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Non-Lagrangian forces for cosmic strings
Volume 143, number 3
PHYSICS LETTERS A
8 January 1990
NON-LAGRANGIAN FORCES FOR COSMIC STRINGS P.S. LETELIER Departamento de Matem(ttica Aplicada, IMECC, Universidade Estadual de Campinas, 13.081 Campinas. SP, Brazil
Received 3 July 1989; revised manuscript received 1 September 1989; accepted for publication 7 November 1989 Communicated by J.P. Vigier
Following a construction that gives the non-Lagrangian forces related with radiation reaction for particles, we present a method to construct non-Lagrangian forces for cosmic strings. The force associated to the Polyakov string is compared with a particular class of non-Lagrangian forces.
Almost all presently known results about cosmic strings [1,2] are obtained using models of strings with no transverse dimensions (thin string approximation [3,4] ). In this a p p r o x i m a t i o n the evolution of the strings is described by the usual N a m b u action and it is characterized by the appearance of conspicuous points like cusps and kinks [ 5 - 7 ] . The astrophysical as well as cosmological applications o f strings are based on the knowledge of the string gravitational fields. Since cusps move with the speed o f light, their gravitational fields will present - from the point of view of gravity - the most interesting features [7]. F o r instance, some o f the structures observed in the sky may be formed by the effect of gravitational repulsion produced near the string cusps
[8]. If one takes either a higher order a p p r o x i m a t i o n of the string transverse dimensions in the string action [9] (thick string) or the effect of the gravitational radiation reaction on the string motion [7] the field equation will suffer corrections that may prevent the appearance o f points traveling with the speed o f light [10]. A first approach to the gravitational back-reaction p r o b l e m can be found in ref. [71. In this note, following a similar derivation that in the case of particles gives radiation reaction forces [ 1 1 ], we derive correction terms for the string equation o f motion. We find that in the lowest order of a p p r o x i m a t i o n the resulting equation is closely re-
lated with the evolution equation for the Polyakov string [ 12 ]. Analogies between particles and strings have been used since the early days o f the string theory to better understand the structure o f the theory of interacting string [ 13-15 ]. As a by-product o f these theories we have the K a l b - R a m o n d fields [ 13-15 ] that play an i m p o r t a n t role in the F r e u n d - R u b i n ansatz [ 16 ] for K a l u z a - K l e i n theories [ 17 ]. Let us first recall some facts about the motion equation for a radiating particle [ 18 ] o f rest mass m in the presence o f an external force fz', Z¢'= ngy" + f U / r n ,
( 1)
g is a coupling constant, n is a form factor that depends on the radiated field: n = ½ for scalar field, n = for vector fields and n = - ~ for the usual linearized gravity [18] (tensor + scalar field): and ~.=~,~'-Z,'+Z~Z~2
,' .
(2)
We have denoted by an overdot the derivation o f the particle world line Z u with respect to the particle proper time. The case o f the actual radiation reaction force due to gravity is much more complicated [ 1 1 ]. We have that terms of the same order as the explicit radiation reaction terms appear implicitly in the equations of m o t i o n through the retardation effects in the metric, they appear with opposite sign and cancel the main contribution of the explicit terms (thereby assuring that in low-velocity a p p r o x i m a tion the main radiation term is o f quadrupole char-
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103
Volume 143, number 3
PHYSICS LETTERS A
8 January 1990
acter rather than dipole character). In this case the ?'~' term that appears in eq. ( I ) is given by [ 19]
d e t e r m i n e F ~. The identity analogous to ( 5 ) for strings is [ 14]
;,q = Z ~ + 3;g"2,~ 2 " ,
V c X ~ ' V A VAA~,= 0 ,
(3)
and n = - ~253 . The form of higher order constributions is quite involved [ 19] and no other terms have been computed in a closed form [ 11,19 ]. Note that 7~'=7,f (or 7~') is characterized by the following properties: (a) 7 u is a four-vector constructed with the derivatives of Z ~ only. (b) It satisfies
7"Z,, = 0 .
(4)
(c) ~,u does not contains linear terms in Z'U. (d) Z ~ = 0 implies that 7u=0, i.e. a flee particle does not radiate. (e) Expressions (2) and (3) cannot be obtained from a usual local variational principle. The identity (4) for ~,u given by either (2) or (3) is a consequence o f
2,'2~=o.
