# T_NAKÄ[uO

## ɁuȂFCURVED SPACE-TIMEvǂށBiPj

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 "Remarks on the Equivalence of Inertial and Gravitational Masses and on the Accuracy of Einstein's Theory of Gravity." CURVED SPACE-TIME ɓ܂B [ȂFCURVED SPACE-TIMEiP-Pj]========================================================== To proceed further in investigation of the structure of the new space-time, where the gravitating mass depends on velocity according to Eq.14, it will be necessary to use a more powerful analytical tool. (14) ɏ]đxˑd͎ʂV\̌𑱍s邽߂ɁA苭͂ȕ͓Ic[gp邱ƂKvłB Letfs consider again a central gravitating body placed in the origin of the XYZ coordinate system and find the Lagrangian for the motion of a small test body. xAXYZWň_ɒu鎿ʂlAȎʂ̉^̂߂ɁAOW֐悤B Newtonfs second law and Newtonfs gravitation law lead to the following equation: (22) where, for simplicity, it was considered that the test body moves only along the X direction. j[g̑2@ƃj[g̖L̖͂@́Aȉ̕Ō: @@$\frac{d}{dt}\left ( m_{i}\frac{dx}{dt} \right )= -m_{g}\frac{\kappa M_{s}}{x^{2}}\; \; \; \; \; (22)$ P̂߂ɁAʂXɉĂƍlB The generalization to spherical coordinates and any arbitrary motion is a simple matter, but unnecessarily clutters the notation. ɍW̎gpƔCӂ̉^ւ̈ʉ͂̂ƂP邪AKvȏɕ\L@𕡎GB By using the expressions from Eq.14 and Eq.15 for the mg and mi and by introducing the proper time d = dt (1-v2/c2) , it is possible to write the following two equations instead ofEq.22. (23) (24) mg mi (14)(15)̕\ƌŗL d = dt (1-v2/c2) pāA(22) ̑Ɉȉ2̕ƂłB @@$\frac{d}{d\tau }\left ( m\frac{dx}{d\tau } \right )= -m\frac{\kappa M_{s}}{x^{2}}\; \; \; \; \; (23)$ @@$\left ( \frac{cdt}{d\tau } \right )^{2}-\left ( \frac{dx}{d\tau } \right )^{2}= c^{2}\; \; \; \; \; \; (24)$ The rest mass m can be factored out from Eq.23 and this signifies that the motion is independent of mass. Î~ m (23) OֈƂďoƂłA͉^ʂƗĂ邱ƂB The new mass equivalence principle made this possible and transformed the Newtonfs gravitational law into a form that is LC compatible. Vʓ͂\ɂāAj[g̏d͖@[csόɕςB However, the domain is still a flat space-time as indicated by Eq.24 and in such a flat spacetime it is not possible to construct the desired Lagrangian. A(24) ɂĎ̈͂܂ȎłÂ悤ȕȎł͖]܂OW֐𑢂邱ƂłȂB To proceed further it is necessary to transfer the considerations to a curved space-time. bi߂߂ɂ́Al@Ȃ֎ĂƂKvłB It is interesting to note that the LC of Newtonfs gravitational law clearly demands a curved space-time. [csςȃj[g̏d͖@Ȃ𖾂炩ɗv_ɒӂNƂƂ́A[B =============================================================================================== @$dt= \frac{d\tau }{\sqrt{1-\frac{v^{2}}{c^{2}}}}\; \to\; \frac{d}{dt}= \sqrt{1-\frac{v^{2}}{c^{2}}}\: \frac{d}{d\tau }$ @$m_{i}= \frac{m }{\sqrt{1-\frac{v^{2}}{c^{2}}}}\; ,\;m_{g}= \sqrt{1-\frac{v^{2}}{c^{2}}}\:m$ Ȃ̂ŁA(22) @ $\sqrt{1-\frac{v^{2}}{c^{2}}}\: \frac{d}{d\tau }\left \{ \left ( \frac{m }{\sqrt{1-\frac{v^{2}}{c^{2}}}} \right ) \sqrt{1-\frac{v^{2}}{c^{2}}}\: \frac{dx}{d\tau }\right \}= - \sqrt{1-\frac{v^{2}}{c^{2}}}\:m\frac{\kappa M_{s}}{x^{2}}$ ƂȂA(23) oĂ܂B܂A @$\left ( \frac{cdt}{d\tau } \right )^{2}-\left ( \frac{dx}{d\tau } \right )^{2}= \frac{1}{1-\frac{v^{2}}{c^{2}}}\left \{ c^{2}-\left ( \frac{dx}{dt} \right )^{2} \right \}= c^{2} \frac{1-\frac{v^{2}}{c^{2}}}{1-\frac{v^{2}}{c^{2}}}= c^{2}$@ Ȃ̂ŁA(24) oĂ܂B [ȂFCURVED SPACE-TIMEiP-Qj]========================================================== The first step in generalization is to replace the flat space proper time by the curved space proper time. ʉ̑1Xebv́Aȋ-ŗLԂȂ-ŗLԂƎւ邱ƂłB As a second step it will be assumed that the hypothesis of locality is valid. 