Tensors and Relativity: Chapter 8

The line element

The line element

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We shall try and introduce coordinates which are appropriate to the problem. Since our choice of coordinates always lead to requirements on the metric functions, we must proceed carefully in order not to lose solutions by making the restrictions too strong.

Spherical symmetry evidently signifies that in the three dimensional space T= constant, all radial coordinates are equivalent and no perpendicular direction is singled out; in spherical polar coordinates we have [ for constant T ]

equation979

The most general ansatz for the line element in the full four- dimensional spacetime is therefore

eqnarray981

We can simplify this line element further using the coordinate transformation

equation983

so that

equation985

and

equation987

Using these results we can be bring the line element into the form

eqnarray989

Notice that this already contains the usual two dimensional spherical surface element. A further transformation

equation991

eliminates the undesired non- diagonal terms. Thus we arrive at the Schwartzschild form of the line element of a spherically symmetric non- rotating body:

equation993