Table Of ContentSelf-gravitating stellar collapse: explicit geodesics and path integration
Jayashree Balakrishna1, Ruxandra Bondarescu2,
Christine Corbett Moran*3
1College of Arts and Sciences, 317 HGA, Harris-Stowe State University, St. Louis, MO, USA
2 University of Zurich, Zurich, Switzerland, CH-8057
3 NSF AAPF Fellow, TAPIR, California Institute of Technology, Pasadena, CA 91125
∗ corresponding author - [email protected]
(Dated: November 14, 2016)
We extend the work of Oppenheimer & Synder to model the gravitational collapse of a star to a
blackholebyincludingquantummechanicaleffects. Wefirstderiveclosed-formsolutionsforclassical
paths followed by a particle on the surface of the collapsing star in Schwarzschild and Kruskal
coordinatesforspace-like,time-likeandlight-likegeodesics. Wenextpresentanapplicationofthese
paths to model the collapse of ultra-light dark matter particles, which necessitates incorporating
6 quantum effects. To do so we treat a particle on the surface of the star as a wavepacket and
1 integrateoverallpossiblepathstakenbytheparticle. ThewaveformiscomputedinSchwarzschild
0 coordinatesandfoundtoexhibitaningoingandanoutgoingcomponent,wheretheformercontains
2 theprobabilityofcollapse,whilethelattercontainstheprobabilitythatthestarwilldisperse. These
calculationspavethewayforinvestigatingthepossibilityofquantumcollapsethatdoesnotleadto
v
black hole formation as well as for exploring the nature of the wavefunction inside r=2M.
o
N
PACSnumbers:
1
1
A. Introduction non-equivalent ways of quantization that often produce
physically different results [18]. Our path-integral ap-
]
c proach approach is simpler, and can be easily compared
q Black holes play a pivotal role in the evolution of the
to the assumption that the particle obeys the relativistic
- universe providing an important test laboratory for gen-
r Schr¨odinger equation.
g eralrelativity. Withinclassicalgeneralrelativity,Oppen-
In the classical model a star is idealised as a collaps-
[ heimer & Synder modelled the gravitational collapse of
ing self-gravitating dust sphere of uniform density and
startoablackholebyapproximatingthestarwithauni-
4 zeropressurewheretheconstituentparticleshavetheat-
formsphereofdust(hereafterO-Smodel)[1]. Thismodel
v tributes of classical dust: each particle is assumed to be
0 providesananalyticsolutionforstellarcollapsethatcon- infinitesimal in size and to interact only gravitationally
5 nectstheSchwarzschildexteriorofastartoacontracting
with other matter. The inclusion of quantum mechani-
2 Friedmann-Robertson-Walker (FRW) interior. Once the
caleffectsliftssomeoftheseassumptionsallowingforthe
4 surface has passed within r =2M, no internal pressures
0 can halt the collapse and all configurations collapse to possibility that some configurations will not collapse to
. blackholesbutwilldisperseorevenformstablenewcon-
1 a point-like singularity at r = 0. The general features
figurations. Quantitatively, quantum treatment is neces-
0 of this toy collapse model have been examined by many
sary in macroscopic stellar collapse when the action S is
5 authors [2–6].
1 of the order ¯h [19]. In our case, the S/¯h ∼ 1 condition
: In the standard formalism from [3], the stellar surface corresponds to mM ∼¯h. For macroscopic black holes of
v is considered to be initially at rest. Here we consider masses M = 1M −109M , this implies that quantum
i (cid:12) (cid:12)
X configurations with all possible initial velocities. We de- treatment is necessary for ultra-light constituent parti-
r rive closed-form solutions for the equations of motion cles of m∼10−10−10−19 eV. In nature, such ultralight
a in Schwarzschild and Kruskal coordinates for space-like, particle could be dark matter. Dark matter comes close
time-like and light-like geodesics. to the attributes of classical dust, and some dark matter
Asanexampleapplicationofourclosed-formsolutions clouds may be dense enough to collapse to black holes.
for the classical O-S model with non-zero initial veloci- Particlesaslightas10−22−10−23 eVhavebeenproposed
ties, we consider the macroscopic collapse of a spheri- as constituents of dark matter halos [20–23].
cally symmetric sphere of dust composed of ultra-light Toapproximatequantumeffectsofthecollapseofsuch
particles. To approximate quantum effects, the radius ahalocomprisedofultra-lightparticles,westartwithan
of the star is approximated by a Gaussian wavepacket initial wavefunction that represents the position of the
that is initially centred far from r = 2M. Its evolu- particle on the stellar surface, and is at first far from
tion is then followed via a simple path integral approach 2M. To study the propagation of the wavefunction, we
that extends the results of Redmount and Suen [7] from integrate over all possible paths taken by the particle.
a relativistic free particle to a particle constrained by At a given time, the outgoing wavefunction comprises
non-trivial gravity. Gravitational collapse incorporating the probability that the star disperses, and the ingoing
approximate treatments of quantum mechanics has been wavefunction the probability that it collapses. We com-
considered by a variety of authors [8–17]. They involve pute the propagator in analytical form for a particle on
2
the surface of the star in the WKB approximation, and uniform density and zero pressure from the perspective
compare this to the limited assumption that the par- of an observer located on the surface of the star. The
ticle obeys the relativistic Schr¨odinger equation. Our motion of the collapsing surface initially at radius r can
i
equations reduce to a free particle case when the mass be parametrized by
of the star is zero; then the solution to the relativistic r
Schr¨odinger equation is the exact representation of the r = i (1+cosη) (3)
2
wave function [7]. In the more general case of a par- (cid:114)
r r
ticle on the surface of a star, this representation is no τ = i i (η+sinη) (4)
2 2M
longer correct. However, the comparison is instructive.
