Non-local (but also non-singular) physics at the last stages of gravitational collapse

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by  Anshul Saini

Non-local (but also non-singular) physics at the last stages of gravitational collapse
Anshul Saini, Dejan Stojkovic
HEPCOS, Department of Physics, SUNY at Buffalo, Buffalo, NY 14260-1500, USA
WestudytheendstagesofgravitationalcollapseofthethinshellofmatteriningoingEddington-
Finkelsteincoordinates. WeusethefunctionalSchrodingerformalism tocapturequantumeffectsin
thenearsingularitylimit. Wefindthatthattheequationsofmotionwhichgovernthebehaviorofthe
collapsing shell near the classical singularity become strongly non-local. This reinforces previous
arguments that quantum gravity in the strong field regime might be non-local. We managed to
solve the non-local equation of motion for the dust shell case, and found an explicit form of the
wavefunction describing the collapsing shell. This wavefunction and the corresponding probability
densityarenon-singularattheorigin,thusindicatingthatquantizationshouldbeabletoridgravity
of singularities, just as it was thecase with the singular Coulomb potential.
4
1
0 I. INTRODUCTION dust shell in ingoing Eddington-Finkelstein coordinates
2 and derive transformations that we will use later. In
section III we apply Gauss-Codazzi method to find the
r What happens at the last stages of the gravitational
a conserved quantity which has a clear interpretation of
collapse of some distribution of matter is still unknown.
M the Hamiltonian. In section IV we quantize this Hamil-
The reason is our lack of a fully fledged theory of quan-
tonianusingthe functionalSchrodingerformalisminthe
tum gravity which will fatefully describe quantum dy-
4
near singularity limit. As we mentioned, the functional
namics in very strong gravitationalfields (e.g. near clas-
Schrodingerequationisnon-local. Inthesamesectionwe
] sical singularities). Since the formulation of quantum
c solvethisequationexplicitlytofindthewavefunctionand
gravity still seems to be far from our reach, we have to
q probability density (which is non-singular). In section V
work with what we have at hand, and try to push it as
-
we repeat the same procedurefor the shell whose energy
r far as possible. Along the way, we might get a glimpse
g density is constant (i.e. spherically symmetric domain
of what the ultimate theory of quantum gravity should
[ wall). In section VI we quantize the domain wall shell
look like.
3 and find again that the equation governing the behavior
Thepurposeofthispaperistostudyquantumaspects
v near the origin is non-local. We do not explicitly solve
of gravitational collapse of a shell of matter in the con-
2 for the wavefuction in this case since the expressions are
8 text of the functional Schrodinger formalism [1–14]. We cumbersome. Finally, we give conclusionsin sectionVII.
1 willworkinEddington-Finkelsteincoordinateswhichare
6 convenientforstudyingthequestionoftheblackholefor-
. mationtill the very endwhere the collapsingmatter dis-
1 II. THE METRIC IN INGOING
0 tribution crossesits ownSchwarzschildradius andstarts EDDINGTON-FINKELSTEIN COORDINATES
4 approaching the classical singularity at the center. The
1 first interesting finding is that the equations of motion
In this section we will setup the metric of a collapsing
v: describing behavior of the collapsing shell near the clas- shell of matter in ingoing Eddington-Finkelstein coordi-
i sicalsingularitybecomenon-local. Ithasbeenarguedfor
X nates. Since this space-time foliation is non-singular at
awhilethat(forvariousreasons)quantumgravityshould
the Schwarzschild radius, it will allow us to study the
r ultimately be a manifestly non-localtheory[8,9, 15–17],
a the gravitationalcollapse as the shell is approaching the
(seealso[18–20]). Ourfindingisastrongindicationthat
classical singularity at the center.
something like that might indeed be true. While the
The radius of a collapsing spherically symmetric shell
functional Schrodinger formalism is not a full theory of
of mater is R. The parameter of evolution is the ingoing
quantumgravity,itshouldhowevercapturesomeaspects
nullcoordinatevrelatedtotheasymptoticSchwarzschild
of it. Non-locality might be one of those important as-
time as
pects.
Non-local equations are notoriously difficult to solve. v =t+r∗ (1)
However, manipulating the equations of motion in the
near singularity limit, we managed to find an explicit where r∗ is the tortoise coordinate. The trajectory of
solution to the non-localSchrodingerequation. Interest- the collapsing shell is then simply r =R(v). The metric
ingly enough, the solution for the wavefunction is non- outside the collapsing shell is
singular at the origin. In fact, the probability density
R
becomes zero exactly at the origin. This indicates that ds2 = 1 s dv2+2dvdr+r2dΩ2, r>R(v). (2)
− − r
quantization can perhaps remove classical singularities (cid:18) (cid:19)
from gravity, as argued from many different points of
By Birkhoff theorem, the interior metric is Minkowski
view [8, 9, 21–27].
