Volume 246, number 1, 2 PHYSICS LETTERS B 23 August 1990

Gravitational radiation from collapsing cosmic string loops

SW. Hawking Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK

Received 24 May 1990

It is shown that a circular loop of cosmic string collapsing at nearly the speed of light to form a black hole cannot emit

more than 29% of its energy in the form of gravitational radiation. It follows from this and the upper limit on the cosmic

gamma ray background, that not more than lo-l2 of cosmic string loops in the early universe can have collapsed to form

black holes.

Recently, arguments have been given that nearly circular loops of cosmic string can collapse to form black holes [1,2]. The observational limits on the gamma rays from the evaporation of such black holes could place important constraints on cosmic string models. However, one can use linearized theory to calculate the energy carried away by gravitational

waves during the collapse of a circular loop to a point. The amount turns out to be infinite. This result might suggest that the back reaction to the emission of gravitational radiation might slow down the collapse of the loop sufficiently that it never reached the rela-

tivistic y of order (Gp)-’ which is required to form a black hole [l]. If this were the case, the loop would collapse to a point, and just unwind. There would be no black holes, and no problem with the gamma ray

background. However, this is not what happens: I shall show

that a collapsing, planar, circular loop will radiate less than 29% of its energy. The assumption that the loop is circular is, in fact, less special than it might seem, because the collapse will tend to make the loop more circular.

During the later stages of the collapse, the loop will be moving highly relativistically. It will therefore be a good approximation to neglect the rest mass of the string, and to consider it as collapsing from infinity at the speed of light. In order to treat this, it is helpful to consider it as a special case of a more general

problem, the collapse of a thin shell of null matter,

moving at the speed of light [3-51. Suppose the infalling shell moves on a null hyper-

surface, N. Because the shell is moving at the speed of light, its gravitation effect will be felt only in the future of N, which is denoted by B;‘(N). In other words, the spacetime will be curved only in 9+(N). Inside the shell, in M-3+(N), the spacetime will be

flat.

To calculate the precise amount of gravitational

radiation emitted in the collapse of the shell, one would have to solve the nonlinear Einstein equations in 2+(N). Attempts have been made to do this numerically for collapsing stars, but not so far for collapsing loops of cosmic string. However, one can put an upper limit on the energy of the gravitational radiation emited, just by considering the flat region. This limit is about 29% of the energy of the string, in the case of a circular loop.

The limit comes from the result that the area of the event horizon, H of a black hole can never decrease with time [6,7]. The event horizon is the boundary of the black hole, the boundary of the region of spacetime from which it is not possible to escape to infinity. If the Cosmic Censorship Hypothesis holds, the event horizon will enclose any closed two-surfaces, C, which are (marginally) trap- ped: that is, the outgoing null rays orthogonal to C have positive or zero convergence, everywhere on C.

36 0370-2693/90/$ 03.50 @ 1990 - Elsevier Science Publishers B.V. (North-Holland)

Volume 246, number 1, 2 PHYSICS LETTERS B 23 August 1990

Consider a two-surface, C, which lies just inside the collapsing shell, N. Because C is in flat space, it is easy to calculate the convergence of the outgoing null rays or thogonal to C. In general, it will be negative. In other words, C will not be t rapped. However, as the outgoing null rays from C pass through the null hypersurface, N, they will be focussed by the effect of the energy-momentum tensor of the shell. Let v be a function which is zero on N and which is such

that nanbgab = 0, where n a ab = g v.b is the null tangent vector to N. Let I a be the tangent vector to the outgoing null geodesics or thogonal to C, normalised so that lan~ = 1, and let m a and r~ be a pair of complex conjugate null vectors with lama = nama = 0, maria = --1 [8]. Then the convergence, p = la;bmar~l b of the null rays will increase by an amount

A p = 41rf~blal b.

Here the energy-momentum tensor of the shell is

Tab = f,,bS ( v ).

The conservation equations imply

f~bn ~ =0 ,

fab;c( lbn c -- m b l n c -- m C m b) = O.

For a shell which represents the limit of matter moving at high speed, like a collapsing loop of cosmic string, the energy-momentum tensor will have the null form:

fob ~ Stlatlh"

The energy density, s, will obey the conservation equation

s d A = constant ,

where d A is the two-surface area of a group of null geodesic generators of N.

Choose coordinates, t, r, 0, ~b in the flat space inside the collapsing shell. Take the shell to be spherical, and given by the equation r = - t . In other words, N will be the past light cone of the origin. Then the surface density of the shell will be

s = f ( O , ¢ ) / r 2.

The case f = constant will correspond to a uniform spherical shell, and f = k S ( O - 7r/2), will correspond to a circular loop of string, col lapsing at the speed

of light. One can measure the initial mass of the shell when it has large radius and the space outside is almost flat. It will be

M = ½ ffsin 0 dO de .

Let C be a two-surface just inside N which is given by the equation

r = - t = h ( O , ¢ ) .

