Analysis of Two-Particle Systems in 2 + 1 Gravity Through Hamiltonian Dynamics
aa r X i v : . [ g r- q c ] N ov Analysis of Two-Particle Systems in 2 + 1 Gravity ThroughHamiltonian Dynamics
Alexandre Yale ∗†‡ , R. B. Mann †‡ and Tadayuki Ohta § November 25, 2010
Abstract
We study the dynamics of particles coupled to gravity in (2 + 1) dimensions. Using the ADM formalism,we derive the general Hamiltonian for an N -body system and analyze the dynamics of a two-particle system.Nonlinear terms are found up to second order in κ in the general case, and to every order in the quasi-staticlimit. The study of 2+1 dimensional gravity [1, 2, 3, 4, 5, 6, 7, 8, 9] has been around for several decades. Its attractionwas rooted in its mathematical simplicity, which afforded some insight into the dynamics and behaviour ofgeneral relativistic gravity. Indeed, in this framework, the Einstein tensor can be expressed in terms of thecurvature tensor, such that vacuum must be locally flat. This framework became even more popular after thediscovery of the BTZ black hole [10, 11] and is now a standard tool practitioners of quantum gravity employ inunderstanding their subject [12, 13, 14].We are concerned here with the problem of N -body dynamics, which is a long-standing one in relativitydue to its notorious difficulty. In lower dimensional settings, this problem becomes much simpler. For example,the general form of the solution has been obtained for lineal gravity [15] for arbitrary N , after which a varietyof exact solutions for N = 2 were obtained in various contexts that include both charge and cosmologicalexpansion/contraction [16, 17, 18, 19], by investigating the Hamiltonian of such a system through canonicalreduction. Several interesting exact solutions to the N -body equilibrium problem [20, 21, 22, 23] in (1+1)dimensions have also been obtained.In this paper, we follow a similar approach in (2+1) gravity to obtain canonical equations of motion to analyzea two-body system. The analysis of the N -body problem in (2+1) dimensions also has an interesting history,beginning with construction of a spinning point-particle solution [4] and then a consideration of the quantumscattering problem [24]. Further developments came upon realizing that the problem could be analyzed from atopological perspective [25], and an implicit solution for the metric and the motion of N interacting particles wasobtained [26]. Based on a mapping from multivalued Minkowskian coordinates to single-valued ones, it becomesexplicit for two particles with any speed and for any number of particles with small speed. It is possible to showthat the collision of point particles in 2+1 AdS spacetime can result in the formation of a black hole [27].The connection between these approaches and more traditional canonical methods as employed in (1+1) [15]and (3+1) dimensions [28] has not been explicated. In this paper, we address this issue. We begin by finding thetotal action corresponding to the system, which consists of the Einstein-Hilbert action for the field as well as aterm corresponding to coupling gravity to matter. A variational approach will then lead to coordinate conditionsand constraints, which will in turn produce the total Hamiltonian as an expansion in powers of the gravitationalcoupling κ . The Hamiltonian will be explicitly calculated to second order in κ for the general two-body case andto every order for the quasi-static limit. Our results agree with previous work in (2+1) dimensions which hasexplored, through geodesics, the equations of motion in the quasi-static approximation [25]. A first step is to derive