Hamiltonian Formalism for Spinning Black Holes in General Relativity

Hamiltonian formalism for spinning black holes in general relativity

Gerhard Schafer¨

Abstract A Hamiltonian treatment of gravitationally interacting spinning black holes is presented based on a tetrad generalization of the Arnowitt-Deser-Misner (ADM) canonical formalism of general relativity. The formalism is valid through linear order in the single spins. For binary systems, higher-order post-Newtonian Hamiltonians are given in explicit analytic forms. A next-to-leading order in spin generalization is presented, others are mentioned. Comparisons between the Hamil- tonian formalisms by ADM, Dirac, and Schwinger are made.

1 Introduction

About half a century after Einstein’s invention of general relativity, Hamiltonian formulations of the theory became available. Dirac developed his formalism in the years 1958 - 1959 [1, 2, 3], Arnowitt, Deser, and Misner (ADM) in the years 1959 - 1960 [4, 5, 6], for a summary see [7], and Schwinger in 1963 [8]. The three formalisms came up from quite different action functionals: Dirac used the Einstein action, ADM the Einstein-Hilbert action in Palatini form, and Schwinger the tetrad- generalized Palatini-based Einstein-Hilbert action. In the following, the three ap- proaches are summarized and compared. Often in the article, the units 16πG = 1 and c = 1 are applied. In the context of his constraint dynamics formalism with various typs of constraints, weakly and strongly vanishing ones, Dirac gave the full Hamiltonian, i.e. before applying the constraint equations and coordinate conditions, in the form, with 0 ∂i denoting partial space-coordinate derivative and i spatial inﬁnity,

Gerhard Schafer¨ Friedrich-Schiller-Universitat¨ Jena, Theoretisch-Physikalisches Institut, Max-Wien-Platz 1, 07743 Jena, Germany, e-mail: [email protected]

1 2 Gerhard Schafer¨ ∫ ∫ 3 −1/2 i j 3 i HD = − d x∂i(g ∂ j(gγ )) + d x(NH − N Hi) I ∫ −1/2 i j 3 i = − dSig ∂ j(gγ ) + d x(NH − N Hi) (1) i0 (correctness of this form within the full non-linear theory was shown only in 1974, by Regge and Teitelboim [9]), whereas ADM and Schwinger gave the Hamiltonian in fully reduced forms, i.e. after applying constraint equations and appropriate coordinate conditions, ∫ I 3 HADM = d x∂i∂ j(gi j − δi jgkk) = dSi∂ j(gi j − δi jgkk), (2) i0 ∫ I 3 i j i j HS = − d x∂i∂ j(gγ ) = − dSi∂ j(gγ ). (3) i0 Though derived under specific coordinate conditions, here the form of the three- i j dimensional metric gi j, (i, j = 1,2,3), with inverse metric γ , is still left general to better illuminate the general expressions behind. All three surface integrals coincide under the assumptions made: asymptotic flat spacetimes with coordinates of the i form N = 1 + O(1/r), gi j = δi j + O(1/r), and N = O(1/r) at spacelike infinity (r → ∞) including the first derivatives of the metric functions to decay as 1/r2. 00 −1/2 i i j The Lagrangian multipliers N ≡ (−g ) and N ≡ γ g0i are respectively coined “lapse” and “shift” functions by Wheeler [10]. The clear identification of HADM with the total energy of the system is one of the merits of ADM in the Hamiltonian approach to general relativity. The constraint equations read,

H = 0 and Hi = 0, (4) with the Hamilton density of weight one, used by ADM and Dirac, ( ) 1 1 H ≡ −g1/2R + g g πi jπkl − (g πi j)2 + H (ADM) 1/2 ik jl i j M g ( 2 ) −1/2 i j 1 i j kl 1 i j 2 = B + ∂i(g ∂ j(gγ )) + g g π π − (gi jπ ) + HM (D), (5) g1/2 ik jl 2 where 1 B ≡ g1/2g g [(γilγ jm − γi jγlm)γkn + 2(γikγlm − γilγmk)γ jn], (6) 4 i j,k lm,n or with Schwinger’s weight-two density,

1/2 i j ik jl i j 2 1/2 g H = Q + ∂i∂ jq + q q Πi jΠkl − (q Πi j) + g HM (S), (7) where 1 1 1 Q ≡ − qmn∂ qkl∂ qkl − q ∂ qkl∂ qmn − qkl∂ ln(q1/2)∂ ln(q1/2), (8) 4 m n 2 ln m k 2 k l Hamiltonian formalism for spinning black holes in general relativity 3

