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Geometric gravitational force on particles moving in a line
Physics Letters B 299 (1993) 24-29 North-Holland
PHYSICS LETTERS B
Geometric gravitational force on particles moving in a line D. Cangemi and R. Jackiw Centerfor Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Received 13 October 1992; revised manuscript received 17 November 1992
In two-dimensional space-time, point particles can experience a geometric, dimension-specific gravity force, which modifies the usual geodesic equation of motion and provides a link between the cosmological constant and the vacuum 0-angle. The description of such a force fits naturally into a gauge theory of gravity based on the extended Poincar6 group, i.e. "string-inspired" dilaton gravity.
Consideration o f ( 1 + 1 )-dimensional - lineal gravity, for pedagogical and perhaps physical purposes, requires inventing gravitational dynamics; the Einstein theory cannot be modeled owing to the vanishing o f the two-dimensional Einstein tensor. Recent activity [ 1,2 ] in this area has focused on scalartensor theories [3], whose dynamical variables are the metric tensor and an additional scalar field, while gravitational field equations involve the Ricci scalar R - the single quantity encoding all local geometric information in two dimensions. Two models, governed by the following geometric actions, Il = f d E x x / ~ r / ( R - 2 ) I 2 = j" d2x x / ~
,
(1)
(qR-2) ,
(2)
are especially noteworthy, in that both can be equivalently described by gauge theoretical actions based on the de Sitter group [ 4 ] for I~, and on the Poincar6 group [ 2 ] with central extension [ 5 ] for I2, the latter describing "string-inspired" dilaton gravity [ 1,2 ]. Moreover, both theories are obtained by various dimensional reductions from three dimensions [3,69 ], and also 11 and 12 can be related to each other by various singular limiting procedures [ 5,7,9 ]. [ In ( 1 )
and (2) r/is a scalar field and 2 the cosmological constant. ] Here we examine the interaction of a point particle with the gravitational field. The usual matter action for a particle of mass m on the worldline x~(z) is constructed from the arc length: Im = -- m ~ ds = - m ~ drx/.fu(r)gu~(x(r) ).~(z) .
(3)
[We set the velocity of light to unity and our Minkowski signature is ( + - ); the overdot denotes differentiation with respect to r, which parametrizes the worldline in an arbitrary way. ] However, the ( l + 1 )dimensional setting provides another possibility, which is specific to this dimension and which enjoys various interesting geometric attributes. Our purpose here is to examine this point-particle-gravity interaction, and in the second half of this letter to show how naturally it fits with the gauge theoretical formulation for I2. Let us begin by observing that the usual geodesic equation of motion, modified by an additional force
l_". ~ r--~B ~ k p _- ~o" ~ , d N XU+N dr
~r This work is supported in part by funds provided by the US Department of Energy (DOE) under contract ~DE-AC0276ER03069, and by the Swiss National Science Foundation. 24
0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
Volume 299, number 1,2
PHYSICS LETTERSB
will lose general covariance if .~ is externally prescribed. ( F is the Christoffel symbol evaluated on the worldline, as also is the metric tensor.) In order to maintain general covariance .~ must be constructed from the dynamical variables of the theory, and experience with electromagnetism (which is not included in the above discussion) shows that general covariance is preserved when .~ has the tensorial properties of/7~2 ~, where -~u~is an anti-symmetric, second-rank tensor. In dimensions greater than two, such a tensor cannot be constructed from particle and/or gravitational variables; it arises when electromagnetic (or other gauge-field) degrees of freedom are dynamically active. However, in two dimensions gravitational variables allow constructing the required tensor:
.~= - e,/--g ~.
~z~xu+d1 .
12~ru~p2t~+.~gU'~x~-g~,,p2~=O .
(6)
The additional interaction term in (6) is similar to that arising from a constant external electromagnetic field in fiat two-dimensional Minkowski space-time. Indeed, just as the latter preserves the Poincar6 symmetry of a non-interacting point particle in flat spacetime, so similarly our covariantly constant field (5) respects general covariance. Moreover, as we now describe, the force law can be derived from a novel contribution to the matter-gravity action. Construction of the addition to the conventional matter-gravity action (3), which gives rise to our force, is geometrically subtle. Consider the two-form ½~u~dxUdx ~, proportional to the volume form V=-½x~-gEu~dxUdx ~, which may also be expressed in terms of the Zweibeine e au as ~a ~b - ~I t~ab~u~ dx u d x~. Since Vis closed, d V= 0, it is exact, at least locally:
V=da.
0a~ 0a~ ~Ox ( x ) - ~ x ~ ( x ) = _ _ ~ S g ~ .
(7a)
Eq. (7a) defines a one-form, whose components are also seen to satisfy
(7b)
Since the right-hand side of (7b) is a tensor, a can be taken as a vector. The covariant action is now constructed from a:
I~ = - . ~ f dzau(x(z))2u(r ) .
(8)
Under coordinate redefinition, both a and x ( r ) change, and it is straightforwardly verified that I a is a scalar. Note that the gauge ambiguity au--,au+ Oufl, which is a consequence of the defining equation (7b), changes the action only be end-point contributions. Variation o f l ~ with respect to x ( t ) produces with the help of (7b) the force in (6):
8I~
(5)
Here e~ (also e~") is the numerical anti-symmetric tensor density, Cot= - 1 (e °t = 1 ) and .~ is a constant fixing the magnitude of the field. Thus, the covariant, two-dimensional equation of motion involving matter and metric variables and generalizing the usual geodesic equation reads
21 January 1993
--.~x/-g(x(z))Eu~2~(r)
.