the action for an N -body system. This action will consist of two parts: the first, I E ,is the Einstein-Hilbert part due to the geometry of spacetime, and the second, I M , is due to matter-gravity ∗ [email protected] † University of Waterloo, Waterloo, Ontario, Canada ‡ Perimeter Institute, Waterloo, Ontario, Canada § Miyagi University of Education, Sendai, Miyagi, Japan oupling. Our derivation will follow the general ideas from the original ADM article [29]. In this formalism, themetric is defined as ds = − N dt + g ij (cid:16) dx i + N i dt (cid:17) (cid:16) dx j + N j dt (cid:17) (1)where N = (cid:0) − g (cid:1) − and N i = g i are the lapse function and shift covector. We also define the extrinsiccurvature K ij = (2 N ) − (cid:0) N i | j + N j | i − g ij, (cid:1) and the canonical momentum conjugate to the metric: π ij = −√ g ( K ij − Kg ij ), where the vertical bar denotes a covariant derivative with respect to the induced metric g ij .Using the Gauss-Codazzi equations, we can perform a 2 + 1 decomposition of the Einstein-Hilbert action bywriting R as a combination of R ≡ R and extrinsic curvatures: I E = 12 κ Z d x p − g R = 12 κ Z d xN √ g (cid:16) R − K + K ij K ij (cid:17) − (2 √ gK ) , + h √ g ( KN i − g ij N ,j ) i ,i = 12 κ Z d xN √ gR − N √ g ( π − π ij π ij ) − π , + 2 h πN i − √ gg ij N ,j i ,i = 12 κ Z d x n π ij g ij, + N R + N i R i − π , − π ij N j − πN i + √ gg ij N ,j ] ,i o . = 12 κ Z d x n π ij g ij, + N R + N i R i o , (2)where R = √ g ( gR + π − π ij π ij ) and R i = 2 π ij | j , and where in the last step we discarded total divergences. Wenow calculate the Lagrangian corresponding to matter-gravity coupling, which will give us the second half of theaction. We consider the Lagrangian which minimally couples N particles to gravity: L M = X a Z ds a ( − m a s(cid:18) − g µν ( z ) dz µa ds a dz νa ds a (cid:19)) . (3)We can rewrite this Lagrangian in a more convenient form by introducing the canonical momentum p aµ = ∂ L M ∂ ˙ z µa conjugate to z aµ as well as Lagrange multipliers λ a which ensure p µa p aµ = − m a : L M = X a Z ds a (cid:18) p aµ dz µa ds a − λ ′ a ( s a )( p aµ p aν g µν ( x ) + m a ) (cid:19) δ (3) ( x − z a ( s a ))= X a (cid:18) p aµ ˙ z µa − λ a ( p aµ p aν g µν ( x ) + m a ) (cid:19) δ (2) ( r a ( x ))= X a (cid:16) p ai ˙ z ia − N p p ai p aj g ij + m a + N i p aj g ij (cid:17) δ (2) ( r a ( x )) (4)where λ a = λ ′ a ds a dz a (because of the δ -function integration) and r a = x − z a , and where in the last step we used(1) as well as the solution to the constraint p µa p aµ = − m a : p a = N i p aj g ij − N p p ai p aj g ij + m a . (5)Combining this with our previous result for the Einstein-Hilbert action, we write the total action I = I E + R L M as I = Z d x X a p ai ˙ z ia δ (2) ( r a ) + 12 κ ( π ij g ij, + N R + N i R i ) ! , (6)where [4, 29] R = 1 √ g ( gR + π − π ij π ij ) − κ X a p p ai p aj g ij + m a δ (2) ( r a ) R i = 2 π ij | j + 2 κ X a p aj g ij δ (2) ( r a ) (7)must both vanish following variations of N and N i . These constraints (7) have a very natural physical in-terpretation. The latter of these is a momentum-balance constraint, indicating that in this (2+1)-dimensionalself-gravitating system the particle momenta must be cancelled by the momentum of the gravitational field. Theformer is an energy-balance constraint, which basically indicates that the energy of the gravitational field (acombination of “potential energy” expressed as the curvature of the spacelike slice and “kinetic energy” consist-ing of the square of the gravitational momentum) must equal the relativistic energy of the particles. e