Π − −1 π − 1 π i j γi j 2 with i j = g ( i j 2 gi j) and q = g , q = g , where g denotes the deter- i j minant of gi j and lowering of indices is with gi j, π = gi jπ . The Dirac and ADM 1/2 i j canonical field momentum is given by πi j = −g (Ki j − Kgi j), with K = γ Ki j, − Γ 0 where Ki j = N i j is the extrinsic curvature of a spacelike hypersurface defined 0 Γ 0 through constant-in-time slice, t = x = const. i j denote Christoffel symbols. −1/2 Schwinger’s canonical field momentum Πi j is just g Ki j. The intrinsic curva- i j i j ture scalar reads R. The expressions gi jdx dx and Ki jdx dx are respectively called first and second fundamental form of a hypersurface. Both tensors gi j and Ki j are symmetric. The momentum density of weight one takes the forms

jk Hi ≡ 2gi jDkπ + HMi (ADM and D)

lm lm lm = −Πlm∂iq + ∂i(2Πlmq ) − ∂l(2Πimq ) + HMi (S), (9) where Di denotes the three-dimensional covariant space derivative. The given densities are densities with respect to three-dimensional coordinate transformations. HM and HMi are matter densities. In 1967, DeWitt [11] and more refined later on in 1974, Regge and Teitelboim [9] explicitly showed that in asymptotically flat spacetimes the Hamiltonian, before applying constraint equations and coordinate conditions, takes the form I ∫ 3 i H = dSi∂ j(gi j − δi jgkk) + d x(NH − N Hi). (10) i0 This general Hamiltonian - not yet to be identified with the energy of the system -, delivers all field equations also including those for the lapse and shift functions after imposing appropriate coordinate conditions. Undoubtedly, originating from the very useful coordinate choice made by ADM - one may call it “maximal isotropic” -, πii ∂ − ∂ ψδ TT = 0, 3 jgi j ig j j = 0 or gi j = i j + hi j , (11) the ADM formalism became the most often applied canonical formalism. The in- πi j TT dependent field variables therein are TT and hi j . Both variables are traceless and tracefree (TT). Already in 1961 Kimura [12] used this formalism for applications. The Poisson bracket reads ( ) δF δG δG δF {F,G} = δ TTkl − , (12) i j δ TT δπkl δ TT δπkl hi j TT hi j TT with 1 ∂ ∂ δ TTkl = (P P + P P − P P ), P = δ − i j , (13) i j 2 il jk ik jl kl i j i j i j ∇2 ∇2 δ TTkl where 1/ denotes the inverse Laplacian. The nonlocality of the TT-operator i j is just the gravitational analogue of the well-known nonlocality of the Coulomb 4 Gerhard Schafer¨ gauge in the electrodynamics. If Schwinger would have chosen coordinate conditions corresponding to those introduced above (ADM also introduced another set of coordinate conditions to which Schwinger adjusted), Π i j φδ i j ii = 0, q = i j + fTT, (14) a similar simple technical formalism for detailed calculations would have resulted Π TT i j with the independent field variables i j and fTT. To our best knowledge, only the paper by Kibble [13] delivers an application of Schwinger’s formalism, apart from Schwinger himself, namely a Hamilton formulation of the Dirac spinor field in gravity. Much later in 1978, Nelson and Teitelboim [14] completed the same task within the tetrad-generalized Dirac formalism [15]. The Poisson bracket in the Schwinger formalism resembles very much the ADM one. In terms of the ADM variables, Schwinger’s fundamental field components take the quite simple form, [16], 1 qi j = (ψ2 − hTThTT)δ − ψhTT + hTThTT. (15) 2 kl kl i j i j ik k j Dirac on the other side had chosen the following coordinate system or gauge, called “maximal slicing” because of the field momentum condition,

1/3 i j π = 0, ∂ j(g g ) = 0. (16)

The corresponding independent field variables are 1 π˜ i j = (πi j − γi jπ)g1/3, g˜ = g−1/3g . (17) 3 i j i j To leading order linear in the metric functions, the Dirac gauge coincides with the ADM gauge. The full reduction of the Dirac-form dynamics to the independent degrees of freedom has been performed by Regge and Teitelboim [9]. The Poisson bracket reads ( ) δF δG δG δF 1 δF δG { } δ˜kl − π˜ i j kl − π˜ kl i j F,G = i j kl kl + ( g˜ g˜ ) i j kl , (18) δg˜i j δπ˜ δg˜i j δπ˜ 3 δπ˜ δπ˜ with 1 1 δ˜kl = (δ kδ l + δ lδ k) − g˜ g˜kl, g˜ g˜ jl = δ l. (19) i j 2 i j i j 3 i j i j i The following quoted results, apart from the last one, are based on the ADM approach. The first post-Newtonian (1PN) equations of motion, usually called Einstein-Infeld-Hoffmann equations of motion, were rederived by Kimura in 1961 [12]. In 1974, Ohta et al. [17] derived the 2PN binary equations with shortcomings corrected by Damour and Schafer¨ [18] only much later. About the same time, in 1985, Schafer¨ succeeded with the dissipative 2.5PN level, [16]. Using for the first time in post-Newtonian calculations the technique of dimensional regularization, the 3PN binary Hamiltonian was obtained in 2001 by Damour, Jaranowski, and Schafer¨ Hamiltonian formalism for spinning black holes in general relativity 5

