There is a static spherically symmetric perfect fluid solution of the EFE where the energy-momentum tensor is ##diag(\rho,p,p,p)## with ##\rho=b\,\left( 2\,b\,{r}^{2}+3\,a\right) /{\left( 2\,b\,{r}^{2}+a\right) }^{2}## and ##p={b}/({2\,b\,{r}^{2}+a})##. a and b are parameters with b>0 and 0<a<1. On the surface ##r=\sqrt{(1-a)/(2b)}\equiv r_{max}## the PF metric coincides with the Schwarzschild exterior, as long as the Schwarzschild parameter m has the value ##m_s= {\sqrt{1-a}\,\left( 1-a\right) }/( {4\,\sqrt{2b}})##.(adsbygoogle = window.adsbygoogle || []).push({});

Calculating ##M_s## the mass/energy total of the PF

[tex]\begin{align*}

M_s &=\ 4\pi\int_0^{r_{max}} r^2\rho\ dr = 4\pi\left[ \frac{\,b\,{r}^{3}}{2\,b\,{r}^{2}+a} \right]_0^{r_{max}}\\

&= \frac{\sqrt{2}\,\pi\,\sqrt{1-a}\,\left( 1-a\right) }{\sqrt{b}}\\

&= 8\pi\ m_s

\end{align*}

[/tex]

This seems most satisfactory but raises the question - what happened to the pressure terms in the EMT ? It appears that the integral of the energy density accounts for all the exterior vacuum curvature. Is this an anomaly or am I right to be surprised ?

(Actually I was very glad when the integral turned out like this - until the question of the pressure appeared).

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# What is gravitating ?

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