Let us consider the General Relativity metric:(adsbygoogle = window.adsbygoogle || []).push({});

[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 begloballyvalidif 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 thephysical context. Parallel-Transport is not so serious an issue in flat spacetime.

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# The General Relativity Metric and Flat Spacetime

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