Quantum evaporation of a naked singularity

0.14 MB·English

by  Rituparno Goswami

IGPG-05/6-8
Quantum evaporation of a naked singularity
Rituparno Goswami , Pankaj S. Joshi , Parampreet Singh
∗ ∗ †
∗Tata Institute for Fundamental Research, Colaba, Mumbai 400005, India and
†Institute for Gravitational Physics and Geometry,
Pennsylvania State University, University Park, PA 16802, USA
We investigate here quantum effects in gravitational collapse of a scalar field model which clas-
sically leads to a naked singularity. We show that non-perturbative semi-classical modifications
near the singularity, based on loop quantum gravity, give rise to a strong outward flux of energy.
This leads to the dissolution of the collapsing cloud before the singularity can form. Quantum
gravitational effects thuscensor naked singularities by avoiding their formation. Further, quantum
gravity induced mass flux has a distinct feature which may lead to a novel observable signature in
astrophysical bursts.
6 PACSnumbers: 04.20Dw,04.60.Pp
0
0
2 Naked singularities are one of the most exotic objects of quantum gravity effects at the late stages of gravita-
predicted by classical general relativity. Unlike their tional collapse. Existence of naked singularities in these
n
black hole siblings, they can be in principle directly ob- modelsiswell-known[10]andoneofthe simplestsetting
a
J served by an external observer. There have been many is to consider an initial configuration of a homogeneous

2 investigations which show that given the initial density andisotropicscalarfieldΦ=Φ(t)withapotentialV(Φ)
2 andpressureprofilesforamattercloud,thereareclasses (given by eq.(6)) and the canonical momentum PΦ. In
of collapse evolutions that lead to naked singularity for- thiscaseithasbeenshownthatfateofthesingularitybe-
2
mation (see e.g. [1] for some recent reviews), subject to ingnakedorcovereddependsontherateofgravitational
v
anenergyconditionandastrophysicallyreasonableequa- collapse [11]. For an appropriately chosen potential, for-
9
2 tionsofstatesuchasdust,perfectfluidsandsuchothers. mation of trapped surfaces can be avoided even as the
1 Thishasledtoextensivedebatesontheirexistence,with collapse progresses,resulting in a naked singularity with
6 apopularideabeingcosmiccensorshipconjectureswhich anoutwardenergyflux,inprincipleobservable. Sincethe
0
forbid classical nakedness [2]. Since naked singularities interior of homogeneous scalar field collapse is described
5
originate in the regime where classical general relativ- by a Friedmann-Robertson-Walker (FRW) metric, tech-
0
/ ity is expected to be replaced by quantum gravity, it niques of loop quantum cosmology can be used to inves-
c
has remained an outstanding problem whether a quan- tigate the way quantum gravity modifies the collapse.
q
- tumtheoryofgravityresolvestheirformation. Also,with Letusconsidertheclassicalcollapseofahomogeneous
r
g thelackofobservablesignaturesfromthePlanckregime, scalar field Φ(t) with potential V(Φ) and the canonical
: nakedsingularitiescouldinfactbeaboonforaquantum momentum for the marginally bound (k = 0) case. The
v
theory of gravity. Because, the singularity being visible, interior metric is given by
i
X any quantum gravitational signature originating in the
2 2 2 2 2 2
r ultra-high curvature regime near a classical singularity ds = dt +a (t) dr +r dΩ (1)
a −
can in principle be observed, thus providing us a rare
(cid:2) (cid:3)
test for quantum gravity. with classical energy density and pressure of the scalar
field,
Oneofthenon-perturbativequantizationsofgravityis
loop quantum gravity [3] whose key predictions include ρ(t)=Φ˙2/2+V(Φ), p(t)=Φ˙2/2 V(Φ) . (2)
Bekenstein-Hawking entropy formula [4]. Its application −
tosymmetryreducedmini-superspacequantizationofho-
The dynamical evolution of the system is obtained from
mogeneous spacetimes is called loop quantum cosmol-
the Einstein equations which yield [11]
ogy [5] whose success includes resolution of the big bang
singularity [6], initial conditions for inflation [7, 8], and R˙2R=F(t,r), ρ=F /κaR2, p= F˙/κR2R˙ (3)
,r
possibleobservablesignaturesincosmicmicrowaveback- −
ground radiation [8]. These techniques have also been Here κ=8πG, and F(t,r)=(κ/3)ρ(t)r3a3 has interpre-
applied to resolve black hole singularity in a scalar field tation of the mass function of the collapsing cloud, with
collapse scenario [9]. F 0 and R(t,r) = ra(t) is the area radius of a shell
≥
Since the dynamics of a generic collapse is very com- labeledbycomovingcoordinater. Inacontinualcollapse
plicatedandtoolstoaddresssuchaprobleminquantum thearearadiusofashellataconstantvalueofcomoving
gravity are still under development, it is useful to work radius r decreases monotonically. The spacetime region
with a simple collapse scenario as of a scalar field. It is trapped or otherwise, depending on the value of mass
serves as a good toy model to gain insights on the role function. If F is greater (less) than R, the the region
2
is trapped (untrapped). The boundary of the trapped and shall be constrained by phenomenological consider-
region is given by F =R. ations.