now consider rewriting this action by choosing convenient coordinate conditions. We define h ij = g ij − δ ij ,where δ ij is the Kronecker delta, and use the orthogonal decomposition h ij = (cid:18) δ ij − ∂ i ∂ j (cid:19) h T + h i,j + h j,i , (8)where 1 / ∆ is the inverse of the flat-space Laplacian. Note that h T and h i can be found to be [29] h T = h ii − h ij,ij h i = 1∆ (cid:20) h ij,j − h kj,kji (cid:21) . (9)With these new definitions in hand, we can write Z d xπ ij ∂ t g ij = Z d x (cid:20) (1 + h T ) π ii ∂ t log(1 + h T ) − π ij,j ∂ t (cid:18) h i − ∂ i h T (cid:19)(cid:21) = Z d x (cid:20) − log(1 + h T ) ∂ t ( π ii (1 + h T )) − π ij,j ∂ t (cid:18) h i − ∂ i h T (cid:19)(cid:21) = Z d x (cid:26) − ∆ log(1 + h T ) ∂ t (cid:20)
1∆ ((1 + h T ) π ii ) (cid:21) − π ij,j ∂ t ( h i − ∂ i h T ) (cid:27) (10)where we discarded total derivatives in arriving at the last expression. Hence, the action (6), with constraints R = 0 and R i = 0 from (7), can be rewritten as I = Z d x X a p ai ˙ z ia δ ( x − z a ) − κ ∆ log(1 + h T ) ∂ (cid:20)
1∆ ((1 + h T ) π ii ) (cid:21) − κ π ij,j ∂ ( h i − ∂ i h T ) . (11)This action suggests that we should adopt the coordinate conditions x = − (cid:16)h h T i π ii (cid:17) x i = h i − ∂ i h T , (12)such that, defining the Hamiltonian density H = − κ ∆ log(1+ h T ) and using x µ,ν = δ µν , the action can be rewrittenas I = Z d x (X a p ai ˙ z ia δ ( x − z a ) − H ) . (13)By imposing proper boundary conditions, we rewrite our coordinate conditions in a more convenient form: π ii = 0 g ij,j = 12 g jj,i (14)where the second equation implies that h i − ∂ i h T = 0, which in turn means that h ij = δ ij h T , and thus, g ij = δ ij (1 + h T ) . (15)In the coordinate system defined by these coordinate conditions and defining φ ≡ log(1 + h T ), the constraintequations R = 0 and R i = 0 lead to∆ φ = − κ X a ( m a + p a e − φ ) δ ( x − z a ) − e φ π ij π ij (16) ∂ j ( π ij e φ ) = − κ X a p ai δ ( x − z a ) , (17)where the Hamiltonian is given by H = − κ R d x ∆ φ and where p ≡ p is the square of the norm of the vector p . The solution satisfying the condition π ii = 0 is π ij = e − φ ( − κ π X a D ijk ( p a ) ∂ k log r a ) , (18)where D ijk ( p a ) = p ai δ jk + p aj δ ik − p ak δ ij . Thus, Equation (16) can be rewritten as∆ φ = − κ X a (cid:16) m a + p a e − φ ( z a ) (cid:17) / δ ( x − z a ) − ((cid:16) κ π (cid:17) X a X b D ijk ( p a ) D ijl ( p b ) ∂ k log r a ∂ l log r b ) e − φ . (19) e can then determine the Hamiltonian for a system of particles, which can be written down in the generalform: H = X a (cid:16) m a + p a e − φ ( z a ) (cid:17) / + κ π Z d x (X a X b D ijk ( p a ) D ijl ( p b ) ∂ k log r a ∂ l log r b ) e − φ . (20)where φ is obtained by solving (19).As will be exemplified in the following section, this Hamiltonian can be solved perturbatively in κ . Writing φ = P κ i φ ( i ) , it is clear that the first term of on the right-hand side of (19) only depends on φ (1) , . . . , φ ( i − whilethe second depends only on φ (1) , . . . , φ ( i − . Thus, we inductively solve for each term in the power series for φ by calculating ∆ φ ( i ) from the known φ (1) , . . . , φ ( i − , and solving it for φ ( i ) before moving onto the ( i + 1) term.Provided that we can solve these ∆ φ differential equations, this method allows us to solve for the Hamiltonianup to any order in κ . To deal with the particles’ divergent self-energies, which tend to be common in this sort of calculation, weintroduce a renormalization scheme similar to that used in quantum field theory, consisting of introducing anenergy scale to our measurement of distances. We