- Email: [email protected]
Gauge theoretic formulation of dilatonic gravity coupled to particles
__ _liii
&A
ELSEVIER
PROCEEDINGS SUPPLEMENTS
Nuclear Physics B (Proc. Suppl.) 88 (2000) 245-248
Gauge theoretic formulation of dilatonic gravity coupled to particles* Victor 0. Rivelles Universidade de S?io Paulo, C. Postal 66318,05315-970, e-mail: [email protected] We discuss the formulation non-Abelian charge.
Instituto de Fisica Sao Paulo, SP, Brazil
of the CGHS model in terms of a topological BF theory coupled to particles carrying
In order to have a better understanding of the quantum properties of four-dimensional gravity theories two-dimensional models have been extensively studied. A particular model where black holes can be formed and quantization can be performed is the Callan, Giddings, Harvey and Strominger (CGHS) model [l]. It is described by the action
where iz is the curvature scalar build with the metric By,,, 4 is the dilaton, A is the cosmological constant and fi is a set of scalar matter fields. If the conformal transformation gfiv = em2@B,, is performed the action (1) takes the form
where q = em2@ and R is the scalar curvature built with the metric gpv. Now the field equation for 77 implies that R = 0 so that the twodimensional space-time is locally flat and there is no black hole solution. While the model described by (1) presents Hawking radiation the model described by (2) has no Hawking radiation. It has been argued that with proper care of the conforma1 transformation no ambiguity exists [2]. Even so the quantization in either form is not free of troubles [3]. An important feature of the action (2) is that it can be cast as a topological gauge theory of the *This
work
is partially
supported
by CNPq.
0920-5632/00/% - see front matter 0 2000 Elsevier Science B.V PI1 SO920-5632(00)00777-S
BF type with a gauge group which is a central extension of the two-dimensional PoincarB group [4]. If matter is coupled in this formulation it should be coupled in a gauge invariant way. A possibility makes use of a formulation of relativistic particles which carry non-Abelian charges [5]. It was applied to the gauge theoretic version of dilatonic gravity [6]. An important aspect is that the curvature equation R = 0 never acquires a source term. We will show that in the formulation where particles carry non-Abelian charge a new gauge invariant coupling does exist for topological BF theories. This new coupling provides a source term for the curvature equation and black hole solutions can then be found [7]. Particles carrying non-Abelian degrees of freedom were originally introduced in the context of QCD [8,5]. Th e y are described by the group element g(T) and a real constant element of the algebra K, 7 being the proper time of the particle. It is useful to introduce the variable Q(T) = 9(M9-l(T),
(3)
which is in the adjoint representation. a covariant derivative can be introduced
As usual
D, = & + ei+‘A,(x(~)).
(4
If we also consider a kinetic term for the relativistic particle then an action which is gauge and reparametrization invariant is [5] S = -m
s
dT&+
s
d~Tr(Kg-1(7)D,g(T)).(5)
This action is also invariant under the transformation K + SKS-1 where S is r independent. All rights reserved.
FCO.Rioelles/Nuclear
246
Physics B (Proc. Suppl.)
This shows that the action (5) is independent of the direction in the internal symmetry space given by K. Varying the action (5) with respect to xp(r) we get a non-Abelian version of the Lorentz force = -eTr(F’1,Q)k”,
rn?p + P&i”?’
(6)
while varying with respect to g(r) we get a covariant conservation equation for the non-Abelian charge Q 2
&(7)W(~) = 0.
+ [A,(z(r)),
d2xTr(q F),
I
(8)
where F = dA+A2 is the curvature two-form corresponding to the connection one-form A and 77 is a zero-form transforming in the co-adjoint representation of the gauge group. Since the structure of the BF theory requires two fields then, besides the coupling involving the gauge field A, we can consider another coupling involving the Lagrange multiplier n. A coupling of the type Tr(r) Q) is gauge invariant but not proper time reparametrization invariant. In order to get a reparametrization invariant action we introduce the worldline einbein e(r) and the respective mass term for the particle. So we can consider an extension of the former actions to S
=
+
e
d2xTr(q F) +
s
+g S s
d2x d2x
1 2m2
+
J
dTTr(Q(T)Ap(x))
b2(x - X(T)) 3
dTe(T) Tr(Q(~)rl) d2(a: - ox)
JdTe(T), s
+
hAh-’
g
-+
hg,
-dhh-I, K + K,
time reparametrization
k’“(T’) =
e’(7’) = de(T),
.YF pv = g
s
(10)
(11)
d7 e(T) Q(T) b2(x - X(T)).