In section II we setup the metric for the collapsing ds2 = dT2+dr2+r2Ω2,r <R(v) (3)
−
2
The interiortime coordinate,T,is relatedto the ingoing For our case of the thin shell of dust the surface stress
null coordinate, v, via the proper time on the shell, τ. energy can be written as
[Note that the proper time τ is different from the same
quantity in Schwarzschild coordinates since the space- Sαβ =σuαuβ, (12)
time foliation is different.] The relations are
where σ is energy per unit area of the shell. For the
constant rest mass of the shell, σ is not constant, and
2
dT dR
= 1+ (4) changes its value as the radius of the shell changes. Our
dτ s dτ extrinsic curvature tensor becomes
(cid:18) (cid:19)
and 1
[K ]=8πσ u u + (3)g . (13)
ij i j ij
2
(cid:18) (cid:19)
2
dv 1 dR dR
= B+ (5) Outside the shell the stress energy tensor [Tn ] will be
dτ B dτ −s dτ  i
(cid:18) (cid:19) zero. Then Eq. (11) gives
 
where dσ
+σum =0. (14)
dτ |m
R
s
B 1 . (6)
≡ − R The metric on the shell is
From Eq. (5), we can get, ds2 = dτ2+R2(τ)(dθ2+sin2θdφ2) (15)
−
R 2 = B (7) where τ is the proper time parameter of the observer
τ (1 B )2 1 locatedonthe shell. Forthis formofthe metric Eq.(14)
− Rv − becomes
From Eq. (4) and Eq. (5),
1
(σ((3)g)2ui)
,i
dv 1 =0 (16)
= (8) ((3)g)12
dT R 2 2R +B
v v
−
Since uτ = 1 and (3)g = R4(τ) it becomes 4πR2σ = µ.
We will use these relatipons when needed to convert from
We can think of µ as the rest mass of the shell which
one to another time coordinate.
always remains constant for the dust shell. Now we can
use Eq. (13) to find equations of motion
III. COLLAPSE OF THE DUST SHELL:
1
CONSERVED QUANTITY [K ]=8πσ u u + (3)g =µ (17)
θθ θ θ θθ
2
(cid:18) (cid:19)
Inthissectionwewillderiveaquantitywhichremains One can find K as
θθ
conserved during the collapse of the dust (pressureless)
shell. WewillthenidentifythisquantitywiththeHamil- K = n = 1n2g = rnr (18)
θθ θ;θ θθ,r
tonian. We will use the Gauss-Codazzi method, also − −2 −
knownas Israelapproachfor surfacelayers(see e.g. [28–
Combining Eq. (17) and Eq. (18) we get
30] and references therein).
Letusconsiderashellwhichisatime-likethree-surface r(nr+ nr−)=µ (19)
separating two regions of space-time. The extrinsic cur- − −
vature, Kij, in this case is discontinuous. The explicit where nr+,nr− are radial components toward exterior
form of the extrinsic curvature can be derived from Ein- and interior region respectively. The components of u
stein equations as and n can be evaluated by using the dot products n.u=
0, n.n=1 and u.u= 1. Towards the exterior regionof
1 −
[Ki ]=8π Si ∂i Sk (9) shell, we get
j j j k
− 2
(cid:18) (cid:19)
n uv+n ur =0 (20)
v r
where S is surface stress-energy defined as
+ǫ R
Sα = lim Tα dn (10) 1 s (uv)2 2uvur =1 (21)
β ǫ→0 β − r −
Z−ǫ (cid:18) (cid:19)
One can show that, in general,
R
1 s (n )2 2n n =1 (22)
Sim|m+[Tni]=0 (11) − r r − r v
(cid:18) (cid:19)
3
We can solve this system of equations to get n = uv. The ultimate goal is to find an action in terms of the
r
±
Substituting this result in Eq. (21) gives ingoing coordinate v, so we write
1 Rs (n )2 2urn =1. (23) dR dv 2 µG
r r S = µ 1 . (33)
(cid:18) − r (cid:19) − eff − s − dv dT − 2R
Z (cid:18) (cid:19)
Now, nr =grνn , which leads to  
ν
Substituting dv/dT fromEq.(8),wearriveatthe desired
R action
nr = 1 s n +n . (24)
r v
− r µG
(cid:18) (cid:19) S = dvµ B 2R R 2 2R +B .
