Canonical Hamiltonian for an extended test body in curved spacetime: To quadratic order in spin

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by  Justin Vines

Canonical Hamiltonian for an extended test body in curved spacetime:
To quadratic order in spin
Justin Vines,1,∗ Daniela Kunst,2,† Jan Steinhoff,1,3,‡ and Tanja Hinderer4,§
1Max-Planck-Institute for Gravitational Physics (Albert-Einstein-Institute),
Am Mu¨hlenberg 1, 14476 Potsdam-Golm, Germany, EU
2ZARM, University of Bremen, Am Fallturm, 28359 Bremen, Germany, EU
3Centro Multidisciplinar de Astrof´ısica — CENTRA, Departamento de F´ısica,
Instituto Superior T´ecnico — IST, Universidade de Lisboa — ULisboa,
Avenida Rovisco Pais 1, 1049-001 Lisboa, Portugal, EU
4Maryland Center for Fundamental Physics & Joint Space-Science Institute,
Department of Physics, University of Maryland, College Park, MD 20742, USA
(Dated: July 19, 2016)
We derive a Hamiltonian for an extended spinning test body in a curved background spacetime,
to quadratic order in the spin, in terms of three-dimensional position, momentum, and spin vari-
ableshavingcanonicalPoissonbrackets. Thisrequiresacarefulanalysisofhowchangesofthespin
supplementary condition are related to shifts of the body’s representative worldline and transfor-
mations of the body’s multipole moments, and we employ bitensor calculus for a precise framing
of this analysis. We apply the result to the case of the Kerr spacetime and thereby compute an
explicit canonical Hamiltonian for the test-body limit of the spinning two-body problem in general
6
relativity, valid for generic orbits and spin orientations, to quadratic order in the test spin. This
1
fullyrelativisticHamiltonianisthenexpandedinpost-NewtonianordersandinpowersoftheKerr
0
spinparameter,allowingcomparisonswiththetest-masslimitsofavailablepost-Newtonianresults.
2
Both the fully relativistic Hamiltonian and the results of its expansion can inform the construction

l of waveform models, especially effective-one-body models, for the analysis of gravitational waves
u
J from compact binaries.

8
1
I. INTRODUCTION ratio(EMR)approximationexpandsaboutthetest-mass

limit but is valid in the strong-field, relativistic regime.
]
c The advent of gravitational wave astronomy—which A synergistic approach is the effective-one-body (EOB)
q
hascommencedwiththefirstdetectionofabinaryblack formalism [9, 10], which incorporates information from
-
r holemerger[1]—promisestoshedlightonmanyprofound thePNlimit, theEMRlimit, andnumericalrelativityin
g
questions in astrophysics and gravitational physics. The an attempt to provide an accurate effective description
[
firstsuchquestionswithinthereachofgravitationalwave of two-body systems throughout the parameter space.

2 observatories will concern the nature of gravity in the In the EMR approximation, the zeroth-order solution
v
strong-field regime and the properties of black holes and is given by a (point) test mass moving along a geodesic
9
neutron stars, as inspiraling and coalescing binary sys- ofabackgroundblackholespacetime—theSchwarzschild
2
5 temsofsuchcompactobjectsshouldbefrequentsources, spacetime ofa nonspinning black holeor theKerr space-
7 both for the advanced generation of ground-based gravi- timeofaspinningblackhole. Correctionstothissolution
0 tational wave detectors [2–4] and for future space-based canproceedintwo(intermingled)directions: Firstly,one
1. detectors [5], as well as for pulsar timing arrays [6]. Un- can compute the perturbation to the gravitational field
0 derstanding in great detail the dynamics of such two- produced by the small body, its self-field, and the resul-
6 body systems, expected to be governed by general rel- tantinfluenceonitsmotion. Thisisthegoalofthe“self-
1 ativity, is thus a cornerstone objective of gravitational force” paradigm, as reviewed e.g. by Refs. [11–13]. Sec-
v: wave physics. ondly, one can compute “finite-size effects” on the small
i A sufficiently accurate and general solution to the rel- body’smotion,duetoitsspinandtointrinsicandtidally
X
ativistic two-body problem will require a synergy of re- induceddeformations. Suchfinite-sizeeffectsintheEMR
r sults from both numerical and analytic computations. limit(neglectingtheself-field)arethefocusofthispaper.
