Black hole formation from collisions of cosmic strings

Jorge Russo

IRGAC - July 2006

Based on R. Iengo and J.R., hepth/0606110, June 2006

And “Handbook on string decay”, JHEP, 0601072

Cosmic fundamental strings

•Recent works indicate that in brane inflation models cosmic strings are copiously produced during the brane collision [Jones, Stoica, Tye (2002); Sarangi, Tye (2003)]. This gives the exciting possibility that there could be fundamental strings of cosmic size. [See also J. Polchinski, hep-th/0412244; A. Davis and T. Kibble, hep-th/0505050]

• The dynamics of cosmic strings could lead to interesting astrophysical events such as gravitational waves or black hole formation. [See e.g. Vilenkin and Shellard (Cambridge U. press 1994)]

•Stability. What is the lifetime of a cosmic fundamental string? Are there long-lived cosmic string configurations?

•What are the rules governing the collision of cosmic fundamental strings?

•What are the chances of black hole formation by a collision of cosmic strings?

I. Decay properties of fundamental strings

[Iengo, J.R., “Handbook on String Decay”, JHEP (2006)]

A highly excited quantum string state can be described in terms of a classical string.

This state is unstable, and it can basically decay in two ways:

1) By splitting into two massive strings.

2) By emitting massless radiation.

•Which is the dominant decay mode?

•What is the lifetime?

•How does it depend on the mass and on the “shape” of the string?

•Does a string become more unstable for larger masses?

•Is there any long-lived string state in the string spectrum?

Computing the lifetime or decay rate of a given massive string state using string theory techniques is extremely complicated. It has been done only in few cases for specially simple states. Here we will derive simple rules to make a good estimate of the decay rate by splitting and massless radiation, and hence the lifetime of an arbitrary string state.

Decay Rate due to breaking

1. Open strings

Probability for an open string to break: it is constant and proportional to g^2

[Dai, Polchinski (1989)]

More precisely, we will propose that the probability of breaking per unit time on a given point is:

T

gP o

o

2

T: period of oscillation of the string

(so probability of breaking at any moment on a given point is T Po = g^2 )

In the “temporal” gauge, the period is essentially the mass:

'2'2

1,

0

00

T

XdMX T

Decay rate = Po times number of “points” times number of “instants”:

220

so

ssopen l

LgP

l

T

l

L

This formula can be checked against the explicit quantum calculation of the decay rate for the open string with maximum angular momentum Jmax [Okada, Tsuchiya (1989)].

One finds

220

)( ,s

quantumopen l

LMMg

in precise agreement with the above estimate.

Decay rates for open Jmax and closed Jmax.

Folded string

The breaking can take place only if at a given time the two folds break at the same point, up to an uncertainty of order ls

23

22 , oss

ss

ssooss

folded ggMl

Lg

Tl

LglPP

l

T

l

L

In particular, for the folded string with Jmax,

212

0max ,s

sJ l

LMlg

Remarkably, it is constant independent of M.

The squashing closed string

LxLz

LyLx

2,sinsinsin2

,cossin2,coscoscos2

0

For generic, it describes an ellipse that rotates around one of its axes and simultaneously performs pulsations, with one of its radii becoming zero at =n .

The solution interpolates between the folded string (= 0)

,0,cossin2,coscos2 zLyLx

,sinsin2,cossin2,0 LzLyx

and the pulsating string (/2

Squashing closed string:

It becomes folded once in a period.

Breaking process is suppressed.

Dominant decay by massless radiation

Decay rates for strings that become folded at an instant of time

The process of breaking is possible only at the instant when the string is folded.

Quantum mechanically, this represents a time interval of order ls

Thus the rate is

Nlg

Ml

Lg

NT

l

s

ssquash

ssfoldedfolded

ssquash

1

1

2

242

The last line applies to the squashing ellipse where L is proportional to M.

This gives a lifetime

Mgl sssquashsquash221

Lifetime for the squashing closed string in D = 10 and D = 4 computed from the one-string loop correction to the propagator.

Decay rate due to massless emission

•Contribution from four sectors NS-NS, R-NS, NS-R and R-R.