(5)
For consistency the external force must satisfy the constraint
f"2,, = O.
(6 )
The terms 7f and ~,~ are the first two members of a familly that shear the same properties ( a ) - ( e ) mentioned above. The generic member of the family is d/+2
/-i
d.s,+2 Z " + Z " k~=o
l dl-k+l
_.
~ dk+2
~
-
which is a consequence o f the more general identity VA X u V B VcX~, -~. I ( V B ) , C A + V C ~ A B - - V4 ?,BC ) = 0
.
( 11)
The quantity V B V c X " is closely related with the extrinsic curvature [ 12,20]. As a m a t t e r o f fact ( 11 ) tells us that VBVcX u are four-vectors perpendicular to the world sheet. ESX¢'_ = V A V A X "u c a n be interpreted as a four-vector pointing along a mean direction on the two-plane spanned by the four-vectors V B V c X " . To find the four-vector F u we notice that ( 2 ) is obtained from a differentiation o f ( 5 ) . Thus, from ( 11 ) we get V4 X ~ FI,LBC'= O,
( 12 )
where Ff~B(. ~- V I ) X I ' I V A D X ° : V B ( . X c ~
AI- V . . I B c . X # .
(
13 )
We have introduced the notation V~sc = VAVeVc, etc. We note that from F~ABC we cannot built an invariant (scalar) under the r e p a r a m e t r i z a t i o n o f the string world sheet - we need an even n u m b e r o f indices referring to the world sheet. Thus differentiating one more time the expression ( 11 ), we obtain VAX'UF~Dt:.Bc=O
,
(
14
Z, ,
(7) where C~ indicate as usual the b i n o m i a l coefficients. The evolution equation for cosmic strings (closed N a m b u strings) equivalent to ( 1 ) is given by [ 14 ] V4 V'~X u = k F ~ + F U / M ,
(10)
where Ff)F~B("- VD~:Bc'X~' + V H X I ' ( V B c X , V n s n X '~ + V E H X ~ W D B c X ~ + Vz)tt X ~ V F . B c X ~ ) .
( 15
,u The quantity FDEBC has the following symmetry.,
(8)
F g t : e c = Ff)EcB •
( 16
where F ¢' is an external force acting on the string o f world sheet X ~, M and k are constants, VA represents the covariant derivative on the world sheet o f metric
Therefore, we have only two different invariants constructed with Ff)EeC,
",' ~p = ill,, O 4 XUO B X ~ ,
(9 )
F~l ( 1) -- p~DDBR ,
( 17 )
A, B. etc. take the values 0 and 1: 0n denotes partial derivation with respect to the two p a r a m e t e r s that describe the world sheet: 1.A= (l.0 1.1 ). NOW following the particle case as a guide we shall
/'f(2) =--FUDBDB "
( 18 )
104
F~(1) and F~'(z ) can be written in the more appealing form
Volume 143, number 3
PHYSICS LETTERS A
F~'( I ) = [~[]XU + VAXU( DX,~Tq V AX'~ + 2V ABX'~VBIIX,~) ,
(19)
F~(2) = VAVq VAXU + VAXU( VscX,~ V B c X ~ "k- 2 V A B X a [ -] V B X a ) .
( 20 )
Using the identities (10) and ( 11 ), and [ 18 ] V BcA X u -- VcBA X u = V DX~,RD,4BC ,
(21)
RABCD = 1 ( ~2A C ~)BD -- ~BC ~AD ) R ,
( 22 )
where RABCD and R are the Riemann-Christoffel curvature tensor and the Ricci scalar respectively, we find that F,~(,) = [ ] D X ' ~ + W ' X " [ ½VA([]X, D X ")
+ 2VABX"VB[]X. ] ,
(23)
Ff(2) =F~(t ) - ½RVqX u .
