Spinning Bodies in General Relativity

J.W. van Holten

Nikhef, Amsterdam NL

November 17, 2015

Abstract A covariant hamiltonian formalism for the dynamics of compact spinning bodies in curved space-time in the test-particle limit is described. The construction allows a large class of hamiltonians accounting for speciﬁc properties and interactions of spinning bodies. The dynamics for a minimal and a speciﬁc non-minimal hamiltonian is discussed. An independent derivation of the equations of motion from an appropriate energy-momentum tensor is provided. It is shown how to derive constants of motion, both background-independent and background-dependent ones.

1. Introduction General Relativity is an important tool in modern astrophysics. Among other applications it is indispensible in modelling the formation and gravitational interactions of neutron stars and black holes as well as the emission of gravitational waves. Such modelling needs to take into account the effects of rotation. In particular compact objects like neutron stars and solar-mass black holes can achieve high spin rates. Two famous examples are the pulsar in the Crab nebula rotating at a rate of 30 Hz and the observed pulsar in the binary system PSR1913+16 rotating at 17 Hz [1], neither of which is extreme: some millisecond pulsars have spin rates ten times higher [2]. The dynamics of spinning compact bodies is therefore an important subject in GR. Much work has been done on this topic in the past. Most recent investigations build on arXiv:1504.04290v2 [gr-qc] 17 Nov 2015 earlier work by Mathisson, Papapetrou and Dixon describing compact bodies in terms of mass-multipole moments [3, 4, 5]. One can identify the motion of the mass monopole with the world-line of a test particle and describe the evolution of the other multipoles with reference to this world-line. It is also possible to cast the equations of motion in hamiltonian form, but this is rather complicated as the canonical momentum is not proportional to the proper velocity and one needs constraints to fix the relations between momentum, proper velocity and spin [6]-[17]. It is however possible to follow an alternative route using a test-particle approximation formulated in terms of non-canonical kinetic momenta and extended with spin degrees of freedom that live on the world-line of the particle [18]. Such models have a straightforward hamiltonian formulation still allowing much freedom in the specification of the dynamics by the choice of a hamiltonian.

1 2. Dynamics of spinning particles The key to this construction is the choice of particle phase space spanned by the position µ µν ξ (τ), the covariant momentum πµ(τ) and the antisymmetric spin-tensor Σ (τ). Here τ is the proper-time parametrizing the particle world-line. There exists a closed set of model-independent Poisson-Dirac brackets for these variables deﬁned by the fundamental brackets [18] 1 {ξµ, π } = δµ, {π , π } = ΣκλR , ν ν µ ν 2 κλµν

µν µ νκ ν µκ (1) {Σ , πλ} = Γλκ Σ − Γλκ Σ ,

Σµν, Σκλ = gµκΣνλ − gµλΣνκ − gνκΣµλ + gνλΣµκ. Remarkably the structure functions of these brackets are the quantities characterizing the geometry of the space-time manifold on which the particle moves: the metric gµν, the λ connection coefficients Γµν and the Riemann tensor Rκλµν. Their properties guarantee the closure of the bracket algebra required by the Jacobi identities for triple brackets. To get a complete specification of the dynamics of the spinning particles the brackets must be supplemented by a proper-time hamiltonian; the minimal choice is the kinetic hamiltonian 1 H = gµνπ π , (2) 0 2m µ ν where m is the particle mass. Other choices are possible and a relevant example will be discussed later on. The equation of motion for any phase-space function F (x, π, Σ) is obtained by computing its bracket with the hamiltonian: dF F˙ = = {F,H } . (3) dτ 0 Defining the proper 4-velocity uµ = ξ˙µ the equations of motion for the fundamental dynamical degrees of freedom become Duµ 1 π = mg uν, =u ˙ µ + Γ µuλuν = ΣκλR µ uν, (4) µ µν Dτ λν 2m κλ ν and DΣµν = Σ˙ µν + uλΓ µΣκν + uλΓ νΣµκ = 0. (5) Dτ λκ λκ The equations show that for a free particle –as defined by the minimal kinetic hamiltonian– the covariant momentum equals the kinetic momentum and that the spin-tensor is covariantly constant. However in the presence of a spin- and curvature-dependent force the world-line is not a geodesic and the four-velocity is not transported parallel to itself. µ Note that the spin-tensor can be decomposed into two space-like vectors Sµ and Z : 1 Σµν = −√ εµνκλu S + uµZν − uνZµ,Sµu = Zµu = 0, (6) −g κ λ µ µ