As expected, we observed that the WKB and relativistic (cid:12)(cid:12)(cid:112)r /2M −1−tanη/2(cid:12)(cid:12)
Schr¨odinger approximations are out of step at early and t = 2Mlog(cid:12)(cid:12)(cid:112) i (cid:12)(cid:12) (5)
late times, and appear to converge towards one another (cid:12) ri/2M −1+tanη/2(cid:12)
(cid:114)
at intermediate times. r (cid:104) r (cid:105)
+ 2M i −1 η+ i (η+sinη)
In Schwarzschild coordinates, we can follow the evolu- 2M 4M
tion of the surface of the star only until the formation
These equations correspond to Eqs. (32.10a) to (32.10c)
of an apparent horizon due to the r = 2M coordinate
of [3]. The star collapses to a singularity in finite proper
singularity. In Kruskal coordinates, one can continue to
time. However, it takes an infinite Schwarzschild t to
study the evolution of the star inside the apparent hori-
reach the apparent horizon at r = 2M and thus an ex-
zon (r < 2M). The paths we have derived here include
ternal observer will never see the star passing its gravi-
time-reversing space-like paths, which allow for the pos-
tational radius (r =2M).
sibility of extraction of information from inside the hori-
In [3] the stellar surface is considered to be initially
zon to the outside. These paths turn inside the horizon
at rest. Here we integrate the equations of motion for
r >0, andheadtowardr =2M. Infuturework, explicit
configurations with all possible initial velocities.
geodesic equations in Kruskal coordinates may be used
as a stepping stone to model behaviour inside r =2M.
The rest of the paper is structured as follows: in Sec-
II. CLASSICAL ACTION AND RADIAL
tionIwedescribetheOppenheimer-Snydermodel. Sec-
GEODESICS IN SCHWARZSCHILD
tion II includes an overview of the classical action and
COORDINATES
paths, a computation of the Oppenheimer-Snyder limit
andthederivationoftheclassicalpathsinSchwarzschild
TherelativisticactionSandtheLagrangianLforthis
coordinates. Section III details the classical paths in
system are
Kruskal coordinates, which comprise a starting point for
futureworkthatexploresthecollapseinsider =2M. As (cid:115)
(cid:90) (cid:90) (cid:18) 2M(cid:19) r˙2
anexampleapplicationoftheclosedformgeodesics,Sec- S =−m dτ =−m dt 1− − ,
cl r 1−2M/r
tionIVdescribesthequantumtreatmentofthedustcol-
lapse applicable to ultra-light particles in Schwarzschild (6)
coordinates. The conclusions follow in Section V. where τ is the proper time, and r˙ = dr/dt. The La-
grangian is
(cid:115)
I. THE OPPENHEIMER-SNYDER MODEL (cid:18) 2M(cid:19) r˙2
L=−m 1− − (7)
r 1−2M/r
In General Relativity, a first approximation to the
The momentum
exterior space-time of any star, planet or black hole
is a spherically symmetric space-time modelled by the ∂L mr˙
p= = (8)
Schwarzschildmetric. ThisisaconsequenceofBirkhoff’s ∂r˙ (cid:114) (cid:104) (cid:105)
(1−2M/r) (1−2M/r)2−r˙2
theorem [3]. The Schwarzschild line element thus takes
the usual form
and the Hamiltonian is
(cid:18) 2M(cid:19) dr2
ds2 =− 1− dt2+ +r2dΩ2, (1) (cid:114) (cid:115) (cid:18) (cid:19)
r 1−2M/r 2M 2M
H =pr˙−L= 1− m2+p2 1− , (9)
r r
where
which reduces to the free particle Hamiltonian in the
dΩ2 =dθ2+sin2θdφ2. (2) M → 0 limit. The equation of motion derived from the
Euler-Lagrange equations is
Throughout this paper we use geometric units with G=
c=1. 3M (cid:18)dr(cid:19)2 (cid:18) 2M(cid:19)d2r M (cid:18) 2M(cid:19)2
− + 1− + 1− =0,
TheOppenheimer-Snyder(O-S)modelfollowsthecol- r2 dt r dt2 r2 r
lapse of a star that is idealized as a dust sphere with (10)
3
which can be integrated to A. Classical Paths in Schwarzschild Coordinates
(cid:18) 2M(cid:19)2(cid:20) 2M (cid:21)
r˙2 = 1− c + (1−c ) . (11) Schwarzschild coordinates allow us to explore the
r 1 r 1
space-time outside r = 2M. The possible paths taken
From this the classical paths are determined by a particle on the surface of a collapsing star are writ-
ten explicitly in Table I. Every point (r ,t ) may be
(cid:90) tf (cid:90) rf r2 reached either via a direct path or via an findfirect path.
dt=t −t =± dr , (12)
f i x (r−2M) Table II contains the boundary regions that determine
ti ri r
whether a path is direct or indirect, and whether it lies
where
inside or outside the light cone.
(cid:112)
x = c r2+2Mr(1−c ). (13) The table uses
r 1 1
(cid:112)
The + sign represents motions from ri to rf > ri, and x(rs)=xs = c1rs2+2Mrs(1−c1), (19)
− sign represents the star collapsing from r to r < r .
i f i √
Theparameterc1 canbeusedtoseparatethespace-time βs = c1xs+c1rs+M(1−c1), (20)
regions
x (cid:18)(r +x )2(cid:19)
α = s +2Mlog(r −2M)−2Mlog s s ,
ds2 >0⇐⇒ c >1 outsidethelightcone (14) s c s r
1 1 s
ds2 =0⇐⇒ c =1 onthelightcone (21)
1
where s=i,f,a.
ds2 <0⇐⇒ c <1 insidethelightcone
1 The direct paths include a logarithmic term when
(15) c > 0 and an arc sin term when c < 0. The indirect
1 1
pathsoccurforr intheb −b region. Theyeachcontain
In the free particle case (M →0 in Eq. (11)), c =r˙2 i 3 4
1
azerovelocitypointwherethepathturnsaround. Math-
represents the velocity squared of the particle.
ematically, the velocity passes through zero (dr/dt = 0
Stellar Surface at Rest Limit atsomeradiusr )whenx(r )=0,whichcorrespondsto
a a
c =−2M/(r −2M). Theindirectpathsreachthefinal
When the surface of the star is initially at rest, initial 1 a
point(r ,t )aftergoingthroughazerovelocitypointat
velocity f f
r (t <t ) with r >r and r >r . The paths from r
(cid:12) (cid:12) a a f a f a i i
(cid:12)dr(cid:12) to r correspond to the positive sign in Eq. (12), while
(cid:12) (cid:12) =0. a
(cid:12)dt(cid:12)r=ri theTopautnhdferrosmtanradttohersfecpoartrhesspwonedmsatokethaennaengaaltoigvye swigitnh.