(24)
Note that F~U(t) as well as Ft(z u ) satisfy VAXUF~,=O
(25)
by construction. Thus, the analogue of the lowest order 7u term for strings is F~(I). Note that F~'(~) is a four-vector constructed with the derivatives of X u only (property (a) for particles). It satisfies (25) ( property ( b ) for particles ), and unlike F~'(2), it does not contain linear terms in ~ X u (property (c) for particles). We also have that ~ X u = O (free string) implies for nonsingular string points (det (~'AB):# 0) that F f ( l ) = 0 (property (d) for particles). Expressions (23) and (24) cannot be obtained from a usual variational principle (property (e) for particles ); we will come back to this point later. The terms F~(~) and F~(2) have a quadrupolar character (fourth order derivatives) as y~. Also, as in the case of particles, the two F f represent the first member of a class of F u. The second member built with six derivatives of the world sheet has the form F~(,) with i= 1..... 9. The explicit form of the F~ is quite involved. To be more precise let us consider the gravitational baekreaction force for cosmic strings. Even though a general expression for this force is unknown, due to the peculiarities of the strings, we can affirm that this force will be represented by nonlocal integrals involving the whole string world-sheet. Note that in the case of particles the higher order con-
8 January 1990
stributions are nonlocal and depend on the whole history of the particle [ 19 ]. We can speculate that whether a finite self-force for strings can be written in an explicit way, it may be a linear combinations of F~' because of the invariance of the theory and the non-Lagrangian character of the known radiation reaction forces. The computations to evaluate the radiation backreaction for strings are made considering a small segment of the whole string [7 ], e.g., a segment with a cusp. In other words, one approximates the string to a point. So the multipolar properties of the strings as well as its reparametrization invariance are not taken into account. As we said before, even for the simpler case of particles closed expressions beyond the quadrupolar moment are not known. The actual computation fo the string backreaction is an open problem and a difficult one. Alas, the approximate computation already mentioned does not give a hint to prove or disprove the above stated conjecture. Also, physical arguments based on segments of strings, in general, are not valid for the whole string for the same reasons that we mentioned before. In summary we believe that the non-Lagrangian forces F f are good candidates to represent back-reaction forces and that the explicit knowledge of FI' may serve as a hint to find these radiation forces. To better understand the corrections introduced by the F~ terms we shall first study the equations of motion of the Polyakov string [12]. A Lagrangian density for the Polyakov strings is given by 5P= x / - det (yAS) ( M + a R X ~ C ] X u ) ,
(26)
where a is another constant. From (21), (22) and (11) we find [ 3 X u D X u = V AsX~VASXu - - R .
(27)
S i n c e x / - d e t (TAB) R can be written in the form OAKA for a two-dimensional space [20] (Gauss-Bonnet theorem in two dimensions), we have that the inclusion of terms like VqXu[~Xu and VABXPVABXItin the action gives the same equation of motion for closed strings ~1. For this reason we have that the Polyakov string is unique if one considers actions de~t For open strings we have that the inclusion of topological terms in the action modifies the motion of the string end points. See for instance ref. [21 ].
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PHYSICS LETTERS A
p e n d i n g on tensorial densities q u a d r a t i c in the second d e r i v a t i v e s o f X " only. In o r d e r to s e p a r a t e the c o n t r i b u t i o n s to the e q u a tion o f m o t i o n o f the d i f f e r e n t parts o f the Lagrangian ( 2 6 ) , let m e d e f i n e h')----- x ~ - - d e I ( z 4 B ) [ Z ] X u [ ] X . ,
8 January 1990
tion gives forces p e r p e n d i c u l a r to the m e m b r a n e world "'tube". I w a n t to t h a n k N. Deruelle, B. Linet and E. Verdaguer for different discussions about cosmic strings.
(28)
References and its a s s o c i a t e d " g e n e r a l i z e d f o r c e " F~p) as del (Z.,~) F~'p)
j_ ~-
02
0 Y?
Or"~Oru OABX.
0
0 Y?
Or A 00 4 X . "
(29)
After s o m e algebra we find #2
I ,, = F ~ ' ~ ~F~,~
-½DX'~X~[]X"
+ 2V,~ X " V '~BX. [] X " .
