# Could Some Black Holes Have Evolved from Wormholes?

Authors: Peter K. F. Kuhfittig
Journal: Scholarly Research Exchange
 (2008)

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## Abstract

One way to explain the present acceleration of the universe is Einstein_s cosmological constant. It is quite likely, in view of some recent studies, that a time-dependent equation of state had caused the Universe to evolve from an earlier phantom-energy model. In that case, traversable wormholes could have formed spontaneously. It is shown in this paper that such wormholes would eventually
have become black holes. This would provide a possible explanation for the huge number of black holes discovered, while any evidence for the existence of wormholes is entirely missing, even though wormholes are just as good, in terms of being a prediction of general relativity, as black holes.


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## Body

## 1. Introduction

Traversable
wormholes, whose possible existence was first conjectured by Morris and Thorne
in 1988 [1], are
shortcuts or tunnels that could in principle be used for traveling to remote
parts of our Universe or to different universes altogether. Using units in
which c=G=1, a wormhole can be described by the general
line element(1)ds2=−e2γ(r)dt2+e2α(r)dr2+r2(dθ2+sin2θdϕ2).The motivation for this idea is
the Schwarzschild line element(2)ds2=−(1−2Mr)dt2+dr21−2M/r+r2(dθ2+sin2θdϕ2),which may be viewed as a black
hole centered at the origin of a (t,r,θ,ϕ) coordinate system. Both wormholes and black
holes are predictions of Einstein's general theory of relativity. The main
difference is that wormholes must necessarily violate certain energy
conditions; more precisely, the stress-energy tensor of the matter source of
gravity violates the weak energy condition [1]. If the matter source was different in the past, then
wormholes could theoretically have been formed naturally. Examples are
equations of state that parameterize certain dark energy models, as we will see
in the next section. Moreover, according to some recent studies, such a
transition is likely to have occurred in the relatively recent past [2, 3].In this paper, we study an equation of state that is
both space and time dependent. It is proposed that if the equation of state
evolved into the present cosmological constant model, then any wormhole that
had previously come into existence would have formed an event horizon, thereby
becoming a black hole. This would provide a possible explanation for the failure
to detect any evidence of wormholes, while black holes appear to be abundant.


## 2. Background

Interest in
traversable wormholes has increased in recent years due to an unexpected
connection, the discovery that our Universe is undergoing an accelerated
expansion [4, 5]. This acceleration, caused
by a negative pressure dark energy, implies that a¨>0 in the Friedmann equation a¨/a=−(4π/3)(ρ+3p). The equation of state is p=−Kρ, K>1/3,
and ρ>0. While the condition K>1/3 is required for an accelerated expansion,
larger values for K are also of interest. In fact, the most appealing
candidate for dark energy is K=1,
corresponding to the cosmological constant Λ [6]. The presence of the cosmological constant has
resulted in a modification of the Einstein field equations by effectively
adding an isotropic and homogeneous source with constant equation of state
[7]. One can therefore
argue, as in [8], that
this model is the primary candidate for the present Universe.Another widely studied possibility is the caseK>1,
referred to as phantom energy [9]. To see why, we need to recall that the set of orthonormal
basis vectors may be interpreted as the proper frame of a set of observers
who remain at rest in the coordinate system. If the basis vectors are denoted
by et, er, eθ,
and eϕ,
then the orthonormal basis vectors are (referring to line element
(1))(3)et^=e−γ(r)et,er^=e−α(r)er,eθ^=r−1eθ,eϕ^=(rsinθ)−1eϕ.In this frame of reference, the
components of the stress-energy tensor Tα^β^ have an immediate physical interpretation: Tt^t^=ρ, Tr^r^=p, Tθ^θ^=Tϕ^ϕ^=pt,
where ρ is the energy density, p the radial pressure, and pt the lateral pressure. The weak energy
condition (WEC) can now be stated as follows: Tα^β^μα^μβ^≥0 for all time-like vectors and, by continuity,
all null vectors. For example, given the radial outgoing null vector (1,1,0,0),
we obtain(4)Tα^β^μα^μβ^=ρ+p≥0.So if the WEC is violated, we
have ρ+p<0. While all classical forms of matter ordinarily
meet this condition, there are situations in quantum field theory, such as the
Casimir effect, that allow such violations [10]. More importantly, in our case this violation occurs
whenever K>1,
thereby meeting the primary prerequisite for the existence of wormholes.Two recent papers [11, 12] discuss wormhole solutions that depend on a variable
equation of state parameter, that is, p/ρ=−K(r), where K(r)>1 for all r.
The variable r refers to the radial coordinate in the line
element(5)ds2=−e2h(r)dt2+11−b(r)/rdr2+r2(dθ2+sin2θdϕ2).In this form of the line
element, h=h(r) is called the redshift function and b=b(r) is the shape function. The minimum
radius r=r0 corresponds to the throat of the
wormhole, where b(r0)=r0.It is shown in [12] that given a specific shape function, it is possible
to determine K(r) and vice versa. It is also assumed that h′(r)≡0,
referred to as the “zero tidal-force solution” in [1].An earlier study [13] assumed that the equation of state is time dependent.
In this paper, we will assume that the equation depends on both r and t,
while remaining independent of direction.Since the equation of state is time dependent, we
assume that the corresponding metric is also time dependent. It is shown in the
next section that such a metric describes a slowly evolving wormhole structure
without assigning specific functions toh, b,
and K.
In particular, the function h in line element (5) need not be a constant.Evolving wormhole geometries are also discussed in
[14, 15].Since we are dealing with a given time-dependent
equation of state, it is natural to consider the consequences of an evolving
equation, particularly one in which the parameterK(r,t) approaches unity. That is the topic of Section 6.