From Eqs. (11), this corresponds to
the vertical motion of a ball under gravity (See Fig. 1).
2M This analogy arises naturally due to the correspondence
c1 =−r −2M. (16) betweentheFriedmannequationandtheequationofmo-
i tionforatestparticleinthegravitationalfieldofapoint
We can recover our equations of motion for this c1 value mass in Newtonian gravity. It can be made only in the
fromtheO-Smodel. Weproceedbytakingthederivative Schwarzschild case because then the interior of the star
of Eq. (3) and Eq. (5) w.r.t. η can be represented by an FRW model, which provides
the link to Newtonian gravity.
dr = −ri sinη =−(cid:112)r(r −r) (17) Consider a ball traveling from point r (initial radius)
dη 2 i i
to point r (final radius) with r below r . It can either
dt = ri2 (cid:114) ri −1(cid:20) cos4(η/2) (cid:21) go down dfirectly from ri to rf (fdirect patih) or it can go
dη 2M 2M (r /2M)cos2(η/2)−1 upfromr ,reachitshighestpoint,whereitsvelocitywill
i i
r2 (cid:114) r be zero, and then fall back down to rf (indirect path).
= i −1, This is shown schematically in Fig. 1(a). Analogously, if
r−2M 2M
a ball has to travel from r to r with r above r , it can
i f f i
whereweusedthatr =r cos2η/2. Wethendividedr/dη go directly from r to r or it can pass above r , reach
i i f f
by dt/dη, and use r = 2M(c −1)/c from Eq. (16) to its zero velocity point above r and return down to r .
i 1 1 f f
recover the equation of motion This case can be seen in Fig. 1(c). The middle figure
(Fig. 1(b)) shows the special case when r = r . Then
dr r−2M(cid:112) i f
=− c r2+2Mr(1−c ). (18) theballcaneitherstayputoritcanbethrownup,reach
dt r2 1 1
itshigherpoint, andfallbackdown. Eachofthesepaths
The negative sign − fits in with our convention for a takesadifferentamountoftimetocomplete. Forafixed
collapsing star. While some paths will have low proba- traveltime∆t,thetwopointsr andr areconnectedby
i f
bility, a path integral approach that includes quantum asinglepaththatiseitheradirectpathoraturn-around
effects requires the consideration of space-like, time-like path. At the turning point of a turn-around path (also
and light-like paths with all possible initial velocities. itshighestpoint)thespeedoftheparticleisalwayszero.
4
Range of r c Equations of Motion
i 1
(cid:16) (cid:17)
2M−b2 c1>0 tf −ti=αf −αi+M(c31c1√−c11)(cid:104)log ββfi(cid:16) (cid:17) (cid:16) (cid:17)(cid:105)
b2−b3 −rf2−M2M <c1<0 tf −ti=αf −αi+ Mc1(3√c−1−c11) arcsin(cid:104)M−(c11−rcf1) −1 −arcs(cid:16)in M−(1c−1rci1) −(cid:17)1 (cid:16) (cid:17)(cid:105)
b3−b4 −ra2−M2M tf −ti=2αa−αf −αi+Mc((cid:104)31c√1−−c11) (cid:16)2arcsin(1)−(cid:17)arcsin M−(cid:16)(c11−rcf1) −1 −(cid:17)a(cid:105)rcsin M−(1c−1rci1) −1
b4−b5 −b42−M2M <c1<0 ti−tf =αf −αi+ Mc1(3√c−1−c11) arc(cid:16)sin (cid:17)M−(c11−rcf1) −1 −arcsin M−(1c−1rci1) −1
b5−∞ c1>0 ti−tf =αf −αi+M(c31c1√−c11)log ββfi
TABLE I: Closed-form solution to Eq. (12) in Schwarzschild coordinates. At each region boundary b (j =2−5) the
j
equations converge to the form given in Table II. All discontinuities cancel resulting in smooth transitions between
regions without singularities.
r c Regional Boundaries
i 1
(cid:16) (cid:17)
b 1 t −t =r −b +2Mlog rf−2M
b12 0 tff −tii = 3f√√M2 (cid:16)1rf3/2−b23/2(cid:17)b+1−22√M2M(cid:0)√rf −√b2(cid:1)+2Mlog(cid:32)(cid:32)11+−(cid:114)(cid:114)22rMMf (cid:33)(cid:33)(cid:16)(cid:16)11−+(cid:113)(cid:113)22bMM2 (cid:17)(cid:17)
rf b2
b3 −rf2−M2M tf −ti =2Mlog(cid:16)1− 2rMf (cid:17)− xcb13 +2Mlog(cid:20)b(3x(bb33+−b23M)2)(cid:21)+M(c31c√1−−c11)(cid:104)arcsin(1)−arcsin(cid:16)M−(1b3−cc11) −1(cid:17)(cid:105)
bb45 −0 b42−M2M ttii−−ttff == x3c√√1fM2−(cid:16)2rMf3/l2o−g(b153/−2(cid:17)2bM+4 )2−√22MM(cid:0)lo√gr(cid:104)fr−(fx(fr√f+−br5f2(cid:1)M)2+)(cid:105)2+MMlog(c31c√1−(cid:32)(cid:32)−c1111+−)(cid:114)(cid:114)(cid:104)−22rMMfar(cid:33)(cid:33)c(cid:16)(cid:16)s11in−+(cid:113)(cid:113)(122)bMM5+(cid:17)(cid:17)arcsin(cid:16)M−(r1f−cc11) −1(cid:17)(cid:105)
rf b5
(cid:16) (cid:17)
b 1 t −t =r −b +2Mlog rf−2M
6 i f f 6 b6−2M
TABLE II: Region boundaries for the equation of motion (Eq. (12)). The first column is the border value of r for
i
each region given the r and c values. It is calculated using the equation of motion in the third column. The light
f 1
cone boundary corresponds to c =1.