( 30 )
T h e P o l y a k o v t e r m F f p ) also satisfies ( 2 5 ) , a n d [ S I X ' = 0 i m p l i e s F f p } = 0 for n o n s i n g u l a r points. Since F}'p) is u n i q u e for L a g r a n g i a n densities quadratic in s e c o n d o r d e r d e r i v a t i v e s o f X C we conclude that n e i t h e r f'{'(~ ) n o r F~'~ j) can be d e r i v e d , in the usual way, f r o m a L a g r a n g i a n density. " and F~p) d e s c r i b e d i f f e r e n t T h e t e r m s F i"l l ) , El(2) possibilities o f self-forces acting on the string. F~'~,~ is the closet a n a l o g u e o f the q u a d r u p o l a r rad i a t i o n r e a c t i o n force 7~. P a r t i c u l a r s o l u t i o n s o f the P o l y a k o v string e q u a t i o n o f m o t i o n are s t u d i e d in ref. [10]. N o t e that always, by c o n s i s t e n c y [ 1 4 ] , we should h a v e also for the external force V.4X"Fu=O. Finally, we w a n t to p o i n t out that the m e t h o d used to d e r i v e F " can also be used to o b t a i n forces for "'strings" o f any d i m e n s i o n s ( m e m b r a n e s ) . In particular, the e x p r e s s i o n s ( 2 0 ) a n d ( 2 1 ) give valid forces for m - d i m e n s i o n a l strings. But, since the identity ( 2 2 ) is valid only for t w o - d i m e n s i o n a l m a n i folds ( o n e - d i m e n s i o n a l string) for h i g h e r - d i m e n /l sional strings ( m e m b r a n e s ) Ff~j) and F~(z~ are no longer related by eq. ( 24 ). T h e m e t h o d by c o n s t r u e ~2 The derivation of (29) in a particular gauge can be found in ref. [ 10 ]. It seems that gauge invariant expressions equivalent to (29) have not appeared before in the literature.
106
[ 1] A. Vilenkin, Phys. Rep. 121 ( 1985 ) 263. [ 2 ] A. Albrecht and N. Turok, Phys. Rev. Len. 54 ( 1985 ) 1868; N. Kaiser and A. Stebbins, Nature 310 ( 1984 ) 391 : J. Traschen, N. Turok and R. Brandenberger, Phys. Rev. D 34 (1986) 919; T. Hara, Prog. Theor. Phys. 75 (1986) 836; N. Turok, Phys. Rev. Left. 55 (1985) 1801: C. Hogan, Nature 320 (1986) 572: J. Gott Ill, Astrophys. J. 288 (1985) 422. [ 3 ] D. F6rster, Nucl. Phys. B 81 ( 1974 ) 84. [4] E. Copeland, M. Hindmarsh and N. Turok, Phys. Rev. Lett. 58 (1987) 1910. [ 5 ] N. Turok, Nucl. Phys. B 242 ( 1984 ) 520. [6] D. Garfinkle and T. Vachaspati, Phys. Rev. D 37 (1988) 257. [7] C. Thompson, Phys. Rev. D 37 (1988) 283. [8] T. Vachaspati, Phys. Rev. D 35 (1987) 1767: Gen. Rel. Grav. 19 (1987) 1053. [9] R. Gregory, in Cosmic strings: the current status, eds. F.S. Acceta and L.M. Krauss (World Scientific, Singapore, 1988). [ 10 ] T.L. Curtright, G.I. Grandour and C.K. Zachos, Phys. Rev. D34(1986) 3811. [ 11] P. Havas, in: Isolated gravitational systems in general relativity, LXVII Corso Soc. Italiana di Fisica (Bologna, 1979) pp. 74-155. [ 12] A. Polyakov, Nucl. Phys. B 268 (1986) 406. [ 13] M. Kalb and P. Ramond, Phys. Rev. D 9 (1974) 2273. [ 14 ] P.S. Letelier, Phys. Rev. D 15 ( 1977 ) 1055. [15] P.S. Letelier, Phys. Rev. D 16 (1977) 322; 18 (1978) 359: J. Math. Phys. 19 (1978) 1898. [16] P.G.O. Freund and M.A. Rubin, Phys. Len. B 97 (1980) 327. [17] M.J. Duff, B.E.W. Nilsson and C.N. Pope, Phys. Rep. 130 (1986) 1. [ 18 ] A.O. Barut and D. Villaroel, J. Phys. A 8 ( 1975 ) 156. [ 19] A. Kfihnel, Ann. Phys. (NY) 28 (1964) 116, and references therein. [20] L.P. Einsenhart, Riemannian geometry (Princeton Univ. Press, Princeton, 1925). [21 ] A.A. Zheltukhin, Yad. Fiz. 34 ( 1981 ) 562 [Sov. J. Nucl. Phys. 34(1981)311].
