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Black Holes and Inflation.

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~
~~~
Annaleri der Physik. 7. Folge, Band 45, Heft 4, 1988, S. 265-270
VEB J. A, Bnrth, Leipzig
Black Holes and Inflation
By P. F. GONZBLEZ-D~AZ
Institut,o de Optica, C.S.I.C.,
Bfadrid, Spain
A b s t r a c t . A model for black hole collapse a i d evaporation in which the black hole is supposed
to be an excited state of one of the Planck black holes pervading the structure of spacetime is diseussed. By assuming a Coleman-W'einberg gravitational effertive potential for a scalar field inside the
collapse matter, it is shown that the black hole state cannot be attained neither through bubble
tunneling nor by the rolling down of the field.
Schwarze LGcher und Inflation
I n h a l t s u b e r s i c h t . Ein Modell des Collapses iind der Verdampfung schwarzer Lijcher wird
diskutiert,, wobei angenommen wird, dalj schwarze Locher angeregte Zustande der Planckschen
schwarzcn Locher sind, verteilt iiber die Struktur der Ranm-Zeit. Unter der Annahme eines effektiven Gravitationspotentials nach Coleman-Weinberg als skalares Feld innerhalb der collabierten
Mnterie kann gezeigt werden, dalJ der ,,schwarze Loch"-Zustand weder durch ,,bubble tunneling"
noch durch ,,rolling down" des Fcldes angenommcn werden kann.
There are some indications suggesting the existence of a deep connection between
the process of black hole formation and evaporation [l] and the grand unification inflationary mechanism assumed to have occurred in the first stages of the evolution of
the universe [2]. Thus, Hawking and Moss [3] pointed out that, in a very similar way
to as the gravitational collaps rapidly approaches a stationary black hole state which
is independent of the metric of the initial collapsing body, the early universe must have
approached rapidly a de Sitter state which is also independent of the initial conditions.
Whereby, these authors assumed the existence of a cosmological "no heir" theorem.
It has been also claimed [4] that the re-heating of the inflationary universe [5] in the
broken symmetry phase can be considered as a process which is equivalent t o the thermal
emission process from a black hole. The main difficulty with this view is that the first
thermalization process must have occurred in a time comparatively much shorter than
that is inverted by a given black hole in evaporating altogether.
The main subject of this letter is to analyze some implications of a new model for
black hole evaporation [6] in relation with the black hole-inflation analogy. I11 this
model, evaporation takes place in a time short enough t o convert the black hole into
a virtual object [7]. This letter contains the suggestion that the conversion of observable energy into vacuum energy (black hole) is carried out via a Coleman-Weinberg
potential whose parameters forbid the phase transition t o occur.
The starting point of our analysis is not new. Consider [8] the Lagrangian of a scalar
field p propagating in a background spacetime with curvature R in the case that the
266
Ann. Physik Leipzig 45 (1988) 4
scalar field is conformally coupled. The action integral for that theory is [8]:
[
X = J ~ Z ~ z ( - g ) l (16nG)-l
/~
R
- p2R/12
+1 8,a.p
+1
2
-
A0v4], (1)
with &
> 0. I n order to preserve the equivalence principle, a change in the quartic-self1
,
interaction was introduced [8] so that At =A, - T,uiM*-z, R = -6 PO2 M * - 2 ~ 2where
M* = ( 3 / 4 3 ~ G )is~ the
/ ~ Planck mass. The key point now is to realize that when the involved matter does not undergo any electroweak or strong interaction, but it is only acted
upon by gravitation (such as it is thought to occur in the latest stages of black hole
1
collapse), the matter potential VE(p) = - -p&,~'
A,*q4becomes:
2
+
i. e . a true gravitational matter potential.