1
Ifwekeepthetimeintervalconstant,wefindthatfora We continue with the ball analogy to estimate the es-
given r there is a point r =b >r , where the velocity capevelocityforourparticle. Inourcase,R isreplaced
f i 4 f E
dr/dt at b = 0. Similarly, there is a point r = b < r by the initial radius of the star r , and v =dr/dτ. The
4 i 3 f i i
such that the velocity dr/dt at r = 0. For any point r kinetic energy of the particle on the surface of the star
f i
between b and b , the only way a particle can reach r equals its potential energy when
3 4 f
in the same time ∆t is if it goes through a turning point
r > r and r > r . The particle cannot reach r > r Mm m(cid:12)(cid:12)dr(cid:12)(cid:12)2
a f a i f i = (cid:12) (cid:12) . (23)
if it’s velocity vanishes at any point between ri and rf. ri 2 (cid:12)dτ(cid:12)r=ri
Themotiontowardsorawayfromr =2M canbecom-
Here
paredtotheverticalmotionofaballundergravitywhen
thrown straight down or up, respectively. Classically, dr (cid:18) 2M(cid:19)−1 dr
= 1− (24)
the star will always collapse to a black hole. Quantum
dτ r dt
mechanically, we include all possible initial velocities of
the surface of the star towards and away from 2M. The with dr/dt=r˙ given by Eq. (11). This results in
wavefunction of a particle on the surface of the star de-
Mm mc Mm
terminesthespreadofthetrajectoryofthesurfaceofthe = 1 + (1−c ), (25)
r 2 r 1
star itself. i i
Escape Velocity. For a particle of mass m that is whose solution is c = 0. So, while all outgoing paths
1
thrown up from the surface of the Earth, an initial up- withc <0willturn-around,outgoingpathswithc ≥0
1 1
wardvelocityv ≥11.2km/secensuresthatthepar- will not have turning points. When c = 0, the turn-
escape 1
ticle escapes Earth gravity. At this speed ing point (dr/dτ| = 0, x = 0) occurs at r → ∞ as
ra a a
expected for an escaped particle.
GM m mv2
E = i , (22) The different paths (Table I) and their various re-
R 2
E gional boundaries (Table II) are summarized below for
where M , and R are the Earth mass and radius. a given (r ,t ) outside r =2M:
E E f f
5
FIG. 1: Schematic of paths. • c =−2M/(r −2M): r = b < r . The mini-
1 f i 3 f
mum value of r where r is reached directly with
i f
(a) r <r x(r )=0 and null final velocity
f i f
dr
| =0.
dt r=rf
Paths with r ≤ b are direct. Paths with r be-
i 3 i
tween b and b are indirect.
3 4
• c =−2M/(b −2M): r = b > r . This path
1 4 i 4 f
startswithzeroinitialvelocity. Allpathswithr >
i
b are direct.
4
III. CLASSICAL PATHS IN KRUSKAL
COORDINATES
Our study is the first to consider space-like paths in
Kruskalcoordinatesforaparticleonthesurfaceofacol-
(b) r =r
f i
lapsing star. They have not been explicitly considered
beforebecausetheyarebelievedtobeoflowprobability.
We conjecture that they play an important role in quan-
tummechanicalcollapseinthesamewaytheprobability
of passing through the potential barier is important in
tunneling. Classically, it can never happen, and yet this
probability cannot be ignored in quantum mechanics.
In this section we derive and discuss the closed-form
solutions to the equation of motion for the space-like,
light-likeandtime-likepathsinKruskalcoordinates. We
find that space-like geodesics have the interesting prop-
(c) rf >ri erty that they can turn around outside r = 2M and
move back in time. Unlike the Schwarzschild t and r,
which are time and space coordinates respectively out-
side r = 2M but switch roles for r < 2M, the Kruskal
coordinate v is always a time coordinate and the coordi-
nateuisalwaysaspacecoordinate. Therelationbetween
theSchwarzschildrandtandtheKruskalvariablesuand
v is given by
(cid:40)(cid:112) r −1er/4Mcosh t r >2M
u= 2M 4M
(cid:112)1− r er/4Msinh t r <2M
2M 4M
(cid:40)(cid:112) r −1er/4Msinh t r >2M
v = 2M 4M
(cid:112)1− r er/4Mcosh t r <2M
2M 4M
with
(cid:16) r (cid:17)
u2−v2 = −1 er/2M (26)
2M
and
• c =1: thepathsarelight-like(ds2 =0)withr =
1 i
v t
b < r and r = b > r . The paths with c >1
1 f i 6 f 1 = tanh , r >2M (27)
are space-like, while those with c <1 are time- u 4M
1
like. u t
= tanh , r <2M.
