content_image/353965.jpg content_image/353966.jpg content_image/353967.jpg content_image/353968.jpg content_image/353962.jpg content_image/353961.jpg content_image/353964.jpg content_image/353963.jpg content_image/353960.jpg content_image/353959.jpg content_image/353958.jpg
spatial variable r ∗ = r ∗ ( r ) defined through 

then we have 

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with u = u ( t, r ∗ ), v = v ( t, r ∗ ) and F ( r ∗ ) := g$_{00}$ ( r ) /f 2 where r is a function of r $^{∗}$. We want to show that u and v satisfy a wave equation. If we consider the combinations (14)+(15) ± 2(16) we arrive at the following equations 

While taking the square roots of the above equations only those choices of the sign are allowed for which the determinant of the Jacobian of the transformation ˜ x := ( v, u, ϑ, ϕ ) −→ x = ( t, r, ϑ, ϕ ) does not vanish identically. Hence, we require 

Clearly, there are four possible choices of the sign. If we consider for example the case 

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we would obtain ∂$_{t}$u = ∂$_{t}$v, ∂$_{r}$$_{∗}$u = ∂$_{r}$$_{∗}$v which implies det( J ) = 0. The same happens if we consider 

On the other hand the choice 

leads to the equations 

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