= m(rst)_{i}\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\; \; \; \; \; \; (15)$ The Einsteinfs mass equivalence between the inertial and the gravitational masses is thus maintained only at rest. ACV^C̊ʂƏd͎ʂ̓͂̂悤ɐÎ~Ԃł̂ݕۂB By substituting the above equations into Eq.9 the result becomes: (16) (9) ̒L̕ウ邱ƂɂČʂ̂悤ɂȂ: @$t_{c}^{2}= \frac{\frac{m(rst)_{i}}{\sqrt{1-v^{2}/c^{2}}}\cdot a\cdot A(rst)\sqrt{1-v^{2}/c^{2}}}{4\pi \cdot \kappa \cdot m(rst)_{g}^{2}\cdot (1-v^{2}/c^{2})}= \frac{m(rst)_{i}\cdot a\cdot A(rst)}{4\pi \cdot \kappa \cdot m(rst)_{g}^{2}\cdot (1-v^{2}/c^{2})}\; \; \; \; \; \; (16)$ This result now fully follows STR and is LC for all velocities. ̌ʂ́A݊SɓꑊΘ_(STR}ɏ]A[cq̂߂ׂĂ̑̑ΉB As a result it will be possible to use the new equivalence principle to build a covariant scalar metric theory of static gravitational field based on Newtonfs gravitational law. ̌ʁAj[g̏d͖@ɊÂÓId͏̋σXJ[vʗ_\z邽߂ɐVgƂłB In the following sections, it will be also assumed that m(rst)g = m(rst)i = m. ȉ̏͂ł́A$m(rst)_{g}= m(rst)_{i}= m$ ƌ􂷁B The different dependence of gravitational mass on velocity may seem strange and counterintuitive at first,however, if this is an universal dependency, all falling bodies will obey it and no violation of the Galileo free fall experiment will be observed. d͎ʂ́iʂƂ́jقȂ鑬xˑ͍ŏ͊Œςɔ悤Ɍ邪AꂪՓIȈˑȂ΂ׂĂ̗̂͂ɏ]AKI̎RʂŴ͊ώ@Ȃ낤B It is also clear that the free falling bodies will follow the geodesic curves regardless of their gravitational mass change. R̂̏dʎʂ̕ωɊւ炸nǂ邱Ƃ܂łB To the best of the authorfs knowledge no mass equivalence tests have been conducted or designed yet to test the mass equivalence for bodies that are moving at very high relative velocities. ҂̒m̂Ԍł́AőΓIȉ^Ă铙͎̎{ꂽƂv悳ꂽƂb͕ĂȂB There seem to be no experimental support for the Einsteinfs mass equivalence other than for bodies at rest or at small relative velocities. Î~ԂႢΑxԈȊOł́AACV^C̎ʓɑ΂IȎxȂ悤łB ================================================================================================ ł͂Q̋^_܂B P Gravitomagnetism Ƃ̂łB @艺Ƃ܂襘HɓĂ܂̂ŁA~߂Ă܂Ad͏d̃AiW[Ƃė闝_ƕƂ܂B d͏Ód̃AiW[Ƃė̂͗ǂ̂łA͓d[cϊƏoĂ܂ˁB ʑΘ_ꑊΘ_Ő̂Hd͔g͓dg̃AiW[ɂȂ̂HƂ^₪oĂĂ܂̂łB ꂩAΘ_IƂ̂͂܂gȂȂƕĂ܂Ał(15)ɖɎĂ܂ˁB ɋƂ(14)̏d͎ʂ̑xˑłB̍ė~Ǝv܂BɂĂ͕ʋLɂčlĂ݂Ǝv܂B ƏŁA MOVING CLOCKS ̏͂I܂̂ŁA𑱂܂傤B [ړ鎞vFMOVING CLOCKSiR|Rj]=========================================================== There are many consequences of this gravitational mass dependency on velocity, such as no gravitational interaction of photons moving in the same direction, as predicted by one of the quantum theories of gravity. ʎqd͘_1ɂė\悤ɁA(ႦΓɓĂqɏd͑ݍpȂȂ)̏d͎ʂ̑xˑKRIȌʂ݂B All particles moving with the velocities close to the speed of light will have no mutual gravitational interaction including neutrinos and gravitons. ̋߂xׂ̂Ă̗qɁAj[gmƃOrg܂ޏd͑ݍpƂ킯ł͂ȂB The difference between the inertial and the gravitational masses has also been predicted elsewhere, based on the thermodynamic considerations. ʂƏd͎ʂ̈Ⴂ́AM͊wIl@ɊÂ̊ϓ_\ꂽB However, the discussion of these interesting topics is beyond the scope of this article and cannot be addressed here any further. A̋[bɊւc_́A{e͈̔͊OłāAȏケŏqׂ邱ƂłȂB Since this articles focuses only on the perihelion advance and the gravitational red shift, as mentioned in the introduction, the confirmation of the new mass equivalence principle will thus be left to the agreement of the derived results with observations. u͂߂ɁvŏqׂƂA̋L̏œ_͋ߓ_ړƏd͐ԕΈڂȂ̂ŁAVʓ̊mF́Aϑɂhʂ̍ӂɔCB To summarize the main conclusion of this section: it can be stated that as viewed from the laboratory coordinate system, the gravitational mass of the moving body, which moves with a constant velocity v, changes with the velocity such that the product mg mi = m2 remains constant. ̏͂̎Ȍт܂Ƃ߂: PIȑvƂƂɈړ镨̂̏d͎ʂ͑xƂƂɕωāA mg mi = m2 ɂB The topic of the next section will be the confirmation of the new mass equivalence principle by calculating the Mercuryfs perihelion advance. ̘͂b́A̋ߓ_ړvZ邱ƂɂVʓ̊mFłB ================================================================================================ ܂Ȁ͂ł͈قȂ镄̓dׂѓdmՓ˂鎞Ԃ߂Ă܂B ̃AiW[őѓdĂȂmL͂ŏՓ˂鎞Ԃ߂ĂÂȂŊʂƏd͎ʂ̑xˑ̈ႢĂłˁB ̏͂ɍsOɂŎꂽ (14) lĂ݂Ǝv܂A肢l@oȂꍇ́Afɂ̎F߂Ďɍsm܂BB

uOC

NbNċC`悤I
OCăNbN΁ÃuOւ̃Nt܂B
OC

^Cg i{j uO^

 ^Cg {@

e jbNl[^

Rg

 jbNl[ {@
Ɂuړ鎞vFMOVING CLOCKSvǂށBiRj T_NAKÄ[uO/BIGLOBEEFuuO
TCYF