[19]. Much simpler to derive was the dissipative 3.5PN level [20]. Based on a postlinear paper by Schafer¨ [21], the ﬁrst post-Minkowskian (1PM) n-body Hamiltonian was achieved in closed form by Ledvinka, Schafer,¨ and Bicˇak´ in 2008 [22]. The general relativistic Hamilton dynamics of compact objects with spin (proper rotation) has found several explicit results. Counting the order of the spin as 1/c (notice the spin of a miximally rotating black hole reading GM2/c), the leading order spin-orbit coupling is decribed by an 1.5PN Hamiltonian [23], the next to leading order one by a 2.5PN Hamiltonian, [24, 25], and the next-to-next to leading order one by a 3.5PN Hamiltonain [26]. The leading order radiation damping from spin-orbit coupling is decribed by an 4PN Hamiltonian [27]. In case of spin(1)-spin(2) coupling, the conservative 2PN Hamiltonian is given in [23], the 3PN Hamiltonian by [28, 25], and the 4PN one by [29]. The 4.5PN Hamiltonian of the leading order radiation damping from spin(1)-spin(2) coupling is given in [30]. Results on the spin(1)-spin(1) coupling are the 2PN and 3PN Hamiltonians for black holes by respectively [23] and [31]. Several leading higher-order-in-spin Hamiltonians were obtained from Kerr metric considerations, [32]. Spinning test-particles in the Kerr metric have been treated by Barausse, Racine, and Buonanno in 2009, [33] using Dirac’s constraint dynamics formalism.

2 Analytic representation of binary black holes – the Brill-Lindquist initial value solution

The model used in this article to describe compact objects are Dirac delta functions. In this section it will be shown that Dirac delta functions can indeed be used to describe black holes in interaction with each other. An isolated black hole with mass m is represented through the line element ( ) ( ) − Gm 2 4 2 1 2rc2 2 2 Gm i j ds = − c dt + 1 + δi jdx dx + Gm 2rc2 1 2rc2 ( ) ( ) − Gm 2 4 1 2Rc2 2 2 Gm i j = − c dt + 1 + δi jdX dX (20) + Gm 2Rc2 1 2Rc2

i 2 i i i 2 i i when isotropic coordinates x with r = x x and X with R = X X are( employed.) 2 R = Gm /r The two coordinate systems are related through the inversion map 2c2 . Obviously, the line element shows isometry under this map. For binary black holes, initially at rest, the metric at that instant of time may have the form   β β 2 ( ) − 1G − 2G 4 1 2 2 α G α G ds2 = − 2r1c 2r2c  c2dt2 + 1 + 1 + 2 dx2 (21) α1G α2G 2 2 1 + 2 + 2 2r1c 2r2c 2r1c 2r2c 6 Gerhard Schafer¨ as shown by Brill and Lindquist in 1963, [34], for the space part of the given metric. The time part of the given metric has been derived later by Jaranowski and Schafer¨ [35]. The coefﬁcients αa and βa, a = 1,2 do depend on the masses and the relative 2 i − i i − i coordinate distance of the two black holes r12 = (x1 x2)(x1 x2). Furthermore, 2 i − i i − i i ra = (x xa)(x xa), where xa are the position vectors of the black holes. Notice, our variables are living in an euclidean space xi which is conformally related with the physical one. The geometrical interpretation of Brill-Lindquist black holes are two Einstein-Rosen bridges, both starting in the same physical space but ending in two different other ones. The energy of the Brill-Lindquist black hole conﬁguration reads

4 I − c ∂ Ψ α α 2 EADM = π dsi i = ( 1 + 2)c , (22) 2 G i0 α α Ψ 1G 2G ∆Ψ i ̸ i i ∆ ≡ ∇2 where = 1 + 2 + 2 , where = 0 for x = x ,x with . The 2r1c 2r2c 1 2 ′i inversion map of the three-metric of black hole, say 1, at its throat reads x1 = i α2 2 4 2 ′i ′i − i i i − i | i − i | x1 1 G /4c r1, where x1 = x x1, x1 = x x1, r1 = x x1 . The three-metric line element dl2 takes the forms ( ) α G α G 4 dl2 = Ψ 4dx2 = 1 + 1 + 2 dx2 2r c2 2r c2 ( 1 ( 2 )) α α 4 Ψ ′4 ′2 1G 2G ′2 = dx = 1 + ′ 2 1 + 2 dx , (23) 2r1c 2r2c