v 4M
• c =0: r = b < r and r = b > r . This Thus the line element in Kruskal coordinates takes the
1 i 2 f i 5 f
path defines the escape velocity. All paths with form
cth1e>pa0rtthicalet atoreesocuatpgeoitnhgewgirlalvnitoyt orefvtehreses,taarll.owing ds2 = 32M3e−r/2M(cid:0)du2−dv2(cid:1)+r2dΩ2. (28)
r
6
The singularity at r = 0 occurs at u2 −v2 = −1. In from the equations in Table IV. Note that u is a func-
f
this paper we consider only the first quadrant u>0 and tion of v and r and u depends on v , which is typ-
f f i i
v > 0. The horizon r = 2M occurs at u = v. Outside ically taken to be zero, and r . Thus if one finds r ,
i i
the horizon u>v, whereas inside the horizon u<v. this determines u . The point r = b(cid:48) corresponds to
i i 4
The classical equation of motion x = 0 with c = −2M/(b(cid:48) − 2M). The c = 0 and
i 1 4 1
c = 1 (lightlike) paths that reach points (u ,v ) in-
1 f f
u¨ (cid:18) 4M2(cid:19) u−vu˙ side the horizon are used to determine the r values b(cid:48)
= 1− (29) i 5
1−u˙2 r2 u2−v2 and b(cid:48)56 respectively. The ri = b(cid:48)6 value corresponds to
x = 0 with c = −2M/(r − 2M) (turning point at
f 1 f
can be derived from the Euler-Lagrange equations r = r < 2M, space-like path, c > 1). The value of
f 1
r =b(cid:48) istobedeterminedbyusingc =−2M/(r −2M)
i 7 1 a
d ∂L ∂L (x = 0,r < r < 2M) in the equation of motion. The
− =0, (30) a a f
dv ∂u˙ ∂u two unknowns b(cid:48) and r are determined by using the
7 a
equationofmotionanditspartialderivativewithrespect
for the Lagrangian
to c . Both equations are given in Table IV. In the re-
1
gionoutsideb(cid:48) (r >b(cid:48)),thereexistnoclassicalpathsto
(cid:115) 7 i 7
(1−u˙2)(1−2M/r) (u ,v ).
L(u,u˙,v)=−4M . (31) f f
(u2−v2) In Fig. 2(a) and (b) we trace direct time-like, di-
rect space-like, and indirect space-like trajectories that
Note that here u˙ =du/dv. start at vi = 0 and pass through a fixed point P(uP =
The equation of motion can be reduced to 9.995,vP = 10.0) with rP ≈ 0.8M, which is located in-
sidetheapparenthorizon. Path 1andpath 2aretime-
du v/u±x /r like, and originate at different points outside r = 2M.
dv = 1±(v/u)(rxr/r), (32) Path 1 (ui = 2.59, c1 = −1.05) lies in the b(cid:48)4 −b(cid:48)5 re-
gion of Table III. Path 2 (u =16.15, c1=0.8) lies in
i
where xr is given by Eq. (13). Direct paths from ri the b(cid:48)5−b(cid:48)56 region, which contains time-like paths with
to r > r have the “+”-sign in the numerator and de- a turning point in the u−v plane (du/dv = 0) that lies
f i
nominator with du/dv > 0. Direct paths from r to r outside the apparent horizon.
i f
withr <r willhavethe“−”-signinthenumeratorand The next region is b(cid:48) − b(cid:48), where each point out-
f i 56 6
denominator. side the horizon is connected to points inside the hori-
Theclassicalpathsdescribingtheevolutionofthestar zon by two space-like paths: one direct and one indi-
outside r = 2M have been discussed in Sec. IIA in rect. Path 3a and path 3b originate at the same
Schwarzschild coordinates. It was shown that there were u = 27.6 (r = 10.5M). Path 3a reaches P directly.
i i
directspace-likeandlight-likepathsaswellasdirectand However, after having passed through P, it turns at
indirect time-like paths. Any two points in (r,t) were r = 0.48M before reaching the horizon like all space-
a
connected by a unique classical path. like paths. Path 3b is indirect turning at r = 0.3M
a
What is new in Kruskal space-time are the turning (ua = 11.3,va = 11.35), and then reaching P on its way
points in the u−v plane for all space-like, light-like and backtowardstheapparenthorizon. Bothpathsalsohave
time-like paths. Spacelike paths (c > 1, |du/dv| > 1) av−uturningpointoutsidetheapparenthorizonwhere
1
turn in time at points when dv/du = 0, whereas time- dv/du=0. Afterhavingpassedthroughitsv−uturning
like paths (c < 1, |du/dv| < 1) turn in space when point, each path moves back in time (the space coordi-
1
du/dv = 0. Additionally, there are indirect space-like nate u continuously decreases, while the time coordinate
and light-like paths that have turning points in r inside v increases up to the turning point and then decreases).
the apparent horizon (here we refer to paths as being in- In the r = b(cid:48) − b(cid:48) region there are two indirect
i 6 7
direct when they turn in r; the light-like paths turn at paths that connect the same point outside the horizon
r = 0.) In contrast to the time-like paths, the in-going to a point inside r = 2M. Paths 4a and 4b con-
space-like paths are not unique. A point outside r =2M nect the point of u = 46.3 to P. They each have a
i
might be connected to a point inside the apparent hori- dv/du = 0 turning point outside the horizon, and also
zon via a direct and an indirect space-like path or via turn in r inside the horizon before reaching P. Path
two indirect space-like paths. 4a turns at r = 0.77147 < r and path 4b turns at
a P
We determine the regions (Table III) and the light- r = 0.79969 < r . It can be seen that the paths are
a P
cone boundaries (Table IV) that delimit the different very close together. As u increases, the paths in this
i
kinds of paths that start at (u ,v =0) outside the hori- region become progressively closer until they merge at
i i
zon and reach (u ,v ) inside the apparent horizon. Out- r =b(cid:48). Path 5showsthesingleindirectpaththatorig-
f f i 7
side the horizon, the light cone boundary values of ‘u ’ inates at u = 46.7 (r = b(cid:48) = 12.13M). Beyond this
i i i 7
for a given u and v can be determined by replacing point, there are no real solutions and hence no way to
f f
t − t in Table II by tanh−1(v /u ). The bound- reach point P.
f i f f
ary values of r (v = 0) with r < 2M, are calculated Fig. 2(c) displays the c values of space-like paths
i i f 1
7
Range of r c Equations of Motion for Paths with r >2M and r <2M.