(9)
Thus while no local expression for the action is available, its x(r)-variation is local, giving rise to an electromagnetic-like, velocity-dependent force, which is free of gauge ambiguity and preserves general covariance. Next, we examine the contribution from I~ to the energy-momentum tensor Twhich is a functional of the matter variable x(r) and function of the field argument x: T~P(x(r) Ix) 2
--
~agap(x~
(Im +1~) .
(10a)
Evaluating the metric variation of I~ is problematic in the absence of an explicit formula for a. However, we can fix the additional term in the energy-momentum tensor by requiring its covariant conservation. In this way Tis found to be 1
TaP(x(t) Ix) = - ~ = ~ x ) r
1
× JdtN--~x
"o~
.p
2
( O x (t)~ ( x - - x ( z ) )
- ½A(x(t) Ix)g'~a(x) ,
(10b)
and we satisfy D,,T~a=0,
(11)
provided that 25
Volume 299, number 1,2
PHYSICS LETTERS B
21 January 1993
produces the following transformations on the connections:
10.4 20x u (x(z)Ix)
= --'~ I d z ~ u ~ " ~ ' ( z ) t ~ 2 ( x - x ( z ) )
"
(12)
ea_..~~a = ( ~ -1 ) ab ( eb + ~.bcOC(D_t_dOb ) , m-,o~= m + d a ,
Eq. (12) is easily solved in the parametrization where x ° ( z) = z, and one finds
a--,a=a--Oae,,beb--½02oJ+dfl+½dOaEabO b .
A = - ~e(x I-x I(t)) +2,
Here ~ is the finite Lorentz transformation with rapidity or:
(13)
where 2 is constant. Thus the metric variation of I~ also is local, giving rise to a cosmological "constant" that jumps by 2 ~ as the particle's trajectory is crossed, a property that is independent of the above parametrization choice. Although the matter dynamics can be presented without commitment to a specific model for gravity, the new interaction (8) fits especially naturally with a gauge theoretical formulation for I2, which we now review [ 5 ]. The structure group for the gauge theory is the extended (1 + 1 )-dimensional Poincar6 group with (anti-hermitian) Lorentz generator J, and translation generators Pa that close upon commutation on a central element, L which commutes with J and P~.
J/gab=~ab cosh a+~ab sinh a .
[Pa, Pb]=eabl.
3
FA--,FA= ~
U=
(o
(15)
acoc!) 1
F = d A + A 2 = f apa + f J + g I = (dea+Eabtoeb)p a + d m J
(16)
It is recognized t h a t f a is proportional to the torsion and f t o the curvature, while g involves the abelian field strength associated with a, supplemented by the volume two-form. A finite group transformation, generated by the gauge function O, o=Oaea + O t J + flI ,
26
.
(21)
- ½0 2
The upper left 3 × 3 block in U comprises the adjoint representation of the conventional Poincar6 group with -eacoc giving the translation, while the fourth row and column arise from the extension. In this representation, the extended algebra possesses a nonsingular, invariant inner product hart,
_
O o) ° _,°
and curvature two-form F,
+ ( d a + ½ea~abeb)I.
(20)
( U - I ) A B F B,
B=0
(14)
[ Latin letters refer to tangent space components and are moved by the fiat metric tensor hab=diag(1, - 1 ). ] Gauge connection one-forms are associated with the generators: Zweibeine e ~ with Pa, spin-connection o9 with J, and an additional connection a with /, giving a Lie-algebra valued connection one-form A, A=e"Pa +OgJ+aI ,
(19)
The curvature components transform similarly to (18 ), except that the inhomogeneous shifts involving parameter differentials are absent. Equivalently, one may collect the curvature components into a multiplet F A = (fa, f g), which then transforms by the adjoint 4 × 4 representation of the extended Poincar6 group:
\oc~c,~eS
[Pa, J]=~.abpb,
(18)
(17)
for which TUb U= h; this allows raising and lowering indices (A, B). A gauge invariant gravity action, equivalent to/2, is constructed with the help of a Lagrange multiplier multiplet qA= (qa, t/2, q3), taken to obey the coadjoint transformation law: 3
rlA--,fli= •
rlBUnA.