2 where 1 √ S = −g ε uνΣκλ,Zµ = Σµνu . (7) µ 2 µνκλ ν Thus Sµ represents the spin proper, reducing in the rest frame to the magnetic components of the spin-tensor, whilst Zµ represents a mass-dipole reducing to the electric components. These components themselves are not covariantly constant but their variations are of higher order in the spin-tensor: 1 √ 1 D S = −gε Rν uρΣκλΣστ ,D Zµ = R uρΣµνΣστ . (8) τ µ 4m µνκλ ρστ τ 2m νρστ Therefore it is not possible to require the permanent vanishing of the mass dipole on the world-line, like it is required in the canonical approach by the Pirani condition [6].

3. Einstein equations The equations of motion (4) and (5) have been derived from a covariant hamiltonian phase-space formulation. However, the same equations can be derived from the Einstein equations for a spinning point particle with an appropriate choice of energy-momentum tensor as we now show. The issue is that the covariant divergence of the Einstein tensor vanishes because of the Bianchi identities: 1 ∇µG = ∇µR − ∇ R = 0. (9) µν µν 2 ν Therefore the energy-momentum tensor which provides the source term of the Einstein equations must have the same property

µ ∇ Tµν = 0. (10) The energy-momentum tensor of a free spinless particle of mass m moving on a world-line Xµ(τ) is given by the proper-time integral [19] m Z T µν = √ dτuµuν δ4 (x − ξ(τ)) . (11) 0 −g The square root is included because we take the δ-distribution to be a scalar density of weight 1/2 such that Z d4y δ4(y − x)f(y) = f(x).

It is then straightforward to establish that after a partial integration m Z Duν ∇ T µν = √ dτ δ4 (x − ξ(τ)) , (12) µ 0 −g Dτ which vanishes if the particle moves on a geodesic. The spin-dependent force in eq. (4) can be taken into account by adding to the energy-momentum tensor a term 1 Z T µν = ∇ √ dτ uµΣνλ + uνΣµλ δ4 (x − ξ(τ)) . (13) 1 λ 2 −g

3 Computing the covariant divergence of this expression now involves commuting two covariant derivatives which provides a term with a Riemann tensor plus a term which –after partial integration– becomes the covariant derivative of the spin-tensor: 1 Z ∇ T µν = √ Rν dτ uµΣκλδ4 (x − ξ(τ)) µ 1 2 −g µκλ (14) 1 Z DΣνλ + ∇ √ dτ δ4 (x − ξ(τ)) . λ 2 −g Dτ

Combining the constributions (12) and (14) to the divergence of the total energy-momentum tensor we get µν µν µν ∇µT = ∇µ (T0 + T1 ) = 0, (15) where the ﬁrst term on the right-hand side of eq. (14) involving an integral over a δ-function combines with the expression on the right-hand side of (12) to form the world-line equation of motion (4), whilst the second term on the right-hand side of eq. (14) involving a derivative of a delta-function vanishes separately because of the spin-tensor equation of motion (5). Thus the equations of motion (4) and (5) are the necessary and suﬃcient conditions for the divergence of the energy-momentum tensor (15) to vanish. This provides an independent argument for the equations of motion of a spinning particle in curved space-time proposed in section 2.

4. Conservation laws By eq. (3) a constant of motion for a spinning particle is a quantity J(x, π, Σ) of which the bracket with the hamiltonian vanishes:

{J, H0} = 0. (16)

There are three background-independent constants of motion. First, the hamiltonian itself: m H = − , (17) 0 2 where m is the mass of the particle; eq. (17) is equivalent to normalizing proper time such µ that uµu = −1. Furthermore there are two constants of motion constructed as quadratic expressions in the spin: 1 1√ I = g g ΣµνΣκλ = S Sµ + Z Zµ,D = −gε ΣµνΣκλ = S Zµ. (18) 2 µκ νλ µ µ 8 µνκλ µ Apart from these generic conservation laws there can be other constants of motion deter- mined by symmetries of the background space-time. More speciﬁcally, a quantity J of the form 1 J = αµπ + β Σµν, (19) µ 2 µν