i 1 i f
(cid:16) (cid:17) (cid:104) (cid:16) (cid:17) (cid:16) (cid:17)(cid:105)
b(cid:48)4−b(cid:48)5 −b(cid:48)42−M2M <c1 <0 −4Mtanh−1(cid:16)uvff(cid:17)=αf −αi+Mc(31c√1−−c11) arc(cid:16)sin(cid:17)M−(c11−rcf1) −1 −arcsin M−(1c−1rci1) −1
bb(cid:48)5(cid:48)56−−bb(cid:48)6(cid:48)7 0c1<=c−1 <ra2−−M2rMf2−M2M −−44MMttaannhh−−11(cid:16)uuvvffff(cid:17)==α2αfa−−ααif+−Mα(ic3+1c1√−Mc11)(lc3o1cg1√−c11β)βfilog(cid:16)ββiβa2f(cid:17)
TABLE III: The equation of motion with r <2M. At the light cone boundaries they converge to the form given in
f
Table IV. All discontinuities cancel each other and there are no singularities anywhere in the equations of motion.
r c Light Cone Boundaries
i 1
(cid:16) (cid:17) (cid:12) (cid:12) (cid:104) (cid:16) (cid:17)(cid:105)
b(cid:48)4 −b(cid:48)42−M2M −4Mtanh−1 uvff =αf −2Mlog(cid:12)(cid:12)1− 2bM(cid:48)4 (cid:12)(cid:12)+M(c31c√1−−c11) −arcsin(1)+arcsin(cid:32)(cid:114)M−(r1f−cc11)(cid:33)(cid:32)−1(cid:114) (cid:33)
b(cid:48)5 0 −4Mtanh−1(cid:16)(cid:16)uvff(cid:17)(cid:17)= 3√√22M (cid:16)(cid:12)rf3/2−b(cid:48)53/2(cid:17)+2√(cid:12)2M(cid:16)√rf −(cid:112)b(cid:48)5(cid:17)+2Mlog(cid:32)1+2r(cid:114)Mf2r−Mf1(cid:33)(cid:32)11−+(cid:114)22bbMM(cid:48)5(cid:48)5 (cid:33)
b(cid:48) 1 −4Mtanh−1 uf =2Mlog(cid:12) 4M2 (cid:12)−r −b(cid:48)
56 (cid:16)vf(cid:17) (cid:12)(cid:16)(rf−2M)((cid:17)b(cid:48)56−2M)(cid:12) f 56 (cid:18) (cid:19)
bb(cid:48)6(cid:48)7 −−rrfa22−−MM22MM −−44MMttaannhh−−11(cid:16)uuvvffff(cid:17)==22Mαal−ogαf2r−Mf α−b71(cid:48) +−Mα(b3c(cid:48)61c1√+−c1M1)l(o3cg1c1√(cid:18)−c11β)bβ(cid:48)7la2oβfg(cid:19),c1rf+βMb(cid:48)6(1−c1)
x2b(cid:48)7+42Mc21bx(cid:48)7b((cid:48)71−c1) + x2f+4M2c21rfxf(1−c1) − 3M2c(211√−cc11)log βbβ(cid:48)7aβf =0
TABLE IV: Equations of motion at light cone boundaries for paths with r <2M. Outside 2M the equations for
f
Table I hold with t −t replaced by 4Mtanh−1(v /u ). Like before, the value c =1 corresponds to a light-like
f i f f 1
path. Note that b(cid:48) is the caustic point where both the path f(−2M/(r −2M),u,v) and its partial derivative
7 a
∂f/∂c vanish.
1
reaching point P (u =9.995,v =10.0) from u ≥46.1 all points on this path must have r > r . By the same
P P i a
at v =0. At first, for each u in the figure there are two token,fortime-likepathsr >2M andr <r . Thiscase
i i a a
c values with which the final point P can be reached. was described in the Schwarzschild analysis.
1
The figure clearly shows the roots (c1 values) coming Allspace-likepathsthatreachpointsinsidethehorizon
closer and closer together until they merge at the caus- subsequently head towards the horizon r =2M (u=v).
tic point r = b(cid:48)7. Beyond this point there are no paths Classically, they end at uf =vf =0. However, quantum
thatreachpointP. Extensionstothisworktodetermine mechanically there may be paths close to the classical
the Kruskal-WKB wavefunction will require an analysis path that bring information from within the black hole
of the caustic at r = b(cid:48)7 to remove any potential diver- horizon to the outside.
gences.
The indirect paths connecting (u ,v ) with r > 2M
i i i
to (u ,v ) with r < 2M penetrate the horizon deeper
P P P
than rP turning at ra <rP before reaching rP (e.g., see IV. QUANTUM TREATMENT IN
the green path in Fig. 2(a)). After turning, all paths SCHWARZSCHILD COORDINATES
must continue towards the apparent horizon reaching it
at u = v = 0. These paths move back in time from a
We consider a collapsing cloud of ultra-light particles
point outside the horizon, where dv/du = 0, after hav-
with mM ∼¯h. Such particles are not localized and thus
ing traveled to that point forward in Kruskal time from
neither is the surface of the star. The position of a par-
(u ,v =0) with r >2M.
i i i ticle on the surface of the star is then approximated by
As discussed before, the turning points in r corre- a wavefunction. In a relativistic path integral approach,
spond to xr = 0 (in the Kruskal paths xr appears in thiswavefunctioncan be computedin theWKBapprox-
Eq. (58)). They occur at r = ra when xa = 0, where imationviaanintegralovertheclassicalaction[24]. Itis
c1=−2M/(ra−2M). Since c1 >1 for space-like paths, necessarytoincludeconfigurationswillallpossibleinitial
clearly ra < 2M (the turning points are inside the hori- velocities. For each point (rf,tf) all classical paths de-
zon). For any r on a path with ra as the turning point, rived above are needed. In this paper, we only perform
(cid:112)
x can be written as 2Mr(r−r )/(2M −r ). To en- the WKB analysis in Schwarzschild coordinates, where
r a a
surethenon-negativityofthetermunderthesquareroot we can only use the paths to r >2M.