## 3. The Metric

In this paper,
we will be dealing with a time-dependent metric describing an  evolving
wormhole(6)ds2=−e2γ(r,t)dt2+e2α(r,t)dr2+r2(dθ2+sin2θdϕ2),where γ and α have continuous partial derivatives, so that γ and α are continuous, as well.In view of line element (5), we have(7)e2α(r,t)=11−b(r,t)/r.So the shape function is now
given by(8)b(r,t)=r(1−e−2α(r,t)).To study the effect of a
gradually changing equation of state, we assume the existence of a fixed throat
at r=r0,
that is, b(r0,t)=r0 for all t.
(In other words, the wormhole is close to being static for relatively long
periods of time.) As a consequence, for all t,(9)limr→r0+α(r,t)=+∞,limr→r0+∂∂rα(r,t)=−∞.As before, γ=γ(r,t) is the redshift function, which must be
everywhere finite to prevent an event horizon at the throat. As in the case of
the Schwarzschild line element, (∂/∂r)γ(r,t)>0. The qualitative features of α and γ are shown in Figure 1.Figure 1
Graph showing the qualitative features ofα(r,t) and γ(r,t).To obtain a traversable wormhole, the shape function
must obey the usual flare-out conditions at the throat, modified to accommodate
the time dependence(10)b(r0,t)=r0,∂∂rb(r0,t)<1∀t.So b(r,t)<r for all t near the throat. Another requirement is
asymptotic flatness: b(r,t)/r→0 as r→∞.The next step is to list the time-dependent components
of the Einstein tensor in the orthonormal frame. (For a derivation, see
Kuhfittig [16].)(11)Gt^t^=2re−2α(r,t)∂∂rα(r,t)+1r2(1−e−2α(r,t)),Gr^r^=2re−2α(r,t)∂∂rγ(r,t)−1r2(1−e−2α(r,t)),Gt^r^=2re−γ(r,t)e−α(r,t)∂∂tα(r,t),(12)Gθ^θ^=Gϕ^ϕ^=−e−2γ(r,t)×[∂2∂t2α(r,t)−∂∂tγ(r,t)∂∂tα(r,t)+(∂∂tα(r,t))2]−e−2α(r,t)[−∂2∂r2γ(r,t)+∂∂rγ(r,t)∂∂rα(r,t)−(∂∂rγ(r,t))2]−1re−2α(r,t)(−∂∂rγ(r,t)+∂∂rα(r,t)).Recall that from the Einstein field equations in the
orthonormal frame,Gα^β^=8πTα^β^,
the components of the Einstein tensor are proportional to the components of the
stress-energy tensor. In particular, Tt^t^=ρ(r,t), Tr^r^=p(r,t), Tθ^θ^=Tϕ^ϕ^=pt(r,t), and Tt^r^=Tr^t^=(1/8π)Gt^r^=g(r,t),
where f(r,t)=−g(r,t) is usually interpreted as the energy flux in
the outward radial direction [17]. For the outgoing null vector (1,1,0,0),
the WEC, Tα^β^μα^μβ^≥0,
now becomes ρ+p±2g≥0.


