Stringy cosmic strings with horizons

Volume 240, number 1,2 PHYSICS LETTERS B 19 April 1990

STRINGY COSMIC STRINGS W I T H H O R I Z O N S

G.W. GIBBONS, M.E. ORTIZ and F. RUIZ RUIZ Department of Applied Mathematics and Theoretical Physics', University of Carnbridge, Silver Street, Cambridge CB3 9EW, UK

Received 21 December 1989

It is shown that any sigma model coupled to gravity with a target space having closed geodesics admits singular cosmic string solutions with event horizons. We also prove that if one makes the general assumption of boost symmetric sources, non-singular solutions of this type cannot exist. The case of the target manifold being the fundamental domain of the modular group PSL(2, ~) is discussed in detail.

There continues to be considerable interest in the gravitational fields of global cosmic strings [ 1-5 ]. For straight, t ime-independent cosmic strings the most general metric compatible with reflection symmetry in the t and x 3 directions is of the form

ds2= - A (x k) dt2 + B ( x k) (dx 3 ) 2-~) ' i l (Xk ) d)( t dx j ,

where i, j, k = 1, 2 and, for the time being, we shall not suppose that A, B or Yv have rotational symmetry. If we assume that the ene rgy-momentum tensor is boost invariant, i.e. T', = T 3, it follows that

[ l n ( A / B ) ]:~;~=O.

For a regular solution with no singularities or horizons, A / B must be everywhere bounded. It follows therefore from the maximum principle that A / B must in fact take a constant value which, by suitably re- scaling t and x 3, can be taken to be 1. In other words boost invariance of the sources implies boost invariance of the metric.

The newtonian gravitational field U of any static metric may be identified with ½ l n ( - g o o ) . The general relativistic version of Poisson's equation [ 6 ] is

U;~:~= 4~G( Too + T~ + T~e + Too).

For a boost invariant source consisting of a scalar field ~~ with metric G.,~(O) on field space and potential

V(Ù) we obtain

' CSIC (Spain) Fellow.

u :%~= - 87rG V ( 0 ) ,

which is negative provided V(O) is positive. Global strings are thus expected to give rise to repulsive gravitational fields. Moreover, unless V(0) vanishes everywhere U will diverge logarithmically at large spatial distances and the metric will not be asymptot- ically fiat. Even if V(0) vanishes identically any bounded U must in fact be constant since U is then harmonic. One therefore does not expect to find regular global string solutions.

For an isolated single string it is reasonable to assume in addit ion rotational symmetry. The metric then becomes

d s : = e x p ( 2 U ) [ - d / 2 + ( d x 3 ) 2 ] -t-A -1 d r 2 + r 2 d0 2 ,

with U and A functions of r only. In the simplest U ( 1 ) case, a complex scalar field 0 is assumed to vary as

= exp (in0) R (r). The non-vanishing stress tensor vierbein components are thus given by

= ½ [ 3 ( d R / d r ) 2 + n 2 R 2 / r 2 ] + V ( R 2 ) , (1)

T~e= ½ [ 3 ( d R / d r ) 2 - n 2 R 2 / r 2] - V(R 2) , (2)

T o 6 = - ½ [ A ( d R / d r ) 2 - n Z R 2 / r 2 l - V ( R 2 ) , (3)

where V(O*O) is the U ( 1 ) invariant potential for the field O. Far from the core one expects R-,y/ with dR~dr negligible. The stress tensor at large distances will then behave like

50 0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland)


2 2 T,~a-~ ~ 2 diag(1, - 1 , - 1 , 1) (4)

in coordinates (t, x 3, r, 0). In flat spacetime the total energy per unit length of such a source diverges. Thus one cannot ignore the gravitational field, even at in- finity, and our previous remarks show that the metric component goo should diverge. Typically one finds that if the core is regular, i.e. R - , 0 as r-*0 then the metric becomes singular at some finite radial proper distance [2-5 ].