a
On the analytic side, two complementary approximation
The equations of motion of a spinning (pole-dipole)
schemesareavailable: thepost-Newtonian(PN)approx-
test body in curved spacetime were first derived by
imation expands about the Newtonian limit but is valid
Mathisson [14, 15] and Papapetrou [16] and were later
for arbitrary mass ratios [7, 8], while the extreme-mass-
extended to include the effects of higher multipoles by
Dixon[17]; see[18]forareview. Theresultantdynamics
of a spinning test body (to pole-dipole order) serves as
∗ [email protected] the basis of the spinning EOB models of Refs. [19–23],
† [email protected] whichemploythecanonicalHamiltonianforapole-dipole
‡ jan.steinhoff@aei.mpg.de;http://jan-steinhoff.de/physics/ particle derived in Ref. [24]. The conservative dynamics
§ [email protected] of these EOB models is defined by the Hamiltonian of
2
an effective spinning test particle in an effective space- effective dynamical mass µ(z,P,S) is given by (5.17),
time which is a deformation of the Kerr spacetime, in
the same way that the original EOB model [9, 10] was µ2 =−P Pµ (1.2)
µ
based on a nonspinning test particle in a deformation of 1
the Schwarzschild spacetime. In both cases, the defor- =m2−PaωabcSbc+ 4ωabcωadeSbcSde
mationsencodefinite-mass-ratioeffectsdeterminedfrom
1
PNcalculationsandvanishasthemassratiogoestozero, + 4RabcdSabScd−(C−1)Ea(Pb)sasb+O(S3).
so that exact (strong-field) results are recovered in the
test-particle limit. We should note that other successful Here, ω =e µ(∇ e ν)e are the Ricci rotation coeffi-
abc a µ b cν
EOB models, including the original spinning EOB mod- cients,C isaconstantresponsecoefficientmeasuringthe
els [25, 26], and more recently, e.g., Refs. [27–29], do not test body’s spin-induced quadrupolar deformation with
incorporatetheexactspinningtest-particlelimit(butdo C =1fortestblackholes,E(P) istheelectricpartofthe
include the geodesic test-mass limit). ab
Weyl/vacuum-Riemann tensor with respect to Pa (5.7),
This paper extends the work of Ref. [24], which was saisthePauli-Lubanskispinvector(5.14),andtheframe
valid to linear order in the spin, or to dipolar order in
components S are determined by the solution (4.9) of
0i
the body’s multipole expansion, to derive a canonical
the Newton-Wigner SSC. The constant mass m(S) is a
√
Hamiltonianforanextendedtestbodyinacurvedback-
function of the likewise constant spin length S = sas
a
ground which is valid to quadrupolar order. We treat
and encodes the moment of inertia [38].
explicitly here spin-induced quadrupoles and all other
Our derivation of the Hamiltonian (and the covari-
spin-squared effects. Our methods also provide signif-
ant action principle which yields it) resolves some pre-
icant simplifications of some of the dipole-order calcu-
vious ambiguities concerning the adjustability of the co-
lations of Ref. [24], as we employ crucial insights from
efficients of the curvature coupling terms in the last line
Ref. [30] on the handling of generic spin supplementary
of (1.2), as we discuss in particular below (2.21) and be-
conditions (SSCs) [conditions which fix a representative
low (5.16).
center-of-mass worldline for the test body] at the level
By specializing to the case where the background
of the action. We highlight how a change of the SSC,
spacetime is Kerr, we arrive at a canonical Hamilto-
corresponding to a shift of the center-of-mass worldline,
nian for the test-body limit of the relativistic spinning
entails transformations of the body’s multipole moments
two-body problem, valid to quadrupolar order in the
andcorrespondingmodificationsoftheactionandHamil-
test body’s multipole expansion. Our results comple-
tonian. Weusebitensorcalculus[11,31,32]toprovidea
ment those of Refs. [39–42], which have also considered
precise and manifestly covariant treatment of the world-
spin-squared effects for test bodies in Kerr, with one no-
line shift. We finally show, extending the linear-in-spin
table new feature of our results being that they allow
result in Ref. [24], how use of the Newton-Wigner SSC
for generic orbits and spin orientations. Of particular
[33–35] allows one to construct a Hamiltonian in terms
interest in this respect are compact (covariant) expres-
ofthree-dimensionalposition,momentum,andspinvari-
sions for the Riemann tensor and its couplings to the
ables with a canonical Poisson bracket structure. Beside
spin which are valid for generic orbits, obtained by ex-
Ref.[24],acanonicalformalismforspinningtest-particles
ploiting the algebraic specialness of Riemann tensor in
ingeneralrelativitywasobtainedin[36]throughadirect
Kerr. Other treatments of spinning test-particle motion
construction of the symplectic structure and in [37] from
in algebraically special spacetimes can be found e.g. in
an ADM canonical formulation.
Refs. [43, 44].