α α α 2 α2 2 r′i Ψ ′ 1G 1 2G i 1 G 1 i i i − i i − i with = 1 + ′ 2 + ′ 4 and r2 = 4 ′2 + r12, r12 = r1 r2 = x2 x1. 2r1c 4r2r1c 4c r1 Hereof, by definition, the rest-mass of black hole 1 comes out to read I c2 α α G m ≡ − ds′∂ ′Ψ ′ = α + 1 2 . (24) 1 π i i 1 2 2 G i0 2r12c The calculations presented above treat the metric functions geometrically as pure vacuum solution without sources. No divergences occur. Infinities only mean infinite distances. In the following it will be shown how to reconstruct the Brill-Lindquist initial value solution with the aid of Dirac delta functions. Let us have a look at the constraint equations with point-mass sources, ( ) 1 1 16πG ( )1/2 g1/2R − πi π j − πiπ j = ∑ m2c2 + γi j p p δ (25) 1/2 j i i j 3 a ai a j a g 2 c a √ √ −gG00 = 16πG −gT 00 (identical with 2 c4 in standard notation), π − ∂ π j πkl∂ 16 G δ 2 j i + igkl = 3 ∑ pai a (26) c a Hamiltonian formalism for spinning black holes in general relativity 7 √ √ (2 −gG0 = 16πG −gT 0). δ is an abbreviation of δ(xi − xi ) with xi the position i c4 ∫ i a a a 3 vector of mass a ( d xδa = 1). Using the ADM gauge in the form ( ) 1 4 g = 1 + ϕ δ + hTT, [3∂ g − ∂ g = 0], i j 8 i j i j j i j i j j 2 πii = 0, or πi j = ∂ π j + ∂ πi − δ ∂ πk + πi j , (27) i j 3 i j k TT TT πi j the constraint equations simplify to the following equation, imposing hi j = TT = pai = 0, ( ) π − 1 ϕ ∆ϕ 16 G δ 1 + = 2 ∑ma a. (28) 8 c a With the aid of the ansatz ( ) α α ϕ 4G 1 2 = 2 + , (29) c r1 r2 after Hadamard partie finie regularization, the constraint equation yields, α α α a bG ̸ ma = a + 2 , a = b, (30) 2rabc   √ ( ) 2 − 2 α − ma + mb c rab  ma + mb ma mb −  a = ma + 1 + 2 + 2 1 . (31) 2 G c rab/G 2c rab/G

The Hamiltonian clearly results in α α 2 2 1 2 HBL = (α1 + α2) c = (m1 + m2) c − G . (32) r12 The d-dimensional generalization of the above treatment runs as follows. The d- metric functions read,

4 1 d − 2 g = ψ d−2 δ , Ψ = 1 + ϕ, i j i j 4 d − 1 ( ) − 4G Γ ( d 2 ) α α ϕ = 2 1 + 2 . (33) c2 d−2 d−2 d−2 π 2 r1 r2

The d-dimensional inverse Laplacian ∆ −1 takes the form, Γ − −1 ((d 2)/2) 2−d −∆ δa = r , (34) 4πd/2 a 8 Gerhard Schafer¨ where Γ denotes the Euler gamma function. The ansatz for Ψ thus reads ( ) G(d − 2)Γ ((d − 2)/2) α α Ψ = 1 + 1 + 2 (35) 2 − π(d−2)/2 d−2 d−2 c (d 1) r1 r2 and the constraint equation takes the form ( ( )) G(d − 2)Γ ((d − 2)/2) α α 1 + 1 + 2 α δ = m δ . (36) 2 − π(d−2)/2 d−2 d−2 a a a a c (d 1) r1 r2 Chosing 1 < d < 2, a fully ﬁnite result comes in the form, ( ) G(d − 2)Γ ((d − 2)/2) α 1 + b α δ = m δ , a ≠ b. (37) 2 − π(d−2)/2 d−2 a a a a c (d 1) r12 Analytic continuation through d = 3 can now be performed without facing divergences.

3 Spin in Minkowski space

Before entering into gravity in asymptotically ﬂat spacetimes, a discussion of spin in Minkowski spacetime seems most convenient. Using canonical variables, the total i angular momentum J = (J ) = (Ji) of a particle with spin vector Sˆ reads

J = Xˆ × P + Sˆ, (38) where Xˆ denotes the canonical position vector and P its canonical linear momentum. The Lorentz boost is given by 1 K ≡ −tP + G = −tP + HXˆ − Sˆ × P (39) H + m √ with the center-of-energy vector G. The free-particle Hamiltonian H = m2 + P2. The center-of-energy position vector is given by 1 X¯ = Xˆ − Sˆ × P, (40) (H + m)H thus, G = HX¯ . The center-of-spin vector, or Newton-Wigner position vector, is de- ﬁned by Xˆ . The Poisson brackets of its components vanish {Xˆ i,Xˆ j} = 0. The center- ¯ ˆ − 1 ˆ × of-energy position vector X = X (H+m)H S P has non-vanishing Poisson brack- ˆ 1 ˆ × ets of its components and the center-of-inertia position vector X = X + (H+m)m S P as well. Using the center-of-inertia position vector, in four-dimensional lan- Hamiltonian formalism for spinning black holes in general relativity 9

µν µν guage, S Pν = 0 holds, for the center-of-energy position vector S¯ nν = 0, nµ = µν µν (−1,0,0,0) is valid, and for the center-of-spin position vector mSˆ nν +Sˆ Pν = 0 happens. The reason for those different spin-supplementary conditions is rooted in the invariance of the total angular momentum against shift of coordinates. For isolated systems, the Poincare´ algebra is valid,

{Pi,H} = {Ji,H} = 0, {Ji,Pj} = εi jk Pk,

{Ji,Jj} = εi jk Jk, {Ji,G j} = εi jk Gk, {Gi,H} = Pi, 1 1 {G ,P } = H δ , {G ,G } = − ε J . (41) i j c2 i j i j c2 i jk k

Gi is not a constant of motion, but Ki is,

dK/dt = ∂K/∂t + {K,H} = −P + {G,H} = 0. (42)