The collapsing interior can be matched at some suit- The change in behavior of the classical geometrical
able boundary r = r to a generalized Vaidya exterior density (1/a3) for scales a . a , can be well approxi-
b
geometry, given as [12], mated by [7] ∗
2 2 2 2
ds = (1 2M(r ,v)/r )dv 2dvdr +r dΩ . (4) 3 2 2
− − v v − v v dj(a)=D(q)a− , q :=a /a , a := jγ/3ℓP (7)
∗ ∗
The Israel-Darmois conditions then lead to [11, 12] p
with
r a(t)=r (v), F(t,r )=2M(r ,v) and
b v b v
2 D(q)=(8/77)6q3/2 7 (q+1)11/4 q 111/4
M(rv,v),rv = F/2rba + rbaa¨ . (5) −| − |
n h 6 i
11q (q+1)7/4 sgn(q 1)q 17/4 . (8)
The form of the potential that leads to a naked sin-
− − − | − |
gularity is determined as follows. The energy density h io
of scalar field can be written in a generic form as ρ = For a a , d (a/a )15a 3 and for a a it behaves
j −
ln 4a n, where n > 0 and l is a proportionality con- classic≪ally∗with∝d a∗ 3. The scale at w≫hic∗h transition
− − j −
≈
stant. Using energy conservation equation, this leads to in the behavior of the geometrical density takes place is
the pressure p = [(n 3)/3] ln 4a n. On subsituting determined by the parameter j.
− −
−
eq.(2) in these we obtain [11] At the fundamental level the dynamics in the loop
quantum regime is discrete, however, recent invsetiga-
Φ= n/κlna, V(Φ)=(1 n/6)ln−4e√κn Φ . (6) tions pertaining to the evolution of coherent states have
− −
ThenitispeasilyseenthatF/R=(κ/3)ln 4a2 nr2. Thus shown that for scales a0 = √γℓP . a . a∗ = jγ/3ℓP,
− − dynamics can be described by modifications to Fried-
in the collapsing phase as a 0, whether or not the p
mann dynamics on a continuous spacetime [14] with the
−→
trapped surfaces form is determined by the value of n.
modified matter Hamiltonian
It is straightforward to check that for 0 < n < 2, if no
trapped surfaces exist initially then no trapped surfaces Φ =dj(a)PΦ2/2+a3V(Φ) (9)
would form till the epoch a(t) = 0 [11], with a(t) = H
1 nt/2√3 2/n. and the modified Friedmann equation
−
The absence of trapped surfaces is accompanied by a
(cid:0)negative pres(cid:1)sure implying that for a constant value of a˙2/a2 =(κ/3)(Φ˙2/2D+V(Φ)) (10)
thecomovingcoordinater,F˙ isnegativeandsothemass
which is obtained by the vanishing of the total Hamil-
contained in the cloud of that radius keeps decreasing.
tonian constraint and the Hamilton’s equations: Φ˙ =
This leadsto a classicaloutwardenergyflux. As the col-
lapseproceeds,thescalefactorvanishesinfinitetimeand dj(a)PΦ, P˙Φ = a3V,Φ(Φ) [7]. These also lead to the
−
modified Klein-Gordon equation
physicaldensitiesblowup,leadingtoanakedsingularity.