•Computing massless decay rate for a generic quantum string state is very complicated. In particular, the covariant vertex operator is not known.

•We have computed the quantum decay rate for every channel for the string Jmax and for the rotating ring.

•There is one more case which can be computed: the average string state.

•There is little chance that other cases (i.e. where the angular momentum is not near maximum or it is not the average) can ever be computed.

Is there any way to estimate the string decay rate by massless emission?

When the string is very massive, one can hope that the radiation can be described by the classical formula of gravitational radiation from a source TX).

But this formula, in the most favorable cases, can only account for NS-NS radiation.

Rotating Ring:[Chialva, Iengo and J.R.]

)exp(,)exp( 21 iLZiLZ

•NS-NS emission is dominant.

•R-NS and NS-R suppressed by 1/M2 , and R-R suppressed by 1/M4.

•NS-NS emission is accurately described by the classical formula: Identity between classical and quantum expressions up to terms of order 1/N.

In general, the classical formula is expected to hold for graviton energies much less than the string mass =O(1/sqrt(’)).

If massless emission with higher energies is suppressed, then the classical formula can be used to compute the total decay rate by radiation emission

In this regime w << O(1/sqrt(’)) we expect that also NS-R, R-NS and R-R are also suppressed.

0.,.,||||||

..

)),((

),(,||

0

2

0

)(222

)(

0.

,

222

22 0

pXedJJJJ

ei

XXXxddT

XXTexdJJdM

g

TR

XpiRLR

d

XpiXiDDD

s

R

Classical formula:

•Frame p = (w,-w,0,...,0)

•Gauge X0 = ’M

•X+ , X- are light-cone coordinates.

In general, ipX+ =O( iw sqrt[N] f(Then the JR,L are exponentially suppressed for high graviton energies w unless there is a saddle point. Two cases:

1. No saddle point (smooth strings): then the classical formula can be applied. Total radiation emission is convergent, and we get g2 M5-D (very small in D > 5!!!)

2. “Cusps” or “kinks” (discontinuities in derivatives of X). For D < 5 OK. (ex. Jmax)

[see Vilenkin, Damour , and Iengo and J.R, “Handbook of string decay” ; Iengo (2006)]

In conclusion, in D = 4 one has the law

massless const. g2 M

Checked against the full quantum string calculation for the ring and for Jmax.

The same law holds for the average string state (computed using full superstring theory [Amati, J.R. (1999); Chialva, Iengo, J.R. (2005)])

New examples of long-lived Closed strings

1. Rotating straight string on M4 x S1

The idea is to consider a closed string that winds around the extra dimension W in such a way no two points get in contact

•In four dimensions, the string looks identical to Jmax

•For n = 1, the solution is classically unbreakable. The decay rate by breaking will be suppressed exponentially, O(exp(-R2/a')).

•The string can decay by radiation, with = g2 M . The radiation is dominated by soft modes with energy w =1/L. Thus

Therefore the string takes a time = M / gs 2 (or L /gs 2 ) to substantially decrease its mass.

2,0,,

,cossin,coscos2222

RnLt

nRWLYLX

'2

1,2

0 sgc

dt

dM

X

W

X

Y

2. Rotating open string which oscillates in extra dimensions

Consider a brane-world model, with a D3 brane placed in three uncompact directions X, Y, Z of our universe. The solution is

It has the same form as the squashing closed string, but here 0 < < pi , whereas for the closed string 0 < < 2 pi.

The solution describes a string rotating in the plane X,Y with the ends attached on the brane W=0. The string can break only at the special times where it lies on the brane W=0, namely tau= n pi. This suppresses the breaking rate process (computed with the rules given before, one gets = const gs)

The dominant decay mode is again gravitational radiation. As in the previous case, we find a long time required for a substantial decrease of the mass of order M / gs

2.

MLLt

LWLYLX

'2,

,sinsinsin,cossin,coscoscos

Black hole Formation from cosmic strings

Hawking (1989); Polnarev, Zembowicz (1989):

Bounds on G from observations of gamma ray background

Rs= 2 G M = 2 G L . Hence cosmic string loops that shrink by a factor 1/G will form black hole. During the evaporation gamma rays are emitted.