For many-particle systems with interaction, it generally holds,

P = ∑pa, J = ∑(ra × pa + sa), (43) a a

√ √ M ≡ H2 − P2, H = M2 + P2, (44)

G 1 G Xˆ = + (J − × P) × P, (45) H M(H + M) H with

{Xˆ i,Xˆ j} = {Pi,P j} = 0, {Xˆ i,P j} = δ i j, {M,Xˆ j} = {M,P j} = {M,H} = 0. (46) √ 2 2 For free particles with spin, one has, H = ∑a ha, with ha = ma + pa, and G = ∑ (h r − 1 s × p ). a a a ha+ma a a

4 Spin and gravity – asymptotic ﬂat spacetimes

The treatment of spin in gravity is most conveniently achieved by the introduction of µ µ a tetrad field ea (µ = 0,1,2,3;a = 0,1,2,3) having the properties ea ebµ = ηab and ηab ′µ b µ eaµ ebν = gµν = gνµ . Local Lorentz transformations are defined by ea = L aeb a η b η with L c abL d = cd. The condition of homogeneous transformation of the spacetime derivative of a physical object ϕ under local Lorentz transformations introduces ab a linear connection ωµ , 10 Gerhard Schafer¨ ∂ 1 ab Dµ ϕ ≡ ∂µ ϕ + ωµ G ϕ, ∂µ ≡ , (47) 2 [ab] ∂xµ ω′ab a b ωcd a ∂ bd with transformation property µ = L cL d µ + L d µ L . The object G[ab] is de- [ab] fined through infinitesimal local Lorentz transformations of ϕ, δϕ = −δξ G[ab]ϕ, with infinitesimal group parameters δξ [ab]. ab The curvature tensor Rµν is defined by

ab Dµ Dν ϕ − Dν Dµ ϕ = Rµν G[ab]ϕ, (48)

ab ab ab ac bd ac bd Rµν = ∂µ ων − ∂ν ωµ + ων ωµ ηcd − ωµ ων ηcd. (49)

The simplest Lagrangian density (apart from the herein trivial cosmological constant) for the gravitational ﬁeld reads, L c µ ν ab ω ∂ C µ G = det(eγ )ea eb Rµν ( ) + µ , (50) µ where the exact divergence ∂µ C is needed to render the variational principle valid for variations with prescribed properties at the boundary of the four-dimensional integration area, The vacuum ﬁeld equations read δL G c ab σ − a cb ρ σ 0 = µ = det(eγ )(2Rµσ eb eµ Rρσ ec eb ), (51) δea

δL G ab ab ⇒ ω ω ∂ν 0 = ab µ = µ (e, e). (52) δωµ

The latter equation reduces the gravitational field to the Einsteinian one with sup- pressed torsion. This property will be kept in the following when treating spinning sources of the gravitational field.∫ 4 The matter action WM = d x[LM + LC] for a spinning classical object can be put into the form, e.g. [36], where the Lagrangian density of the dynamical part reads, ∫ [( ) ] µ θ ab 1 ab dz 1 d L = dτ pµ − S ωµ + S δ (53) M 2 ab dτ 2 ab dτ (4) and where the constraints part is given by ∫ [ ] λ L = dτ λ a pbS + λ Λ [i]a p − 3 (p2 + m2) δ . (54) C 1 ab 2[i] a 2 (4) ∫ δ δ δ µ − µ 4 δ τ (4) is a four-dimensional Dirac delta function, (4) = (x z ),( d x (4) = 1), µ is a proper time variable, pµ and z denote respectively the four-dimensional kinetic Hamiltonian formalism for spinning black holes in general relativity 11 momentum form and position vector of the particle and Sab is its spin tensor. The θ ab θ ab Λ a ΛCb − θ ba angle variables with d = C d = d are anholonomic ones. Capital indices refer to body-fixed Lorentz-frame coordinates. λ1, λ2, and λ3 are Lagrangian multipliers. The equations of motion resulting hereof reads, DS ab = 0, (55) Dτ

µ Dpµ 1 ρ ab µ dz µ = − Rµρ u S , u ≡ = λ p . (56) Dτ 2 ab dτ 3 They completely correspond to the divergence freeness or dynamics of the Tulczy- jew stress-energy tensor for pole-dipole particles, [ ] ∫ ( ) √ µν τ λ µ ν δ (µ ν)α δ N gT = d 3 p p (4) + u S (4) . (57) ||α

The index symbol || denotes covariant four-dimensional derivative.

