Simplifying The Components
On each hypersurface of constant, constant and constant (i.e., on each radial line), should only depend on (by spherical symmetry). Hence is a function of a single variable:
A similar argument applied to shows that:
On the hypersurfaces of constant and constant, it is required that the metric be that of a 2-sphere:
Choosing one of these hypersurfaces (the one with radius, say), the metric components restricted to this hypersurface (which we denote by and ) should be unchanged under rotations through and (again, by spherical symmetry). Comparing the forms of the metric on this hypersurface gives:
which immediately yields:
- and
But this is required to hold on each hypersurface; hence,
- and
Thus, the metric can be put in the form:
with and as yet undetermined functions of . Note that if or is equal to zero at some point, the metric would be singular at that point.
Read more about this topic: Deriving The Schwarzschild Solution
Famous quotes containing the word components:
“Hence, a generative grammar must be a system of rules that can iterate to generate an indefinitely large number of structures. This system of rules can be analyzed into the three major components of a generative grammar: the syntactic, phonological, and semantic components.”
—Noam Chomsky (b. 1928)