(12)
This will allows us to find black hole solutions The field equation for the Lagrange multiplier @“D,q
= e
dTQ(T) 62(x -
X(T))
C?(T),
[7]. is (13)
I
while the equation ified to $+e[A,(x(r)),
for non-Abelian
Q]i’l+g[rl(z(r)),
charge is mod-
Q]e(r)
= 9@4)
generalizing the conservation equation (7) to the BF theory. The field equation obtained by varying the worldline einbein is g
J
d2x Tr(Q(~)q(x))
6”(x-x(T))+im’
= 0.(15)
For the two dimensional dilatonic gravity theory we choose the gauge group as the central extension of the Poincark group [4]
(16)
where Pa is the translation generator, J is the Lorentz transformation generator and 2 is a central element of the group. The supersymmetric extension of (16) was performed in [9]. When we consider the algebra (16) we can expand the one form gauge potential in terms of the generators of the algebra
A =eaPa +wJ+AZ.
q~+ hrjh-‘,
-p(T),
with all remaining fields being reparametrization scalars. The new coupling now implies that the connection is no longer flat and has as source the non-Abelian charge Q
(9)
where e and g are independent coupling constants. The action (9) is invariant under gauge transformations A
and proper
(7)
These equations are known as the Wong equations [8]. Consider now a two-dimensional BF topological field theory S =
88 (2000) 245-248
(17)
The fields ea, w and A are going to be identified with the zweibein, the spin connection and an Abelian gauge field, respectively. The Lagrange multiplier n can be expanded as (18)
VO. Rivelles/Nuclear
Physics B (Proc. Suppl.)
with components q”, ~2 and 773with r/2 being proportional to the dilaton in (2). Then the curvature two-form F has components Fa(P)
=
dea + webeba,
F(J)
=
dw,
F(Z)
=
dA + ieaebcab.
Finally the equation of motion (15) gives a constraint among the non-Abelian charge, the Lagrange multiplier on the worldline and the particle mass
J --&El)
9
charge
Q can be ex-
Q = Q”P, + Q3J + QzZ. The field equations
(20)
for the gauge fields (12) are
@V(dPet + +&ba)
JPY Fd w +gJ EFV +-eaebeab (dpav ;pv, +gJ
+
dTe(T) Qa(7) 6’(z - Z(T)) = 0,
g
d7-e(7) Q3(7) 6’(a: -z(T))
= 0,
(22)
dTe(T)
=
0,
(23)
Qz(T)~'(~:
while the field equations pliers (13) are EPV(8”% +
e
J J J
e
dT
+
Q~(T)
-X(T))
for the Lagrange
multi-
X(T))S?‘(T)
0,
(24)
ezEabqb)
b2(X
-X(T))~@(T)
=
0,
(25)
=
0.
(26)
@"d"7l3
+
e
dr Q3(7-) 6’(z -X(T))?(T)
The equations of motion charge (14) are
dQ”
x +
+ %%$3 -
for the
non-Abelian
wpQb)if
gEb”(v’Q3- v3Qb)e(T) = 0, d&z -dT
Eab(ee;Qb9
dQ3 o d7=.
X(T)
- gv”Qbe(T))
(27) = 008)
+Qzrl3)6'(X
-X(T))
(30)
=
0, t(T)
=
7.
Let us consider first the gauge field sector. In the gravitational sector we will choose a diagonal zweibein e: = ey = 0 with the non vanishing components satisfying ez = (e:)-l, and a vanishing space component of the connection w1 = 0. For the Abelian gauge field we will choose the axial gauge AI = 0. eqs.(21-23) reduce to
he: +woe: = &fJJo
=
Q3r/2
We will discuss some solutions of the above equations. In order to solve (21-30) we have to perform several gauge fixings. We also have to choose a space-time trajectory for the particle. We will look for static solutions so we use Rindler like coordinates (x, t). For simplicity let us consider the proper time gauge for the particle e(~) = 1 and set the particle at rest in the origin
+v3fabeL)
-
(Q% + .