(23)
B=0
It is then easy to show that the gravitational dynamics of the gauge invariant action I~,
Volume 299, n u m b e r 1,2
1'2 = f a ~
qAF A ,
PHYSICS LETTERS B
(24)
0
(27)
qa _~~ a = ( ~/{- 1) ab( qb q_ ~bcoc )
[analogous t o f a in (20) w i t h f s e t to unity and g to
and of I2 coincide. Note in particular that no cosmological constant is present in (24); in the gauge theoretical formulation it is not a parameter in the lagrangian, but a value taken on by a variable, here q3 ~. A geometric formulation of the gravity-matter system is provided by (1/2~zG)Iz+Im+I~, where G is the gravitational coupling strength. But we seek a gauge theoretical description; so a gauge theoretical point-particle action is now constructed, following a variation of the Grignani and Nardelli method [ 8 ]. We shall arrive again at Im+ I~, once a gauge freedom is fixed. Let us begin with the particle action in flat spacetime, where the vector potential for the additional force is explicitly given by l ~ b x b ( r ) . The matter action, in first-order form and in the absence of curvature, reads /matter = f dr [ (pa - k~EabXb)yca+ ½N(p2--m 2) ] . (25) All quantities depend on the parametrization r; when p and the Lagrange multiplier N are eliminated, Im~tt¢~coincides with Im + I~ in the absence of gravity. The theory is Poincar6 invariant, hut owing to the additional interaction, the translation algebra acquires a central extension. To couple Imatt~ to gravity, we view p~ as the first two components of a coadjoint tangent space fourvector PA, with vanishing last component P3 = 0; consequently p~ transforms according to Pa "~ ff a = Pb'A[ba
21 January 1993
(26)
[analogous to ~/~ in (23) with ~/3 set to zero]. The position variable x"(r) is replaced by the tangent space coordinate qa(r), taken to be the first two components of an adjoint four-vector qA with third component set to unity and fourth component to l (qahabqb--C), where c is a fixed number (this implies qAhAsqS=c). The consequent transformation law reads a~ This mechanism for generating the cosmological constant is here dictated by the group structure. In other contexts it has been inserted "by hand", see ref. [ 10].
l(qahabq b-c) ]' The v-derivative of the position variable is promoted to a covariant derivative: (D~q) ~= ~a + Eab(q b c o u
_
eb)jcu .
(28)
Here is introduced the worldline on the manifold x ( r ) ; the space-time indices in the gravitational potentials are saturated with ~u. The transformation law for the covariant derivative, which follows from (18) and (27), is (D~q)~ (~g-')ab(Dvq)b.
(29)
Finally, further terms are added to achieve gauge invariance. Implementing the above steps results in the following action ~2.
I" +I'~= ~ dr
[(Pa+l,~abqb)(Drq)
a
-½N(pE+mE)--.~(au-lq~e~u)5c u] .
(30)
The dynamical quantities to be varied independently in (30) are the particle variables p (r), q (r) and x (r), the gravitational gauge potentials e, co and a, evaluated on x ( r ) , and the Lagrange multiplier N(r). Performing the gauge transformations ( 18 ), (26), (27) and (29) on (30) demonstrates that the action is invariant, apart from end-point contributions ~3.
I'm+l'~--,I'~+I'~--~ f dr d (fl-lq~O~).
(31)
Note that owing to the gauge variation (27) of q~, one can always pass to a gauge where qa vanishes, thereby simplifying the theory, see below. The total action in the gauge group formulation I= ~
1
ZT~t./
1'2 +Ira + I ~
(32)
is now varied to derive equations of motion. Variation of the gravitational Lagrange multipliers ~/a determines the geometry, and here matter variables do not enter. We find, as in pure gravity, 32 The various sign changes from (25) are necessitated by the fact that the Zweihein enters the symplectic term pD~q with a "twist" by Cab;compare (41 ). #3 We have not found a manifestly covariant formulation with non-vanishing ~ .
27
Volume 299, number 1,2 5I O= - -
571a(X) 51
O= - O=
PHYSICS LETTERSB
~fa=de'~+e%09eb=O,
(33)
=~f=dw=0,
(34)
8rh(x) 81 =~g=da+½eaeabeb=O. - 8713(x)
21 January 1993
be set to zero in the action, leaving a gauge fixed expression: Ilq=o =
1 I~+ 2nG
fdv[pa(J
f. a be OuX' U )
(41)
- ½N(P 2 + m z) - .~au2U ] .
(35)
The first of these enforces vanishing torsion and permits evaluation of the spin-connection: 09= e a habe u" Oueb / det e .
(36)
The second requires vanishing curvature, and is solved by e~=~,
Since here a satisfies (35), the matter part of (41) is the first-order form of Im+ I , , given in (3) and (8). Finally we recall the equations for the gravitational Lagrange multipliers qA, which follow from varying the gravitational potentials, always at q = 0: 5I q=o 8e~,(x) =0 OU71a -]- (.abO)# ?1b -]- 713~-abe bu
(37)
which implies that 09 in (36) vanishes. Finally the third evaluates the vector potential a, precisely in the way required by our particle gravity interaction, compare (7a). With the trivial geometry enforced by (33), (34), (36) and (37) we have (apart from a pure gauge contribution)
fil
81
Next we examine the particle equations obtained by varying x (z) and q (~) after eliminating p ( z ) and N ( z ) . We present these only at q=O, which results from a gauge choice, as explained above:
q=O
=0
I
5x"(r) ~=o
d 12u + 12,~Fgp2~=.~u~
~d~ 8I
5qa(.r )
q=o
.
(39)
=0
Owing to the trivial geometry (37), (38), the remaining equations reduce as follows, in the parametrization x ° ( r ) = z . The matter equation of motion (39) becomes d
m2 u
dt x/~ - v 2 1 2 ' U c P ( Oo,eap - F gpe~ + eab09,~ ) = - 2 ' ~ ( e au.~ua+ . ~ a b e b a ) .
-
(46)
.~eu~"
[v--2~(t) ], and is solved by (40)
We have used (39) to simplify (40), which is then identically satisfied: the left side vanishes (there is no torsion), the right side also vanishes when it is recalled that ~u~ = - .~ x / ~ u ~ • Thus, varying q does not produce an independent equation here, and q may 28
(44)
(45)
2nG?13=½.~e(xl-x'(t))-½2=-½A.