Since no trapped surfaces form during collapse, the out-
ward energy flux shall in principle be observable. How- Φ¨ + 3a˙/a D˙(q)/D(q) Φ˙ +D(q)V,Φ(Φ) =0 . (11)
−
ever,nearthe singularitywhenenergy density is close to (cid:16) (cid:17)
Planckian values, this classical picture has to be modi- Since at classical scales (a a ) D 1, the modified
≫ ∗ ≈
fiedandweneedtoinvestigatethescenarioincorporating dynamical equations reduce to the standard Friedmann
quantumgravitymodificationstotheclassicaldynamics. dynamical equations. For scales a.a , the Φ˙ term acts
∗
Let us hence consider the non-perturbative semi- likeafrictionaltermforacollapsingphase. Wenotethat
classical modifications based on loop quantum gravity sincesemi-classicalmodificationsforinhomogeneouscase
for the interior. The underlying geometry for the FRW are still not known, we cannot do a complete quantum
spacetime in loop quantum cosmology is discrete and analysisofinteriorandexterior. Theexteriorisassumed
both the scale factor and the inverse scale factor op- to remain classical. Further, as a continuous spacetime
erators have discrete eigenvalues [13]. In particular, canbe approximatedtill scale factora0, the matching of
there exists a critical scale a = jγ/3ℓP below which interiorandexteriorspacetimesremainsvalidduringthe
∗
the eigenvalues of the inverse scale factor become pro- semi-classical evolution.
p
portional to the positive powers of scale factor. Here Themodifiedenergydensityandpressureofthescalar
γ 0.2375 is the Barbero-Immirzi parameter [4], ℓP is fieldinthesemi-classicalregimecanbesimilarlyobtained
≈
Plancklengthandjisahalf-integerfreeparameterwhich from the eigenvalues of density operator and using the
arises because inverse scale factor operator is computed stress-energy conservation equation [15]
bytracingoverSU(2)holonomiesinanirreduciblespinj
representation. The value of this parameter is arbitrary ρeff =dj(a) Φ =Φ˙2/2+D(q)V(Φ) (12)
H
3
3 dynamics is governed by the modified Friedmann and
Klein-Gordon equations. The scalar field which experi-
10000
2.5 100 enced anti-friction in classical regime, now experiences
friction leading to decrease of Φ˙.
1
ρ The slowingdownofΦ decreasesthe rateofcollapse
2 0.01 •
andformationofsingularityisdelayed. Eventuallywhen
0.0001
R 1.5 1e-06 scale factor becomes smaller than a0 this leads to break-
down of continuum spacetime approximation and semi-
1e-08 0 0.5 1 1.5 2 2.5 3 3.5 4
t classical dynamics. Discrete quantum geometry emerges
1
atthisscale[14]andthedynamicscanonlybedescribed
by quantum difference equation. The naked singularity
0.5
isthus avoidedtillthe scalefactoratwhichacontinuous
spacetime exists.
0 0 1 2 t 3 4 5 Weshowtheevolutionofarearadiusintimeascollapse
proceeds in Fig.1. The semi-classical evolution (solid
FIG. 1: Evolution of area radius with time. The classical curve)closelyfollowsclassicaltrajectory(dashed)tillthe
evolution (dashed) leads to naked singularity in finite time
time t . Within a finite time after t , the classical col-
whereas insemi-classical evolution (solid) it isavoided. Inset ∗ ∗
lapseleadstoavanishingRandnakedsingularity. How-
showsevolutionofenergydensity(inPlanckunits)withtime.
The parameters chosen are n=1.9 and j =100. ever,the area radius never vanishes in the loop modified
semi-classical dynamics and the naked singularity does
not form as long as the continuum spacetime approxi-
and mation holds. The inset of Fig.1 shows the evolution of
2 1 D˙(q) Φ˙2 D˙(q) energy density in Planck units. Classical energy density
peff = 1 D(q)V(Φ) V(Φ). (dashedcurve)blowsup whereasit remains finite andin
−3 (a˙/a) D(q) 2 − −3(a˙/a)
(cid:20) (cid:21) fact decreases in the semi-classical regime.