eff v v v
− − − 2R −
Substituting n from Eq. (22), we get Z (cid:18) q (cid:19)
v p (34)
The corresponding Lagrangianis
1+ 1 Rs n 2
nr = − r r . (25)
(cid:0) 2nr (cid:1) L= µ B 2R µG R 2 2R +B . (35)
v v v
− − − 2R −
Using Eq. (23) and Eq. (25), we find (cid:18)p q (cid:19)
We neededanexplicitformofthe Lagrangiansothatwe
can define the canonicalmomentum and find the Hamil-
R
nr = 1 s +(ur)2. (26) tonian in terms of momentum. Canonical momentum in
±s(cid:18) − r (cid:19) this case is defined as Π= ∂L , which yields
∂Rv
We willignorenegativesignbecauseatthe shell ur =R˙, 1 µG (R 1)
v
so the relation becomes Π=µ + − . (36)
√B−2Rv 2R Rv2−2Rv+B!
nr+ = 1 Rs +R˙2. (27) Finally,Hamiltoniancorrespopndingtothe Lagrangianin
s − R Eq. (35)
(cid:18) (cid:19)
We caneasilyobtainnr− fromnr− by substituting M = 1 µG 1
H =µ(B R ) .
0 − v √B−2Rv − 2R Rv2−2Rv+B!
(37)
nr− = 1+R˙2. (28) p
wherethe velocityR shouldbe eliminatedby the useof
v
p Eq. (36). We emphasis that so far we did not use any
Substituting nr+ and nr− into Eq. (19), we get
approximations,so the Hamiltonian in Eq. (37) is exact.
R
r 1 s +R˙2 1+R˙2 =µ (29)
− s(cid:18) − R (cid:19) −p ! IV. QSUHAENLTLUINMTCHOELLLAIMPSITEOOFFRTH→E0DUST
which can be simplified to give
Themaingoalofthethispaperistoseewhathappens
1 µ2 at the last stages of the collapse of the shell, i.e. when
M =µ 1+R 2 2 (30)
τ − 2R R 0. SincewehaveanexplicitHamiltonianofthesys-
→
(cid:0) (cid:1) tem, we can apply the functional Schrodinger formalism
ThequantityM inEq.(30)isanintegralofmotion,and and study quantum effects near the classical singularity.
has a clear interpretationof the totalenergy. It contains In the framework of the functional Schrodinger formal-
therestmassoftheshellµ,thekineticenergyrepresented ism,we willsimply write downthe Schrodingerequation
by Rτ, and gravitational self-energy µ2/(2R). We will for the wave-functional Ψ[R(v)], and try to solve it.
therefore identify it with the Hamiltonian of the system. We first derive the behavior of R near R = 0. From
τ
We now express the Rτ in terms of RT to obtain Eq. (30), we have
1 µ M µG 2
M =µ (31) R = + 1. (38)
1−RT2 − 2R! τ s(cid:18) µ 2R(cid:19) −
We can then write dowpn an effective action that gives FromhereweseethatR µG asRisapproachingzero.
τ ≈ 2R
Eq. (31) as its Hamiltonian as Substituting this result in Eq.(7), we find
µG 1 µG 2
S = µ 1 R 2 (32) R (39)
eff T v
− − − 2R ≈−2 R
Z (cid:18)q (cid:19) (cid:18) (cid:19)
4
Thus,therateatwhichthedustshellcollapsesnearR= We can take care of the operator ordering as
0 diverges as R 1 .