It is known [9] that in a theory with conformal coupling -Rp2//i2, the effective
gravitational constant is given by
G-1
- G-1 - 4
eff TC(G(0)' - cT2)/3,
(3)
for the temperature-dependent version of theory (1).I n Eqn. ( 3 ) c is a constant of order
unity which will depend on the involved symmetry, and o(0) is the zero-temperature
vacuum expectation value of the field q. If matter is acted upon by gravitational forces,
we obtain from ( 2 ) and ( 3 )
Gef f
-
T;
(4)
>
where T , is the gravitational temperature, and A, = $ P : M * - ~ . I n the case of a black
hole, T , = I t h . If matter is also affected by forces other than gravity, expression (4)
1
becomes inapplicable and A, > -,U;M*-~.I n this case, the largest deviation that may
2
be expected for GPff with respect t o the Newtonian value must take place in current
GUTS in which o(0) 1015 GeV. Hence, for any T 5 T,, the Newtonian value of the
gravitational constant should be only modified in the eighth decimal place.
For a dimensional analysis, one can then write the action integral inside a black
hole in the form:
-
-
+
Sbh J d4z(-g)l/' [T&(8g/8x)2 . . .] ,
(5)
in which we have only written the term depending explicitely on the metric g ; other
terms in the integral can be ignored in our discussion. The order of magnitude of the
change induced in integral (5) due to metric fluctuations A g over a spacetime region with
. . . . Hence, the order
a dimension of the order L4 should be then AXbh (T,,,Ldg)'
of magnitude of metric fluctuations results [lo]:
-
+
4 g N (TbhL)-l.
(6)
Thus, the estimated fluctuations in the metric depend not only on the scale of the
region of observatlon L but also on the black hole temperature in that region. Hence,
if the mass of the black hole is M , the minimum possible resolution limit is of the order
TGhl GM in the region near the horizon. Note that we have taken as Tbh the Hawking
temperature in spite of the gravitation constant near the horizon is not G. This assumption is justified by the following argument [6]. The key point in the Hawking theory for
-
267
P. F. GONZ~LEZ-D~Az,
Black Holes and Inflation
black hole emission is t o consider that just outside the horizon there is a crowding of
a n infinite number of waves, and hence short wavelength dominate. Because it,s frequency was arbitrarily large, the waves would propagate by geometric optics through
the centre of the body and out on the pass null infinity. However if the resolution limit
near the horizon is of the order of the black hole radius, geometric optics must become
inapplicable and, instead, wave optics should be the appropriate theory t o describe propagation. This situation was analyzed in Ref. [7] and it was obtained that the expectation value of the stress tensor in the steady state region is:
where N N GM2 is the normalized black hole entropy, x is the surface gravity, and
(T/L”)H
is the Hawking stress tensor. Thus, the effect of introducing a minimum resolution limit of the order QM leads t o a picture in which the black hole behaves like a n
ensemble of N independent black bodies, all emitting a t the temperature T,, = xGf
2n N (GM)-l, which is just the Hawking temperature.
One of the consequences of the above discussion is that the effective gravitational
constant for a black hole with mass M is G;,h,- G2M2. Hence large black holes will
seem to have an extreme gravitational physics in spite of the curvature goes as G-2M-2
which can be quite small for large holes. This difficulty is nevertheless ruled out as the
e
G-1M-2 E (where e is
energy of e.g. a photon on the hole surface is given by [i’]
the energy of the same photon as determined by an observer out from the hole) so that
it should be acted upon by a gravitational force -GtFfMer-2 N G-lM-Is of the order
of the corresponding Newton force.
Since the black hole lifetimes is now given by tbh GM (i.e. just the time the emitted radiation lasts in traveling the region of the collapsing body just outside the horizon)
and its dimension is of the order of the resolution limit, any black hole should actually
be a virtual entity, such as are the Planck black holes pervading all the spacetime topological structure [lo, 111. Thus, the gravitational collapse could be re-interpreted as the
process by which one of these Planck holes is excited t o have a mass M and occupy a
volume G3M3 [7]. This locally excited vacuum state would spontaneously decay by
--G-l. That process would ultimately imply the
emitting thermal radiation a t a rate
inflation of a given region of spacetime from a volume G312 to a volume G3M3.