This problem does not arise for a local string because the term which causes the difficulty depends on ~0/00. For a local string this is replaced by a covariant derivative which vanishes far from the core. Another way in which the problem is eliminated is to consider sigma model strings [7]. These have stress tensors very similar to ( 1 ) - ( 3 ) but V(~) vanishes identically. Far from the core (and perhaps everywhere) the first two terms on the right-hand side o f (2) and ( 3 ) cancel one another because 0 is assumed to tend to a holomorphic function of the transverse spatial isothermal coordinate z = X 1 + ix 2 =

p exp(i0), with

A - l d r 2 + r 2 d02=~"~ 2 d g d z ,

r ~ .Qp ,

and

r

p=ex

Thus in the holomorphic case the exterior newtonian gravitational field vanishes.

If one assumes that (4) is true everywhere, which amounts to assuming that the map ~(r, 0) maps spacetime into a circle of radius q in field space, one can, as pointed out by Cohen and Kaplan [ 3], solve the Einstein equations exactly. The result is

_ Z ( l + w ) / 2 dtZ + z ( l - W ) ( d x 3 ) 2

+ exp( -,uGn2z 2) Z("a-l)/2(dz2+dO 2) , (5)

where / t= 4nGt/2 and w is an arbitrary real constant. Very recently Harari and Polychronakos [8] have shown that the special case w= 1 gives a regular Killing horizon, which is associated with the repulsive nature of the central cosmic string generating the

field. In some aspects the situation resembles the cos- mological event horizon in de Sitter spacetime which is associated with the repulsive gravitational field due to the stress tensor T ~ = 8nGV(O)g~p.

The purpose of the present paper is to generalize the Harar i -Polychronakos construction to cover the case of gravity coupled to a general sigma model, and see how it may be applied to low energy superstring theory.

The special case of (5) with w= 1 may be cast in the form

d s 2 = - [ 1 . - 2 / / n 2 ln(r/ro) ] d f 2 + ( d x 3) 2

dr 2 + + r 2 d 0 2 , (6)

1 - - 2 / / n 2 l n ( r / r o )

where ro is a constant. The static metric is singular at r = 0 but at r=ro exp( 1/2/~n 2) has a regular Killing horizon. Note that the metric ( 5 ) is only boost invariant in the t, x 3 plane if w=0 . In particular the Harar i -Polychronakos metric (6) is not boost invariant. In fact the argument we gave above based on the harmonic property of ln(A/B) can be extended to cover the case of a regular horizon with a regular in- terior inside which ln(A/B) is a bounded function. Thus there can be no non-singular solution with a horizon, provided that the source is boost symmetric. This was already known in the particular example of the rotationally symmetric U( 1 ) global string [4,5 ]. However, we see that it holds much more generally, since the argument we have just given depends only on the boost invariance of the source.

For the particular case of a scalar source, these re- sults are consistent with a general no-hair theorem for gravity coupled to a general sigma model with fields 0 A and a positive potential V(~)>~0 which is in ad- dition assumed to be concave, i.e. VAVBViS a positive semi-definite symmetric matrix, where 7A denotes covariant derivative with respect to the sigma model metric GAg(0).

To establish our no-hair result we employ a general sigma model identity - the so-called Bochner formula. A derivation of this formula and an explana- tion of its significance in a notation suitable for phy- sicists can be found in the excellent article by Misner [9] on harmonic maps. Similar methods have been previously used in the theory of cosmic strings in ref. [5].

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The energy density Tfor a sigma model coupled to gravity is given by

• OX~ ~ x l~ ~ .

Then

T:":.=OA~:~OA.~

+ OA'~(6AI~R'ec~--~'e~R'~cRDO('70D~)o1~.e

+ 0 # ( 0 ~ : ' : , ) .... (7)

where all covariant derivatives are covariant with respect to the spacetime metric g ~ and the target-space metric GA~ in the manner described in ref. [9]. The field equation for the scalar field reads

0"4;":/7 = G'4BV B V . ( 8 )

We now integrate (7) over the region of spacetime inside an event horizon over a finite slice in (t, x3) - space using the divergence theorem. If the fields are assumed to be independent of time, the boundary term

f J 0M (JM

must vanish, where v,~ is the null generator of the event horizon, which coincides with the Killing vector O/Ot on the horizon. Here dA is the area element of the event horizon. We now substitute the Einstein field equations

R./s = 8rcG [ G AnOA~OBls+ g~I~ 1"( 0 ) ] ,

and the equation of motion (8) in the right-hand side of (7). Under the assumption that the sectional cur- vature RAeco of the target manifold is negative or zero the right-hand side is positive definite and vanishes only for constant fields. It follows that there can be no non-singular static solutions with horizons for scalar fields satisfying our assumptions on V(0). Note that for the U ( 1 ) Mexican hat potential the convex- ity assumption is not in fact true.