Collision of cosmic strings may also form black holes.

How do fundamental strings interact? Fundamental strings can split and join. This leads to two types of interactions:

1. Joining of strings

2. Interconnection of two strings

For open strings, this process corresponds to the u-channel open string diagram.

Intercommutation of open strings

Construction of the string solution after splitting or joining at = 0:

Basic rule: demand continuity of all X i and first time derivative dXi/d at tau=0.

This uniquely determines the outgoing string solution:

•Left and right parts are piecewise identical to the original string.

•Only periodicity changes, because the string length has changed.

Example: Joining of two open strings I and II with EI = EII

ME

ME

ME

EM

RLIRL

ME

ME

IIRLIIRL

III

RLIRL

RLIIRL

IIIIII

I

IIII

sifsXsX

sifEMsXsX

EEIf

sifsXsX

sifsXsX

2))(()(

)/()(

)2()(

)2()(

,,

,,

23

2,,

22,,

Splitting of Open string with Jmax

In a generic situation, when an open string breaks, one kink on each of the outgoing strings is formed, which travels back and forth along the string [Iengo, J.R., JHEP (2003)]

Classical breaking for the squashing string

The ellipse becomes folded at = 0. We assume that the string breaks into two pieces at the points a and a .

The outgoing strings, their masses and momenta can be determined by requiring continuity of X and the first time derivative of X at = 0. Then we compute E, p, J for the strings I and II by integrating the canonical momenta associated with E,p, J in the intervals:

I ) from 0 to a and from a to 2 .

II ) from a and from a .

We find

)sin

)1(('

4)(

)sin

('

4)(

2

22

2

2222

2

22

2

2222

a

aL

pEaM

aa

LpEaM

IIIII

III

•Independent of

•Defines MI = f(MII)

•The strings I and II rotate and pulsate.

•They have two kinks which travel along the string.

•At periodic times, they become folded.

t = 0.1 t = 0.5

Evolution of the squashing closed string after splitting in the space ( y, z, t )

Curve of classical splitting MI=f(MII) and maxima of (MI,MII) for the squashing string

M

MMIII

III

MMM0,

),()(

Black hole formation

Simplest example of a cosmic string:

A straight open string with maximum angular momentum

It can be rendered long lived by the higher dimensional extensions as discussed before.

Remarkably, we will find that the joining or interconnection of two strings of equal and opposite maximal angular momentum generically leads to the formation of black holes, provided their relative velocity is small enough.

Black hole formation by Joining of strings

Consider two open strings with maximum angular momentum, described by the solutions

They have JI = - JII = L2

As the strings rotate, the end = 0 of the string I touches the end = of the string II at tau = n .

The resulting string has J=0. The solution after the joining is

sincos,coscos2

sincos,coscos

LYLLX

LYLX

IIII

II

sL

sY

sifsL

sifsL

LsX

XXX

L

L

LL

2sin2

)(

2cos2

2cos2

)(

)()(),(

23

2

22

Outside these intervals XL,R are defined by replacing s by s = s – [s/2].

Joining of Jmax + anti-Jmax

[Iengo and J.R., hep-th/0606110]

Single open string becoming a closed string by joining its endpoints

Note that the string reduces to a point due to its own classical evolution in flat spacetime.

Now let us take into account gravitational effects.

When the string contracts by a factor of order 1/G , gravitational collapse should be inevitable and a horizon will form [Hawking, Polnarev,Zembowicz, Vilenkin].

One important question is whether the open string could radiate out most of its energy before its size becomes smaller than the Schwarzschild radius.

There are two decay channels:

1. the radiation channel, where the string emits a graviton.

2. the massive channel, where the string breaks into two pieces.

Massive channel. Because of momentum conservation, in the present case after the breaking each piece will carry a momentum in the inward direction. Therefore the system cannot lose energy by breaking. Turning on gravity, the attractive nature of gravitational forces reinforces the fact that each piece will follow an inward collapse.

Radiation channel: it can be estimated using the formula for the decay rate given before.