2XebvƂāAǏÓƉ肳B This means that the gravity can be locally transformed out by a free fall of the coordinate system with the body and that at any instant the coordinate transformation from the moving to the laboratory coordinate system is LC. ́Ad͂̂̍Wn̎RɂċǏnɕϊłAǂȏuԂłAړnWnւ̍Wω[csςł邱ƂӖB Finally, it will be considered that since the central gravitating mass Ms is stationary and spherically symmetric, the metric of the curved space-time will also be spherically symmetric and will be time independent. ŌɁASɈ͌̎ Ms ܂苅Ώ̂ł邱ƂƁAȂ̌vʂΏ̂ł莞ԂɓƗłƍlB To find the laboratory coordinate system Lagrangian for the motion of a test body in this static gravitational field,the Lagrangian can be considered in the following general form as described in appendix A: (25) where the functions A(x) and B(x) are any arbitrary functions of coordinate x satisfying certain conditions as also described in appendix A. ̐ÓId͏Ŏ̂̉^̎WnOW֐邽߂ɁÃOW֐͕t^ A Œ߂ʌōl邱Ƃł: @@$L=e^{A(x)}\left ( \frac{cdt}{d\tau } \right )^{2}-e^{B(x)}\left ( \frac{dx}{d\tau } \right )^{2}\; \; \; \; \; (25)$ t^ A łLq悤ɁA֐ A(x) B(x) ͓̏󋵂𖞑ĂW x ̔Cӊ֐łB A beautiful theorem that proves this assertion quite generally for all spacetime coordinates can be found in the literature. ʓIɂׂĂ̎Wł̎咣ؖ藝͕ŎƂłB The exponential factors in Eq.25 represent the metric coefficients. (25) ɂwq́Avʂ̌WӖB The Lagrangian in Eq.25 has now enough free coefficients to accommodate Newtonfs gravitational law. (25) ̃OW֐́Aj[g̏d͖@߂̂ɏ\ȎRWB To find the equations of motion and the space-time metric, it is necessary to solve the following set of equations: (26)(27) and the Euler-Lagrange (EL) equations for both the time and the space coordinates that follow from the Lagrangian:(28) ^Ǝvʂ̕t邽߁Aȉ̕QKvB @@$c^{2}=e^{A(x)}\left ( \frac{cdt}{d\tau } \right )^{2}-e^{B(x)}\left ( \frac{dx}{d\tau } \right )^{2}\; \; \; \; \; (26)$ @@$\frac{d^{2}x}{dt^{2}}= -\frac{\kappa M_{s}}{x^{2}}\; \; \; \; \; (27)$ OWA玞W߂邽߂̃IC[-OW(EL)͎̒ʂB @@$\frac{d}{d\tau }\left ( \frac{\partial L}{\partial \frac{dt}{d\tau }} \right )= \frac{\partial L}{\partial t}\; ,\; \; \; \frac{d}{d\tau }\left ( \frac{\partial L}{\partial \frac{dx}{d\tau }} \right )= \frac{\partial L}{\partial x}\; \; \; \; \; (28)$ Eq.26 is the relativistic coordinate constraint and Eq.27 is Eq.23 where the rest mass was factored out. (26) ͑Θ_IȍWŁA(27) (23) ̐Î~ʂqƂĊO֏ôłB More complex force dependence on x is also possible to consider here. xɈˑ蕡Gȗ͂Aōl̂\B For example, a static cosmology with the known radial mass distribution would lead to a different function for the second derivative. Ƃ΁Aǂmꂽʂ̕ˏ󕪕zɂÓIF_́AQ̂߂ɈقȂ֐𓱂ƂɂȂB =============================================================================================== ƂƂΌɂǂ蒅ƂȂ̂łAƔꂽ̂ŁA͂̕ӂŁBB

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