Download Stringhe, buchi neri e coerenza quantistica

Stringhe, buchi neri
e coerenza quantistica
Gabriele Veneziano
Has string theory solved the
information paradox?
Agreement between BH-entropy and counting of states in the
case of extremal (BPS) BH (Strominger-Vafa) @ weak coupling
Result can be extended to strong coupling thanks to SUSY
Decay of quasi extremal BHs follows Hawking iff one traces
over initial brane configuration (density matrix)
Amati-Russo result for strings and Hagedorn T.
Questions:
What happens if one starts from a pure state? At weak
coupling it does not work, but may work at strong coupling
Are there corrections to the pure thermal spectrum?
How does all this extend to more ordinary (say Kerr) BHs?
Outline
1.
Black holes, fundamental strings, “stringholes”
2.
Unitary string-hole production in SuperPlanckian collisions (hep-th/0410166)
3.
Summary
Outline
1.
Black holes, fundamental strings, “stringholes”
2.
Unitary string-hole production in SuperPlanckian collisions (hep-th/0410166)
3.
Summary
String vs Black-Hole entropy
D= d+1, h = c = numerical factors =1
ls = string length
Tree-level string entropy
Counting states (FV, BM (‘69), HW (‘70))
Sst = M/Ms = L/ls
= No. of string bits in the total string length
=> Hagedorn temperature in ST: THag ~ Ms
NB: no coupling, no G appears!
Black-Hole entropy
SBH = M RS = (RS/lP)D-2 ~ M(D-2)/(D-3)
(GM = RSD-3 , Gh = lPD-2 , 1/TBH = d S/dM = RS /h)
to be contrasted with previous
Sst = M/Ms = L/ls
*****
For any D:
Sst /SBH > 1 @ small M, Sst /SBH < 1 @ large M
Where do the two entropies meet? Obviously at
RS = ls i.e. at TBH = Ms = THag !
“string holes” = states satisfying this entropy
matching condition
The condition RS = ls looks strongly D-dependent
(Cf. dep. of RS on M)
However, using string unification @ the string scale,
(lP /ls)D-2 = gs2 ~ αGUT
entropy matching occurs, at any D, for
M = Ms gs-2 = MP gs-2(1-1/(D-2))
and the common value of Sst and SBH is simply
Ssh = gs-2 ~ 1/αGUT (~ 102 )
In string theory gs2 is actually a field, the dilaton,
and its value is free in PT.
Consider the (M, gs2) plane
Playing with the correspondence
M/Ms
Much more difficult
to establish except
for extremal case
R S > ls
Black Holes (= Strings? )
R S < ls
Strings ≠ BH
RS = l s ,
“string hole” curve
many properties match here
gs2
Safe conclusion since these strings are larger than RS
M
D=4
Horowitz & Polchinski, ‘97, ‘98
Damour & GV, ‘00
strong gravity
effects
weak gravity
effects
S~M
g0s2
gs2
Collapse @ fixed M. Gravitational binding can increase (log of)
density of states from linear to quadratic in the physical mass.
Turning string entropy into BH entropy
S
Black hole
String (naïve)
gs-2
M = gs-2 Ms= Msh
M
Evaporation at fixed gs or how to turn a BH
into a string (Bowick, Smolin,.. 1987)
M/Ms
trajectory of evaporating BH
Black Holes
RS = l s
Strings
string-holes
gs2
Is singularity at the end of evaporation avoided thanks to ls?
Outline
1.
Black holes, fundamental strings, “stringholes”
2.
Unitary string-hole production in SuperPlanckian collisions (hep-th/0410166)
3.
Summary
The problem at hand
(Super)-Planckian-energy collisions of light particles
within a consistent quantum theory of gravity
(superstring theory) in d = D-1 “large” dimensions
Since superstring theory is essentially an S-matrix
theory, the description will be, naturally,
quantum and unitary
Three reasons for going about it
1.
Weinberg’s: “Because I can!”
2.
Theoretical: Information paradox


3.
Is the S-matrix description breaking down in some
regime? Is quantum coherence lost?
And, if not, is the final state (close to) thermal?
“Phenomenological”: finding signatures of
string/quantum gravity @ future colliders:

In KK models with large extra dimensions;

In brane-world scenarios; in general:

If we can lower the true QG scale down to the TeV
NB. Future colliders at best marginal for producing BHs!