## 4. The Equation of State

Since (6)
describes a slowly evolving wormhole, the equation of state should have a
time-dependent parameter K(r,t).
As in [12], K depends on the radial coordinate, but not on
the direction. To be compatible with the wormhole geometry in Section 3,
Lobo suggested the inclusion of a term analogous to the flux term
[18]. One such
possibility is(13)p(r,t)=−K(r,t)[ρ(r,t)+2g(r,t)],where |2g(r,t)|<ρ(r,t). The last condition implies that α(r,t) changes slowly enough; in fact, g(r,t) can be identically zero. If K(r,t)>1, this equation of state describes a generalized
phantom-energy model, as we will see in (18). As a result, the case K=1 would still correspond to a cosmological
constant.Since the notion of dark or phantom energy applies
only to a homogeneous distribution of matter in the Universe, while wormhole
spacetimes are necessarily inhomogeneous, we adopt the point of view in Sushkov
[19]: extended to
spherically symmetric wormhole geometries, the pressure appearing in the
equation of state is now a negative radial pressure, while the transverse
pressure is determined from the field equations.Given the evolving equation of state (13), suppose at
some point in the past, the equation of state parameter K had actually crossed the phantom divide, so
that, for a time, K(r,t)>1.
In fact, according to [2, 3], it is quite likely that such a transition had taken
place in the relatively recent past. Suppose further that K was decreasing, that is, (∂/∂t)K(r,t)<0 with K(r,t)→1 at a time closer to the present, and that K decreased fast enough during this time
interval to compensate for the very gradual increase in the energy density
characteristic of phantom energy, allowing us to assume that(14)∂∂tp(r,t)≥0.With this information we can now
determine the sign of g(r,t):(15)∂∂tp(r,t)=18π∂Gr^r^∂t=18πe−2α(r,t)[∂∂tα(r,t)((−2)2r∂∂rγ(r,t)−2r2)+2r∂∂t(∂∂rγ(r,t))]≥0.Since the pressure would change
at a finite rate, it follows that (∂/∂t)α(r,t) is finite. Also,(16)∂∂tα(r,t)≤0∀r,so that g(r,t)≤0,
as well. The reason is that, judging from Figure 1, the rate of change of the
slope of γ,
that is,(17)∂∂t(∂∂rγ(r,t))is likely to be vanishingly
small. We will return to this point in Section 6.Returning to (13), observe that(18)Tα^β^μα^μβ^=ρ+p+2g=ρ(r,t)−K(r,t)[ρ(r,t)+2g(r,t)]+2g(r,t)=[ρ(r,t)+2g(r,t)][−K(r,t)+1]<0,since K(r,t)>1 and |2g(r,t)|<ρ(r,t).
So the WEC is violated, as one would expect in a phantom-energy scenario. We
would also expect the flare-out conditions to be met. That is the topic of the
next section.


## 5. The Flare-out Conditions

From the
Einstein field equationsGα^β^=8πTα^β^  and the above equation of state, we
have(19)Gt^t^=8πρ,Gr^r^=8π[−K(r,t)][ρ(r,t)+2g(r,t)].Using (11), we obtain the
following system of equations:(20)Gt^t^=8πTt^t^=2re−2α(r,t)∂∂rα(r,t)+1r2(1−e−2α(r,t)),Gr^r^=8πTr^r^=8π[−K(r,t)][ρ(r,t)+2g(r,t)]=2re−2α(r,t)∂∂rγ(r,t)−1r2(1−e−2α(r,t)).Substituting the expressions for ρ(r,t) and 2g(r,t) yields(21)−K(r,t)2re−2α(r,t)∂∂rα(r,t)=K(r,t)1r2(1−e−2α(r,t))+2re−2α(r,t)∂∂rγ(r,t)−1r2(1−e−2α(r,t))+K(r,t)4re−γ(r,t)e−α(r,t)∂∂tα(r,t).The following rearrangement will
be needed again in Section6:(22)∂∂rα(r,t)+2e−γ(r,t)eα(r,t)∂∂tα(r,t)=−12r(e2α(r,t)−1)+12r1K(r,t)(e2α(r,t)−1)−1K(r,t)∂∂rγ(r,t).For the purpose of analysis,
however, a more convenient form is the following:(23)2(∂/∂r)α(r,t)e2α(r,t)−1=−2(∂/∂r)γ(r,t)K(r,t)(e2α(r,t)−1)−1r(1−1K(r,t))−4e−γ(r,t)eα(r,t)(∂/∂t)α(r,t)e2α(r,t)−1.At this point, we
define(24)F(r,t)=−2(∂/∂r)γ(r,t)K(r,t)(e2α(r,t)−1)−1r(1−1K(r,t))−4e−γ(r,t)eα(r,t)(∂/∂t)α(r,t)e2α(r,t)−1.The form ∫du/(eu−1)=ln(eu−1)−u now yields(25)ln(e2α(r,t)−1)−2α(r,t)=∫crF(r′,t)dr′,where c is an arbitrary constant. Recalling that (∂/∂t)α(r,t) is finite, observe that for any fixed t, F(r,t) is sectionally continuous for r≥r0,
so that the integral exists for c≥r0. Thus we may write(26)e2α(r,t)−1=e2α(r,t)+∫crF(r′,t)dr′,whence(27)1−e−2α(r,t)=e∫crF(r′,t)dr′.From (8) we have b(r,t)=re∫crF(r′,t)dr′. So the requirement b(r0,t)=r0 now determines the arbitrary constant c=r0. Thus,(28)b(r,t)=re∫r0rF(r′,t)dr′.Differentiating, we get for r=r0(29)∂∂rb(r0,t)=e∫r0r0F(r,t)dr+r0e∫r0r0F(r,t)dr×[−2(∂/∂r)γ(r0,t)K(r0,t)(e2α(r0,t)−1)−1r0(1−1K(r0,t))−4e−γ(r0,t)eα(r0,t)(∂/∂t)α(r0,t)e2α(r0,t)−1]=1K(r0,t)<1,by the assumption K(r,t)>1.The line element now becomes(30)ds2=−e2γ(r,t)dt2+dr21−e∫r0rF(r′,t)dr′+r2(dθ2+sin2θdϕ2).Since the flare-out conditions
have been satisfied, the line element describes a traversable wormhole.As an illustration, ifα is time independent, γ(r,t)≡0, and K(r,t)=K,
a constant, then(31)e∫r0rF(r′,t)dr′=e∫r0r[−(1/r′)(1−1/K)]dr′=(r0r)1−1/K,e2α(r)=11−(r0/r)1−1/K,which is Lobo's solution
[20].At this point, the following remark is in order: since
we are only interested in the possible existence of wormholes, it is sufficient
to note that to complete the description, the wormhole material should be cut
off at somer=a and joined to an external Schwarzschild
spacetime. (See [20–22]
for details.) This junction will make the space asymptotically flat, a critical
feature referred to in the next section.