The upshot of our arguments so far is that one cannot obtain regular solutions with event horizons unless, possibly, one relaxes the assumption of boost invariance. Nevertheless we still feel that solutions of the Harari-Polychronakos type are sufficiently interesting to merit further investigation. After all, it is

quite conceivable that the assumption of boost invariance, which follows from the simple field theoretic models we have been considering, breaks down near the core of the string. The situation is perhaps analo- gous to that in liquid crystal physics, where sigma model solutions with singularities are used to model topological defects. Even though the models break down near the core region of the defect they give a good description at large scales [ 10,11 ].

We next generalize the solution of Harari and Polychronakos and find that if the target manifold M of the sigma model possesses closed geodesics and the potential is assumed to vanish identically then the metric (5) is an exact solution of the Einstein and sigma model field equations. Thus if we take w= 1 we obtain a solution with a regular event horizon.

The proof of this result is straightforward. Let 04(2) be a closed geodesic in M, with 0~<2~< 2n and 0 1 (0) = 0 '~ (2n), the length of the geodesic being 2nt/. Then, since a geodesic is a particular case of harmonic maps [ 12 ], the field

0 t = 0 1 ( n 0 ) , n = l , 2 ..... d i m M ,

is a solution of the equations of motion with stress tensor given by (4).

Of course many target spaces admit closed geodesics, but of particular current interest in the case when G.,~ is the Poincar6 metric of constant negative cur- vature on two-dimensional hyperbolic space ~z or on the quotient ofN z by a discrete subgroup F of the iso- metry group PSL(2, ~). I f 0 is the dilaton and a the axion, i.e. the dual of the 3-form field strength, the metric on M is given by

GA#~ dO A d 0 ~ = d 0 2 + e x p ( 2 0 ) da 2 .

Sigma models of this sort inevitably arise in low energy supergravity models with "no-scale" SU ( 1, 1 ) invariant scalar fields. Unfortunately H 2 has no closed geodesics and our generalized Harari-Polychronakos construction is not possible. On the other hand, holomorphic solutions are possible and these are supersymmetric - possessing Killing spinors. They are discussed in ref. [ 13 ].

To obtain more interesting solutions we consider as the target manifold not H 2 but rather H2/PSL(2, 7/). This is the fundamental domain, D, of the modular group PSL(2, 7]), and may be thought of as spec-

52


ifying the conformal structure of a two-dimensional torus, or equivalent ly a latt ice in the euclidean plane. I f r, Im r > 0, is the modula r parameter , the metr ic is

d r d f GAB dO A dO B= - 4 - - (9)

( r - f ) 2 '

and two points r; r ' related by

a r + b r t ~ _ _ _

cr+d '

are identif ied, where a, b, c, d are in 7 and satisfy a d - c b = 1. The fundamenta l domain D is non-compact but of finite total area. It contains two conical or orbifold points r = i and r = exp ( 2 ~ i / 3 ) (or their im- ages) at which the target mani fo ld D is not smooth. The deficit angle at these points is ~ and 4~/3 respectively.

The domain D may also be in terpreted as giving a two-parameter family of un imodula r 2 × 2 matr ices hm~. I f r = r ~ +i r2 then

r2 r! I r l 2 "

The point r = i is associated with a square latt ice or torus with h , ,n=6 .... The point r = e x p ( 2 1 r i / 3 ) is associated with a t r iangular lattice. In both cases the matr ix torus or latt ice hm, has an addi t ional symmetry, Z2 X 772 or $3 respectively.

Sigma models of this type also appear in low energy superstr ing theory as a result of compact i f ica- tion, the metr ic (9) being referred to as the Zamo- lodchikov metr ic [14]. The sigma field r measures the size of the internal space, which has metr ic h,,n(r) dy 'ndy n. Greene et al. [15] have found cosmic strings solutions of the corresponding field equations. Their solut ions are locally holomorphic , being of the form

are not holomorphic and hence not supersymmetric . Since they contain a regular event horizon with a non- vanishing tempera ture this last fact is perhaps no ac- cident since spacet imes with horizons with non-zero tempera ture do not in general admi t Kill ing spinors. Moreover, one expects on rather general grounds that supersymmetry should be broken at non-zero temperature .