Integrating this, we find that the total radiated energy from the initial configuration until the string becomes a point is given by

This mass is of the same order of the initial mass. This shows that a black hole will be formed before the string becomes a point.

Formation of the circular pulsating string by quantum scattering

An old question which led to many studies is whether black holes can be formed by quantum scattering of gravitons.

We have argued that the circular pulsating string inevitable collapses into a black hole. So we can ask the question whether there is a non-zero cross section for formation of the circular pulsating string starting from graviton scattering.

We have shown that, indeed, the circular pulsating string can be obtained from the quantum scattering of two gravitons with a rate = g2 exp(-c M 2).

It is small, but different from zero. This process provides an example of a first-principle calculation based on string perturbation theory of black hole formation.

1202

12 )1(,)0()( sgcMM

Black hole formation by interconnection of strings

Interconnection process is very common in 3+1 dimensions, where two infinitely long strings always cross for generic initial data.

For finite-size strings, the collision has a cross section of the order of the square of the length of the string.

Experiment: send two straight rotating strings against each other, with random position for the CM coordinates and random relative orientation. After repeating the experiment Ne times, we ask how many of the resulting string configurations are black holes.

We will consider several conditions for black hole formation:

1) One of the two final strings completely lies inside its Schwarzschild radius Rs at some time during the evolution.

2) At some time the average size <R> of the string lies inside its Schwarzschild radius.

3) A segment of the string lies within the Schwarzschild radius.

In our study, the reduction to a small size just follows by the shrinking of the string that results from flat space evolution, without taking into account gravity. Gravitational forces become very strong when the string size approach Rs and should enhance the evolution towards the collapse.

We write down the solutions for the strings I and II, with parameters A,B and parametrizing the CM coordinate of the string II and its relative orientation.

We assume that the two strings interconnect at tau =0. The two strings of equal length I and II recombine forming two strings a,b (or c,d) of different lengths forming some kink.

The solution is explicitly constructed in [Iengo, J.R. hepth/0606110].

The results are summarized in the following table

N bh = Number of black hole events in Ne string collisions

Ne G Nbh (Rav<Rs) Nbh (R()<Rs)

10000 10-2 1900 – 2000 1100 – 1200

10000 10-3 300 – 320 95 –110

10000 10-4 40 – 46 1 – 3

50000 10-5 20 – 30 0 – 4

50000 10-6 3 – 5 0

II

I

Evolution of a cosmic string after intercomutation of two cosmic strings with

JI = - JII

The string shrinks to a minimum size which in some cases it can be inside

the Schwarzschild radius

Inevitable collapse for Jmax + antiJmax

First consider strings with zero transverse linear momentum

•Joining: the string shrinks to a point.

•Interconnection: a finite segment of the interconnected string shrinks to a point at some

This means that a finite fraction of the mass will shrink to zero size and therefore will form a black hole.

Now, consider the case in which the interconnecting strings have equal and opposite linear momentum along the transverse Z direction.

In this case the interconnected strings will in general also stretch in the Z direction (periodically in tau, if one forgets gravity) and therefore the finite fraction of the string described above will not shrink exactly to zero size at 0. In order to conclude that a black hole will still form we have to compare the elongation in Z with the Schwarzschild radius Rs.

Maximum elongation in Z is Z =O( T v), where T = L is the period of the motion, and v is the relative velocity between the centers of mass.

So Z = L v < 2 G L or v < G

The ratio of the relative velocity between the strings v to the velocity of light (c=1) should be smaller than G times some number of order 1.

Conclusions

• Computing decay rates in superstring theory for arbitrary string states is extremely complicated except for a few special cases. However, by using simple rules we can estimate the decay rate of any initial open or closed string state. These rules reproduce the quantum calculation in all specific cases that have been computed.

• The classical dynamics of string interactions after splitting, joining or interconnection processes can be studied in terms of analytic formulas. We find a variety of features such as kinks, foldings, collapse of segments or full collapse of the string.

• This allows to study in quite explicit way the dynamics of cosmic string collisions and possible gravitational collapse.

• The collision of two straight strings with total J=0 always produces black holes if the relative CM velocity v < G