Let us consider the metric given by the following line element:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]{ds}^{2}{=}{dr}^{2}{+}{r}^{2}{(}{d}{\theta}^{2}{+}{Sin}^{2}{(}{\theta}{)}{d}{\phi}^{2}{)}[/tex]

For a particular value of r we have two independent coordinates in the embedded surface--and the surface has been embedded in a 3D space.For several values of r we have several 2D spheres embedded in the same 3D space.

Now lets take Schwarzschild's Metric:

[tex]{ds}^{2}{=}{(}{1}{-}\frac{2m}{r}{)}{dt}^{2}{-}{(}{1}{-}\frac{2m}{r}{)}^{-1}{dr}^{2}{-}{r}^{2}{(}{d}{\theta}^{2}{+}{Sin}^{2}{(}{\theta}{)}{d}{\phi}^{2}{)}[/tex]

For each value of "t" we have a 3D surface embedded in a 4D space having the coordinates(t,r,theta,phi)

For different values of t we have several 3D surfaces [time slices each having a constant value of t] embedded in 4D space. In an interval of time the observer passes though a multitude of such surfaces in the same 4D space.There is no need at all to have a 5D space for the purpose of embedding in General Relativity.

The line element in GR has four coordinates, in general ---one relating to time and three relating to space.

A surface constraint like t= constant or r= const reduces the number of independent variables to three. If we want to embed these surfaces 4 dimensions are enough.

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# 5D Space,Higher Dimensions etc.

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