first introduce the density of particle i to be n i ( r ) = a πr ( r + a ) (21)where a is an intrinsic length scale depending on the particle’s energy, and where we will be taking the limit as a goes to zero. Notice that for any value of a , R U ndV = 1 and n ( r ) goes to a δ function as a becomes small.Moreover, the equation ∆ g i = n i ( r ) has the solution (up to a term with vanishing Laplacian) g i = 12 π log (cid:16) r i a (cid:17) ≡ π h ( r i ) , (22)as opposed to simply g ∝ log( r ) were we to use the δ function. Then, the self-energy of a particle will belog(1) = 0. a is to be thought of as the gravitational length scale of the interaction, which we assume to be onthe order of the Planck length. Since a is dimensionful, we can take the limit of it being a very small quantity,while at the same time choosing units such that a = 1. Then, for a non-zero r , log(1 + r/a ) = log(1 + r ) ≈ log( r ).We will consider a system of two particles of masses m and m , in the centre of inertia frame where p = − p = p , using the notation E a ≡ p m a + p a with, again, p a ≡ p a . We also use shorthand notation r i = | x − z i | and r = | z − z | .We first note that, in this frame, X a D ijk ( p a ) ∂ k h ( r a ) = ( p i ∂ j + p j ∂ i − δ ij p k ∂ k )( h ( r ) − h ( r )) (23)such that the Hamiltonian (20) will have the form H = X a (cid:16) m a + p a e − φ ( z a ) (cid:17) / + κp π Z d x [ ∇ ( h ( r ) − h ( r )) · ∇ ( h ( r ) − h ( r ))] e − φ . (24)We will be using a perturbative approach to third order in κ , and therefore calculate: φ = κφ (1) + κ φ (2) + κ φ (3) e − φ = 1 − κφ (1) − κ (cid:18) φ (2) − φ (1) φ (1) (cid:19)(cid:16) m a + p a e − φ (cid:17) / = E a − κ p a E a φ (1) − κ (cid:18) p a E a (cid:18) φ (2) − φ (1) φ (1) (cid:19) + ( p a ) E a φ (1) φ (1) (cid:19) . (25)Note that in the limit κ →
0, we wish to retrieve H = P a p m a + p a , and therefore require φ to be of order at east κ . The contribution to the Hamiltonian will be separately calculated for each order in κ . We find:∆ φ (1) = − E n ( r ) + E n ( r )]∆ φ (2) = p (cid:20) E φ (1) ( z ) n ( r ) + 1 E φ (1) ( z ) n ( r ) (cid:21) − p π ∇ ( h ( r ) − h ( r )) · ∇ ( h ( r ) − h ( r ))= − p π (cid:20)(cid:18) E E + 1 (cid:19) h ( r ) n ( r ) + (cid:18) E E + 1 (cid:19) h ( r ) n ( r ) (cid:21) − p π ∆ [ h ( r ) − h ( r )] ∆ φ (3) = p E (cid:20) φ (2) ( z ) − φ (1) ( z ) + p E φ (1) ( z ) (cid:21) n ( r ) + (1 ↔
2) + p π [ ∇ ( h ( r ) − h ( r )) · ∇ ( h ( r ) − h ( r ))]= p E (cid:20) − p π h ( r ) (cid:18) E E (cid:19) − p π h ( r ) − π E h ( r ) + p E π E h ( r ) (cid:21) n ( r )+ (1 ↔ − p π [ ∇ ( h ( r ) − h ( r )) · ∇ ( h ( r ) − h ( r ))] ( E h ( r ) + E h ( r )) (26)where we used the solutions φ (1) = − π [ E h ( r ) + E h ( r )] φ (2) = − p π (cid:20)(cid:18) E E + 1 (cid:19) h ( r ) h ( r ) + (cid:18) E E + 1 (cid:19) h ( r ) h ( r ) (cid:21) − p π ( h ( r ) − h ( r )) (27)The Hamiltonian H = − R d x ∆ φ can now be found as an expansion in κ : H (0) = E + E H (1) = p π (cid:20)(cid:18) E E + 1 (cid:19) log( r ) + (cid:18) E E + 1 (cid:19) log( r ) (cid:21) H (2) = p E log( r ) (cid:26) p π (cid:18) E E (cid:19) + p π + 14 π E − p E π E (cid:27) + (1 ↔ p π ( E + E ) log( r ) . (28)where we used equations (39) and (40) and took the limit h ( r ) → log( r ). In the case of equal masses, E = E ,our Hamiltonian is greatly simplified: H = 2 E + 2 κp π log( r ) + κ p π E log( r ) ( p + 2 E ) (29)Simplifying further, the massless approximation gives: H = 2 p + 2 κπ p log r + 3 κ p π log r (30)which we notice as being the second-order expansion of − πκ log r W (cid:0) − κπ p log r (cid:1) where W