The result for the canonical Hamiltonian can be sum-
While much of our analysis and many of our inter-
marized as follows. In a spacetime with coordinates
mediate results are fully covariant, our final result for
xµ = (t,xi) and an orthonormal frame (or tetrad)
the canonical Hamiltonian in Kerr depends on a choice
e µ = (e µ,e µ), the (reduced) phase space for a spin-
a 0 i of coordinates and a choice of tetrad. We find that a
ning test body consists of the spatial coordinates zi of
comparison with PN results can be relatively easily ac-
its representative worldline zµ, their canonically conju-
complishedbyusingBoyer-Lindquistcoordinatesandthe
gate momenta P , and the frame spatial components
i “quasi-isotropic” tetrad of [24], though we also trace the
S = 1(cid:15) S = 1(cid:15) e µe νS of the spin tensor S ,
i 2 ijk jk 2 ijk j k µν µν relationship between this tetrad and the one used by
given as functions of the time coordinate t and obeying
Carter [45, 46] which diagonalizes the electric and mag-
the canonical Poisson brackets (4.17). The Hamiltonian
netic components of the Riemann tensor. Other choices
H(z,P,S) is defined by
of coordinates and tetrads are likely to yield other use-
(cid:113) ful forms of the Hamiltonian, as in [47], which showed
H =−Pt =N µ2+γijPiPj−NiPi, (1.1) that numerical evolution of the (linear-in-spin) Hamilto-
niansystemisimprovedbyusingKerr-Schildcoordinates
where P = (P ,P ) is the 4D canonical momentum andanassociatedtetrad. Weprovidehereallresults,in-
µ t i
whose time component is the minus the Hamiltonian, cluding the Ricci rotation coefficients and the Riemann
where N, Ni, and γij are the lapse, shift, and inverse tensor components, to explicitly compute the Hamilto-
spatial metric as in (4.18), and where the (canonical) nian(1.1)inKerrforthetwotetradsof[24]and[45,46].
3
[We should note that explicitly expressing H requires al- II. EQUATIONS OF MOTION AND ACTION
gebraically solving Eqs. (1.1) and (1.2), since µ depends PRINCIPLES
on P , but this is easily done working perturbatively in
t
the test spin; see (7.31).] The motion of an extended test body in curved
spacetime is described, in a multipolar approximation,
While our Hamiltonian is valid only to zeroth order by the Mathisson-Papapetrou-Dixon (MPD) equations
inthemassratioandtospin-squared/quadrupolarorder [14, 16, 17], which are given to quadrupolar order by
in the test body’s multipole expansion, it is valid to all
PN orders and to all orders in the spin parameter of the Dpµ 1 1
=− Rµ z˙νSαβ − ∇µR Jνραβ, (2.1)
Kerr black hole. Expanding the Hamiltonian in powers ds 2 ναβ 6 νραβ
of the Kerr spin and in PN orders allows us to make
DSµν 4
comparisons with the test-mass limits of the results of =2p[µz˙ν]+ R[µ Jν]ραβ, (2.2)
high-order PN calculations, notably, those of Refs. [30, ds 3 ραβ
48–52] for next-to-leading-order spin-squared couplings, where pµ is the linear momentum vector, Sµν is the an-
Refs. [37, 53–56] for next-to-next-to-leading-order spin-
tisymmetric angular momentum (or spin) tensor, and
orbitinteractions, andRefs.[50,57–60]forleading-order Jµναβ is the quadrupole tensor, all of which are tensors
couplings at third- and fourth-orders in the spins. We defined along a representative worldline zµ(s) with tan-
find complete agreement with the test-mass limits of all gent z˙µ = dzµ/ds, where s is an arbitrary parameter.
available(complete)PNresults. Weremarkthatthefull
Our sign convention for the Riemann tensor is given by
finite-mass-ratioPNresultsfortheleading-PN-orderspin 2∇ ∇ V = R dV , and we use the (−,+,+,+) met-
[a b] c abc d
couplings for binary black holes, through fourth order in
ricsignature. ThequadrupoleJµναβ maydependoncer-
the spins, can all be “deduced” from the results in the
taininternaldegreesoffreedomofthebody,or(asinthe
test-mass limit.
caseofaspin-inducedquadrupole,oranadiabatictidally
induced quadrupole) it may be determined by only pµ,
While our final spinning test-body Hamiltonian is ex- Sµν, and the local geometry along zµ. In the latter case,
pressed in terms of canonical variables defined by the
Eqs. (2.1) and (2.2) completely determine the evolution
Newton-Wigner SSC, we provide the explicit translation of pµ and Sµν along a given worldline zµ; however, they
into variables defined by other SSCs, and in particular
do not single out a choice of worldline.