=0
) .
We recognize in (44) our previous eq. (12), determining the cosmological "constant" and solved in (13):
1
81
(42)
(43)
~0u713 =2nG.~ ~ dr ~ u , Y C ~ Z ( x - x ( z )
(38)
•~u~ =- ~(Oua~ - O~au) = - ' ~ E u . "
b "c2 "V habe,~x Eu,x N
dz~Z(x_x(z))
q=o = 0=:>0u 712.jI-qa Eab e uO_b
809Ax) 5au(x)
a u = ½~u~x",
f
=2nG
xl(t)=Xl
+x/(t-2°)2+m2/~
(47)
2,
where 2u are integration constants related to pu, the energy po and momentum, P~. pu= mxU(t)
+.~%x~(
t)= ~
~.
(48)
Volume 299, number 1,2
PHYSICS LETTERS B
[Note that the energy is also given by the spatial integral of T OOin (10b), apart from an infinite quantity proportional to 2. However, the spatial integral of T °~ is not P~; f~_~ dx I T °~ is not time-independent even though T u~ is conserved. Rather, because T ~ remains non-vanishing at x ~= + oo, one must add t f~oodx ~ ( d / d x ~) T ~1 to f-~oo cbc I T m to achieve a time-independent quantity, which then coincides with pl.] Eq. (42) becomes, in view of (45) and (47)
Ou~la + lrGA~ua = "Y-'Ua,
(49)
where Y-u~is
J-ua=IrG[Pa-.~EaoXb(t) ] O u e ( x l - x l ( t )
(50)
) .
This has the solution ?]a ~"
n G ( Pa - .~eabxb)e( X l -- x ( t ) )
(51)
+ rcG26ab(Xb--x b) ,
with Xob arbitrary constants. Finally, eq. (43) for q2 reduces to
21 January 1993
(2) The additional interaction gives a central extension to the translation algebra in the matter sector, so that the Poincar6 group is realized in the same manner as in gravity. However, in the absence of this interaction, when the Poincar6 group action on the matter variables loses the extension, the theory is still invariant; indeed, surface terms like those in (31 ) are even absent. Evidently invariance of a theory does not require that the symmetry group be realized in the same way on all variables. (3) In higher dimensions our construction is not available; an alternative is the following *~: The closed d-dimensional volume form defines a ( d - 1 )-form as in (7a). The dual *a is then a one-form which can be coupled to 2 u in any dimension - giving rise to force involving d*a. However, such an interaction does not appear well-defined because a is determined by (7a) only up to an exact ( d - 1 )-form (gauge ambiguity), but such an ambiguity leaves d*a undetermined. ~4 We thank B. Zwiebach for a discussion on this point.
0/t?]2 dr ?]aEalJ=O .
(52)
References When rla is taken from (51 ), this is solved by --2f/2 = M -
zcG
~G2(xU-x~)
2
- -~- [(PU--.~EU,x")2--mZ]e(xl--xl(t))
.
(53) In the "string-inspired" gravity model [1,2], the "physical" metric is g u J ( - 2qz) and exhibits in the absence of matter the geometry of a black hole with mass M a n d location )Co.We see that inclusion of matter, with the additional interaction that we have introduced, merely shifts the black hole parameters. We conclude with the following observations: ( 1 ) The additional interaction results in an intriguing connection between the gauge-theoretic 0-angle characterizing the vacuum state o f the quantized theory and the cosmological constant: In the gravity sector o f the theory the rl3-a interaction gives rise to the cosmological "constant" proportional to ~/3,on the other hand in the matter sector a plays the role of a background electric field, which is known to produce the 0-angle in ( 1 + 1 )-dimensional gauge theories [111.
[ 1 ] C. Callan, S. Giddings, A. Harvey and A. Strominger, Phys. Rev. D45 (1992) 1005. [2] H. Verlinde, in: Sixth Marcel Grossmann Meeting on General relativity, ed. M. Sato (World Scientific, Singapore, 1992). [3] C. Teitelboim, Phys. Lett. B 126 (1983) 41, in: Quantum theory of gravity, ed. S. Christensen (Adam Hilger, Bristol, 1984); R. Jackiw, in: Quantum theory of gravity, ed. S. Christensen (Adam Hilger, Bristol, 1984); Nucl. Phys. B 252 ( 1985 ) 343. [4] T. Fukiyama and K. Kamimura, Phys. Lett. B 160 (1985) 259; K. Isler and C. Trugenberger, Phys. Rev. Lett. 63 (1989) 834; A. Chamseddine and D. Wyler, Phys. Lett. B 228 (1989) 75. [5] D. Cangemi and R. Jackiw, Phys. Rev. Lett. 69 (1992) 233. [6] D. Cangemi, MIT preprint CTP~2124 (July 1992). [ 7 ] A. Achficarro, Tufts University preprint (July 1992 ). [8] G. Grignani and G. Nardelli, Perugia University preprint DFUPG-57-1992 (August 1992). [9] S.-K. Kim, K.-S. Sob and J.-H. Yee, Seoul University preprints SNUTP-92-67, 81 (1992). [ 10] A. Aurilia, H. Nicolai and P. Townsend, Nucl. Phys. B 176 (1980) 509. [ 11 ] S. Coleman, Ann. Phys. (NY) 101 (1976) 239.