With very few exceptions, we do not have much of
a handle on string theory in extreme regimes
(strong coupling, strong curvature)
Superplanckian collisions may help as gedanken
experiments by providing:



A way to find out how QST is able to reproduce
expectations from CGR at large distances
A way to find out how QST modifies gravity at short
distances
Two complementary approaches (>1987):
1.
2.
Gross-Mende + Mende-Ooguri (1987-1990)
‘t-Hooft; Muzinich & Soldate; Amati, Ciafaloni & GV;
Verlinde & V.; FPVV… Arcioni, de Haro, ‘t-Hooft;
…(1987-’05)
Gross-Mende-Ooguri (GMO)

Genus by genus (i.e. loop by loop) calculation (GM,
1987-’88) of elastic scattering at very high energy
and fixed sc. angle θ (h = number of loops):
(from complex saddle
trajectory)
All genus resummation (MO, 1990) only justified in an
energy window, fails at infinite energy
(gs << 1)
Small, probably too conservative (result given below)
Amati, Ciafaloni, GV (ACV) et al.




Work in energy-impact parameter space, A(E,b)
Can go to arbitrarily high energy provided b is also
increased accordingly,
One then goes over to A(E, q~ θ E) by FT and trusts
(leading?) contributions coming from the above region of b.
In this way one can reach the regime of fixed θ << 1
In gravity, fixed θ scattering at very high E is dominated
by large distance physics (opposite of QCD!). Reason
explained below..
GMO vs ACV et al.
-t
UNPHYSICAL
le
g
n
a
d
e
x
Fi
)
O
M
(G
Overlap (small θ)
low energy
M2P
Fixed t
(ACV et al)
s
WITHOUT STRING THEORY
b
Corr’s to eik. ~ (RS/b)2(D-3)
I SCATTERING
θ ~ (RS/b)D-3
θ ~ 2π
lP
II COLLAPSE
RS(E)
CGR arguments for Collapse @ b < RS

Penrose 1974 (unpublished)

CTS arguments:
1.
Eardley and Giddings, gr-qc/0201034,
2.
Giddings and Rychkov, hep-th/0409131
Finite-size effects:


1.
Yurtsever, 1988
2.
Kohlprath and GV, gr-qc/0203093
In string theory the collapse criterion should be amended!
We shall take the string coupling fixed and very small (gs << 1). Cf. our defs.
b
WITH STRING THEORY
I
corr’s to eik. ~ (RS/(b+ls))2(D-3)
II
ls
large
small
BH
III
lP
Eth = Msgs-2 = Msh
lP
ls
RS(E)
Three regimes in super-Planckian scattering



I) Small angle scattering (relatively easy)
II) Large angle and collapse (very hard, all attempts have
failed so far)
III) Stringy (easy again) This is where GMO and ACV can
be compared with amazingly good agreement given the
completely different approaches (q~ θ E)
Cf. tree level fixed t vs. fixed, small θ
Approximate (but exactly unitary)
S-matrix in regions I and III
Operator formula encoding previous ACV results:
(E, p)
Xu
b+ΔX
(E, -p)
Xd
b
Inserting tree-level amplitude (and forgetting ^s!) we get
Therefore, for b >> bI (Region I), we can forget about C, C+ . In
this region we also find:
Inserting this phase shift, and going over to scattering
angle θ, we find a saddle point at
corresponding precisely to the relation between impact
parameter and scattering angle in the (AS) metric of a
relativistic particle: clearly, fixed θ , large E probe large b


This also explains why
Because of eikonal exponentiation, Re δ also gives the
average loop-number. Thus the total huge momentum
transfer q = θ E is shared among Re δ gravitons to give:
meaning that the process is soft at large bs