0
wvfabvb
dTQ,(T)b’(cc
@"(&72
+
(21)
d2x
1
(19) +2
Similarly the non-Abelian panded as
247
88 (2000) 245-248
doAO + eze:
= =
=
gQ"W, gQ%L gQsd(x), gQzW).
(31) (32) (33) (34)
If no matter is present (e = g = 0) then we find flat space-time as the only solution. Now consider the situation when matter is present. If Q3 = 0 and Q” # 0 then the space-time has torsion but no curvature. If Q3 # 0 and Q” = 0 then the space-time has curvature but no torsion. Let us consider the last case. Take Q2 and Q3 as constants (as we shall see below Q2 and Q3 constants and Q” = 0 is a solution of (27-29) ). Then we find as solution of (31-34) wo
=
gQ&),
e:= (6- 2gQ31xb4,
A0
=
-cc + gQzc(x)
+ -4,
(35)
where & and .a are integration const,a.nt,s. The space-time described by (35) 11~ a black hole and the curvature sca1a.r is given b\- (33) R = gQd(x), Notice t,hat gQ.1 ca.n now IX>understood
FCO.RiveNes/Nuclear
248
as the black hole mass and it is essential 9 # 0. For the Lagrange multiplier choices reduce eqs.(24-26) to
w0rl1 =
u0rl0 + r/34
&r)2
eho
-
=
=
sector
Physics B (Proc. Suppl.) 88 (2000) 245-248
to have
the gauge
0,
-e&d(x),
eh = 0, &rl3 = -eQd(z).
(36)
In the presence of matter with Qs and Qs constants and Q” = 0 we find, using (35) 770 =
;(” -
172
fee -
=
%Q&l)$ eQzc(z)
~1 = 0, + Z,
71s=
REFERENCES
-eQ&LW
1.
where E is another integration constant and E(X) is the step function. The appearance of the step function in the solution for the Lagrange multiplier fields signals that there are topological restrictions to the motion of particles. It is remarkable that the would be cosmological constant 77s is now a step function changing sign at the position of the particle. The dilaton 7)~ still has its linear term but has also acquired a step function. We can however set Q2 = 0 and still have a linear dilaton and the black hole (35), which is independent of Qs. For the non-Abelian charge the equations (2729), after the gauge choice, reduce to
e&Iwo - d&$3 @ - 4Q0w - Q&) C&t- eQ’e8 + g(Q’n’ Q,” -
- Qd) = 0, - dQ”v3 - Q311’) = 0, - Q1qo) = 0,
Qs = 0.
(38)
In the presence of matter with Q” = 0 and Q2 and Qs constants we find using (35) and (37) that Q2 and Qs are constants as we had anticipated. Finally the constraint equation (30) becomes, after the gauge choices,
dQa(~ha(~> 0) + +
Q3(Th’2(T,
Qs(~)r/s (7, O)] + $x2
= 0.
0)
In the presence of matter with Q” = 0 (39) is ill defined since according to (37) ns and 77s have a discontinuity at x = 0. We then take Q2 = 0 and (39) becomes gQsE+ $m2 = 0 giving a constraint among the integration constants Qs and I? and the mass m. Recalling that the curvature scalar is R = gQd(cc) we can interpret the black hole mass as being due to the non-Abelian charge of the particle Qs or to its mass m. Some local solutions for the gauge theoretic version of dilatonic gravity theories with nonAbelian sources have been presented. Presently we are investigating the constraint structure of the model and its Hamiltonian formulation. We also plan to study global aspects of the model.
2. 3.
4. 5. 6. 7. 8. 9.