4 is a constant of motion provided

κ ∇µαν + ∇ναµ = 0, ∇λβµν = Rµνλκα . (20) These equations have solutions if there exists a Killing vector with covariant components αµ, as one can then ﬁnd βµν in terms of its curl: 1 β = (∇ α − ∇ α ) . (21) µν 2 µ ν ν µ 5. Spin-dependent forces An important aspect of the brackets (1) is their closure independent of the choice of hamiltonian. The kinetic hamiltonian H0 is the minimal choice, but it can easily be extended with additional interactions. In particular one can introduce a spin-dependent gravitational Stern-Gerlach force [18, 20] proportional to the gradient of the curvature by including a spin-spin interaction with coupling constant κ in the hamiltonian: κ H = H + H ,H = R ΣµνΣκλ. (22) 0 SG SG 4 µνκλ In combination with the brackets (1) this produces the equations of motion Duµ 1 κ π = mg uν, = ΣκλR µ uν − ΣρσΣκλ∇ R , (23) µ µν Dτ 2m κλ ν 4m µ ρσκλ and DΣµν = κΣρσ R µ Σνλ − R ν Σµλ . (24) Dτ ρσ λ ρσ λ Thus the spin-tensor is no longer covariantly constant, but experiences a non-linear spin- dependent force itself. As for the minimal case the equations of motion derived from this non-minimal hamiltonian can be obtained equivalently from the Einstein equations with an appropriately extended energy-momentum tensor. Deﬁne 1 Z 1 T µν = ∇ ∇ dτ ΣµλΣκν + ΣνλΣκµ √ δ4 (x − X) SG 2 κ λ −g (25) 1 Z 1 + dτ Σρσ R νΣλµ + R µΣλν √ δ4 (x − X) . 4 ρσλ ρσλ −g Again computing the divergence and using the Ricci identity when commuting covariant derivatives one gets a term involving a δ-function with support on the world-line and a term involving the gradient of this δ-function: 1 Z 1 ∇ T µν = dτ ∇νR ΣρσΣκλ √ δ4 (x − X) µ SG 4 ρσκλ −g (26) 1 Z 1 + ∇ dτ Σρσ R λΣκν − R νΣκλ √ δ4 (x − X) . 2 λ ρσκ ρσκ −g

5 µν These terms are to be combined with the similar terms from the divergence of T0 and µν T1 in eqs. (12) and (14). It follows that µν µν µν ∇µ (T0 + T1 + κTSG ) = 0 (27) if and only if the equations of motion (23) and (24) are satisﬁed. Remarkably all of the conservation laws established for the minimal case carry over unchanged to the non-minimal extension with Stern-Gerlach interactions. This is true not only for the generic constants of motion (H,I,D) but also for the background-dependent quantities J associated with Killing vectors. To prove this it suﬃces to observe that as a consequence of the Bianchi-identities [18] 1 {J, H } = κ ΣµνΣρσ − αλ∇ R + R λβ = 0. (28) SG 4 λ ρσµν ρσµ λν This holds for all values of the coupling constant κ.

6. Final remarks In this work a relativistic spinning-particle dynamics without constraints has been formulated. It was ﬁrst derived from a purely hamiltonian phase-space approach [18], but an alternative derivation based on the Einstein equations with a suitable energy-momentum tensor has been presented here. It is of some interest to observe, that the spin equation of motion (5), (24) also implies the vanishing divergence of another 3-index tensor

µνλ µλν µνλ µνλ M = −M = M1 + κMSG (29) Z 1 Z 1 = dτ uµΣνλ √ δ4(x − ξ) + 2κ∇ dτ ΣµκΣνλ √ δ4(x − ξ), −g κ −g Indeed Z DΣνλ 1 ∇ M µνλ = dτ − κΣρσ R ν Σλκ − R λ Σνκ √ δ4(x − ξ) = 0. (30) µ Dτ ρσ κ ρσ κ −g Obviously this tensor is closely related in structure to the usual orbital angular momentum tensor in General Relativity. A constraint-free spinning-particle dynamics is quite convenient for the analysis of relativistic dynamical systems. We have applied it to some astrophysical problems of practical interest, like the motion of compact spinning bodies in black-hole space-times. Results will be discussed elsewehere [21]. The construction of the corresponding energy-momentum tensors and the corresponding Einstein equations will also enable the calculation of gravi- taional waves emitted by spinning bodies, and the evaluation of radiation reaction eﬀects [22, 23, 24]. These applications require further investigations.