by the more physically motivated “covariant” or Tulczy- Afullydeterminedsystemforevolvingpµ,Sµν,andzµ
jew SSC [17, 61, 62]. Future work in developing effective
can be obtained by enforcing a spin supplementary con-
Hamiltoniansforthespinningtwo-bodyproblemislikely dition(SSC),oftheformS fν =0. Thiscorrespondsto
µν
to benefit from a detailed analysis of how to expound
demanding that the body have a vanishing mass dipole
uponthistranslation—relatingdifferentdefinitionsofpo- about zµ in the local Lorentz frame defined by a time-
sition,momentum,spin,andquadrupolevariables—with like vector field fµ. The worldline zµ follows the body’s
explicit connections to the definitions used in other ap- center of mass as measured in the frame of fµ.
proachestothespinningtwo-bodyproblem,including(i)
The most common and physically sensible choice for
the effective action approaches to spin effects in PN the- the SSC is to use the body’s rest frame, i.e. fµ = pµ,
ory(seee.g.[30,58]),(ii)self-forcecalculationsandtheir
yielding the “covariant” (or Tulczyjew [17, 61, 62]) SSC:
uses in determining EOB potentials (see e.g. [63–65]),
and (iii) extracting appropriate measures of the mass, S˜ p˜ν =0, (2.3)
µν
spin, and other multipole moments of black holes and
fluid bodies in numerical relativity simulations (see e.g. wherewedenotequantitiesdefinedbythecovariantSSC
[66]). with a tilde. We also later insert tildes on indices for
the tangent space at the point z˜, as in S˜ p˜ν˜ = 0, to
µ˜ν˜
We begin in Sec. II by discussing constrained action distinguish them from unadorned indices for the tangent
principles for a spinning test body, summarizing how a space at the point z, but we avoid the clutter of tilded
formulation in terms of generic-SSC variables is related indices unless it is necessary. Another useful choice, due
to one in terms of covariant-SSC variables. Section III to its utility in achieving 3D canonical variables, is the
appliesresultsfrombitensorcalculustoderivethetrans- class of Newton-Wigner SSCs [33–35], defined in terms
formation properties summarized in Sec. II. We use the of an arbitrary unit timelike vector field e0µ by
Newton-Wigner SSC to achieve canonical variables in
(cid:32) (cid:33)
Sec. IV. We specialize to a spin-induced quadrupole and S pν +e ν =0. (2.4)
decompose the couplings to the Weyl/vacuum-Riemann µν (cid:112)−p2 0
tensor in terms of its electric and magnetic parts in
Sec. V. We specialize to the Kerr spacetime and its al- WediscussfirstinSec.IIAanexplicitactionprinciple
gebraically special Riemann tensor in Sec. VI, and we for the quadrupolar MPD equations which enforces the
collect the necessary results and perform the PN expan- covariant SSC. In Sec. IIB, we discuss how to generalize
sion in Sec. VII. We conclude in Sec. VIII. toanactionwhichenforcesagenericSSC,andwefollow
4
the change of variables that relates quantities defined by corresponds to a residual freedom to choose the timelike
the covariant SSC to those defined by a generic SSC. componentΛ˜ µ ofthetetrad[69]. Aconsistentandphys-
0
In Secs. II-III, we use unadorned symbols for quantities ically sensible choice is Λ˜ µ =p˜µ/(cid:112)−p˜2; see also [70].
0
definedbyagenericSSC,andthesebecomethosedefined
byaNewton-WignerSSCinSec.IV,withtildesdenoting
covariant-SSC quantities throughout.
B. Action for a generic SSC
Havinginhandtheaction(2.5)whichyieldstheMPD
A. Action for the covariant SSC
equationswhileenforcingthecovariantSSC,wenowturn
to generalizing this action to accommodate an arbitrary
An explicit action functional which yields the
SSC. We will find, following Refs. [30, 69], that this can
quadrupolar MPD equations (2.1, 2.2) while enforcing
be accomplished by a judicious change of variables in
the covariant SSC is given by [38, 67]
the action (2.5). We arrive at a new action which yields
(cid:90) (cid:20) 1 (cid:21) the same form (2.1, 2.2) of the MPD equations for mo-
S[p˜,S˜,z˜,Λ˜]= ds p˜µz˜˙µ+ 2S˜µνΩ˜µν −H˜D . (2.5) mentspandS alongtheworldlinez definedbyageneric
SSC,butwhichentailsadditionalcurvaturecouplingsnot
The independent degrees of freedom to be varied here present/relevantforthecaseofthecovariantSSC,which
are the momentum p˜µ, the spin S˜µν, the worldline z˜µ, modifytherelationshipbetweenthequadrupoleJ (orthe
and a “body-fixed” orthonormal tetrad Λ˜ µ along the effective dynamical mass M) and p, S and z.