PHYSICS LETTERS A
8 January 1990
NON-LAGRANGIAN FORCES FOR COSMIC STRINGS P.S. LETELIER Departamento de Matem(ttica Aplicada, IMECC, Universidade Estadual de Campinas, 13.081 Campinas. SP, Brazil
Received 3 July 1989; revised manuscript received 1 September 1989; accepted for publication 7 November 1989 Communicated by J.P. Vigier
Following a construction that gives the non-Lagrangian forces related with radiation reaction for particles, we present a method to construct non-Lagrangian forces for cosmic strings. The force associated to the Polyakov string is compared with a particular class of non-Lagrangian forces.
Almost all presently known results about cosmic strings [1,2] are obtained using models of strings with no transverse dimensions (thin string approximation [3,4] ). In this a p p r o x i m a t i o n the evolution of the strings is described by the usual N a m b u action and it is characterized by the appearance of conspicuous points like cusps and kinks [ 5 - 7 ] . The astrophysical as well as cosmological applications o f strings are based on the knowledge of the string gravitational fields. Since cusps move with the speed o f light, their gravitational fields will present - from the point of view of gravity - the most interesting features [7]. F o r instance, some o f the structures observed in the sky may be formed by the effect of gravitational repulsion produced near the string cusps
[8]. If one takes either a higher order a p p r o x i m a t i o n of the string transverse dimensions in the string action [9] (thick string) or the effect of the gravitational radiation reaction on the string motion [7] the field equation will suffer corrections that may prevent the appearance o f points traveling with the speed o f light [10]. A first approach to the gravitational back-reaction p r o b l e m can be found in ref. [71. In this note, following a similar derivation that in the case of particles gives radiation reaction forces [ 1 1 ], we derive correction terms for the string equation o f motion. We find that in the lowest order of a p p r o x i m a t i o n the resulting equation is closely re-
lated with the evolution equation for the Polyakov string [ 12 ]. Analogies between particles and strings have been used since the early days o f the string theory to better understand the structure o f the theory of interacting string [ 13-15 ]. As a by-product o f these theories we have the K a l b - R a m o n d fields [ 13-15 ] that play an i m p o r t a n t role in the F r e u n d - R u b i n ansatz [ 16 ] for K a l u z a - K l e i n theories [ 17 ]. Let us first recall some facts about the motion equation for a radiating particle [ 18 ] o f rest mass m in the presence o f an external force fz', Z¢'= ngy" + f U / r n ,
( 1)
g is a coupling constant, n is a form factor that depends on the radiated field: n = ½ for scalar field, n = for vector fields and n = - ~ for the usual linearized gravity [18] (tensor + scalar field): and ~.=~,~'-Z,'+Z~Z~2
,' .
(2)
We have denoted by an overdot the derivation o f the particle world line Z u with respect to the particle proper time. The case o f the actual radiation reaction force due to gravity is much more complicated [ 1 1 ]. We have that terms of the same order as the explicit radiation reaction terms appear implicitly in the equations of m o t i o n through the retardation effects in the metric, they appear with opposite sign and cancel the main contribution of the explicit terms (thereby assuring that in low-velocity a p p r o x i m a tion the main radiation term is o f quadrupole char-
0375-9601/90/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland)
103
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PHYSICS LETTERS A
8 January 1990
acter rather than dipole character). In this case the ?'~' term that appears in eq. ( I ) is given by [ 19]
d e t e r m i n e F ~. The identity analogous to ( 5 ) for strings is [ 14]
;,q = Z ~ + 3;g"2,~ 2 " ,
V c X ~ ' V A VAA~,= 0 ,
(3)
and n = - ~253 . The form of higher order constributions is quite involved [ 19] and no other terms have been computed in a closed form [ 11,19 ]. Note that 7~'=7,f (or 7~') is characterized by the following properties: (a) 7 u is a four-vector constructed with the derivatives of Z ~ only. (b) It satisfies
7"Z,, = 0 .
(4)
(c) ~,u does not contains linear terms in Z'U. (d) Z ~ = 0 implies that 7u=0, i.e. a flee particle does not radiate. (e) Expressions (2) and (3) cannot be obtained from a usual local variational principle. The identity (4) for ~,u given by either (2) or (3) is a consequence o f
2,'2~=o.