29
PHYSICS LETTERS B
Geometric gravitational force on particles moving in a line D. Cangemi and R. Jackiw Centerfor Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Received 13 October 1992; revised manuscript received 17 November 1992
In two-dimensional space-time, point particles can experience a geometric, dimension-specific gravity force, which modifies the usual geodesic equation of motion and provides a link between the cosmological constant and the vacuum 0-angle. The description of such a force fits naturally into a gauge theory of gravity based on the extended Poincar6 group, i.e. "string-inspired" dilaton gravity.
Consideration o f ( 1 + 1 )-dimensional - lineal gravity, for pedagogical and perhaps physical purposes, requires inventing gravitational dynamics; the Einstein theory cannot be modeled owing to the vanishing o f the two-dimensional Einstein tensor. Recent activity [ 1,2 ] in this area has focused on scalartensor theories [3], whose dynamical variables are the metric tensor and an additional scalar field, while gravitational field equations involve the Ricci scalar R - the single quantity encoding all local geometric information in two dimensions. Two models, governed by the following geometric actions, Il = f d E x x / ~ r / ( R - 2 ) I 2 = j" d2x x / ~
,
(1)
(qR-2) ,
(2)
are especially noteworthy, in that both can be equivalently described by gauge theoretical actions based on the de Sitter group [ 4 ] for I~, and on the Poincar6 group [ 2 ] with central extension [ 5 ] for I2, the latter describing "string-inspired" dilaton gravity [ 1,2 ]. Moreover, both theories are obtained by various dimensional reductions from three dimensions [3,69 ], and also 11 and 12 can be related to each other by various singular limiting procedures [ 5,7,9 ]. [ In ( 1 )
and (2) r/is a scalar field and 2 the cosmological constant. ] Here we examine the interaction of a point particle with the gravitational field. The usual matter action for a particle of mass m on the worldline x~(z) is constructed from the arc length: Im = -- m ~ ds = - m ~ drx/.fu(r)gu~(x(r) ).~(z) .
(3)
[We set the velocity of light to unity and our Minkowski signature is ( + - ); the overdot denotes differentiation with respect to r, which parametrizes the worldline in an arbitrary way. ] However, the ( l + 1 )dimensional setting provides another possibility, which is specific to this dimension and which enjoys various interesting geometric attributes. Our purpose here is to examine this point-particle-gravity interaction, and in the second half of this letter to show how naturally it fits with the gauge theoretical formulation for I2. Let us begin by observing that the usual geodesic equation of motion, modified by an additional force
l_". ~ r--~B ~ k p _- ~o" ~ , d N XU+N dr
~r This work is supported in part by funds provided by the US Department of Energy (DOE) under contract ~DE-AC0276ER03069, and by the Swiss National Science Foundation. 24
0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
Volume 299, number 1,2
PHYSICS LETTERSB
will lose general covariance if .~ is externally prescribed. ( F is the Christoffel symbol evaluated on the worldline, as also is the metric tensor.) In order to maintain general covariance .~ must be constructed from the dynamical variables of the theory, and experience with electromagnetism (which is not included in the above discussion) shows that general covariance is preserved when .~ has the tensorial properties of/7~2 ~, where -~u~is an anti-symmetric, second-rank tensor. In dimensions greater than two, such a tensor cannot be constructed from particle and/or gravitational variables; it arises when electromagnetic (or other gauge-field) degrees of freedom are dynamically active. However, in two dimensions gravitational variables allow constructing the required tensor:
.~= - e,/--g ~.
~z~xu+d1 .
12~ru~p2t~+.~gU'~x~-g~,,p2~=O .
(6)
The additional interaction term in (6) is similar to that arising from a constant external electromagnetic field in fiat two-dimensional Minkowski space-time. Indeed, just as the latter preserves the Poincar6 symmetry of a non-interacting point particle in flat spacetime, so similarly our covariantly constant field (5) respects general covariance. Moreover, as we now describe, the force law can be derived from a novel contribution to the matter-gravity action. Construction of the addition to the conventional matter-gravity action (3), which gives rise to our force, is geometrically subtle. Consider the two-form ½~u~dxUdx ~, proportional to the volume form V=-½x~-gEu~dxUdx ~, which may also be expressed in terms of the Zweibeine e au as ~a ~b - ~I t~ab~u~ dx u d x~. Since Vis closed, d V= 0, it is exact, at least locally:
V=da.
0a~ 0a~ ~Ox ( x ) - ~ x ~ ( x ) = _ _ ~ S g ~ .
(7a)
Eq. (7a) defines a one-form, whose components are also seen to satisfy
(7b)
Since the right-hand side of (7b) is a tensor, a can be taken as a vector. The covariant action is now constructed from a:
I~ = - . ~ f dzau(x(z))2u(r ) .
(8)
Under coordinate redefinition, both a and x ( r ) change, and it is straightforwardly verified that I a is a scalar. Note that the gauge ambiguity au--,au+ Oufl, which is a consequence of the defining equation (7b), changes the action only be end-point contributions. Variation o f l ~ with respect to x ( t ) produces with the help of (7b) the force in (6):
8I~
(5)
Here e~ (also e~") is the numerical anti-symmetric tensor density, Cot= - 1 (e °t = 1 ) and .~ is a constant fixing the magnitude of the field. Thus, the covariant, two-dimensional equation of motion involving matter and metric variables and generalizing the usual geodesic equation reads
21 January 1993
--.~x/-g(x(z))Eu~2~(r)
.