A
worldline, satisfying ηABΛ˜ µΛ˜ ν =gµν, from which the For the first step of the change of variables, follow-
A B
(antisymmetric)angularvelocitytensorΩ˜µν isdefinedas ing[30],wetransformthecovariant-SSCtetradΛ˜ µ into
A
a new (intermediate) tetrad Λ¯ µ by applying a local
A
DΛ˜Aν Lorentz transformation Lµ which boosts the direction
Ω˜µν =Λ˜ µ . (2.6) ν
A ds of the momentum p˜µ into an arbitrary unit timelike vec-
tor vµ:
The“DiracHamiltonian”H˜ consistsonlyofconstraints
D
(not involving derivatives) with Lagrange multipliers, Λ¯ µ =Lµ Λ˜ ν, Lµ =δµ− 2vµp˜ν +(cid:112)−p˜2 ωµων ,
and the action (2.5) is thus reparametrization-invariant: A ν A ν ν (cid:112)−p˜2 −p˜ρωρ
(2.9)
H˜ =2χ˜µS˜ p˜ν + λ(cid:104)p˜2+M˜2(p˜,S˜,z˜)(cid:105). (2.7) where ωµ = p˜µ/(cid:112)−p˜2+vµ. As shown in [30], if this is
D µν(cid:112)−p˜2 2
accompanied by the following transformation of the spin
tensor,
The Lagrange multiplier χ˜µ enforces the covariant SSC,
S˜ p˜ν = 0, while the Lagrange multiplier λ enforces the
µν S˜µνv S¯µνp˜
“mass-shell constraint”, p˜2 = −M˜2. The “dynamical S¯µν =S˜µν +2p˜[µξ˜ν], ξ˜µ =− ν = ν (2.10)
mass” function M˜(p˜,S˜,z˜) includes, in addition to rest- −p˜ρωρ −p˜2
mass or other internal energy contributions, couplings
then the rotational kinematic term in the action trans-
betweenthebody’smultipolesandthebackgroundspace-
time curvature. Taking M˜ to depend on z˜only through forms according to
the metric and the Riemann tensor evaluated at z˜leads
1 1 Dp˜
to the quadrupolar MPD equations (2.1, 2.2), with the S˜ Ω˜µν = S¯ Ω¯µν −ξ˜µ µ, (2.11)
2 µν 2 µν ds
quadrupole given by
J˜µναβ = 3p˜ρz˜˙ρ ∂M˜2 . (2.8) where Ω¯µν =Λ¯AµDdΛ¯sAν. From its definition (2.10), and
p˜2 ∂Rµναβ fromS˜ p˜ν =0,thenewspintensorS¯µν satisfiesthenew
µν
We will return in Sec. V to discuss the specific form of SSC S¯µνων = 0. If the original tetrad satisfied Λ˜0µ =
M˜ which corresponds to a spin-induced quadrupole, but p˜µ/(cid:112)−p˜2, then the new tetrad will have Λ¯ µ = vµ, and
0
for now we leave it as a general function of p˜, S˜, and the the new SSC will read
metric and the Riemann tensor at z˜.
(cid:32) (cid:33)
That the variation of the action (2.5) yields the MPD p˜ν
S¯ +Λ¯ ν ≡C =0. (2.12)
equations (2.1, 2.2) with (2.8) is shown in Appendix A. µν (cid:112) 0 µ
−p˜2
The MPD equations and the covariant SSC can then be
used to solve for the relationship between the tangent
This is the “spin gauge constraint” discussed by [69],
z˜˙µ and the momentum p˜µ, thus yielding complete evo- in which the timelike component Λ¯ µ of the body-fixed
lution equations for p˜µ, S˜µν, and z˜µ [39, 58, 68]. One 0
tetrad plays the role of a gauge field parametrizing a
finds, however, that the Lagrange multiplier χ˜µ cannot
generic choice of SSC defined by C = 0. We obtain the
beeliminatedfromtheequationofmotionfor Λ˜Aµ. This “spingaugeinvariant”actionfunctµionalpresentedin[69]
5
by using (2.10) and (2.11) in (2.5) and modifying the χ haveused(2.9)and(2.10)torelatebacktocovariant-SSC
constraint to match (2.12): quantities. Finally, in order to obtain a canonical form
fortheaction(andonewhichyieldstheMPDequations),
S[p˜,S¯,z˜,Λ¯]=
wewillfindthatwemusttransformtoanewmomentum
(cid:90) (cid:20) 1 S¯µνp˜ Dp˜ (cid:21) pµ at z according to
ds p˜ z˜˙µ+ S¯ Ω¯µν − ν µ −H¯ ,
µ 2 µν −p˜2 ds D (cid:18) (cid:19)
1 1
H¯ =χ¯µC + λ(p˜2+M˜2), (2.13) pµ =gµµ˜ p˜µ˜− 2Rµ˜ν˜α˜β˜S¯α˜β˜ξ˜ν˜+ 2Rµ˜α˜ν˜β˜p˜ν˜ξ˜α˜ξ˜β˜
D µ 2 (cid:18) (cid:19)
1 1
=gµ p˜µ˜− Rµ˜ S˜α˜β˜ξ˜ν˜− Rµ˜ p˜ν˜ξ˜α˜ξ˜β˜ .