## 6. Implications

As we have
seen, sinceK(r,t)>1,
line element (6) describes a traversable wormhole as long as the qualitative
features in Figure 1 are met, resulting in a violation of the WEC. It is
therefore conceivable that wormholes had formed spontaneously during the
phantom-energy phase. Moreover, a possible mechanism for the formation of such
wormholes is discussed in [23].In this section, we study the consequences of our
assumption thatK(r,t)→1 sufficiently fast some time in the past. (As
noted earlier, this limiting case corresponds to a cosmological constant.)
Before doing so, however, let us recall that for an “arbitrary” wormhole
the rate of change of the slope of γ is likely to be vanishingly small, leading to
inequality (16). In addition,(32)limr→r0+α(r,t)=+∞,limr→r0+∂∂rα(r,t)=−∞∀t,while γ=γ(r0,t) must be finite to prevent an event horizon.
Also, limr→∞α(r,t)=limr→∞γ(r,t)=0 for all t.
Finally, (∂/∂r)γ(r,t)>0 (see Figure 1).In (22), as K(r,t)→1,
we are left with(33)∂∂rα(r,t)+2e−γ(r,t)eα(r,t)∂∂tα(r,t)=−∂∂rγ(r,t).As r→r0+,
the left side goes to −∞ (since (∂/∂t)α(r,t) is finite and nonpositive), so that the right
side yields(34)limr→r0+∂∂rγ(r,t)=+∞.Since the gradient of the
redshift function is related to the tidal forces [1], (∂/∂r)γ(r0,t) must remain finite. (In other words, γ must not have a vertical tangent at (r0,γ(r0,t)).)
So for (34) to hold, γ=γ(r,t) must approach the vertical line r=r0 asymptotically. In other words, limr→r0+γ(r,t)=−∞, thereby producing an event horizon at the
throat. The outcome is a black hole, since, in asymptotically flat spacetimes,
a black hole is essentially characterized by the impossibility of escaping from
a certain prescribed region to future null infinity. More formally, the
boundary of J−1(ℐ+),
the causal past of future null infinity, must be a region that does not include
the entire spacetime. But this requirement is met because our starting point is
a well-defined region, the throat of a wormhole.In summary, this paper discusses a wormhole geometry
supported by a generalized form of phantom energy; the equation of state isp(r,t)=−K(r,t)[ρ(r,t)+2g(r,t)],
where K(r,t)>1 for all t.
It is quite likely that the equation of state had crossed the phantom divide in
the relatively recent past. If, in addition, the equation of state evolved so
that K(r,t)→1 closer to the present, the result is a
spacetime with a cosmological constant, a primary candidate for a model of the
present Universe. It is shown that any existing wormhole would form an event
horizon at the throat. Assuming that wormholes could have formed naturally
during the phantom-energy phase, this outcome provides at least a possible explanation for the abundance of black holes and the complete lack of
wormholes, even though wormholes are just as good a prediction of general
relativity as black holes.



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