v ∝ R2 −1
In this limit the Hamiltonian in Eq. (37) can be ap- 1
Aˆ= 1 (ΠˆR+RΠˆ) , (48)
proximated as − µ2G
(cid:20) (cid:21)
1 µG 1 so that
H =µ( R ) (40)
v
− " 2|Rv | − 2RRv# Aˆ−1 =1 1 (ΠˆR+RΠˆ). (49)
p − µ2G
which gives
In terms of derivatives, Aˆ−1 is
H Gµ 2
R =2 (41)
v µ − 2R i 2iR ∂
(cid:18) (cid:19) Aˆ−1 =(1+ )+ (50)
µ2G µ2G∂R
For R 0, we can ignore the constant term H/µ, and
→
we will again get Eq. (39). Let’sdefinetheactionofanoperatorAˆasϕ=Aˆψ,which
In the limit R 0, the canonical momentum reduces means ψ = Aˆ−1ϕ, where ϕ is just some function which
→
to
gives the wavefunction ψ upon action of the operator
Aˆ. Explicit action of Aˆ−1 on ϕ converts the equation
Π=µ 1 + µG (42) Aˆ−1ϕ=ψ into a linear differential equation
" 2 Rv 2R#
| | dϕ 1 iµ2G
p + (1 iµ2G)ϕ+ ψ =0 (51)
Expressing Rv in terms of Π in Hamiltonian (40) we get dR 2R − 2R
R 2ΠR −1 µ2G This equation can be solved to give
H = − 1 + . (43)
G (cid:20) − µ2G(cid:21) 2R iµ2G R−(1+i2µ2G)ψdR
ϕ= (52)
TheHamiltonianinEq.(43)governstheevolutionofthe − 2 (1−iµ2G)
R R 2
collapsingdustshellin vicinity ofR=0. As inthe stan-
dardquantizationprocedure,wepromotethemomentum Since ϕ=Aˆψ we obtain the action of Aˆ as
Π into an operator
∂ Aˆ= iµ2G R−(1+i2µ2G)(.)dR (53)
Π=−i¯h∂R. (44) − 2 R R(1−i2µ2G)
We can now write the functional Schrodinger equation where (.) is the placeholder for the function on which Aˆ
for the wave-functional ψ[R(v)]
is acting.
Let’s concentrate on the stationary solutions to
∂ψ
Hψ =i¯h (45) Eq. (45) in the form of
∂v
and try to solve it. Unfortunately, the structure of the ψ(R,v)=ψ(R)eiEv/h¯ (54)
Hamiltonian(43)issuchthattheusualtreatmentisprac-
where v is the time evolution parameter, and E is the
tically impossible. The main problem is that the differ-
energy eignevalue. The time independent Schrodinger
ential operator in Hamiltonian (43) is non-local. This
equation becomes Hψ = Eψ. The Hamiltonian in
finding represents a strong support for suggestions that
Eq. (43) in terms of the operator Aˆ becomes
quantumgravitymightbe ultimately a non-localtheory.
While finding solutions to non-local equations is very
RAˆ µ2G
difficult, we will show that it is possible to define a pro-
H = + (55)
cedure(similartothe oneoutlinedin[5])whichwilllead G 2R
to the solution of Eq.(45). We first isolate the non-local
Accounting for the ordering of operators, this Hamilto-
operator Aˆ from the Hamiltonian (43)
nian becomes
2ΠR −1 1 µ2G
Aˆ= 1 (46) H = RAˆ+AˆR + . (56)
− µ2G −2G 2R
(cid:20) (cid:21) (cid:16) (cid:17)
Its inverse is The Schrodinger equation (45) becomes
Aˆ−1 =1 2ΠR (47) −1 RAˆ+AˆR ψ+ µ2Gψ =Eψ (57)
− µ2G 2G 2R
(cid:16) (cid:17)
5
When Aˆ operates on R we get Now we can move all the terms to one side and separate
the term with the integral
α
Aˆ(R)= R−21(1−α) R−12(1+α)RdR (58)
−2
Z
which yields
αR+β
Aˆ(R)= − (59)
3 α
whereα=iµ2Gandβ isaninte−grationconstant. Soour R−21(1+α)ψdR= 4GR−1+2α 1 αR−β µ2G +E ψ
α 2G α 3 − 2R
equation becomes Z (cid:20) (cid:18) − (cid:19) (cid:21)
(61)
We cannowdifferentiatethis equationwith respecttoR
1 αR β µ2G
− RAˆψ+ − ψ + ψ =Eψ (60) to remove integration. Differentiation yields
2G α 3 2R
(cid:18) − (cid:19)
2R21(1−α) 2µ2G2R−12(3+α)+ 4GE 2β R−21(1+α) ψ′ = (62)
" α−3 − (cid:18) α − α(α−3)(cid:19) #
α 1 2GE(α+1) β(α+1)
− +1 R−21(1+α) µ2G2(α+3)R−21(5+α)+ R−12(3+α) ψ
α 3 − α − α(α 3)
(cid:20)(cid:18) − (cid:19) (cid:18) − (cid:19) (cid:21)
This can be written as V. COLLAPSE OF AN INFINITELY THIN
SPHERICAL DOMAIN WALL: CONSERVED
dψ a1+a2R+a3R2 QUANTITY
= dR (63)
ψ a R+a R2+a R3
Z 4 5 6
It is a logical possibility that the non-local behavior
where a = µ2G2(α+3) , a = 2GE(α+1) β(α+1) , in the near singularity region that we found in the pre-
1 − 2 α − α(α−3)
vious section is an artifact of the example that we were
a = 1+ α−1, a = 2µ2G2, a =(cid:16) 4GE 2β an(cid:17)d
3 α+3 4 − 5 α − α(α−3) studying, i.e. the dust shellof matter. In this section we
a = 2 This integral can be solv(cid:16)ed for general v(cid:17)alues will repeat the procedure for the shell of matter whose
6 α−3
of constants. However, since we are working in the limit energy per unit area, σ, is constant, which is the situa-
of R 0, we keep only the leading order terms tionrepresentedbyasphericallysymmetricdomainwall.