(9)
Thus while no local expression for the action is available, its x(r)-variation is local, giving rise to an electromagnetic-like, velocity-dependent force, which is free of gauge ambiguity and preserves general covariance. Next, we examine the contribution from I~ to the energy-momentum tensor Twhich is a functional of the matter variable x(r) and function of the field argument x: T~P(x(r) Ix) 2
--
~agap(x~
(Im +1~) .
(10a)
Evaluating the metric variation of I~ is problematic in the absence of an explicit formula for a. However, we can fix the additional term in the energy-momentum tensor by requiring its covariant conservation. In this way Tis found to be 1
TaP(x(t) Ix) = - ~ = ~ x ) r
1
× JdtN--~x
"o~
.p
2
( O x (t)~ ( x - - x ( z ) )
- ½A(x(t) Ix)g'~a(x) ,
(10b)
and we satisfy D,,T~a=0,
(11)
provided that 25
Volume 299, number 1,2
PHYSICS LETTERS B
21 January 1993
produces the following transformations on the connections:
10.4 20x u (x(z)Ix)
= --'~ I d z ~ u ~ " ~ ' ( z ) t ~ 2 ( x - x ( z ) )
"
(12)
ea_..~~a = ( ~ -1 ) ab ( eb + ~.bcOC(D_t_dOb ) , m-,o~= m + d a ,
Eq. (12) is easily solved in the parametrization where x ° ( z) = z, and one finds
a--,a=a--Oae,,beb--½02oJ+dfl+½dOaEabO b .
A = - ~e(x I-x I(t)) +2,
Here ~ is the finite Lorentz transformation with rapidity or:
(13)
where 2 is constant. Thus the metric variation of I~ also is local, giving rise to a cosmological "constant" that jumps by 2 ~ as the particle's trajectory is crossed, a property that is independent of the above parametrization choice. Although the matter dynamics can be presented without commitment to a specific model for gravity, the new interaction (8) fits especially naturally with a gauge theoretical formulation for I2, which we now review [ 5 ]. The structure group for the gauge theory is the extended (1 + 1 )-dimensional Poincar6 group with (anti-hermitian) Lorentz generator J, and translation generators Pa that close upon commutation on a central element, L which commutes with J and P~.
J/gab=~ab cosh a+~ab sinh a .
[Pa, Pb]=eabl.
3
FA--,FA= ~
U=
(o
(15)
acoc!) 1
F = d A + A 2 = f apa + f J + g I = (dea+Eabtoeb)p a + d m J
(16)
It is recognized t h a t f a is proportional to the torsion and f t o the curvature, while g involves the abelian field strength associated with a, supplemented by the volume two-form. A finite group transformation, generated by the gauge function O, o=Oaea + O t J + flI ,
26
.
(21)
- ½0 2
The upper left 3 × 3 block in U comprises the adjoint representation of the conventional Poincar6 group with -eacoc giving the translation, while the fourth row and column arise from the extension. In this representation, the extended algebra possesses a nonsingular, invariant inner product hart,
_
O o) ° _,°
and curvature two-form F,
+ ( d a + ½ea~abeb)I.
(20)
( U - I ) A B F B,
B=0
(14)
[ Latin letters refer to tangent space components and are moved by the fiat metric tensor hab=diag(1, - 1 ). ] Gauge connection one-forms are associated with the generators: Zweibeine e ~ with Pa, spin-connection o9 with J, and an additional connection a with /, giving a Lie-algebra valued connection one-form A, A=e"Pa +OgJ+aI ,
(19)
The curvature components transform similarly to (18 ), except that the inhomogeneous shifts involving parameter differentials are absent. Equivalently, one may collect the curvature components into a multiplet F A = (fa, f g), which then transforms by the adjoint 4 × 4 representation of the extended Poincar6 group:
\oc~c,~eS
[Pa, J]=~.abpb,
(18)
(17)
for which TUb U= h; this allows raising and lowering indices (A, B). A gauge invariant gravity action, equivalent to/2, is constructed with the help of a Lagrange multiplier multiplet qA= (qa, t/2, q3), taken to obey the coadjoint transformation law: 3
rlA--,fli= •
rlBUnA.