(13)
The phenomena of delay and avoidance of the naked
Itis thenstraightforwardtocheckthatpeff is generically
singularity in continuous spacetime is accompanied by a
negative for a . a and for a a it becomes very
∗ ≪ ∗ burst of matter to the exterior. If the mass function at
strong. For example, at a a0, peff 9ρeff. This is
much stronger than its clas∼sical count≈erp−art p = [(n scales a ≫ a∗ is Fi and its difference with mass of the
3)/3]ρwith0<n<2. Thusweexpectastrongbursto−f cloud for a<a∗ is ∆F =Fi−F, then the mass loss can
be computed as
outwardenergyfluxinthesemi-classicalregime. Further,
for a a , D(q) 1 and the Klein-Gordon equation 1
yields≪Φ˙ ∗a12. H≪ence from the eq. (12) we easily see ∆F = 1 ρeffd−j . (15)
thatthee∝ffectivedensity,insteadofblowingup,becomes F(ai) " − ln−4a3i−n#
extremely small and remains finite.
Fora<a ,asthe scalefactordecreases,the energyden-
Themodifiedmassfunctionofthecollapsingcloudcan
∗
sity and mass in the interior decrease and the negative
be evaluated using eq.(3) and eq.(10),
pressure strongly increases. This leads to a strong burst
F =(κ/3)(d−j1Φ˙2/2+a3V(Φ))r3 . (14) of matter. The absence of trapped surfaces enables the
quantumgravityinducedbursttopropagateviathegen-
In the regime a ∼ a0, d−j1Φ˙2 becomes proportional to eralized Vaidya exterior to an observer at infinity. The
a12, the potential term becomes negligible and thus the evolutionofmassfunctionisshowninFig.2. Inthesemi-
mass function becomes vanishingly small at small scale classical regime, ∆F/F approaches unity very rapidly.
i
factors. This feature is independent of the choice of parameter
The picture emerging from loop quantum modifica- j. The choice of potential causes mass loss to exterior
tions to collapse is thus following. in classical collapse also, but it is much smaller and in
Before the area radius of the collapsing shell reaches anycasetheclassicaldescriptioncannotbetrustedaten-
•
R =ra att=t , collapseproceeds as per classicaldy- ergydensitygreaterthanPlanck,whenwemustconsider
na∗micsa∗ndassma∗llerscalefactorsareapproachedΦ˙ and quantum effects as above.
the energy density ρ a n increase. The mass function Interestingly, for a given collapsing configuration, the
−
is proportional to an∝3 and (as 0 < n < 2) it decreases scale at which the strong outward flux initiates depends
−
with decreasing scale factor so there is a mass loss to on the loop parameter j which controls a . If j is large
∗
the exterior, which is also understood from existence of thenburstoccursatanearlierarearadiusandviceversa.
negative classical pressure. The inset of Fig.2 shows the mass loss ratio for different
As the collapsing cloud reaches R , the geometric values of j. For all choices, ∆F/F 1, but the out-
i
den•sity classically given by a 3, modifi∗es to d and the going flux profile changes. The loop q→uantum burst has
− j
4
2.5 singularity, the final fate of naked singularity must be
1
decided by using full quantum evolution. Even in such
0.95
0.9 cases we have valuable insights from semi-classical loop
2 0.85
∆F/Fi 0.8 quantum effects with the possibility of phenomenologi-
0.75 cally constraining the j parameter.
0.7
1.5 0.65
0.6 0 100 200 300 400 500 600 In the toy model considered, we showed that the clas-
R
F sical outcome and evolution of collapse is radically al-
1 tered by the non-perturbative modifications to the dy-
namics. Our considerations are of course within the
mini-superspacesetting, andthe generalcaseofinhomo-
0.5
geneities and anisotropies remains open. However, the
possibility of such observable signatures in astrophysical
0 bursts, as originating from quantum gravityregime near
0 1 2 3 4 5 6
R singularityisintriguing,indicatingthatgravitationalcol-
FIG.2: Evolutionofmassfunctionwitharearadiusforsame lapse scenario can be used as probes to test quantum
parametersasinFig.1. Loopquantumevolution(solid)leads gravity models.
to dissolution of all the mass of the collapsing shell. Dashed
Acknowledgments: We thank A. Ashtekar, M. Bo-
curveshowsclassical trajectory. Insetshowsmasslossprofile
for j = 106 (outer), j = 5.0×105 (middle) and j = 105 jowaldandR.Maartensforusefulcomments. PSJthanks
(inner). BharatKapadia for variousdiscussions. PS is supported
by Eberly research funds of Penn State and NSF grant
PHY-00-90091.