is the Lambert W function.Note that the Hamiltonian in 1+1-dim dilaton gravity was also expressed in terms of the W function [18]. We can also use this formalism to describe exactly the Hamiltonian dynamics of a system in the quasi-staticlimit. In the following, we will therefore retain all orders in the coupling constant κ and only the lowest (linear)order in p . A. Bellini et. al. treated the geodesic equations in this approximation [25]. For simplicity weshall consider only the equal-mass case. To simplify the notation, we will, in contrast to the previous section,return to a δ mass distribution and a log( r ) potential. We will explicitly keep the divergent terms log r andlog r , where r ii = | x i − x i | , to display their cancellation at every order in κ . From (20) the Hamiltonian in thisapproximation is given by H =2 m + p m e κπ m log r (cid:16) e κπ m log r + e κπ m log r (cid:17) + κp π Z d x ( ∇ (log r − log r ) · ∇ (log r − log r )) e κπ m (log r +log r ) . (31) here φ s = − κπ P a log r a is a static solution to (19) and where m = m = m and p = − p = p . The integralof the 3rd term on the right hand side of (31) is Z d x [ ∇ (log r − log r ) · ∇ (log r − log r )] e κπ m (log r +log r ) = Z d x [ ∇ (log r + log r ) · ∇ (log r + log r ) − ∇ log r · ∇ log r )] e κπ m (log r +log r ) = − π κm h e κπ m (log r +log r ) + e κπ m (log r +log r ) − i − Z d x ( ∇ log r · ∇ log r ) e κπ m (log r +log r ) . (32)The first term in the preceding expression cancels the second (divergent) term in eq. (31). Using the integrationformula Z d x ( ∇ log r · ∇ log r )(log r + log r ) n = − n +1 n + 1 π (log r ) n +1 , (33)we can rewrite the Hamiltonian (31) as H = 2 m + p m − κp π Z d x ( ∇ log r · ∇ log r ) e κπ m (log r +log r ) = 2 m + p m e κmπ log r = 2 m + p m ( r ) κmπ (34)which indeed agrees with equation (29) in the quasi-static limit. This Hamiltonian gives us the canonicalequations of motion ˙ r = ∂H∂ p = 2 m ( r ) κmπ p (35)˙ p = − ∂H∂ r = − κπ p ( r ) κmπ − r , (36)which lead to the equations of motion in the second order¨ r = κmπ r ( r · ˙ r ) − r ( ˙ r ) r . (37)This equation is identical with Eq.(4.7) in [25]: ¨ η r = 4 GM ( ˙ η r ) η r , where η r = η x + i η y , G = κ π and M = 2 m . We have presented here the general expression (20) for the Hamiltonian describing N particles coupled to(2+1) dimensional gravity. Beginning with a derivation a general form for the action corresponding to N -bodydynamics, we employed the ADM formalism and proper coordinate conditions to find this general Hamiltonian.Our results are complementary to those that employ topological methods [25, 26, 27].Our result (20) has the advantage that it can be studied in a wide variety of physical regimes, includingsmall and large mass, small and large momenta, and small and large gravitational coupling. We explicitlydemonstrated this in two cases. First we obtained an expansion to second-order in the gravitational couplingconstant κ for the two-particle system. Second we also considered a quasi-static, two-particle equal-mass systemto every order in κ , in which case our results agree with those previously found in the literature [25, 26].A number of interesting problems remain for future consideration. An obvious thing to try is to incorporateadditional couplings to electromagnetism and a cosmological constant. A study of the quantization of theHamiltonian (20) should also afford interesting insight into the nature of (2+1) quantum gravity coupled tomatter. Acknowledgements
This work was supported by the Natural Sciences and Engineering Council of Canada. Appendix: Fourier Integrals