(23)
B=0
It is then easy to show that the gravitational dynamics of the gauge invariant action I~,
Volume 299, n u m b e r 1,2
1'2 = f a ~
qAF A ,
PHYSICS LETTERS B
(24)
0
(27)
qa _~~ a = ( ~/{- 1) ab( qb q_ ~bcoc )
[analogous t o f a in (20) w i t h f s e t to unity and g to
and of I2 coincide. Note in particular that no cosmological constant is present in (24); in the gauge theoretical formulation it is not a parameter in the lagrangian, but a value taken on by a variable, here q3 ~. A geometric formulation of the gravity-matter system is provided by (1/2~zG)Iz+Im+I~, where G is the gravitational coupling strength. But we seek a gauge theoretical description; so a gauge theoretical point-particle action is now constructed, following a variation of the Grignani and Nardelli method [ 8 ]. We shall arrive again at Im+ I~, once a gauge freedom is fixed. Let us begin with the particle action in flat spacetime, where the vector potential for the additional force is explicitly given by l ~ b x b ( r ) . The matter action, in first-order form and in the absence of curvature, reads /matter = f dr [ (pa - k~EabXb)yca+ ½N(p2--m 2) ] . (25) All quantities depend on the parametrization r; when p and the Lagrange multiplier N are eliminated, Im~tt¢~coincides with Im + I~ in the absence of gravity. The theory is Poincar6 invariant, hut owing to the additional interaction, the translation algebra acquires a central extension. To couple Imatt~ to gravity, we view p~ as the first two components of a coadjoint tangent space fourvector PA, with vanishing last component P3 = 0; consequently p~ transforms according to Pa "~ ff a = Pb'A[ba
21 January 1993
(26)
[analogous to ~/~ in (23) with ~/3 set to zero]. The position variable x"(r) is replaced by the tangent space coordinate qa(r), taken to be the first two components of an adjoint four-vector qA with third component set to unity and fourth component to l (qahabqb--C), where c is a fixed number (this implies qAhAsqS=c). The consequent transformation law reads a~ This mechanism for generating the cosmological constant is here dictated by the group structure. In other contexts it has been inserted "by hand", see ref. [ 10].
l(qahabq b-c) ]' The v-derivative of the position variable is promoted to a covariant derivative: (D~q) ~= ~a + Eab(q b c o u
_
eb)jcu .
(28)
Here is introduced the worldline on the manifold x ( r ) ; the space-time indices in the gravitational potentials are saturated with ~u. The transformation law for the covariant derivative, which follows from (18) and (27), is (D~q)~ (~g-')ab(Dvq)b.
(29)
Finally, further terms are added to achieve gauge invariance. Implementing the above steps results in the following action ~2.
I" +I'~= ~ dr
[(Pa+l,~abqb)(Drq)
a
-½N(pE+mE)--.~(au-lq~e~u)5c u] .
(30)
The dynamical quantities to be varied independently in (30) are the particle variables p (r), q (r) and x (r), the gravitational gauge potentials e, co and a, evaluated on x ( r ) , and the Lagrange multiplier N(r). Performing the gauge transformations ( 18 ), (26), (27) and (29) on (30) demonstrates that the action is invariant, apart from end-point contributions ~3.
I'm+l'~--,I'~+I'~--~ f dr d (fl-lq~O~).
(31)
Note that owing to the gauge variation (27) of q~, one can always pass to a gauge where qa vanishes, thereby simplifying the theory, see below. The total action in the gauge group formulation I= ~
1
ZT~t./
1'2 +Ira + I ~
(32)
is now varied to derive equations of motion. Variation of the gravitational Lagrange multipliers ~/a determines the geometry, and here matter variables do not enter. We find, as in pure gravity, 32 The various sign changes from (25) are necessitated by the fact that the Zweihein enters the symplectic term pD~q with a "twist" by Cab;compare (41 ). #3 We have not found a manifestly covariant formulation with non-vanishing ~ .
27
Volume 299, number 1,2 5I O= - -
571a(X) 51
O= - O=
PHYSICS LETTERSB
~fa=de'~+e%09eb=O,
(33)
=~f=dw=0,
(34)
8rh(x) 81 =~g=da+½eaeabeb=O. - 8713(x)
21 January 1993
be set to zero in the action, leaving a gauge fixed expression: Ilq=o =
1 I~+ 2nG
fdv[pa(J
f. a be OuX' U )
(41)
- ½N(P 2 + m z) - .~au2U ] .
(35)
The first of these enforces vanishing torsion and permits evaluation of the spin-connection: 09= e a habe u" Oueb / det e .
(36)
The second requires vanishing curvature, and is solved by e~=~,
Since here a satisfies (35), the matter part of (41) is the first-order form of Im+ I , , given in (3) and (8). Finally we recall the equations for the gravitational Lagrange multipliers qA, which follow from varying the gravitational potentials, always at q = 0: 5I q=o 8e~,(x) =0 OU71a -]- (.abO)# ?1b -]- 713~-abe bu
(37)
which implies that 09 in (36) vanishes. Finally the third evaluates the vector potential a, precisely in the way required by our particle gravity interaction, compare (7a). With the trivial geometry enforced by (33), (34), (36) and (37) we have (apart from a pure gauge contribution)
fil
81
Next we examine the particle equations obtained by varying x (z) and q (~) after eliminating p ( z ) and N ( z ) . We present these only at q=O, which results from a gauge choice, as explained above:
q=O
=0
I
5x"(r) ~=o
d 12u + 12,~Fgp2~=.~u~
~d~ 8I
5qa(.r )
q=o
.
(39)
=0
Owing to the trivial geometry (37), (38), the remaining equations reduce as follows, in the parametrization x ° ( r ) = z . The matter equation of motion (39) becomes d
m2 u
dt x/~ - v 2 1 2 ' U c P ( Oo,eap - F gpe~ + eab09,~ ) = - 2 ' ~ ( e au.~ua+ . ~ a b e b a ) .
-
(46)
.~eu~"
[v--2~(t) ], and is solved by (40)
We have used (39) to simplify (40), which is then identically satisfied: the left side vanishes (there is no torsion), the right side also vanishes when it is recalled that ~u~ = - .~ x / ~ u ~ • Thus, varying q does not produce an independent equation here, and q may 28
(44)
(45)
2nG?13=½.~e(xl-x'(t))-½2=-½A.