where the original covariant-SSC dynamical mass M˜, µ˜ 2 ν˜α˜β˜ 2 α˜ν˜β˜
given as a function of the original spin S˜µν, is expressed (2.17)
intermsofthenewspinS¯µν viaS˜µν =P˜µP˜νS¯αβ,which
α β
followsfrom(2.10), whereP˜µ =δµ−p˜µp˜ /p˜2 isthepro- With these transformations, as shown in the following
α α α section, the action (2.13) becomes
jector orthogonal to p˜µ. The action (2.13) can also be
obtained by a “minimal coupling to gravity” of the one S[p,S,z,Λ]= (2.18)
derived in the context of special relativity in [69].
(cid:90) (cid:20) 1 (cid:21)
In the case of flat spacetime, Ref. [69] demonstrated ds p z˙µ+ S Ωµν −H +O(S3) ,
that C is a first class constraint, and thus a genera- µ 2 µν D
µ
(cid:32) (cid:33)
tor of infinitesimal gauge transformations, and that the pν λ
action (2.13) is invariant under these “spin gauge trans- HD =χµSµν (cid:112)−p2 +Λ0ν + 2(p2+M2),
formations”. These transformations induce infinitesimal
Lorentz transformations of the tetrad Λ¯0µ and corre- M2 =M˜2−Rµναβpµξν(Sαβ +pαξβ), (2.19)
spondingshifts,S¯µν →S¯µν+2p˜[µξ˜ν],ofthespin,similar
to (2.9) and (2.10), while leaving the momentum p˜µ and DΛAν
where Ωµν =Λ µ and
the worldline z˜µ invariant. It is important to note that A ds
theworldlinez˜µherecorrespondstotheworldlinedefined
Sµνp
bythecovariantSSC,S˜ p˜ν =0,andbytheMPDequa- ξµ =− ν, (2.20)
µν −p2
tions for p˜µ and S˜µν. It is not the worldline defined by
the generic SSC C =S¯ (p˜ν/(cid:112)−p˜2+Λ¯ ν)=0 and the which is the deviation vector at z pointing to z˜, at least
µ µν 0
MPD equations for p˜µ and S¯µν; the equations of motion to quadratic order in spin, as ξµ = −gµ ξ˜µ˜ + O(S3).
µ˜
forp˜µ andS¯µν resultingfromtheaction(2.13)areinfact The original dynamical mass M˜ is expressed in terms
not the MPD equations. The action (2.13) would yield of the new variables, to O(S2) accuracy, by using the
the MPD equations if the Dp˜µ/ds term were removed. same functional form of M˜ as for the original covariant-
We can transform the action (2.13) into a form which SSCvariablesbutwiththespinreplacedbyitsprojection
doesyieldtheMPDequationsbymakingfurtherchanges PµPνSαβ orthogonal to pµ, where Pµ = δµ −pµp /p2.
α β ν ν ν
of variables, including a shift of the worldline to that AsshowninAppendixA,theequationsofmotionresult-
defined by the new generic SSC. We will see that the ing from the action (2.18) are the MPD equations (2.1,
necessary worldline shift, from z˜(s) to a new worldline 2.2) [+O(S3)] with the quadrupole given by
z(s), to quadratic order in the spin, is given by moving
a unit interval along the affinely parametrized geodesic Jµναβ = 3pρz˙ρ ∂M2 (2.21)
whose initial tangent is the vector ξ˜at z˜given by (2.10). p2 ∂R
µναβ
In other words, z is the “exponential map” of ξ˜at z˜, =gµ gν gα gβ J˜µ˜ν˜α˜β˜
µ˜ ν˜ α˜ β˜
z =expz˜ξ˜, ξ˜µ˜ = S¯−µ˜ν˜p˜p˜2ν˜, (2.14) − 3ppρ2z˙ρ(cid:16)p[µξν]p[αξβ]+p[µjν]αβ +p[αjβ]µν(cid:17)+O(S3),
andξ˜µ˜ isthe“deviationvector”atz˜pointingtoz. Here, jναβ = 1(cid:16)ξνSαβ −ξ[νSαβ](cid:17).
