This is probably a naive question and I'm missing something really simple. The [Schwarzschild solution](https://arxiv.org/abs/physics/9905030) has been constructed in consideration of the following requirements:

1. The field equations $ \frac{\partial \Gamma\_{\mu\nu}^\alpha}{\partial x^\alpha}+\Gamma\_{\mu\beta}^\alpha \Gamma\_{\nu\alpha}^\beta=0. $
2. The coordinate condition $Det(g\_{\mu \nu}) = -1.$
3. All the metrical components are independent of time $x^4$.
4. The equations $g\_{\rho 4}=g\_{4\rho}=0$ hold exactly for $\rho=1,2,3$.
5. The solution is spatially symmetric with respect to the origin of the coordinate system in the sense that one finds again the same solution when $x\_1, x\_2, x\_3$ are subjected to an orthogonal transformation (rotation).
6. The $g\_{\mu\nu}$ vanish at infinity, with the exception of the following four limits different from zero: $$g\_{44}=1,~g\_{11}=g\_{22}=g\_{33}=-1.$$

The problem is to determine a line element with coefficients such that the 1) ... 6) are satisfied, with the exception of the point $x\_1=x\_2=x\_3=0$, the location of a point mass, where the requirements are undefined.

Now, it appears to me that the line element $$ds^2=(dx^4)^2-(dx^1)^2-(dx^2)^2-(dx^3)^2$$ satisfies the requirements 1) ... 6). Then, what condition or requirement prevents the line element from being like above, everywhere in the vicinity of a point mass?