(5)
For consistency the external force must satisfy the constraint
f"2,, = O.
(6 )
The terms 7f and ~,~ are the first two members of a familly that shear the same properties ( a ) - ( e ) mentioned above. The generic member of the family is d/+2
/-i
d.s,+2 Z " + Z " k~=o
l dl-k+l
_.
~ dk+2
~
-
which is a consequence o f the more general identity VA X u V B VcX~, -~. I ( V B ) , C A + V C ~ A B - - V4 ?,BC ) = 0
.
( 11)
The quantity V B V c X " is closely related with the extrinsic curvature [ 12,20]. As a m a t t e r o f fact ( 11 ) tells us that VBVcX u are four-vectors perpendicular to the world sheet. ESX¢'_ = V A V A X "u c a n be interpreted as a four-vector pointing along a mean direction on the two-plane spanned by the four-vectors V B V c X " . To find the four-vector F u we notice that ( 2 ) is obtained from a differentiation o f ( 5 ) . Thus, from ( 11 ) we get V4 X ~ FI,LBC'= O,
( 12 )
where Ff~B(. ~- V I ) X I ' I V A D X ° : V B ( . X c ~
AI- V . . I B c . X # .
(
13 )
We have introduced the notation V~sc = VAVeVc, etc. We note that from F~ABC we cannot built an invariant (scalar) under the r e p a r a m e t r i z a t i o n o f the string world sheet - we need an even n u m b e r o f indices referring to the world sheet. Thus differentiating one more time the expression ( 11 ), we obtain VAX'UF~Dt:.Bc=O
,
(
14
Z, ,
(7) where C~ indicate as usual the b i n o m i a l coefficients. The evolution equation for cosmic strings (closed N a m b u strings) equivalent to ( 1 ) is given by [ 14 ] V4 V'~X u = k F ~ + F U / M ,
(10)
where Ff)F~B("- VD~:Bc'X~' + V H X I ' ( V B c X , V n s n X '~ + V E H X ~ W D B c X ~ + Vz)tt X ~ V F . B c X ~ ) .
( 15
,u The quantity FDEBC has the following symmetry.,
(8)
F g t : e c = Ff)EcB •
( 16
where F ¢' is an external force acting on the string o f world sheet X ~, M and k are constants, VA represents the covariant derivative on the world sheet o f metric
Therefore, we have only two different invariants constructed with Ff)EeC,
",' ~p = ill,, O 4 XUO B X ~ ,
(9 )
F~l ( 1) -- p~DDBR ,
( 17 )
A, B. etc. take the values 0 and 1: 0n denotes partial derivation with respect to the two p a r a m e t e r s that describe the world sheet: 1.A= (l.0 1.1 ). NOW following the particle case as a guide we shall
/'f(2) =--FUDBDB "
( 18 )
104
F~(1) and F~'(z ) can be written in the more appealing form
Volume 143, number 3
PHYSICS LETTERS A
F~'( I ) = [~[]XU + VAXU( DX,~Tq V AX'~ + 2V ABX'~VBIIX,~) ,
(19)
F~(2) = VAVq VAXU + VAXU( VscX,~ V B c X ~ "k- 2 V A B X a [ -] V B X a ) .
( 20 )
Using the identities (10) and ( 11 ), and [ 18 ] V BcA X u -- VcBA X u = V DX~,RD,4BC ,
(21)
RABCD = 1 ( ~2A C ~)BD -- ~BC ~AD ) R ,
( 22 )
where RABCD and R are the Riemann-Christoffel curvature tensor and the Ricci scalar respectively, we find that F,~(,) = [ ] D X ' ~ + W ' X " [ ½VA([]X, D X ")
+ 2VABX"VB[]X. ] ,
(23)
Ff(2) =F~(t ) - ½RVqX u .
