Solutions in the Vacuum Regions

Within either vacuum region the consistency condition in Equation (7) becomes a mere identity, so that we are left with only two equations, in which we replace both $\rho(r)$ and $P(r)$ by zero,

$$1-2\left[r\lambda'(r)\right] = \,{\rm e}^{2\lambda(r)},$$ (13)

$$1+2\left[r\nu'(r)\right] = \,{\rm e}^{2\lambda(r)}.$$ (14)

This immediately implies that $\lambda'(r)+\nu'(r)=0$, and hence that $\lambda(r)+\nu(r)=A$, where $A$ is a dimensionless integration constant. The first of these two equations involves only $\lambda(r)$, and can also be written as

  $$\left[r\,{\rm e}^{-2\lambda(r)}\right]' = 1,$$ (15)

which can be immediately integrated to

  $$\,{\rm e}^{-2\lambda(r)} = 1-\frac{R}{r},$$ (16)

where $R$ is an integration constant with dimensions of length.

We must now discriminate between the inner and outer vacuum regions. In the outer vacuum region we must get flat space at radial infinity, which requires that both $\lambda(r)$ and $\nu(r)$ go to zero for $r\to\infty$. This in turn implies that $A=0$ in the outer vacuum region, thus leading to $\nu(r)=-\lambda(r)$. As is well known, the condition that the Newtonian limit be realized at radial infinity requires that $R=r_{M}$, the Schwarzschild radius $r_{M}=2MG/c^{2}$ associated to the asymptotic gravitational mass $M$ of the system. Thus we arrive at the time-honored Schwarzschild solution [#!Schwarzschild!#,#!Wald!#] in the outer vacuum region,

$$\lambda_{s}(r) = -\, \frac{1}{2}\, \ln\!\left(\frac{r-r_{M}}{r}\right),$$ (17)

$$\nu_{s}(r) = \frac{1}{2}\, \ln\!\left(\frac{r-r_{M}}{r}\right),$$ (18)

where the subscript $s$ denotes the outer vacuum region. Note that there is a limitation on the values of the parameters $r_{2}$ and $r_{M}$ describing the distribution of matter, because these expressions have a singular behavior at $r=r_{M}$. We must have $r_{M}<r_{2}$ to ensure that there is no event horizon formed outside the distribution of matter.

In the inner vacuum region there are no asymptotic conditions to be applied, and thus the integration constants $A$ and $R$ will have to be left undetermined, to be determined later on via the boundary conditions at the interfaces between the vacuum and the matter, as we come in from radial infinity towards the origin. For convenience we will put $R=-r_{\mu}$, and write the solution in the inner vacuum region as

$$\lambda_{i}(r) = -\, \frac{1}{2}\, \ln\!\left(\frac{r+r_{\mu}}{r}\right),$$ (19)

$$\nu_{i}(r) = A + \frac{1}{2}\, \ln\!\left(\frac{r+r_{\mu}}{r}\right),$$ (20)

where the subscript $i$ denotes the inner vacuum region. Note that the value of $r_{\mu}$ determines the singularity structure of this solution within the inner vacuum region. If $r_{\mu}<0$ then there is a singularity at the strictly positive radial position $r=-r_{\mu}$, corresponding to the formation of an event horizon at that position. If $r_{\mu}=0$ then there are no singularities at all within this region. If $r_{\mu}>0$ then there is only one point of singularity, located at the origin $r=0$. We will show later on that we do indeed have that $r_{\mu}>0$.

We therefore have the complete analytical solutions in the inner and outer vacuum regions, which contain one input parameter of the problem, the mass $M$ associated to the Schwarzschild radius $r_{M}$, and two integration constants still to be determined, $A$ and $r_{\mu}$.