=0
) .
We recognize in (44) our previous eq. (12), determining the cosmological "constant" and solved in (13):
1
81
(42)
(43)
~0u713 =2nG.~ ~ dr ~ u , Y C ~ Z ( x - x ( z )
(38)
•~u~ =- ~(Oua~ - O~au) = - ' ~ E u . "
b "c2 "V habe,~x Eu,x N
dz~Z(x_x(z))
q=o = 0=:>0u 712.jI-qa Eab e uO_b
809Ax) 5au(x)
a u = ½~u~x",
f
=2nG
xl(t)=Xl
+x/(t-2°)2+m2/~
(47)
2,
where 2u are integration constants related to pu, the energy po and momentum, P~. pu= mxU(t)
+.~%x~(
t)= ~
~.
(48)
Volume 299, number 1,2
PHYSICS LETTERS B
[Note that the energy is also given by the spatial integral of T OOin (10b), apart from an infinite quantity proportional to 2. However, the spatial integral of T °~ is not P~; f~_~ dx I T °~ is not time-independent even though T u~ is conserved. Rather, because T ~ remains non-vanishing at x ~= + oo, one must add t f~oodx ~ ( d / d x ~) T ~1 to f-~oo cbc I T m to achieve a time-independent quantity, which then coincides with pl.] Eq. (42) becomes, in view of (45) and (47)
Ou~la + lrGA~ua = "Y-'Ua,
(49)
where Y-u~is
J-ua=IrG[Pa-.~EaoXb(t) ] O u e ( x l - x l ( t )
(50)
) .
This has the solution ?]a ~"
n G ( Pa - .~eabxb)e( X l -- x ( t ) )
(51)
+ rcG26ab(Xb--x b) ,
with Xob arbitrary constants. Finally, eq. (43) for q2 reduces to
21 January 1993
(2) The additional interaction gives a central extension to the translation algebra in the matter sector, so that the Poincar6 group is realized in the same manner as in gravity. However, in the absence of this interaction, when the Poincar6 group action on the matter variables loses the extension, the theory is still invariant; indeed, surface terms like those in (31 ) are even absent. Evidently invariance of a theory does not require that the symmetry group be realized in the same way on all variables. (3) In higher dimensions our construction is not available; an alternative is the following *~: The closed d-dimensional volume form defines a ( d - 1 )-form as in (7a). The dual *a is then a one-form which can be coupled to 2 u in any dimension - giving rise to force involving d*a. However, such an interaction does not appear well-defined because a is determined by (7a) only up to an exact ( d - 1 )-form (gauge ambiguity), but such an ambiguity leaves d*a undetermined. ~4 We thank B. Zwiebach for a discussion on this point.
0/t?]2 dr ?]aEalJ=O .
(52)
References When rla is taken from (51 ), this is solved by --2f/2 = M -
zcG
~G2(xU-x~)
2
- -~- [(PU--.~EU,x")2--mZ]e(xl--xl(t))
.
(53) In the "string-inspired" gravity model [1,2], the "physical" metric is g u J ( - 2qz) and exhibits in the absence of matter the geometry of a black hole with mass M a n d location )Co.We see that inclusion of matter, with the additional interaction that we have introduced, merely shifts the black hole parameters. We conclude with the following observations: ( 1 ) The additional interaction results in an intriguing connection between the gauge-theoretic 0-angle characterizing the vacuum state o f the quantized theory and the cosmological constant: In the gravity sector o f the theory the rl3-a interaction gives rise to the cosmological "constant" proportional to ~/3,on the other hand in the matter sector a plays the role of a background electric field, which is known to produce the 0-angle in ( 1 + 1 )-dimensional gauge theories [111.
[ 1 ] C. Callan, S. Giddings, A. Harvey and A. Strominger, Phys. Rev. D45 (1992) 1005. [2] H. Verlinde, in: Sixth Marcel Grossmann Meeting on General relativity, ed. M. Sato (World Scientific, Singapore, 1992). [3] C. Teitelboim, Phys. Lett. B 126 (1983) 41, in: Quantum theory of gravity, ed. S. Christensen (Adam Hilger, Bristol, 1984); R. Jackiw, in: Quantum theory of gravity, ed. S. Christensen (Adam Hilger, Bristol, 1984); Nucl. Phys. B 252 ( 1985 ) 343. [4] T. Fukiyama and K. Kamimura, Phys. Lett. B 160 (1985) 259; K. Isler and C. Trugenberger, Phys. Rev. Lett. 63 (1989) 834; A. Chamseddine and D. Wyler, Phys. Lett. B 228 (1989) 75. [5] D. Cangemi and R. Jackiw, Phys. Rev. Lett. 69 (1992) 233. [6] D. Cangemi, MIT preprint CTP~2124 (July 1992). [ 7 ] A. Achficarro, Tufts University preprint (July 1992 ). [8] G. Grignani and G. Nardelli, Perugia University preprint DFUPG-57-1992 (August 1992). [9] S.-K. Kim, K.-S. Sob and J.-H. Yee, Seoul University preprints SNUTP-92-67, 81 (1992). [ 10] A. Aurilia, H. Nicolai and P. Townsend, Nucl. Phys. B 176 (1980) 509. [ 11 ] S. Coleman, Ann. Phys. (NY) 101 (1976) 239.
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