≈
We will find that the near singularity behavior is quali-
a1 tatively the same even in this case, i.e. the Hamiltonian
lnψ = dR+constant (64)
a R becomes non-local in R 0 limit.
Z 4 →
In the case of the domain wall, mass M is also a con-
Solving this equation and substituting the values of the servedquantity. However,therelationrelatingmasswith
constants, we find the solution for the wavefunction R and R is now given as
τ
ψ =λR3+i2µ2G. (65) M = 1 1+R 2+ B+R 2 4πσR2 (67)
τ τ
2
where λ is a constant. The corresponding probability (cid:18)q q (cid:19)
density P =ψ∗ψ is where B 1 2GM/R. This expressionis implicit as it
≡ −
contains M in B. The explicit relation of M in terms of
ψ 2 =λ2R3. (66) R and R can be written as
| | τ
This result is very important. It demonstrates that the
probability density associated with the wavefunction ψ M =4πσR2 1 Rτ2 2πGσR (68)
− −
whichdescribesthecollapseoftheshellofmatterisnon- (cid:20)q (cid:21)
singular near the classical singularity. In fact, the prob-
Using the relation between T and τ (Eq. (4)), we get
ability density in Eq. (66) vanishes exactly at R = 0. It
is remarkable that a simple quantum treatment of the
1
gravitational collapse indicates that classical singularity M =4πσR2 2πGσR (69)
at the center can be removed. " 1 R 2 − #
T
−
p
6
The effective action which can reproduce above relation, Solving this equation for R yields R 2h2 where
v | v |≈ R4
i.e. givesthe correctmass conservationlaw, canbe writ- h= H whichisthesameastheabovederivedbehavior.
4πσ
ten as Theexactformofcanonicalmomentumcanbewritten
as
S = 4πσ R2 1 R 2 2πGσR (70)
eff T
− − − 1 2πσGR(R 1)
Z (cid:20)q (cid:21) Π= 4πσR2 − v − (78)
We wantto convertthe T coordinate into the infalling v − "√B−2Rv − Rv2−2Rv+B#
coordinate as
which in R 0 limit can be appproximated as
→
2
∂R ∂v
S = 4πσ R2 1 2πGσR 1
eff − s − ∂v ∂T −  Π=4πσR2 +2πσGR (79)
Z (cid:18) (cid:19) " 2 Rv #
 (71) | |
After substituting the expression for dv, we arrive at From this we can get Rpin terms of π as
dT v
1 π −2
Seff =−4πσ dvR2 B−2Rv−2πσR Rv2−2Rv+B . Rv =−2 4πσR2 −2πσGR (80)
Z (cid:18) q (cid:19) h i
p (72)
The Hamiltonian given by Eq.(75) can now be approxi-
The corresponding Lagrangiancan be written as
mated as
Leff =−4πσR2 B−2Rv−2πσR Rv2−2Rv+B H =4πσR2(−Rv) √B 1 2R (81)
(cid:20)p q (73(cid:21)) (cid:20) − v(cid:21)
Canonical momentum is defined as Π = ∂L , so we ob- Submitting the value of R from Eq. (80) gives
∂Rv v
tain
R
H = (82)
Π= 4πσR2 −1 2πσGR(Rv−1) (74) − 1− 8π3σπ2GR3
− "√B−2Rv − Rv2−2Rv+B# TheHamiltonianinEq.(cid:0)(82)governsth(cid:1)eevolutionofthe
p collapsingsphericaldomain wallin vicinity ofR=0. As
Since the Hamiltonian is H = (πR L), substituting
v
− before, we promote the momentum Π into an operator
the R in terms of Π gives
v
∂
1 2πσGR Π= i¯h . (83)
H =4πσR2(B R ) − ∂R
v
− "√B−2Rv − Rv2−2Rv+B# We can now write the functional Schrodinger equation
(75)
p
∂ψ
Hψ =i¯h (84)
∂v
VI. QUANTUM COLLAPSE OF THE DOMAIN
WALL IN THE LIMIT OF R→0 and try to solve it. However, as in our previous case of
the dust shell, we see that the differential operator in
Since we have an explicit Hamiltonian of the system, Hamiltonian (82) is non-local. This fact reinforces an
wecanapplythefunctionalSchrodingerformalismagain indication that quantum gravity should be ultimately a
and study quantum effects near the classical singularity. non-local theory.