4.1 Hamiltonian for self-gravitating spinning compact objects

Variation of the matter action with respect to the Lagrangian multipliers λ1, λ2, and λ3 results in the relations, respectively, γk j µ pk S ji j nS ≡ n Sµ = = g nS , (58) i i np i j

a p(i) p Λ [ j](0) = Λ [ j](i) , Λ [0]a = − , (59) p(0) m

√ ≡ µ − 2 γi j γik δ i np n pµ = m + pi p j , gk j = j, (60)

µ where n = (1,−Ni)/N, nµ = (−N,0,0,0). To ﬁx the tetrad ﬁeld, the so-called time gauge (Schwinger’s coining, but introduced by Dirac earlier) proves extremely useful,

i µ 1 N e = nµ , i.e. e0 = , ei = − . (61) (0) (0) N (0) N Then 12 Gerhard Schafer¨

(m) gi j = ei e(m) j. (62) The matter Lagrangian density is split into three parts, a kinetic part with time derivative of the matter and ﬁeld variables, respectively denoted LMK and LGK, and a constraint part, LMC, L L L − H matter iH matter MK, GK, MC = N + N i . (63)

The matter energy and momentum densities read, p nS H matter = −npδ − Ki j i j δ − (nSkδ) , (64) np ;k

( ) H matter j δ 1γmk δ δ (kγl)m pknSl δ i = (pi + Ki jnS ) + Sik + i . (65) 2 np ;m The semicolon denotes covariant three-dimensional derivative; later on, the comma will denote partial derivative. The transformation to canonical matter variables, in- dicated by hats, is given by

nSi p γk jSˆ zi = zˆi − , nS = − k ji , (66) m − np i m

p nS p nS S = Sˆ − i j + j i , (67) i j i j m − np m − np

( ) ( j) p(k) p Λ [i]( j) = Λˆ [i](k) δ + , (68) k j m(m − np)

( ) 1 p(knS j) p = pˆ − K nS j − Aˆkle e( j) + S + Γ k j, (69) i i i j ( j)k l,i 2 k j np i where

1 mp(inS j) g g Aˆkl = Sˆ + (70) ik jl 2 i j np(m − np) and

ab 2 S Sab = Sˆ(i)( j)Sˆ(i)( j) = 2Sˆ(i)Sˆ(i) = 2s = const, (71)

Λˆ (i)Λˆ [k]( j) δ [k] = i j, (72) Hamiltonian formalism for spinning black holes in general relativity 13

θˆ (i)( j) ≡ Λˆ (i) Λˆ [k]( j) − θˆ ( j)(i) d [k] d = d . (73)

Putting LMK + LGK = LˆMK + LˆGK + (td), one ﬁnds,

1 (i)( j) Lˆ = pˆ zˆ˙iδ + Sˆ θˆ˙ δ, (74) MK i 2 (i)( j)

Lˆ ˆi j (k)δ GK = A e(k)ie j,0 . (75)

Adding the Lagrangian of gravity, LG, results in a new Lagrangian for gravity, also see Deser and Isham [37],

Lˆ L πi j î jδ (k) L − E GK + G = [2 + A ]e(k)ie j,0 + GC i,i (76) with L − H field iH field GC = N + N i (77) and [ ] 1 1 ( )2 H field = −√ gR + g πi j − g g πikπ jl , (78) g 2 i j i j kl

H ﬁeld π jk i = 2gi j ;k, (79) as well as,

Ei = gi j, j − g j j,i. (80)

The application of the crucial spatially symmetric time gauge for the tetrads introduced by Kibble, [13],

e(i) j = ei j = e ji, (81)

√ ei je jk = gik, ei j = (gkl)(matrix root!), (82) reduces the tetrads to the metric functions. The partial derivatives of the tetrads result in the expressions

(k) kl 1 kl (kl) e e = B g µ + g µ , B = B , (83) (k)i j,µ i j kl, 2 i j, i j [i j] where (··) and [··] denote symmetrization and antisymmetrization, respectively, and 14 Gerhard Schafer¨ ∂ ∂ kl em j − emi 2Bi j = emi em j . (84) ∂gkl ∂gkl From the new gravity action the canonical ﬁeld momentum is easily read off to be 1 πi j = πi j + Aˆ(i j)δ + Bi j Aˆ[kl]δ. (85) can 2 kl The ADM spacetime coordinate conditions do take now the following forms, − πii 3gi j, j g j j,i = 0, can = 0, (86)

Ψ 4δ TT πi j πi j πi jTT gi j = i j + hi j , can = ˜can + can , (87)

TT πi jTT with the transverse-traceless objects hi j and can ,

TT πiiTT TT πi jTT hii = can = hi j, j = can, j = 0, (88)

i j and longitudinal one π˜can, 2 π˜ i j = V i +V j − δ V k . (89) can can, j can,i 3 i j can,k The constraint equations read H ﬁeld H matter H ﬁeld H matter + = 0, i + i = 0. (90)

Finally, the total action in canonical form is given by, [36], ∫ ∫ [ ] 1 W = d4x πi jTThTT + dt pˆ zˆ˙i + Sˆ θˆ˙ (i)( j) − E , (91) can i j,0 i 2 (i)( j) H with E = dSiEi, and the Hamiltonian reads, ∫ [ ] ≡ − 3 ∆Ψ i ˆ TT πi jTT E HADM = 8 d x zˆ , pˆi,S(i)( j),hi j , can , (92) with Poisson bracket commutation relations

i {zˆ , pˆ j} = δi j, {Sˆ(i),Sˆ( j)} = εi jkSˆ(k), { TT πklTT ′ } δ TTklδ − ′ hi j (x,t), can (x ,t) = i j (x x ). (93) Hamiltonian formalism for spinning black holes in general relativity 15 5 Post-Newtonian(PN) Hamiltonians

In this section, the spin will be counted of order (1/c)0 and not 1/c as in the Intro- duction. For non-spinning compact objects, the binary dynamics is known up to the 3.5PN order,

2 2 H(t) = m1c + m2c + HN + H1PN

+ H2PN + H3PN + ...