At first glance, this picture could however seem utterly implausible. It seems clear
that a so remarkable astrophysical event such as it should be the explosion of the centre
of a galaxy or even that of a much less massive black hole, could not occur inadvertently.
Recent infrared and submillimetre spectroscopic evidence [ 12) points clearly towards
the existence of a massive black hole in the centre of our galaxy. We will show that, a t
least in the present astrophysical realms, the gravitational collapse may only proceed
just up to the limit of the event horizon but it does not finally end in a true black hole
state ; therefore, the considered effect of conversion of observable energy into vacuum
energy appears to be forbidden. This could well justify the measured dynamics and
distribution of matter around the galactic centre and other suspected astrophysic black
holes, and would avoid the catastrophic energy bursts implied by the above model.
Our argument is as follows. We consider first how may go the process which inflates
a vacuum local spacetime region with Planck dimension up t o an excited vacuum confined region of dimension -GM. By swallowing the mass M of the collapse body, the
initial vacuum energy density W M * ~must then evolve to reach a presumed value of
the order of the black hole energy density -IM*~M--$. We remark that such a process
can be represented by defining an order parameter qi (scalar field) which evolves accor-
-
-
n;r-
2G8
Ann. Physik Leipzig 45 (1988) 4
ding t o a Coleman-Weinberg effective potential with the temperature-dependent approximate form :
-
Note that a t 9- 0, Vbh(O,T )N M*4, and a t p- M*, Vbh(M*,T ) M*2T2M*61W2if we take for T the black hole temperature. It is worth noticing that potential
(8) corresponds t o the temperature-dependent approximate version of the one-loop effective potential obtained from the classical gravitational potential (2) when we add
= 0, d2V,*/dcp21,=, = 2pi for a Higgs
the normalization conditions [I31 dV,*/dcp
field with spontaneous symmetry breaking (r = (3/4nG)’I2, and set the coupling constant
3e4/1Gn2 = ,LL:/M*~,
with pi = M*2. Hence, once the observable energy is swallowed
by a closed trapped surface, it is being continuously converted into vacuum energy via
a process which is equivalent t o the mechanism assumed in the new inflationary scenario
for the very early universe [14, 151. This analogy implies that the matter collapsing
to form a black hole must undergo a phase transition from a symmetric phase cp = 0
to a broken symmetry phase cp
M* where the scalar field will oscillate [5] so that all
vacuum energy is rapidly thermalized through the emission process suggested in this
paper for black holes.
During most of the time inverted in the transition the field cp remains near cp = 0
where there only exists a quantum gravitational field with the largest possible fluctuating curvature. As cp reaches the broken symmetry phase a t cp M*, such a gravitational field splits into the known fundamental forces.
We define now the “Hubble constant’’ for the black hole as
IpLo
-
-
H
=
(2Vb”(0)Jf*-2)’/2= -.M*
2
-
1019 GeV.
(9)
Therefore, black hole collapse in vacuum would produce an exponential expansion
which is not long enough to generate a spacetime region so flat as that is observed in
the universe as a whole, and would create energy fluctuations 6M/M larger than those
that are thought to have emerged in the early universe [IG]. These difficulties are avoided in the primordial Coleman-Weinberg version of the inflationary universe [8] where
the effective potential is obtained from (2) in the same way as (8) but with p,,( +M*)
GeV.
We investigate now the high-temperature behaviour of the phase transition induced
by potential (8) by assuming a minimal XU(5) model with three fermion generations
[17]. I n the X U ( 5 ) theory the interaction Lagrangian of the Higgs field is L =
1
I
- -a(tr @2)2
-b tr W , where @ is the Higgs scalar field in the adjoint represen4
2
tation, and a and b are coupling constants. At T
0 the boson mass in this theory is
[18]:
-
+
+
5
+
rn2 = d2V/dp21,=, = (Tg2
30
where the constants g, a, b are assumed t o depend on the temperature. This depeiideiice
can be studied by means of the renormalization group equations [19]. For the effective
potential (8) the boundary conditions are: g2(T = M * , q = 0) = 167~/75,a ( T = M * ,
cp = 0) == b(T = M*, v = 0 ) 2: 126g4/128 = 4 . 4 10-2.