f
8
(dirce1ct>p0ath) Scl ±im√c1−1(cid:104)xfc−1xi − Mc(11√−cc11)log(cid:16)ββfi(cid:17)(cid:105)
∂2Scl ∓imri2rf2(cid:40)(cid:18)xi+4M2cr21ix(i1−c1)(cid:19)−(cid:32)xf+4M2c21rfxf(1−c1)(cid:33)−3M2(c115−/2c1)log(cid:18)ββfi(cid:19)(cid:41)−1
∂ri∂rf 2XiXf(ri−2M)(rf−2M)(c1−1)3/2
c1 <0 S ±im√c −1(cid:104)xf−xi − xf−xi (cid:16)arcsin(cid:16) −c1rf −1(cid:17)−arcsin(cid:16) −c1ri −1(cid:17)(cid:17)(cid:105)
(direct path) cl 1 c1 c1 M(1−c1) M(1−c1)
∂2Scl ∓imri2rf2(cid:40)(cid:18)xi+4M2cr21ix(i1−c1)(cid:19)−(cid:32)xf+4M2c21rfxf(1−c1)(cid:33)+32Mc21(√1−−cc11)(cid:104)arcsin(cid:16)Mc(11−rfc1)−1(cid:17)−arcsin(cid:16)M(c11−ric1)−1(cid:17)(cid:105)(cid:41)−1
∂ri∂rf 2XiXf(ri−2M)(rf−2M)(c1−1)3/2
c =− 2M √ (cid:104) (cid:16) (cid:16) (cid:17) (cid:16) (cid:17)(cid:17)(cid:105)
(in1directrap−a2tMh) Scl ±im c1−1 −xfc−1xi − Mc(11√−cc11) 2arcsin(1)−arcsin 2rrai −1 −arcsin 2rraf −1
∂2Scl +imri2rf2(cid:40)(cid:18)xi+4M2cr21ix(i1−c1)(cid:19)−(cid:32)xf+4M2c21rfxf(1−c1)(cid:33)−32Mc21(√1−−cc11)(cid:20)arcsin(cid:18)2rraf−1(cid:19)−arcsin(cid:16)2rrai−1(cid:17)−2arcsin(1)(cid:21)(cid:41)−1
∂ri∂rf 2XiXf(ri−2M)(rf−2M)(c1−1)3/2
TABLE V: The classical action in various regions. The + sign is chosen when r >r , and the − sign when r <r .
f i f i
A. WKB Approximation: Closed-form Propagator, with r >>2M.
c
Numerical Wavefunction
The action from Eq. (6) is rewritten using Eqs. (11)
and (13) as
In order to study the wavefunction we use a WKB
approximation of the propagator
(cid:90) √ (cid:90) (rf,tf) r
G(r,t;ri,ti)= Drexp[iS/¯h], (33) Scl(ri,ti;rf,tf)=±im c1−1 drx(r) (37)
C (ri,ti)
where S is the action associated with each path. The
set of paths C include space-like, time-like and light-like fordirectpathsconnecting(r ,t )to(r ,t ). The+sign
i i f f
paths. corresponds to r > r and the − sign corresponds to
f i
The WKB approximation involves expanding about r <r .
f i
the classical action S to include paths that slightly de-
cl
Indirectpathsconnecting(r ,t )to(r ,t )throughthe
viate from the classical paths. The propagator is dom- i i f f
turning point (r ,t ) with r > r and r > r are de-
inated by paths near the classical trajectory between a a a i a f
scribed by the classical action
(r ,t ) and (r,t) and is approximated by the WKB ex-
i i
pression [24]
(cid:115) i ∂2S (cid:18) S (cid:19) √ (cid:34)(cid:90) (ra,ta) r (cid:90) (rf,tf) r (cid:35)
GWKB(r,t;ri,ti)= 2π¯h∂ri∂rclf exp i ¯hcl . (34) Scl =im c1−1 (ri,ti) drx(r) − (ra,ta) drx(r) .
(38)
TheWKBwavefunctionisobtainedfromtheintegration
In Schwarzschild coordinates, the wavefunction is con-
of the propagator
tinuous across all regions outside r =2M. The presence
(cid:90) ∞
ofther =2M coordinatesingularityprohibitstheinclu-
Ψ (r ,t )= G (r ,t ;r ,t )Ψ(r ,t )dr .
WKB f f WKB f f i i i i i
sion of paths at or beyond the horizon region. Closed-
0
(35) form expressions for S and ∂2S /(∂r ∂r ) are given in
cl cl i f
The initial wavefunction that describes a particle on Table V. Both functions are continuous across all re-
thesurfaceofastarofradiusri istakentobeaGaussian gions outside r =2M. Once c1 is known for every point
centered about rc (r,t), we have all the ingredients to find the propagator
(cid:34) (r −r )2m2(cid:35) in Eq. (34) using the Scl, and ∂2Scl/∂ri∂rf expressions
Ψ(r ,t =0)=exp − i c (36) fromTable V.Thepropagatoristhenintegratedtofind
i i ¯h2 the wavefunction.
9
the wavefunction in the WKB approximation converges
16 to the relativistic Schr¨odinger solution, which is the ex-
14 act solution for this case. In [7] some disagreement was
observed, but while they attempted a physical interpre-
12
tation, it is likely due to numerical error.
10
P When M →0, the Hamiltonian from (9) reduces to
v 8
(cid:112)
6 H = p2+m2, (39)
4
where p is now the momentum operator. The Hamilto-
2 nian is nonpolynomial in p. The square root term thus
0 corresponds to a non-local momentum operator that is
0 2 4 6 8 10 12
r interpreted as acting on any wavefunction [7]
16 (cid:90) ∞
Path 5 Ψ(x,t)= dkeikxφ(k,t). (40)
14 Path 4a
−∞
Path 4b
12 Path 3a
Path 3b P to give
10 Path 2 x
v 8 Prf a=t h2 M1 HΨ(x,t)=(cid:90) ∞ dkeikx(cid:112)¯h2k2+m2φ(k,t). (41)
6 −∞
4
This integrates to
2
(cid:18) (cid:19)
i∆t(cid:112)
00 5 10 u 15 20 25 φ(k,t)=A(k,x0)exp − ¯h ¯h2k2+m2 . (42)
When ∆t = 0, we recover the original wavefunction,
whichischosentobeaGaussiancentredaroundtheori-
1.66 gin,
Root 1
1.65 Root 2 (cid:18) m2x2(cid:19)
Ψ(x ,t )=N exp − 0 ,
c1 0 0 ¯h2
1.64
and so
1.63 Ψ(x,t) = N (cid:90) ∞ dk(cid:90) ∞ dx exp(cid:0)−m2x2/¯h2(cid:1)(43)
2π 0 0
1.6426 .1 46.2 46.3 46u.i4 46.5 46.6 46.7 × exp(−ik∞∆x)ex−p∞(cid:18)−i∆t(cid:112)¯h2k2+m2(cid:19).