C.G. Callan, S.B. Giddings, J.A. Harvey and A. Strominger, Phys. Rev. D45 (1992) R1005. M. Cadoni and S. Mignemi, Phys. Lett. B358 (1995) 217. D. Cangemi, R. Jackiw and B. Zwiebach, Ann. Phys. 245 (1996) 408; S. Cassemiro F.F. and V.O. Rivelles, Phys. Lett. B452 (1999) 234. D. Cangemi and R. Jackiw, Phys. Rev. Lett. 69 (1992) 233. A.P. Balachandran, S. Borchardt and A. Stern, Phys. Rev. D17 (1978) 3247. J.P. Lupi, A. Restuccia and J. Stephany, Phys. Rev. D54 (1996) 3861. M.M. Leite and V.O. Rivelles, Phys. Lett. B392 (1997) 305. S. K. Wong, Nuovo Cimento 65A (1970) 689. V. 0. Rivelles, Phys. Lett. B321 (1994) 189; D. Cangemi and M. Leblanc, Nucl. Phys. B420 (1994) 363; M. M. Leite and V. 0. Rivelles, Class. Quantum Grav. 12 (1995) 627.
&A
ELSEVIER
PROCEEDINGS SUPPLEMENTS
Nuclear Physics B (Proc. Suppl.) 88 (2000) 245-248
Gauge theoretic formulation of dilatonic gravity coupled to particles* Victor 0. Rivelles Universidade de S?io Paulo, C. Postal 66318,05315-970, e-mail: [email protected] We discuss the formulation non-Abelian charge.
Instituto de Fisica Sao Paulo, SP, Brazil
of the CGHS model in terms of a topological BF theory coupled to particles carrying
In order to have a better understanding of the quantum properties of four-dimensional gravity theories two-dimensional models have been extensively studied. A particular model where black holes can be formed and quantization can be performed is the Callan, Giddings, Harvey and Strominger (CGHS) model [l]. It is described by the action
where iz is the curvature scalar build with the metric By,,, 4 is the dilaton, A is the cosmological constant and fi is a set of scalar matter fields. If the conformal transformation gfiv = em2@B,, is performed the action (1) takes the form
where q = em2@ and R is the scalar curvature built with the metric gpv. Now the field equation for 77 implies that R = 0 so that the twodimensional space-time is locally flat and there is no black hole solution. While the model described by (1) presents Hawking radiation the model described by (2) has no Hawking radiation. It has been argued that with proper care of the conforma1 transformation no ambiguity exists [2]. Even so the quantization in either form is not free of troubles [3]. An important feature of the action (2) is that it can be cast as a topological gauge theory of the *This
work
is partially
supported
by CNPq.
0920-5632/00/% - see front matter 0 2000 Elsevier Science B.V PI1 SO920-5632(00)00777-S
BF type with a gauge group which is a central extension of the two-dimensional PoincarB group [4]. If matter is coupled in this formulation it should be coupled in a gauge invariant way. A possibility makes use of a formulation of relativistic particles which carry non-Abelian charges [5]. It was applied to the gauge theoretic version of dilatonic gravity [6]. An important aspect is that the curvature equation R = 0 never acquires a source term. We will show that in the formulation where particles carry non-Abelian charge a new gauge invariant coupling does exist for topological BF theories. This new coupling provides a source term for the curvature equation and black hole solutions can then be found [7]. Particles carrying non-Abelian degrees of freedom were originally introduced in the context of QCD [8,5]. Th e y are described by the group element g(T) and a real constant element of the algebra K, 7 being the proper time of the particle. It is useful to introduce the variable Q(T) = 9(M9-l(T),
(3)
which is in the adjoint representation. a covariant derivative can be introduced
As usual
D, = & + ei+‘A,(x(~)).
(4
If we also consider a kinetic term for the relativistic particle then an action which is gauge and reparametrization invariant is [5] S = -m
s
dT&+
s
d~Tr(Kg-1(7)D,g(T)).(5)
This action is also invariant under the transformation K + SKS-1 where S is r independent. All rights reserved.
FCO.Rioelles/Nuclear
246
Physics B (Proc. Suppl.)
This shows that the action (5) is independent of the direction in the internal symmetry space given by K. Varying the action (5) with respect to xp(r) we get a non-Abelian version of the Lorentz force = -eTr(F’1,Q)k”,
rn?p + P&i”?’
(6)
while varying with respect to g(r) we get a covariant conservation equation for the non-Abelian charge Q 2
&(7)W(~) = 0.