a distinct signature, at a a the flux decreases for a
∼ ∗
shortperiodandthenrapidlyincreases. Since thecausal
structure of classicalspacetime is such that trapped sur-
faceformationisavoided,thisquantumgravitationalsig-
nature can be in principle observed by an external ob- [1] A. Krolak, Prog. Theor. Phys.Suppl. 136, 45 (1999); P.
serverasaslightdimmingandsubsequentbrighteningof S.Joshi, Pramana55, 529(2000); T.Harada, H.Iguchi,
and K.Nakao, Prog.Theor.Phys. 107 (2002) 449.
the collapsing star. This peculiar phenomena is directly
[2] R.Penrose,Riv.NuovoCimento,Num.Sp.1,252(1969).
related to the peak in the function d (a), and depends
j [3] See for eg., A. Ashtekar and J. Lewandowski, Class.
solely onthe value ofparameterj. If we comparethis to Quant. Grav. 21 R53 (2004).
other phenomenological applications [7, 8, 9], this effect [4] A. Ashtekaret al., Phys. Rev.Lett. 80, 904 (1998).
could not be masked by the role of other loop quantum [5] M. Bojowald, gr-qc/0505057.
parameters in a more general setting. This phenomena [6] M. Bojowald, Phys. Rev.Lett. 86, 5227 (2001).
is thus a direct probe to measure j and an observer can [7] M. Bojowald, Phys. Rev.Lett. 89, 261301 (2002).
[8] S. Tsujikawa, P. Singh, & R. Maartens, Class. Quant.
estimatetheloopquantumparameterj byobservingthe
Grav. 21, 5767 (2004).
flux profile of the burst based on this mechanism and [9] M. Bojowald et al., Phys. Rev.Lett. 95 (2005) 091302.
measuring the variation in luminosity of the collapsing [10] D. Chistodoulou, Ann. of Maths. 140, 607 (1994); 149,
cloud. 183 (1999); M. W. Choptuik, Phys. Rev. Lett. 70, 9
During such a burst most of the mass is ejected and (1993); A. M. Abrahams and C. R. Evans, Phys. Rev.
this may dissolve the singularity. Thus non-perturbative Lett.70, 2980 (1993); M.D.Roberts,Gen.Relat. Grav.
21, 907 (1989); J. Traschen, Phys. Rev. D 50, 7144
semi-classical modifications may not allow formation of
(1994); P. R. Brady, Phys. Rev. D 51, 4168 (1995); C.
nakedsingularityasthecollapsingcloudevaporatesaway
Gundlach, Phys. Rev.Lett. 75, 3214 (1995).
due to super-negative pressures in the late regime. It
[11] P. S. Joshi & R. Goswami, gr-qc/0410144; R. Giambo
has been demonstrated that these super-negative pres-
gr-qc/0501013.
sureswouldexistforarbitrarymatterconfigurations[15] [12] A. Wang & Y. Wu, Gen. Rel. Grav. 31 (1), 107 (1999);
whichimpliesthatresultsobtainedherewouldholdeven P. S. Joshi & I. H. Dwivedi, Class.Quant.Grav. 16, 41
in a more general setting [16]. Loop quantum effects (1999).
then imply a quantum gravitational cosmic censorship, [13] A. Ashtekar, M. Bojowald & J. Lewandowski, Adv.
Theor. Math. Phys. 7, 233 (2003).
alleviating the naked singularity problem. We note that
[14] M.Bojowald, P.Singh&A.Skirzewski,Phys.Rev.D70,
the semi-classical effects do not show that the singular-
083517 (2004); P. Singh, K. Vandersloot, Phys. Rev.
ity is absent, it is only avoided till scale factor a0, be- D72, 084004 (2005).
lowwhichthesemi-classicaldynamicsandmatchingmay [15] P. Singh, Class. Quant.Grav. 22, 4203 (2005).
break down. If for a given choice of initial data, semi- [16] P.S.Joshi, I.H.Dwivedi, Phys.Rev.D 47, 5357 (1993); P.
classical dynamics is unable to completely dissolve the S. Joshi, R. Goswami, Phys.Rev.D 69, 064027 (2004).