¯h
√ √
FIG. 2: (a) v vs r and (b) v vs u are shown for The normalization |N|2 =m 2/(h¯ π) is time indepen-
space-like and time-like paths with v =0 that pass dent.
i
through P(u =9.995,v =10.0) of r =0.8, a point In terms of the propagator
P P P
located inside the apparent horizon. (c) The c values
1 (cid:90) ∞
for indirect space-like paths in the region between b(cid:48)
6 Ψ(x ,t )= dx G(x ,t ;x ,t )Ψ(x ,t ), (44)
and b(cid:48) from which point P can be reached are displayed f f 0 f f 0 0 0 0
7 −∞
for each u . No paths exist beyond r =b(cid:48), which is
i 7
where
where the two roots merge into one.
(cid:90) ∞ dk
G(x,t;x ,t ) = exp(ik∆x)
0 0 2π
−∞
B. Relativistic Schro¨dinger Solution (cid:20) (cid:21)
i∆t(cid:112)
×exp − ¯h2k2+m2 ,
¯h
1. Free Particle
m(i∆t+(cid:15))
= lim √ K (mλ1/2/¯h).(45)
(cid:15)→0+ π¯h λ(cid:15) 1 (cid:15)
The M → 0 limit describes a relativistic free particle
that is no longer confined by the gravity of the sphere Here K is the modified Bessel function and λ ≡
1 (cid:15)
of dust. This case was first studied by Redmount and ∆x2+(i∆t+(cid:15))2. When ∆t=0 the propagator reduces
Suen in [7]. We revisit this limit to investigate whether to δ(∆x) as expected.
10
0.3
Re[Ψ] 0.12 t=5
0.2 Im[Ψ] t=10
|Ψ|2 0.1 t=15
0.1 t=20
0.08 t=25
2
0 |B t=30
K0.06
W t=35
Ψ
−0.1 | t=40
0.04
t=45
−0.2
0.02
0
0 2 4 6 8 10 12
x
f 0 20 40 60
r
f
FIG. 3: Re[Ψ], Im[Ψ] and |Ψ|2 are shown for the free
particle case at t=10h¯/m as a function of xf, which FIG. 4: The time evolution of WKB |Ψ(rf,tf)|2 is
also has units of ¯h/m. The solid lines show the shown from t=5M to t=45M in the case of
Schr¨odingersolution, whiledottedlinesanddashedlines mM/¯h=1. We can see that the star collapses to a
show the WKB approximation for (cid:15)=0.05, (cid:15)=0.005 black hole.
respectively. It can be seen that as (cid:15)→0 the WKB
approximation converges to the Schr¨odinger solution.
where p is the momentum operator and
2M
The WKB propagator for the relativistic free particle B =B(M,r)=1− . (49)
is then computed using Eq. (34) [7] r
The whole wavefunction can then be written as
(cid:115) (cid:34) (cid:35)
m (i∆t+(cid:15))2 mλ1/2
G = exp − (cid:15) , (46) (cid:90)
WKB 2π ¯hλ3(cid:15)/2 ¯h Ψ(r,t)= driG(r,t;ri,ti)Ψ(ri,τi), (50)
where WKB wavefunction is given by
where the propagator is given by
(cid:90) ∞ (cid:90) ∞ dk
Ψ (x,t)= dx G (x,t;x ,t )Ψ(x ,t ).
WKB 0 WKB 0 0 0 0 G(r,t;r ,t ) = exp(ik∆r) (51)
−∞ i i 2π
−∞
(47)
(cid:20) (cid:21)
∆t(cid:112)
The integrals from Eq. (44) and Eq. (47) are evaluated × exp −i ¯h2k2B2+m2B .
numerically. The results are shown in Fig. 3. It can be ¯h
seen that the wavefunction in the WKB approximation
Like before, the propagator can then be integrated ex-
converges to the Schr¨odinger solution in all parts of the
actly to obtain
light cone. When no star M is present, the relativistic
Schr¨odingerisexactasoriginallydiscussedinRedmount m(i∆t+(cid:15))
and Suen [7]. However, when they performed the same G(x,t;xi,ti)=(cid:15)l→im0π¯hB1/2λ1/2K1(mB1/2λ1(cid:15)/2/¯h),(52)
numericalcomparison,theyfoundsomedisagreementbe- (cid:15)
tween the two solutions. This we believe to be due to where K is the modified Bessel function and
1
numerical error of [7]. We observe convergence (see Fig.
3.) of the wavefunction in the WKB approximation to (cid:18)∆r(cid:19)2
λ = +(i∆t+(cid:15))2. (53)
the Schr¨odinger solution in the M →0 limit. (cid:15) B
When r → 2M the propagator vanishes. Thus this so-
2. Non-zero mass lution is inaccurate at late times and cannot model the
final stages of the collapse of the dust sphere.
WhileinthefreeparticlecasetheSchr¨odingersolution
was exact, when M (cid:54)= 0 it becomes a rough approxima-
tion,whichfailsasweapproachr =2M. Comparingthe C. Numerical Results
two solutions is still instructive. If we follow the same
procedure as for the free particle above, and re-write the We integrate Eq. (35) numerically for the Gaussian
Hamiltonian from Eq. (9) as wavefunction from Eq. (36) centered about r = 30M.
c
Classically, the star collapses to r = 2M in infinite
(cid:112)
H = B2p2+Bm2, (48) Schwarzschild time. However, quantum mechanically