To describe the solutions in detail we need to describe the closed geodesics on D in detail . Let us be- gin with some simple cases. Geodesics on the upper half-plane correspond to circles whose radii lie on the real axis. Choosing D as the region bounded by the lines rl = -+ 1 and the circle I rl = 1 (see fig. 1 ) we must f ind a sequence of such circular arcs which lie in D and satisfy suitable j u m p condi t ions on the boundary of D. Note that the boundary lines have no geometric

//

• , . . . . . . . . . . ' , ~ U / / / A

V//g/ ' / / / ~ / i x / l l A U// /~; / / ,~ / /

i i

. . . . . . . . . / i . . . . . . . . .

. . . . . . . . . . / / / / / / / ~ I / A / . / l ' / / / / / ~ " "

i

j ( r ) = z ,

w h e r e j ( r ) is the elliptic modula r function. The form of the spacet ime (which is not a x i s y m m e t r i c ) , i s boost - invar iant and since locally ho lomorphic it admits, at least locally, Kill ing spinors, i.e. it is supersymmetric. Apart from the orbifold singularities it fits into the sigma model string f ramework given in ref. [ 7 ] and extended to supersymmetry in ref. [ 13 ].

By contrast the solut ions we wish to consider here

- I /2 0 1/2

Fig. 1. Fundamental domain of the modular group PSL(2, Z). The closed geodesic has been drawn following the method ex- plained in the text and corresponds to N = 3.

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Volume 240, number 1.2 PHYSICS LETTERS B 19 April 1990

significance; they are chosen for convenience . Since the l ines ~, = +_ ½ are ident if ied, a geodesic h i t t ing one vertical b o u n d a r y re-enters D m o v i n g in the same vertical sense at the same vertical height on the op- posite boundary ; the centre of the circular arc is thus shifted along the real axis by one unit . By contrast , since the two circular b o u n d a r y arcs with z~ > 0 and z, < 0 are ident i f ied in the opposi te sense, geodesics h i t t ing one circular b o u n d a r y re-enter D m o v i n g in the opposi te hor izonta l sense at the same vert ical height.

These s imple rules are sufficient to show the exis- tence of some ( in fact, inf ini te ly m a n y ) closed geodesics. For example: Start at an integer po in t (N, 0) on the posi t ive real axis and cons t ruc t a circle with radius Nx/N 2-- 1 centered on that point . It will cut the circle I ~l = 1 at right angles be tween 0 < ~, < ½ as long as N>~ 3. The so-const ructed circle will hit z~ = ½ at ~2 = Nx/N-~-~. The associated geodesic will re-enter D at z, = - ½, z2 = N ~ - ~ as a circle centered at ( N - 1,

0) with the same radius Nx/N 7-- 1, and will leave D at ~, = l , z2 = ~ . Proceeding in the same way we ob ta in 2 N + 1 arcs with centres on the real axis at z~ = N , N - 1, ..., - N . We have now ob ta ined a closed geodesic consis t ing of 2 N + 1 circular segments. The geodesic is symmet r i c about the imag ina ry axis on which it reaches its greatest vert ical height, and inter- sects i tself 2 N - 1 t imes, N - 1 on the imag ina ry axis z, = 0 and N t i m e s on ~, = + ½.

A s imilar cons t ruc t ion will work if one starts not with an integer point bu t a half-integer, i.e. at z~ = ( 2 N - 1 ) / 2 , z2=0, with N : 3 , 4 . . . . . In this case the geodesic a t ta ins its greatest vert ical height on the

l ine ~, = +_ ½. This exhausts the possibi l i t ies if the geodesic is to hit the b o u n d a r y circle I zl = 1 at right angles.

All sigma fields m app ing spacet ime into closed geodesics cons t ruc ted in this m a n n e r will give rise to cosmic strings with event hor izons as solut ions of the low energy field equa t ions of f u n d a m e n t a l superstrings, p rov ided we are prepared to tolerate a singular core.

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Stringy cosmic strings with horizons