Lesson: while in QCD it is better to get a large transverse
momentum via the exchange of as few gluons as possible, in
QSG it is better to share it among as many gravitons as
possible!
We still have to take into account the operators in δ
Physically, they describe “diffractive excitation” (DE) via
graviton exchange. For lack of time I will not discuss this
complication (unless you ask..)
Region III
Let us forget for a moment that Im δ ≠ 0, C and C+
The saddle point condition now gives the relation:
corresponding to deflection from an homogeneous beam
of transverse size lsY:
b
b
EL
θ <<1
DE
θ =gs
θ =1
D=4
ls Y
fixed
θ curves
BH
max θ @
this E
lP
ls Y
RS(E)
Region III: diffractive excitation
We will skip it once more
Region III: effects of Im δ ≠ 0
The operators C and C+ are now “activated” , recall:
The elastic amplitude, <0|S|0>, is now suppressed as
exp(-2 Im δ) and therefore:
NB: exponent is of order -(s/M*2) => -gs-2 = -Ssh @ E= Eth
(M* = MP in D=4, M* > MP for D>4)
(see below for a possible origin of M* in QSG)
Which final states saturate unitarity?
Recall once more:
The final state, S|0>, is a coherent state of quanta
associated with C, C+. What are they? In order to arrive at
above expression for S one had to use the AGK rules of
Gribov’s Reggeon Calculus: C (C+) annihilates (creates) a cut
gravi-Reggeon*) (CGR) The probability of producing n CGRs
obeys a Poisson distribution with an average given by:
*) The GR is the stringy graviton; a CGR is whatever is dual to
it in the sense of old DHS duality
At this point we can compute the average energy of a
final state associated with a single CGR:
We have thus found that the final-state energies obey a sort
of «anti-scaling» law
This antiscaling is very unlike what we are familiar with in HEP
It is however similar to what we expect in BH physics!
In particular: For D=4 Teff ~ THaw even at E < Eth , while for
D>4 Teff --> THaw for E --> Eth
Ms/gs2
window
BH
Ms/gs
E
E-1/(D-3)~TH
Ms
Ms
~E-1
<E>cgr
MD M* =Ms/gs
Ms/gs2
E
Typical final state via the optical theorem
Unitarity cut through 5 CGRs
1.
Our results show that, at least (much) below Eth, there is
no loss of quantum coherence. However the spectra are not
thermal
2.
When we go above Eth we can no-longer neglect “classical”
corrections. They correspond to interactions among our
CGRs: hopefully, they will turn their Poisson distribution
into a (approximately) thermal one for their decay
products, but there is no reason to expect a breakdown of
unitarity
3.
The diffractive states may carry out conserved global
quantum numbers (if there are any)
How does s/M*2 turn into SBH ~ E RS above Eth ? A
possible answer is that ls-dependence at b, RS < ls,
turns into RS-dependence at b < ls < RS. If so:
Finally, what’s the origin of M*? After use of EE’s:
stress-energy
tensor
Summarizing the main points
 We have been able to recast the main results of ACV
in the form of an approximate, but exactly unitary, Smatrix whose range of validity covers a large region of the
kinematic energy--angular-momentum plane;
 We have studied the nature of the dominant final states in
a window of energy and impact parameter at whose
boundary we expect black-hole formation to begin;
 We have found a sort of precocious black-hole behaviour, in
particular an ``anti-scaling" dependence of the average
energy of the final particles from the initial energy, quite
reminiscent of the inverse relation between black-hole
mass and temperature;
 This anti-scaling behaviour introduces, through the variable
x = ω E/ M*2, a new energy scale M* = Ms /gs, whose
physical origin we have tried to trace back
 These results may have a twofold application:
• a conceptual one within the search for an explicit
resolution of the information paradox,
• a more phenomenological one in the context of the
string/quantum-gravity signals expected at colliders in
models with large extra dimensions.