+ [A,(z(r)),
d2xTr(q F),
I
(8)
where F = dA+A2 is the curvature two-form corresponding to the connection one-form A and 77 is a zero-form transforming in the co-adjoint representation of the gauge group. Since the structure of the BF theory requires two fields then, besides the coupling involving the gauge field A, we can consider another coupling involving the Lagrange multiplier n. A coupling of the type Tr(r) Q) is gauge invariant but not proper time reparametrization invariant. In order to get a reparametrization invariant action we introduce the worldline einbein e(r) and the respective mass term for the particle. So we can consider an extension of the former actions to S
=
+
e
d2xTr(q F) +
s
+g S s
d2x d2x
1 2m2
+
J
dTTr(Q(T)Ap(x))
b2(x - X(T)) 3
dTe(T) Tr(Q(~)rl) d2(a: - ox)
JdTe(T), s
+
hAh-’
g
-+
hg,
-dhh-I, K + K,
time reparametrization
k’“(T’) =
e’(7’) = de(T),
.YF pv = g
s
(10)
(11)
d7 e(T) Q(T) b2(x - X(T)).
(12)
This will allows us to find black hole solutions The field equation for the Lagrange multiplier @“D,q
= e
dTQ(T) 62(x -
X(T))
C?(T),
[7]. is (13)
I
while the equation ified to $+e[A,(x(r)),
for non-Abelian
Q]i’l+g[rl(z(r)),
charge is mod-
Q]e(r)
= 9@4)
generalizing the conservation equation (7) to the BF theory. The field equation obtained by varying the worldline einbein is g
J
d2x Tr(Q(~)q(x))
6”(x-x(T))+im’
= 0.(15)
For the two dimensional dilatonic gravity theory we choose the gauge group as the central extension of the Poincark group [4]
(16)
where Pa is the translation generator, J is the Lorentz transformation generator and 2 is a central element of the group. The supersymmetric extension of (16) was performed in [9]. When we consider the algebra (16) we can expand the one form gauge potential in terms of the generators of the algebra
A =eaPa +wJ+AZ.
q~+ hrjh-‘,
-p(T),
with all remaining fields being reparametrization scalars. The new coupling now implies that the connection is no longer flat and has as source the non-Abelian charge Q
(9)
where e and g are independent coupling constants. The action (9) is invariant under gauge transformations A
and proper
(7)
These equations are known as the Wong equations [8]. Consider now a two-dimensional BF topological field theory S =
88 (2000) 245-248
(17)
The fields ea, w and A are going to be identified with the zweibein, the spin connection and an Abelian gauge field, respectively. The Lagrange multiplier n can be expanded as (18)
VO. Rivelles/Nuclear
Physics B (Proc. Suppl.)
with components q”, ~2 and 773with r/2 being proportional to the dilaton in (2). Then the curvature two-form F has components Fa(P)
=
dea + webeba,
F(J)
=
dw,
F(Z)
=
dA + ieaebcab.
Finally the equation of motion (15) gives a constraint among the non-Abelian charge, the Lagrange multiplier on the worldline and the particle mass
J --&El)
9
charge
Q can be ex-
Q = Q”P, + Q3J + QzZ. The field equations
(20)
for the gauge fields (12) are
@V(dPet + +&ba)
JPY Fd w +gJ EFV +-eaebeab (dpav ;pv, +gJ
+
dTe(T) Qa(7) 6’(z - Z(T)) = 0,
g
d7-e(7) Q3(7) 6’(a: -z(T))
= 0,
(22)
dTe(T)
=
0,
(23)
Qz(T)~'(~:
while the field equations pliers (13) are EPV(8”% +
e
J J J
e
dT
+
Q~(T)
-X(T))
for the Lagrange
multi-
X(T))S?‘(T)
0,
(24)
ezEabqb)
b2(X
-X(T))~@(T)
=
0,
(25)
=
0.
(26)
@"d"7l3
+
e
dr Q3(7-) 6’(z -X(T))?(T)
The equations of motion charge (14) are
dQ”
x +
+ %%$3 -
for the
non-Abelian
wpQb)if
gEb”(v’Q3- v3Qb)e(T) = 0, d&z -dT
Eab(ee;Qb9
dQ3 o d7=.
X(T)
- gv”Qbe(T))
(27) = 008)
+Qzrl3)6'(X
-X(T))
(30)
=
0, t(T)
=
7.