+ H2.5PN(t) + H3.5PN(t) + ..., (94) where the 2.5PN and 3.5PN Hamiltonians are non-autonomous dissipative ones, 2 [20]. Introducing the following quantities, Hˆ = (H − Mc )/µ, µ = m1m2/M, M = m1 + m2, ν = µ/M with 0 ≤ ν ≤ 1/4 (test particle case ν = 0, equal mass case ν = 1/4), p = p1/µ, r = r12 = |x1 − x2|, pr = (n · p), q = (x1 − x2)/GM, and n = n12 = q/|q|, in the center-of-mass frame, p1 + p2 = 0, the following expressions hold,

p2 1 Hˆ = − , (95) N 2 q 1 1 1 1 c2Hˆ = (3ν − 1)p4 − [(3 + ν)p2 + ν p2] + , (96) 1PN 8 2 r q 2q2 1 c4Hˆ = (1 − 5ν + 5ν2)p6 2PN 16 1 1 + [(5 − 20ν − 3ν2)p4 − 2ν2 p2 p2 − 3ν2 p4] 8 r r q 1 1 1 1 + [(5 + 8ν)p2 + 3ν p2] − (1 + 3ν) , (97) 2 r q2 4 q3 1 c6Hˆ = (−5 + 35ν − 70ν2 + 35ν3)p8 3PN 128 [ 1 + (−7 + 42ν − 53ν2 − 5ν3)p6 + (2 − 3ν)ν2 p2 p4 16 r ] 1 + 3(1 − ν)ν2 p4 p2 − 5ν3 p6 r r q [ 1 1 + (−27 + 136ν + 109ν2)p4 + (17 + 30ν)ν p2 p2 16 16 r ] 1 1 + (5 + 43ν)ν p4 12 r q2 16 Gerhard Schafer¨ [( ( ) ) 25 1 335 23 + − + π2 − ν − ν2 p2 8 64 48 8 ( ) ] 85 3 7 1 + − − π2 − ν ν p2 16 64 4 r q3 [ ( ) ] 1 109 21 1 + + − π2 ν . (98) 8 12 32 q4

To save space, from the disspative Hamiltonians only the leading one is given, [16], [ ] 2 nin j d3Qˆ (tˆ) c5Hˆ (tˆ) = p p − i j . (99) 2.5PN 5 i j q dtˆ3

′i ′ j ′2 Here Qˆi j(tˆ) = ν(q q − δi jq /3) and its time derivatives (tˆ = t/GM) are allowed to be eliminated using the equations of motion. Only after the performance of the phase-space derivatives, the primed variables are allowed to be identiﬁed with the unprimed ones. For convenience, the spin-gravity interaction Hamiltonians are given in the non- center-of-mass frame. To simplify notation, S ≡ Sˆ will be put. The leading order spin-orbit Hamiltonian reads [ ] 1PN G 1 × · 3mb − HSO = 2 ∑ ∑ 2 (Sa nab) pa 2pb . (100) c a b≠ a rab 2ma

The leading order spin(1)-spin(2) Hamiltonian takes the form, G 1 H1PN = ∑ ∑ [3(S · n )(S · n ) − (S · S )] (101) S1S2 2 3 a ab b ab a b c a b≠ a 2rab and the leading order spin(1)-spin(1) dynamics is given by, going beyond linear order in spin, G 1 H1PN = [3(S · n )(S · n ) − (S · S )]. (102) S1S1 2 3 1 12 1 12 1 1 c 2r12 The next-to-leading order spin-orbit Hamiltonain reads, [ [ 2 · 2PN G 5m2p1 3(p1 p2) H = − ((p1 × S1) · n12) + SO c4r2 8m3 4m2 1 1 ] 2 · · · 2 − 3p2 3(p1 n12)(p2 n12) 3(p2 n12) + 2 + 4m1m2 4m 2m1m2 [ 1 ] (p1 · p2) 3(p1 · n12)(p2 · n12) + ((p2 × S1) · n12) + (103) m1m2 m1m2 Hamiltonian formalism for spinning black holes in general relativity 17 [ ]] · · × · 2(p2 n12) − 3(p1 n12) + ((p1 S1) p2) 2 m1m2 4m1 [ [ ] 2 2 G − × · 11m2 5m2 + 4 3 ((p1 S1) n12) + c r 2 m1 [ ]] 15m + ((p × S ) · n ) 6m + 2 + (1 ↔ 2) (104) 2 1 12 1 2 and the next-to-leading order spin(1)-spin(2) Hamiltonian is given by

H2PN = (G/2m m c4r3)[3((p × S ) · n )((p × S ) · n )/2 S1S2 1 2 1 1 12 2 2 12 + 6((p2 × S1) · n12)((p1 × S2) · n12)

− 15(S1 · n12)(S2 · n12)(p1 · n12)(p2 · n12)