Again from the conserved quantity M we can derive We will now follow the procedure we outlined in the
section (IV). We isolate the non-local operator
M 2
|Rτ |=s 4πσR2 +2πσGR −1 (76) Aˆ= 1 Π −1 (85)
(cid:18) (cid:19) − 8π3σ2GR3
(cid:20) (cid:21)
FromhereweseethatRv ≈−(4π2σMR22)2 asRisapproach- For convenience, we set the constant α= 16π21σ2G which
ing zero. Thus,the rateatwhich the dust shellcollapses gives
near R = 0 diverges as R 1 , in contrast with the
dust shell where the divergven∝ceRw4as quadratic. Aˆ= 1 (86)
In the same limit the Hamiltonian in Eq. (75) can be 1 2α Π
− R3
approximated as
Now, we can define (cid:0) (cid:1)
1 2πσGR
H =4πσR2( R ) + (77) Π
− v " 2|Rv | |Rv | # Aˆ−1 =1−2αR3 (87)
p
7
We take care of the ordering problem as by an infalling observer. Since gravity is the by far the
weakestforceinnature,weexpectthatquantummechan-
Aˆ−1 =1 α Π 1 + 1 Π (88) ics will significantly modify classical behavior of gravity
− R3 R3 only in the strong field regimes, e.g. near classical sin-
(cid:20) (cid:21)
gularities. In the absence of a fully fledged theory of
which also makes this operator unitary. In terms of quantum gravity, we worked in the context of the func-
derivatives, we have tional Schrodinger formalism applied to a simple grav-
itational system - collapsing shell of matter. We used
2 ∂ 3
Aˆ−1 =1+iα (89) the Eddington-Finkelstein space-time foliation which is
R3∂R − R4 convenient for studying the question of the black hole
(cid:18) (cid:19)
formation till the very end where the collapsing shell
Letusagaindefineψ =Aˆ−1ϕ,whichleadstodifferential crossesitsownSchwarzschildradiusandstartsapproach-
equation ing the classical singularity at the center. We derived
the conserved quantity with the clear interpretation as
∂ϕ R3 3 R3
+ ϕ ψ =0 (90) the Hamiltonianofthe systemandquantizedthe theory.
∂R 2iα − 2R − 2iα In the R 0 limit, we found that the equation which
(cid:18) (cid:19)
→
describes the quantum evolution of the collapsing shell
This linear differential equation can be solved to give
is strongly non-local. Non-local terms which are usually
suppressed in large distance limit, become dominant in
ϕ= R3/2e−8Ri4α R3/2e8Ri4αψdR (91) the near singularity limit. This conforms some earlier
2iα speculations and related studies. As an important step
Z
forward,wemanagedtosolvethisnon-localequationex-
Since ϕ=Aˆψ, this gives the operator Aˆ as plicitly and found the form of the wavefuction. Remark-
ably, the wavefunctionand its correspondingprobability
Aˆ= R3/2e−8Ri4α R3/2e8Ri4α(.)dR (92) density are non-singular at R → 0. This is an indica-
2iα tion that quantization can remove classical singularities
Z
from gravity, just as it was the case with the singular
where(.)istheplaceholderforthefunctiononwhichAˆis electromagnetic Coulomb potential.
acting. Inprinciple,onecanfollowtheprocedureweout-
lined in section IV and solve the non-local Schrodinger
equationlike. However,inthiscasecalculationsaremuch
more cumbersome because of presence of additional ex-
ponential in the operator Aˆ and will not be shown here. Acknowledgment
VII. CONCLUSIONS
In this paper we studied quantum aspects of the grav- DS acknowledges the financial support from NSF,
itational collapse near the classical singularity as seen awardnumber PHY-1066278.
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