- #1

Anamitra

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Let us consider the General Relativity metric:

[tex]{ds}^{2}{=}{g}_{00}{dt}^{2}{-}{g}_{11}{{dx}_{1}}^{2}{-}{g}_{22}{{dx}_{2}}^{2}{-}{{g}_{33}}{{dx}_{3}}^{2}[/tex] ---------------- (1)

Using the substitutions:

[tex]{dT}{=}\sqrt{{g}_{00}}{dt}[/tex]

[tex]{dX}_{1}{=}\sqrt{{g}_{11}}{dx}_{1}[/tex]

[tex]{dX}_{2}{=}\sqrt{{g}_{22}}{dx}_{2}[/tex]

[tex]{dX}_{3}{=}\sqrt{{g}_{33}}{dx}_{3}[/tex]

We have,

[tex]{ds}^{2}{=}{dT}^{2}{-}{{dX}_{1}}^{2}{-}{{dX}_{2}}^{2}{-}{{dX}_{3}}^{2}[/tex] ------------ (2)

The above metric corresponds to flat spacetime.

Now let us consider the following integrals:

[tex]{T}{-}{T}{0}{=}\int\sqrt{{g}_{00}}{dt}[/tex]

[along lines for which coordinate values of x1,x2 and x3 are constant.

[tex]{X}_{1}{-}{(}{X}_{1}{)}_{0}{=}\int\sqrt{{g}_{11}}{dx}_{1}[/tex]

[x2,x3 and t are held constant for the above integral]

[tex]{X}_{2}{-}{(}{X}_{2}{)}_{0}{=}\int\sqrt{{g}_{22}}{dx}_{2}[/tex]

[x1,x3 and t are held constant for the above integral]

[tex]{X}_{3}{-}{(}{X}_{3}{)}_{0}{=}\int\sqrt{{g}_{33}}{dx}_{3}[/tex]

[x1,x2 and t are held constant for the above evaluation]

[The previous four integral on the RHS are definite integrals having limits between t0 and t1,x1(0) and x1,x2(0) and x2,x3(0) and x3]

We are simply using physical distances between the coordinate labels to get our new coordinate system.

The flat spacetime metric given by relation (2) seems to be

[tex]{ds}^{2}{=}{g}_{00}{dt}^{2}{-}{g}_{11}{{dx}_{1}}^{2}{-}{g}_{22}{{dx}_{2}}^{2}{-}{{g}_{33}}{{dx}_{3}}^{2}[/tex] ---------------- (1)

Using the substitutions:

[tex]{dT}{=}\sqrt{{g}_{00}}{dt}[/tex]

[tex]{dX}_{1}{=}\sqrt{{g}_{11}}{dx}_{1}[/tex]

[tex]{dX}_{2}{=}\sqrt{{g}_{22}}{dx}_{2}[/tex]

[tex]{dX}_{3}{=}\sqrt{{g}_{33}}{dx}_{3}[/tex]

We have,

[tex]{ds}^{2}{=}{dT}^{2}{-}{{dX}_{1}}^{2}{-}{{dX}_{2}}^{2}{-}{{dX}_{3}}^{2}[/tex] ------------ (2)

The above metric corresponds to flat spacetime.

Now let us consider the following integrals:

[tex]{T}{-}{T}{0}{=}\int\sqrt{{g}_{00}}{dt}[/tex]

[along lines for which coordinate values of x1,x2 and x3 are constant.

[tex]{X}_{1}{-}{(}{X}_{1}{)}_{0}{=}\int\sqrt{{g}_{11}}{dx}_{1}[/tex]

[x2,x3 and t are held constant for the above integral]

[tex]{X}_{2}{-}{(}{X}_{2}{)}_{0}{=}\int\sqrt{{g}_{22}}{dx}_{2}[/tex]

[x1,x3 and t are held constant for the above integral]

[tex]{X}_{3}{-}{(}{X}_{3}{)}_{0}{=}\int\sqrt{{g}_{33}}{dx}_{3}[/tex]

[x1,x2 and t are held constant for the above evaluation]

[The previous four integral on the RHS are definite integrals having limits between t0 and t1,x1(0) and x1,x2(0) and x2,x3(0) and x3]

We are simply using physical distances between the coordinate labels to get our new coordinate system.

The flat spacetime metric given by relation (2) seems to be

*globally*valid*if the above integrals exist*.We may describle spacetime globally with the variables T,X1,X2 and X3 having the metric equation(2) Non-local velocities[in cosmology or elsewhere] should not be a problem since we have a flat spacetime in the*physical context*. Parallel-Transport is not so serious an issue in flat spacetime.
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