(24)
Note that F~U(t) as well as Ft(z u ) satisfy VAXUF~,=O
(25)
by construction. Thus, the analogue of the lowest order 7u term for strings is F~(I). Note that F~'(~) is a four-vector constructed with the derivatives of X u only (property (a) for particles). It satisfies (25) ( property ( b ) for particles ), and unlike F~'(2), it does not contain linear terms in ~ X u (property (c) for particles). We also have that ~ X u = O (free string) implies for nonsingular string points (det (~'AB):# 0) that F f ( l ) = 0 (property (d) for particles). Expressions (23) and (24) cannot be obtained from a usual variational principle (property (e) for particles ); we will come back to this point later. The terms F~(~) and F~(2) have a quadrupolar character (fourth order derivatives) as y~. Also, as in the case of particles, the two F f represent the first member of a class of F u. The second member built with six derivatives of the world sheet has the form F~(,) with i= 1..... 9. The explicit form of the F~ is quite involved. To be more precise let us consider the gravitational baekreaction force for cosmic strings. Even though a general expression for this force is unknown, due to the peculiarities of the strings, we can affirm that this force will be represented by nonlocal integrals involving the whole string world-sheet. Note that in the case of particles the higher order con-
8 January 1990
stributions are nonlocal and depend on the whole history of the particle [ 19 ]. We can speculate that whether a finite self-force for strings can be written in an explicit way, it may be a linear combinations of F~' because of the invariance of the theory and the non-Lagrangian character of the known radiation reaction forces. The computations to evaluate the radiation backreaction for strings are made considering a small segment of the whole string [7 ], e.g., a segment with a cusp. In other words, one approximates the string to a point. So the multipolar properties of the strings as well as its reparametrization invariance are not taken into account. As we said before, even for the simpler case of particles closed expressions beyond the quadrupolar moment are not known. The actual computation fo the string backreaction is an open problem and a difficult one. Alas, the approximate computation already mentioned does not give a hint to prove or disprove the above stated conjecture. Also, physical arguments based on segments of strings, in general, are not valid for the whole string for the same reasons that we mentioned before. In summary we believe that the non-Lagrangian forces F f are good candidates to represent back-reaction forces and that the explicit knowledge of FI' may serve as a hint to find these radiation forces. To better understand the corrections introduced by the F~ terms we shall first study the equations of motion of the Polyakov string [12]. A Lagrangian density for the Polyakov strings is given by 5P= x / - det (yAS) ( M + a R X ~ C ] X u ) ,
(26)
where a is another constant. From (21), (22) and (11) we find [ 3 X u D X u = V AsX~VASXu - - R .
(27)
S i n c e x / - d e t (TAB) R can be written in the form OAKA for a two-dimensional space [20] (Gauss-Bonnet theorem in two dimensions), we have that the inclusion of terms like VqXu[~Xu and VABXPVABXItin the action gives the same equation of motion for closed strings ~1. For this reason we have that the Polyakov string is unique if one considers actions de~t For open strings we have that the inclusion of topological terms in the action modifies the motion of the string end points. See for instance ref. [21 ].
105
Volume 143, number 3
PHYSICS LETTERS A
p e n d i n g on tensorial densities q u a d r a t i c in the second d e r i v a t i v e s o f X " only. In o r d e r to s e p a r a t e the c o n t r i b u t i o n s to the e q u a tion o f m o t i o n o f the d i f f e r e n t parts o f the Lagrangian ( 2 6 ) , let m e d e f i n e h')----- x ~ - - d e I ( z 4 B ) [ Z ] X u [ ] X . ,
8 January 1990
tion gives forces p e r p e n d i c u l a r to the m e m b r a n e world "'tube". I w a n t to t h a n k N. Deruelle, B. Linet and E. Verdaguer for different discussions about cosmic strings.
(28)
References and its a s s o c i a t e d " g e n e r a l i z e d f o r c e " F~p) as del (Z.,~) F~'p)
j_ ~-
02
0 Y?
Or"~Oru OABX.
0
0 Y?
Or A 00 4 X . "
(29)
After s o m e algebra we find #2
I ,, = F ~ ' ~ ~F~,~
-½DX'~X~[]X"
+ 2V,~ X " V '~BX. [] X " .