This appendix will contain proofs of some integration formulas used in section 3, which are done through Fouriertransforms. We begin by noticing that − π Z d k ∆ 1 k e ik · x = δ ( x ) (38)and therefore, the two-dimensional Fourier transform of log( r ) is − πk . We use this to show that, for n ≥ Z d x ∆(log r . . . log r n ) = 1(2 π ) n Z d x ∆ Z n Y i =1 d k i k i e i P k i · x e − i P i k i · z i = 1(2 π ) n − Z n Y i =1 d k i k i (cid:16)X k i (cid:17) δ (cid:16)X k i (cid:17) e − i P i k i · z i = 0 , (39)meaning that the integral of the laplacian of any polynomial function in log r will vanish. Similarly, we can show Z d x ( ∇ log r · ∇ log r ) log r = 1(2 π ) Z d xd k d k d k ( k · k ) k k k e i ( k + k + k ) · x e − i ( k · z + k · z + k · z ) = 12 π Z d k d k d k ( k · k ) k k k δ ( k + k + k ) e − i ( k · z + k · z + k · z ) = 14 π Z d k d k d k (cid:18) k k − k k − k k (cid:19) δ ( k + k + k ) e − i ( k · z + k · z + k · z ) = π ( log r log r − log r log r − log r log r ) . which implies that Z d x ( ∇ log r · ∇ log r ) log r = − π (log r ) Z d x ( ∇ log r · ∇ log r ) log r = − π (log r ) (40)These integrals will not change when we make the replacement log( r ) → log(1 + r/a ) for suitably small a , asthey get little contribution for the small r part of the integral. This can be seen by noticing that replacing thebounds of integration by R ǫ makes the integrals vanish. References [1] P. Menotti, “Gravity in 2+1 Dimensions” (1994) [gr-qc/9412017][2] A. Staruszkiewicz, Acts. Phys. Polon. 24, 734(1963)[3] H. Leutwyler, Nuovo Cimento 42A 159 (1966)[4] S. Deser and R. Jackiw and G. ’T Hooft, Ann. Phys. 152, 220 (1984)[5] J.R. Gott and M. Alpert, Ren. Rel. Grav. 16, 243 (1984)[6] S. Giddings and K. Kuchar, Gen. Rel. Grav 16, 751 (1984)[7] G. Clement, Int. J. Theor. Phys. 24, 267 (1985)[8] G. Clement, Ann. Phys. 201, 241 (1990)[9] G. Grignani and C. Lee, Ann. Phys. 196, 386 (1989)[10] M. Ba˜nados and C. Teitelboim and J. Zanello, The Black Hole in Three Dimensional Space Time, Phys.Rev. Lett. 69 1849-1851 (1992) [hep-th/9204099][11] M. Ba˜nados and M. Henneaux and C. Teitelboim and J. Zanelli, Geometry of the (2+1) Black Hole, Phys.Rev. D48 1506-1525 (1993) [gr-qc/9302012][12] M. Banados, Three-dimensional quantum geometry and black holes [hep-th/9901148][13] S. Carlip, The (2+1)-Dimensional Black Hole, Class.Quant.Grav. 12 (1995) 2853-2880 [gr-qc/9506079]
14] S. Carlip, Conformal Field Theory, (2+1)-Dimensional Gravity, and the BTZ Black Hole, Class.Quant.Grav.22 (2005) R85-R124 [gr-qc/0503022][15] T. Ohta and R. B. Mann, Canonical reduction of two-dimensional gravity for particle dynamics, Class.Quantum Grav. (2001) 3427.[22] R. Kerner and R.B. Mann, Class.Quant.Grav. (2003) L133.[23] R. Kerner and R.B. Mann, Class.Quant.Grav. (2004) 5789.[24] ’t Hooft G 1998 Comm. Math. Phys. 117 685; Deser S and Jackiw R 1998 Comm. Math. Phys. 118 495; P.de Sousa-Gerbert, Nucl. Phys. B346 (1990) 440.[25] A. Bellini and P. Valtancoli, “Exact Quasi-static particle scattering in 2+1 gravity”, Phys. Lett.
B348 ,44-50 (1995)[26] A. Bellini, M. Ciafaloni, and P. Valtancoli, Phys.Lett.
B357 (1995) 532; A. Bellini, M. Ciafaloni, and P.Valtancoli, Nucl.Phys.
B462 (1996) 453.[27] H.J. Matschull, Class. Quant. Grav. (1999) 1069.[28] T. Ohta, H. Okamura, T. Kimura, and K. Hiida Prog. Theor. Phys. (1974) 1598; T. Ohta and T. Kimura1989 Classical and Quantum Gravity
Ch.6 (in Japanese) (Tokyo: MacGrawhill).[29] R. Arnowitt, S. Deser, C.W. Misner.
Gravitation: an introduction to current research , Louis Witten ed.,chapter 7, pp 227–265 (Wiley 1962), Louis Witten ed.,chapter 7, pp 227–265 (Wiley 1962)