PHYSICAL REVIEW D VOLUME 40, NUMBER 6 15 SEPTEMBER 1989
Tidal force from cosmic strings
David Garfinkle Department of Physics, University of Florida, Gainesville, Florida 32611
(Received 22 May 1989)
We find the tidal force from the class of cosmic-string traveling waves of Vachaspati. The tidal force from a straight string vanishes. In contract the tidal force from the curved strings treated here actually diverges as the string is approached. This tends to change the implications of a recent paper by Clarke, Ellis, and Vickers on the relation between string curvature and tidal force.
I. INTRODUCTION
Recent work has treated the gravitational field of a cosmic string. Most of the work has been done on weakly gravitating strings; though there have been some treatments of strings that are strongly gravitating. The Riemann tensor of a straight string vanishes and so a straight string gives rise to no tidal force. Therefore one might suspect that the tidal force of a curved string is small provided that the curvature of the string is small. In a recent paper1 Clark, Ellis, and Vickers assume an upper bound on the tidal force of a closed loop of cosmic string and derive a lower bound on the length of the loop.
It is unclear what the implications of this result are. Clarke, Ellis, and Vickers adopt the point of view that the upper bound on the tidal force should be "small"; thus the lower bound on the length of the loop should be large and therefore there is a difficulty with the usual cosmic-string scenario which assumes the existence of small loops as well as large ones. This point of view was challenged by Unruh, Hayward, Israel, and McManus. 2
These authors argue that the tidal force near a loop of cosmic string is large. They consider strongly gravitating strings and neglect the string thickness. Their model for the spacetime of such an object is a four-dimensional manifold with conical singularities on a two-dimensional timelike surface. They assume that the stress-energy tensor is bounded outside of the string surface and show that the Weyl tensor diverges as the surface is approached. The divergent part of the Weyl tensor depends on the extrinsic curvature of the string surface. Since the tidal force produced by these strings is large, the resulting bound on the length of the loop given by the result of Ref. 1 is small and there is no inconsistency with the cosmic-string scenario.
In this paper we will find the Riemann tensor of a class of weakly gravitating cosmic-string configurations first studied by Vachaspati.3 These configurations are traveling waves propagating along the string. In Ref. 3 the metrics corresponding to each of these string configurations was found. We will calculate the Riemann tensor corresponding to these metrics and examine the relation between the tidal force, the distance from the string, and the extrinsic curvature of the string.
40
II. RIEMANN TENSOR OF THE STRING CONFIGURATIONS
The history of a cosmic string is given by specifying its spacetime coordinates zM as a function of two parameters r and a. Define the quantities F^ v , 0 , and Q^y by
F^_dz^_dz^_dz^_dzZ_ ^ (1)
dr da da dr
0 = ( - i - F ^ v ) 1 / 2 , (2)
The action for the string is the Nambu action4
1 = -tiJQdrda , (4)
where [i is the mass per unit length of the string. From the action it follows that the string's equation of motion is5
e M a „ e \ = o , (5) and the string's stress-energy tensor is6,7
T^v(xa)=fifdrdaeQ^Q\d\xa-za(rya)) . (6)
The class of solutions to Eq. (5) found in Ref. 3 are
Z° = T, zl = a, z2=f(a-r), z3=g(a~r) . (7)
Here / and g are arbitrary functions. Let r]^ be the background flat metric and define the quantities k^, N^y
N^M^py and 0 as
fcM=aM(x-f),
N^d^-f'U-M^ ,
N^d^z-g'ix-m^,
M^N^Nv+N^N, ,
pcos6=y—f{x —t) ,
p sind=z —g {x — t) .
