7.3. BLACK HOLES 435
2) black holes are innate: they are neither born to explosion ofcosmic objects, nor born
to gravitational collapsing; and3) black holes are filled and incompressible, and if the matter field is non-homogeneously
distributed in a black hole, then there must be sub-blackholes in the interior of the
black hole.We prove this theorem in three steps as follows.
Step 1. Closedness of black holes.First, it is classical that all matter, including photons,
cannot escape from a black hole when they are within the Schwarzschild radius.
Step 2.Now we need to show that all external energy cannot enter intothe interior of a
black hole. By the energy-momentum conservation, we have
(7.3.9)
∂E
∂ τ+divP= 0 ,whereEandPare the energy and momentum densities. Take the volume integral of (7.3.9)
onB={x∈R^3 |Rs<|x|<R 1 }:
(7.3.10)
∫B[
∂E
∂ τ+divP]
dΩ= 0 , dΩ=√
gdrdθdφ,where divPis as in (7.1.60), and
g=det(gij) =g 11 g 22 g 33 =αr^4 sin^2 θ, α=(
1 −
2 MG
c^2 r)− 1
.
LetEbe the total energy inB, then the Gauss formula, we have
∫B∂E
∂ τdΩ=d
dtE,
∫BdivPdΩ=∫SR 1√
α(R 1 )PrdSR 1 −lim
r→Rs∫Sr√
αPrdSr.HereSr={x∈R^3 | |x|=r}. In view of (7.3.10) we deduce that
(7.3.11)
dE
dt
=lim
r→Rs∫Sr√
αPrdSr−√
α(R 1 )∫SR 1PrdSR 1.The equality (7.3.11) can be rewritten as(7.3.12) lim
r→Rs
∫SrPrdSr=lim
r→Rs1
√
α(r)[
dE
dt+
√
α(R 1 )∫SR 1PrdSR 1]
= 0.
This together with no escaping of particles from the interior of the black hole shows that
lim
r→R+sPr= 0.