~ The temperature-dependence of
g2 is given by the renormalization group equation g2 = 3n2/5t, where t = ln(T/A) a t
cp = 0. I n the theory considered here the normalization parameter A = 1.46 x 1015GeV.
According to Linde [19], the phase transition may only start a t the temperature T,
P. P.GONZ~LEZ-DL~Z,
Black Holes and Inflation
269
a t which the boson mass squared m2(T)becomes negative. We have determined the
value of T , for which
65 da
47 d b
3n2
--+
--=-*
30 dt
30 dt
4t2
From Eqn. (ll),the renormalization group equations for dafdt and dbfdt, and the
above boundary conditions, we have obtained T,(m = 0) = 8 x 10l6GeV. Thus, although according to Sher [I71 and Billoire and Tamvakis [21] the probability of bubble
nucleation in the present theory would become only appreciable a t temperatures smaller
than 10l6 GeV, phase transition could only take place due t o the rolling down of the
scalar field starting a t a temperature around 8 x 10l6 GeV. Since T , < H , all hightemperature effects are irrelevant t o the theory of the phase transition. During the time
interval required for the phase transition to start, the temperature falls rapidly t o
T , N T,ecHITcN 4x
GeV, a temperature certainly smaller than the corresponding Hawking temperature in the de Sitter space, T H N 10l8 GeV. As potential (8)
is itself gravitational in nature, one cannot also invoke the effects of nonvanishing
curvature additional terms in the Lagrangian t o allow phase transition [3]. Thus,
astrophysical black holes cannot be formed and, therefore, do not evaporate away.
Only in the case of very primordial black holes, phase transition could perhaps take
place due to the effects of the rapid expansion of the universe.
A point is worth mentioning a t this moment. Although it is true that potential (8)
corresponds t o a genuine XU(5) potential which is suitable for describing GUT (note
that theat potential should predict proton lifetimes much larger than present theoretical
estimates), its curvature-dependent form,
V ( R ,T )_N (16n GR,)-l
2
(with R, = -6M*2)cannot represent observable gravitational forces. It has been shown
[20] that, in order t o obtain an action integral (including higher derivative terms) that
can represent usual gravitational interactions, one ought t o proceed in exactly the same
way as for getting Eqn. (8), unless we set now 3e4f32n2 = pifM*2. The q-dependent
form of the corresponding effective potential reads :
which does not match potential (8) unless a t q = M*. Thus, although there is only one
potential t o describe classical gravitation (i.e. Eqn. (2)), the quantum version of this
potential admits a t least two completely different gravitational theories for q < M*.
I n its most general form (in which the mass parameter p, is not fixed) theory (8) describes
the behaviour of the gravitational field inside black holes and, probably, of the very
early universe [8]; theory (13) can be used t o predict the behaviour of the gravitational
field in the presence of any other kind of force. Because these two theories do coincide
a t q = M* (i.e. when S U ( 5 ) is broken into e.g. X U ( 3 ) x X U ( 2 ) X U(l)), these considerations suggest a n alternative strategy for the unification of the fundamental forces :
grand unification of electromagnetic, weak and strong forces leads to the XU(5)-gravity ;
current gravity could not be described in a unified scheme together with the other forces,
a t least according to usual frameworks for unification.
The author is indebted t o C. Siguenza, T. GONZALEZ-CORTES
and D. GONZBLEZSIGUENZA
for help and encouragement.
270
Ann. Physik Leipzig 45 (1988) 4
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M.; SASAKI,
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"211 BILLOIRE,
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Bei der Redaktion eingegangen am 24. Marz 19%.
Anschr. d. Verf. : Dr. P. F. GONZLLEZ-D~AZ
Instituto de Optica, C.S.I.C.,
Serrano 121, 28006 Madrid, Spain
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