The convergence p of the outgoing null rays orthogonal to C will be given by an expression that is homogeneous of degree - 1 in h and is covariant on the two-sphere, involving up to two derivatives of h. Thus

p = aV2h - t + b h V h - l • V h - l + ch 1.

One could determine the numbers, a, b, c, by direct calculation. However, as they are universal, it is easier to fix them by applying the above equation to special cases. One special case is the two-surface formed by the intersection of the light cone N with a null hyper- plane P. This has

h = ho/(1 +cos 0).

The null geodesics or thogonal to this surface lie in the null hyperplane, and so have p = 0. Thus

a = ¢ = - b .

Another special case is that in which h is constant. Then p = - h - ' . Thus c = - 1 .

In the case of a spherical shell, the energy density per solid angle,f , will be a constant. Thus a marginal ly t rapped surface C will form at the intersection of the light cone N with a surface of constant time, t = - h = -4rrf . In the case of the collapse of a circular loop of string, f will be zero apar t from a delta function on the equator of the sphere. This means that C will be a two-surface on the past light cone, N, such that the outgoing null rays or thogonal to C have zero convergence everywhere, apar t from on the equator, 0 - -~r /2 , where C will have a corner. In fact, C will be the intersection of the past light cone, N, with the two null hyperplanes, P+ and P_, given by

h = ho/(1 +cos 0).

These intersect in the two plane, 0 = ~-/2, t = - h o , where ho is the radius of the loop at which the t rapped

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Volume 246, number 1, 2 PHYSICS LETTERS B 23 August 1990

surface first appears. The outgoing null rays

orthogonal to C will lie in the null hyperplanes, P+ and P_, and will therefore have zero convergence, apart from at the corner where P+, P_ and N intersect.

At this corner,

p = - 2 h o ~ 6 ( 0 - ~ / 2 ) .

Thus, for the rays to be focussed by the energy of

the string to have zero convergence, ho = 2rrk. The area of this trapped surface will be the sam~

as that of two flat disks of radius ho. Thus, the area of the trapped surface will be 27rh~. By the singularity

theorems [9,10,7], the existence of a trapped surface will imply that the loop must collapse to a singularity: it cannot just unwind and disappear. If the Cosmic Censorship Hypothesis holds, and we have every reason to believe it does, both the singularity and the trapped surface will be hidden inside a black hole [7]. This means that the event horizon, H, of the black

hole will intersect the light cone, N, before the trap- ped surface appears. Thus the area Az of the intersec- tion of the event horizon with the light cone, will be larger than A1, the area of the trapped surface C. By the area theorem for black holes, otherwise known

as the Second Law of Black Hole Mechanics [6,7,11], the area of the event horizon cannot decrease with time. Thus, A3, the final area of the event horizon will be greater than A2, which, in turn, will be greater

than A1. According to the No hair theorem [12-14,6,7], the

black hole will settle down to a stationary state described by a Kerr solution. In the present case, it will be a Kerr solution with zero angular momentum,

that is, a Schwarzschild solution, because the collaps- ing matter is not rotating. The area of the event horizon of such a black hole is 167rM 2. Thus

M r > x/ A ~ / 1 6 ~ = ho/ x/8,

where Mr is the mass of the final black hole. On the

other hand, the initial energy of the collapsing loop will be

Mi = wk = ho/ 2.

Thus, at most, a fraction ( 1 - 1 / ~ ) or about 29% of

the original energy of the loop can be radiated in gravitational waves during the collapse. This shows that the back reaction to the emission of gravitational radiation cannot prevent the formation of black holes, and that a black hole so formed will have a mass of

the order of the energy of the loop that collapsed. It follows from this that if even only 10 -~2 of the loops in the very early universe collapse to produce little

black holes, those holes would generate more than the observed background of gamma rays [1].

I am very grateful to Wemer Israel and Gary

Gibbons for discussions.

References

[1] S.W. Hawking, Phys. Lett. B 231 (1989) 237. [2] A. Polnarev and R. Zemboricz, Formation of primordial

black holes by cosmic strings, preprint CAMK-194 (1988). [3] G.W. Gibbons, PhD. Thesis, Cambridge University

(1973). [4] R. Penrose, N.Y. Acad. Sci. 224 (1973) 125. [5] P.K. Tod, Class. Quant. Gray. 2 (1985) L65. [6] S.W. Hawking, Math. Phys. 25 (1972) 152. [7] S.W. Hawking and G.F.R. Ellis, Large scale structure of

spacetime (Cambridge U.P., Cambridge, 1973). [8] S. Newman and R. Penrose, J. Math. Phys. 3 (1962) 566. [9] R. Penrose, Phys. Rev. Lett. 14 (1965) 57.

[10] S.W. Hawking and R. Penrose, Proc. R. Soc. Lond. A 314 (1970) 529.

[11] J.M. Bardeen, B. Carter and S.W. Hawking, Commun. Math. Phys. 31 (1973) 161.

[12] W. Israel, Phys. Rev. 164 (1967) 1776. [13] B. Carter, Phys. Rev. Lett. 26 (1971) 331. [14] D.C. Robinson, Phys. Rev. Lett. U 34 (1975) 905.

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