2
we have inserted tildes on indices for the tangent space
at z˜ to distinguish them from unadorned indices for the Thecontributionsinvolvingξµ arisefromtheshift(2.19)
tangentspaceatz. WecanthendefineanewtetradΛAµ of the effective dynamical mass, which introduces new
and spin Sµν at z by parallel transporting Λ¯Aµ˜ and S¯µ˜ν˜ curvaturecouplingsarisingfromtheuseofagenericSSC
along the geodesic from z˜: rather than the covariant SSC. These couplings vanish
(on the constraint surface) for the case of the covariant
Λ µ =gµ Λ¯ µ˜ =gµ Lµ˜ Λ˜ ν˜, (2.15)
A µ˜ A µ˜ ν˜ A SSC, and one can see that the generic action (2.18) re-
(cid:16) (cid:17)
Sµν =gµ gν S¯µ˜ν˜ =gµ gν S˜µ˜ν˜+2p˜[µ˜ξ˜ν˜] , (2.16) duces to the covariant-SSC action (2.5) when the gauge
µ˜ ν˜ µ˜ ν˜ (cid:112)
field Λ µ is taken to be pµ/ −p2. These results (unlike
0
where gµ (z,z˜) is the parallel propagator [11, 32] along those in Sec. V) are valid for both vacuum and nonvac-
µ˜
thegeodesicfromz˜toz,andwherethesecondequalities uum spacetimes.
6
The new curvature couplings in (2.19) arise here from
thetransformations(2.16)and(2.17)ofpµ andSµν,and
these arise, as shown in the following section, from de-
mandingthatthetransformationfromthecovariant-SSC
action (2.5) to the generic-SSC action (2.18) preserves
the canonical forms of the kinematic terms (the terms
withs-derivatives). Thiscoincideswithensuringthatthe
genericaction(2.18)alsoyieldstheMPDequations(2.1,
2.2), asisshowninAppendixA,whichprovidesaphysi-
cal justification of the action (2.18) via the derivation of
the MPD equationsfrom stress-energy conservation [17].
Forfurtherinsightsintothetransformationlaws(2.16) FIG. 1. Along the worldline z(s) defined by a generic SSC,
with tangent z˙µ(s), we have the deviation vector field ξµ(s),
and (2.17), we can note: the final expressions of (2.16)
which points (via the exponential map) to the worldline z˜(s)
and (2.17) for pµ and Sµν are the results of solving the
defined by the covariant SSC, with tangent z˜˙µ˜(s).
MPD equations along the geodesic connecting z˜ to z,
with p˜µ˜ and S˜µ˜ν˜ as initial data at z˜, through O(S2).
We can also note: the holonomy of the MPD equations Differentiating the second relation in (3.1),
around a loop of size S/p is the identity map through
O(S2); see Eq. (4.14) of Ref. [71] with κ=1/2. In both Dξµ =−z˙ν∇ ∇µσ−z˜˙µ˜∇ ∇µσ, (3.2)
of these statements, the quadrupole terms in the MPD ds ν µ˜
equationsdonotcontributeatthestatedorders. Itseems
and solving for the tangent to z˜(s) yields
clearthatthetransformations(2.16)and(2.17)ofpµ and
Sµν underashiftoftheworldlineshouldfollowfromtheir Dξµ
z˜˙µ˜ =Kµ˜ z˙µ+Hµ˜ , (3.3)
definitions in terms of the body’s stress-energy tensor µ µ ds
given by Dixon [17], and likewise for the transformation
where
(2.21) of Jµναβ. While making this connection explicit
wouldrequireacarefulanalysisoftheroleofthesurfaces Hµ˜ =−(∇ ∇µσ)−1 =gµ˜ +O(ξ2), (3.4)
µ µ˜ µ
of integration in Dixon’s definitions, our analysis of the (cid:20) (cid:21)
1
effective action here avoids this complication. Kµ˜ =Hµ˜ ∇ ∇νσ =gµ˜ δν − Rν ξαξβ +O(ξ3)
µ ν µ ν µ 2 αµβ
We will use the generic action (2.18) as our starting
point in Sec. IV, where we specialize to the Newton-
are the “Jacobi propagators” [17, 72–74], with the sec-
WignerSSC.First,inSec.III,weprovideaderivationof
ond equalities giving their expansions in powers of the
how the covariant-SSC action (2.5) is transformed into
deviation vector [73]. Thus,
the generic-SSC action (2.18) via the transformations
(2.17), (2.16), (2.14), and (2.15) of p, S, z, and Λ, whose (cid:18) Dξµ 1 (cid:19)
z˜˙µ˜ =gµ˜ z˙µ+ − Rµ z˙νξαξβ +O(ξ3) ,
inverses are (3.14), (3.15), (3.1, 3.13), and (3.16) below. µ ds 2 ανβ
(3.5)
which gives the tangent to the covariant-SSC worldline
III. COVARIANT SHIFT OF THE WORLDLINE z˜(s) in terms of the generic-SSC worldline z(s) and the
deviation vector ξµ(s) along z(s).