( 30 )
T h e P o l y a k o v t e r m F f p ) also satisfies ( 2 5 ) , a n d [ S I X ' = 0 i m p l i e s F f p } = 0 for n o n s i n g u l a r points. Since F}'p) is u n i q u e for L a g r a n g i a n densities quadratic in s e c o n d o r d e r d e r i v a t i v e s o f X C we conclude that n e i t h e r f'{'(~ ) n o r F~'~ j) can be d e r i v e d , in the usual way, f r o m a L a g r a n g i a n density. " and F~p) d e s c r i b e d i f f e r e n t T h e t e r m s F i"l l ) , El(2) possibilities o f self-forces acting on the string. F~'~,~ is the closet a n a l o g u e o f the q u a d r u p o l a r rad i a t i o n r e a c t i o n force 7~. P a r t i c u l a r s o l u t i o n s o f the P o l y a k o v string e q u a t i o n o f m o t i o n are s t u d i e d in ref. [10]. N o t e that always, by c o n s i s t e n c y [ 1 4 ] , we should h a v e also for the external force V.4X"Fu=O. Finally, we w a n t to p o i n t out that the m e t h o d used to d e r i v e F " can also be used to o b t a i n forces for "'strings" o f any d i m e n s i o n s ( m e m b r a n e s ) . In particular, the e x p r e s s i o n s ( 2 0 ) a n d ( 2 1 ) give valid forces for m - d i m e n s i o n a l strings. But, since the identity ( 2 2 ) is valid only for t w o - d i m e n s i o n a l m a n i folds ( o n e - d i m e n s i o n a l string) for h i g h e r - d i m e n /l sional strings ( m e m b r a n e s ) Ff~j) and F~(z~ are no longer related by eq. ( 24 ). T h e m e t h o d by c o n s t r u e ~2 The derivation of (29) in a particular gauge can be found in ref. [ 10 ]. It seems that gauge invariant expressions equivalent to (29) have not appeared before in the literature.
106
[ 1] A. Vilenkin, Phys. Rep. 121 ( 1985 ) 263. [ 2 ] A. Albrecht and N. Turok, Phys. Rev. Len. 54 ( 1985 ) 1868; N. Kaiser and A. Stebbins, Nature 310 ( 1984 ) 391 : J. Traschen, N. Turok and R. Brandenberger, Phys. Rev. D 34 (1986) 919; T. Hara, Prog. Theor. Phys. 75 (1986) 836; N. Turok, Phys. Rev. Left. 55 (1985) 1801: C. Hogan, Nature 320 (1986) 572: J. Gott Ill, Astrophys. J. 288 (1985) 422. [ 3 ] D. F6rster, Nucl. Phys. B 81 ( 1974 ) 84. [4] E. Copeland, M. Hindmarsh and N. Turok, Phys. Rev. Lett. 58 (1987) 1910. [ 5 ] N. Turok, Nucl. Phys. B 242 ( 1984 ) 520. [6] D. Garfinkle and T. Vachaspati, Phys. Rev. D 37 (1988) 257. [7] C. Thompson, Phys. Rev. D 37 (1988) 283. [8] T. Vachaspati, Phys. Rev. D 35 (1987) 1767: Gen. Rel. Grav. 19 (1987) 1053. [9] R. Gregory, in Cosmic strings: the current status, eds. F.S. Acceta and L.M. Krauss (World Scientific, Singapore, 1988). [ 10 ] T.L. Curtright, G.I. Grandour and C.K. Zachos, Phys. Rev. D34(1986) 3811. [ 11] P. Havas, in: Isolated gravitational systems in general relativity, LXVII Corso Soc. Italiana di Fisica (Bologna, 1979) pp. 74-155. [ 12] A. Polyakov, Nucl. Phys. B 268 (1986) 406. [ 13] M. Kalb and P. Ramond, Phys. Rev. D 9 (1974) 2273. [ 14 ] P.S. Letelier, Phys. Rev. D 15 ( 1977 ) 1055. [15] P.S. Letelier, Phys. Rev. D 16 (1977) 322; 18 (1978) 359: J. Math. Phys. 19 (1978) 1898. [16] P.G.O. Freund and M.A. Rubin, Phys. Len. B 97 (1980) 327. [17] M.J. Duff, B.E.W. Nilsson and C.N. Pope, Phys. Rep. 130 (1986) 1. [ 18 ] A.O. Barut and D. Villaroel, J. Phys. A 8 ( 1975 ) 156. [ 19] A. Kfihnel, Ann. Phys. (NY) 28 (1964) 116, and references therein. [20] L.P. Einsenhart, Riemannian geometry (Princeton Univ. Press, Princeton, 1925). [21 ] A.A. Zheltukhin, Yad. Fiz. 34 ( 1981 ) 562 [Sov. J. Nucl. Phys. 34(1981)311].