Let us consider first the gauge field sector. In the gravitational sector we will choose a diagonal zweibein e: = ey = 0 with the non vanishing components satisfying ez = (e:)-l, and a vanishing space component of the connection w1 = 0. For the Abelian gauge field we will choose the axial gauge AI = 0. eqs.(21-23) reduce to
he: +woe: = &fJJo
=
Q3r/2
We will discuss some solutions of the above equations. In order to solve (21-30) we have to perform several gauge fixings. We also have to choose a space-time trajectory for the particle. We will look for static solutions so we use Rindler like coordinates (x, t). For simplicity let us consider the proper time gauge for the particle e(~) = 1 and set the particle at rest in the origin
+v3fabeL)
-
(Q% + .
0
wvfabvb
dTQ,(T)b’(cc
@"(&72
+
(21)
d2x
1
(19) +2
Similarly the non-Abelian panded as
247
88 (2000) 245-248
doAO + eze:
= =
=
gQ"W, gQ%L gQsd(x), gQzW).
(31) (32) (33) (34)
If no matter is present (e = g = 0) then we find flat space-time as the only solution. Now consider the situation when matter is present. If Q3 = 0 and Q” # 0 then the space-time has torsion but no curvature. If Q3 # 0 and Q” = 0 then the space-time has curvature but no torsion. Let us consider the last case. Take Q2 and Q3 as constants (as we shall see below Q2 and Q3 constants and Q” = 0 is a solution of (27-29) ). Then we find as solution of (31-34) wo
=
gQ&),
e:= (6- 2gQ31xb4,
A0
=
-cc + gQzc(x)
+ -4,
(35)
where & and .a are integration const,a.nt,s. The space-time described by (35) 11~ a black hole and the curvature sca1a.r is given b\- (33) R = gQd(x), Notice t,hat gQ.1 ca.n now IX>understood
FCO.RiveNes/Nuclear
248
as the black hole mass and it is essential 9 # 0. For the Lagrange multiplier choices reduce eqs.(24-26) to
w0rl1 =
u0rl0 + r/34
&r)2
eho
-
=
=
sector
Physics B (Proc. Suppl.) 88 (2000) 245-248
to have
the gauge
0,
-e&d(x),
eh = 0, &rl3 = -eQd(z).
(36)
In the presence of matter with Qs and Qs constants and Q” = 0 we find, using (35) 770 =
;(” -
172
fee -
=
%Q&l)$ eQzc(z)
~1 = 0, + Z,
71s=
REFERENCES
-eQ&LW
1.
where E is another integration constant and E(X) is the step function. The appearance of the step function in the solution for the Lagrange multiplier fields signals that there are topological restrictions to the motion of particles. It is remarkable that the would be cosmological constant 77s is now a step function changing sign at the position of the particle. The dilaton 7)~ still has its linear term but has also acquired a step function. We can however set Q2 = 0 and still have a linear dilaton and the black hole (35), which is independent of Qs. For the non-Abelian charge the equations (2729), after the gauge choice, reduce to
e&Iwo - d&$3 @ - 4Q0w - Q&) C&t- eQ’e8 + g(Q’n’ Q,” -
- Qd) = 0, - dQ”v3 - Q311’) = 0, - Q1qo) = 0,
Qs = 0.
(38)
In the presence of matter with Q” = 0 and Q2 and Qs constants we find using (35) and (37) that Q2 and Qs are constants as we had anticipated. Finally the constraint equation (30) becomes, after the gauge choices,
dQa(~ha(~> 0) + +
Q3(Th’2(T,
Qs(~)r/s (7, O)] + $x2
= 0.
0)
In the presence of matter with Q” = 0 (39) is ill defined since according to (37) ns and 77s have a discontinuity at x = 0. We then take Q2 = 0 and (39) becomes gQsE+ $m2 = 0 giving a constraint among the integration constants Qs and I? and the mass m. Recalling that the curvature scalar is R = gQd(cc) we can interpret the black hole mass as being due to the non-Abelian charge of the particle Qs or to its mass m. Some local solutions for the gauge theoretic version of dilatonic gravity theories with nonAbelian sources have been presented. Presently we are investigating the constraint structure of the model and its Hamiltonian formulation. We also plan to study global aspects of the model.
2. 3.
4. 5. 6. 7. 8. 9.
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