− 3(S1 · n12)(S2 · n12)(p1 · p2) + 3(S1 · p2)(S2 · n12)(p1 · n12)

+ 3(S2 · p1)(S1 · n12)(p2 · n12) + 3(S1 · p1)(S2 · n12)(p2 · n12)

+ 3(S2 · p2)(S1 · n12)(p1 · n12) − 3(S1 · S2)(p1 · n12)(p2 · n12)

+ (S1 · p1)(S2 · p2) − (S1 · p2)(S2 · p1)/2 + (S1 · S2)(p1 · p2)/2] 2 3 − × · × · + (3/2m1r )[ ((p1 S1) n12)((p1 S2) n12) 2 + (S1 · S2)(p1 · n12) − (S1 · n12)(S2 · p1)(p1 · n12)] 2 3 − × · × · + (3/2m2r )[ ((p2 S2) n12)((p2 S1) n12) 2 + (S1 · S2)(p2 · n12) − (S2 · n12)(S1 · p2)(p2 · n12)] 2 4 4 + (6G (m1 + m2)/c r )[(S1 · S2) − 2(S1 · n12)(S2 · n12)]. (105)

Finally, the next-to-leading order spin(1)-spin(1) dynamics reads, [ 2PN G − 5m2 · 2 m2 2 2 − 21m2 · 2 2 HS S = 4 3 3 (p1 S1) + 3 p1S1 3 (p1 n) S1 1 1 c r 4m1 m1 8m1 − 3m2 2 · 2 15m2 · · · − 3 2 2 3 p1 (S1 n) + 3 (p1 n)(S1 n)(p1 S1) 4m m p2S1 8m1 4m1 1 2 9 2 · 2 − 1 · 2 − 9 · · 2 + 4m m p2 (S1 n) 2 (p1 p2)S1 2 (p1 p2)(S1 n) 1 2 4m1 4m1 3 · · − 3 · · · + 2 (p1 S1)(p2 S1) 2 (p1 n)(p2 S1)(S1 n) 2m1 2m1 − 3 (p · n)(p · S )(S · n) + 15 (p · n)(p · n)S2 2m2 2 1 1 1 4m2 1 2 1 1 1 ] − 15 · · · 2 2 (p1 n)(p2 n)(S1 n) 4m1 [ ( ) ( ) ] 2 ( ) G m2 m2 2 2 2m2 2 − 5 1 + (S1 · n) − S + 4 1 + (S1 · n) . (106) 2c4r4 m1 1 m1

Also this Hamiltonian goes beyond linear order in spin. For its derivation an ex- tension of the Tulczyjew stress-energy tensor for pole-dipole particles was needed, [31]. As further examples, the next-to-leading order spin-orbit center-of-energy vec- 18 Gerhard Schafer¨ tor is given,

2 2PN −∑ Pa × GSO = a 3 (Pa Sa) [ 8ma ]

mb 5xa+xb +∑ ∑ ̸ ((Pa × Sa) · nab) − 5(Pa × Sa) a b=a 4marab rab [ 1 3 1 +∑ ∑ ̸ (Pb × Sa) − (nab × Sa)(Pb · nab) a b=a rab 2 2 ]

xa+xb −((Pa × Sa) · nab) , (107) rab as well as the spin(1)-spin(2) one, { } 2PN 1 · · − · xa · Sa GSS = ∑ ∑ [3(Sa nab)(Sb nab) (Sa Sb)] 3 + (Sb nab) 2 . (108) 2 a b≠ a rab rab

To summarize, for binary systems, and in part for many-body systems too, the following PN Hamiltonians are known fully explicitly,

H = HN + H1PN + H2PN + H2.5PN + H3PN + H3.5PN 1PN 2PN 3PN 3.5PN + HSO + HSO + HSO + HSO + H1PN + H2PN + H3PN + H3.5PN S1S2 S1S2 S1S2 S1S2 + H1PN + H1PN + H2PN + H2PN S1S1 S2S2 S1S1 S2S2 + H 3 + H 3 + H 2 + H 2 p1S2 p2S1 p1S1S2 p2S2S1 + H 3 + H 3 + H 2 2 . (109) S1S2 S1S2 S1S2 Also the n-body Hamiltonian through linear order in Newton’s gravitational constant G is known H = H1PM, as well as the test-spin Hamiltonian in the Kerr metric. For selfgravitating objects, the Hamiltonians primarily come out in the form TT H = H[p,q,h ,πTT]. The transition to a Routhian description of the type H = H[p,q,hTT,h˙TT] then allows the derivation of an autonomous Hamiltonian for the conservative dynamics in the form H = H[p,q,hTT(x; p,q),h˙TT(x; p,q)] = H(p,q) as well as a non-autonomous one given by H(t) = H[p,q,hTT(x; p′,q′),h˙TT(x; p′,q′)] = H(p,q; p′,q′). The presented dynamical systems have found derivations with other methods too. Particularly the Effective Field Theory method by Goldberger and Rothstein, [38], has proven very powerful. For details the reader is referred to the literature.

Acknowledgment

The author thanks Stanley Deser for useful discussions. Hamiltonian formalism for spinning black holes in general relativity 19 References

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