Acknowledgement The author is indebted to G. d’Ambrosi and S. Satish Kumar for useful discussions and collaboration. The work reported is part of the research progamme Gravitational Physics of the Foundation for Fundamental Research of Matter (FOM).

6 References

[1] J.H. Taylor and J.M. Weisberg The relativistic binary pulsar b1913+16: Thirty years of observations and analysis, Binary Radio Pulsars, ASP Conf. Series 328 (2005), 25; arXiv:astro-ph/0407149

[2] D. Chakrabarty, E.H. Morgan et al. Nuclear-powered millisecond pulsars and the maximum spin frequency of neutron stars, Nature 424 (6944): 42 (2003); arXiv:astro-ph/0307029

[3] M. Mathisson Neue mechanik materieller systeme, Acta Phys. Pol. 6 (1937), 163

[4] A. Papapetrou Spinning test particles ingeneral relativity I, Proc. Roy. Soc. London A209 (1951), 248

[5] W. Dixon Dynamics of extended bodies in general relativity. I. Momentum and angular momentum, Proc. Roy. Soc. London A314 (1970), 499

[6] F. Pirani On the physical signiﬁcance of the Riemann tensor, Acta Phys. Pol. 15 (1956), 389

[7] R.M. Wald Gravitational spin interaction, Phys. Rev. D6 (1972), 406

[8] A.J. Hanson and T. Regge The Relativistic Spherical top, Ann. Phys. (NY) 87 (1974), 498

[9] I. Bailey and W. Israel Lagrangian Dynamics of Spinning Particles and Polarized Media in General Relativity, Comm. Math. Phys. 42 (1975), 65

[10] I. Khriplovich Particle with internal angular momentum in a gravitational ﬁeld, Sov. Phys. JETP 69 (1989), 217

[11] O. Semerak Spinning test particles in a Kerr ﬁeld. I, Mon. Not. Roy. Astron. Soc. 308 (1999), 863

[12] G. Schaefer Gravitomagnetic eﬀects, Gen. Rel. Grav. 36 (2004), 2223

[13] K. Kyrian and O. Semerak Spinning test particles in a Kerr ﬁeld. II, Mon. Not. Roy. Astron. Soc. 382 (2007), 1922

7 [14] E. Barausse, E. Racine and A. Buonanno Hamiltonian of a spinning test-particle in curved spacetime, Phys. Rev. D80 (2009), 104025; arXiv:0907.4745 [gr-qc]

[15] R. Plyatsko, O. Stephanyshyn and M. Fenyk Mathisson-Papapetrou-Dixon equations in the Schwarzschild and Kerr backgrounds, Class. Quantum Grav. 28 (2011), 195025; arXiv:1110.1967 [gr-qc]

[16] J. Steinhoﬀ Canonical formulation of spin in general relativity, Ann. Phys. (Berlin) 523 (2011), 296; arXiv:1106.4203 [gr-qc]

[17] L.F. Costa and J. Natario Center of mass, spin supplementary conditions and the momentum of spinning particles, Fundamental Theories of Physics 179 (2015), 215; arXiv:1410.6443 [gr-qc]

[18] G. d’Ambrosi, S. Satish Kumar and J.W. van Holten Covariant hamiltonian spin dynamics in curved space-time, Phys. Lett. B743 (2015), 478; arXiv:1501.04879 [gr-qc]

[19] S. Weinberg Gravitation and Cosmology (J. Wiley, 1972)

[20] I. Khriplovich and A. Pomeransky Equations of motion for spinning relativistic particle in external ﬁelds, Surveys High En. Phys. 14 (1999), 145; arXiv:gr-qc/9809069

[21] G. d’Ambrosi, S. Satish Kumar, J. van de Vis and J.W. van Holten Spinning bodies in curved space-time, preprint Nikhef-2015-041

[22] Y. Mino, M. Sasaki and T. Tanaka Gravitational radiation reaction to a particle motion, Phys. Rev. D55 (1997), 3457; arXiv:gr-qc/9606018

[23] T. Quinn and R. Wald An axiomatic approach to electromagnetic and gravitational radiation reaction of particles in curved spacetime, Phys. Rev. D56 (1997), 3381; arXiv:gr-qc/9610053

[24] E. Poisson The motion of point particles in curved space-time, Living Rev. Rel. 7 (2004), 6; arXiv:gr-qc/0306052