(8)
(9)
(10)
(11)
(12a)
(12b)
Then it follows from Eq. (6) that the stress-energy tensor of the solution given in Eq. (7) is
T^=/n(M^-V^)8(y -fix -t))8(z -g(x -t)) . (13)
1801 © 1989 The American Physical Society
1802 DAVID GARFINKLE - 40
Writing the spacetime metric as g,,=~,,,+h,,, the weak-field Einstein equation (in an appropriate gauge) is
As shown in Ref. 3 the solution to Eq. (14) corresponding to the stress-energy tensor of Eq. (13) is
where R is a positive constant. We now calculate the Riemann tensor corresponding
to the metric in Eq. (15). In weak-field gravity the Riemann tensor is given by
Rpoav =aaa[/3hplv -ava[/3hpla . (16)
Therefore we first calculate a,aah,, Using Eq. (15) we find
Using Eqs. (8)-(12) to evaluate the derivatives of p and M,, we find
Here the argument of f and g is x -t. Now define the tensor X,, by
Xp,= ( ~ " c o s B - ~ " s ~ ~ ~ ) ( N , N , - & , & , )
+2( f " ~ i n O + ~ " c o s O ) ~ , , & , , . (22)
Then collecting the results of Eqs. (17)-(21) some straightforward but tedious algebra yields
We now express the Riemann tensor of the string in terms of the shape tensor of the string surface. For a sur- face Z with intrinsic metric q,,, there is a tensor nA,, such that
for any normal 6, to the surface 2. The tensor nh,, is called the shape tensor of the surface Z and is an analog of extrinsic curvature defined for surfaces of any dimen- sion. It follows from Eq. (24) that
The intrinsic metric of the string surface is
- 4,, - TpY - MpI, . (26)
Evaluating IIA,, by using Eqs. (20) and (26) in Eq. (25) we find
nA,,= -k,k,( A ) . (27)
Now define the tensor WFVp, by
Then it follows from Eq. (22) that
W,,, = -k,kuX,p .
Thus we find that the Riemann tensor is given by
Equations (28) and (30) give an expression for the Riemann tensor in terms of the quantity p and the intrin- sic metric and extrinsic curvature of the string surface. The quantity p is the spatial distance (on a t =const sur- face) between the field point and the string.
In comparing the results of this paper to those of Ref. 2, it is important to note that the shape tensor measures the physical curvature near the string, rather than the curvature on the zero thickness string. This is because (a) the shape tensor measures only the curvature of the string surface in flat space, not the contribution of the string's gravitational field to its curvature; and (b) the weak-field approximation breaks down for sufficiently small p. It is expected that this breakdown of the weak- field approximation is due in part to the fact that zero thickness strings are treated. For the finite thickness strings of the usual cosmic-string scenario it is thought that the weak-field approximation is valid everywhere.
Let the Cartesian components of nA,, be of order 1 /L. Then L can be regarded as a radius of curvature of the string traveling wave. It then follows from Eqs. (28) and (30) that the Cartesian components of R,,, are of order p / (pL) . These values of the curvature tensor com- ponents and the radius of curvature satisfy the bounds of Ref. 1 for any values of p , L, and p (within the domain of validity of the weak-field approximation). Thus in this case (and probably in general) the result of Clarke, Ellis, and Vickers places no restriction on the size of a cosmic string. Instead this result simply expresses one aspect of the relation between the string's extrinsic curvature and its tidal force.
ACKNOWLEDGMENTS
I would like to thank Alex Vilenkin, Pablo Laguna, and Robert M. Wald for helpful discussions. This work was supported in part by NSF Grant No. PHY-8500498 to the University of Florida.
40 TIDAL FORCE FROM COSMIC STRINGS 1803
'C. J. S. Clarke, G. F. R. Ellis, and J. A. Vickers, Report No. SA. Vilenkin, Phys. Rep. 121, 265 (1985). SISSA Ref. 88 Astro, 1988 (unpublished). 6 ~ . Vachaspati and A. Vilenkin, Phys. Rev. D 31, 3052 (1985).
2 ~ . G. Unruh, G. Hayward, W. Israel, and D. McManus, Phys. 'D. Garfinkle and C. M. Will, Phys. Rev. D 35, 1124 (1987). Rev. Lett. 62, 2897 (1989). 8 ~ . O'Neill, Semi-Riemannian Geometry (Academic, New York,
3 ~ . Vachaspati, Nucl. Phys. B277, 593 (1986). 1983). 4 ~ . B. Nielsen and P. Olesen, Nucl. Phys. B61,45 (1973).
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