Letustakethemomentump˜ atz˜toberelatedtothe
Wenowshowhowtoconsidertheshiftoftheworldline µ˜
new momentum p at z by
and the transformations of quantities defined along the µ
worldline, in a manifestly covariant manner, using the p˜ =g µ(p +δp ), (3.6)
µ˜ µ˜ µ µ
language of bitensors [11, 17, 32, 72, 73]. An alternative
derivation is presented in Appendix B. where δp is an O(S2) correction to be determined, an-
µ
It will be convenient to start with the worldline z(s) ticipating that ξ =O(S). Then,
defined by a generic SSC and shift to the worldline z˜(s)
definedbythecovariantSSC.Ingeneral,anewworldline Dp˜µ˜ =g µDpµ +(cid:0)z˙ν∇ g µ+z˜˙ν˜∇ g µ(cid:1)p +O(S2)
z˜(s) can be specified by a deviation vector field ξµ(s) ds µ˜ ds ν µ˜ ν˜ µ˜ µ
(cid:18) (cid:19)
Dp
along an old worldline z(s), according to =g µ µ −R α p z˙νξβ +O(S2) , (3.7)
µ˜ ds µ νβ α
z˜=exp ξ ⇔ ξµ =−∇µσ(z,z˜), (3.1)
z where we have used (3.5) and the expansions of the
derivatives of the parallel propagator [11],
where σ(z,z˜) is Synge’s world function [11, 32], giving
half the squared proper interval along the geodesic con- 1
∇ g α =− g µR α ξβ +O(ξ2), (3.8)
nectingz toz˜. Thepointz˜isreachedbytravelingaunit ν µ˜ 2 µ˜ µ νβ
intervalalongtheaffinelyparametrizedgeodesicstarting 1
at z with tangent ξµ, as in Fig. 1. ∇ν˜gµ˜α =−2gµ˜µgν˜νRµανβξβ +O(ξ2). (3.9)
7
Similarly,takingtheintermediatebody-fixedtetradΛ¯ µ˜ with Ωµν and Ω¯µ˜ν˜ as defined in (2.6) and below (2.11).
A
of (2.9) at z˜ to be related to the new tetrad at z by Finally,theintermediatespinS¯µ˜ν˜of(2.10)atz˜isparallel
parallel transport, Λ¯ µ˜ =gµ˜ Λ µ, we have transported into the new spin Sµν at z, as in (2.16).
A µ A
DΛ¯ µ˜ (cid:18)DΛ µ (cid:19)
A =gµ˜ A −Rµ Λ αz˙νξβ +O(S2) ,
ds µ ds ανβ A
(3.10)
and thus,
Putting everything together, we find that the kine-
(cid:16) (cid:17)
Ω¯µ˜ν˜ =gµ˜ gν˜ Ωµν +Rµν z˙αξβ +O(S2) , (3.11) matic terms of the action (2.5) transform according to
µ ν αβ
1 1 S¯µ˜ν˜p˜ Dp˜
p˜ z˜˙µ˜+ S˜ Ω˜µ˜ν˜ =p˜ z˜˙µ˜+ S¯ Ω¯µ˜ν˜− ν˜ µ˜
µ˜ 2 µ˜ν˜ µ˜ 2 µ˜ν˜ −p˜2 ds
(cid:18) 1 1 Sβγp (cid:19)
= p +δp + R αβS ξν − R ν p ξαξβ −R ν p ξα γ z˙µ
µ µ 2 µν αβ 2 µα β ν µα β ν −p2
1 (cid:18) Sµνp (cid:19)Dp D
+ S Ωµν − ξµ+ ν µ + (p ξµ)+O(S3). (3.12)
2 µν −p2 ds ds µ
Weseethatwecanremovethelasttwotermsbychoosing (2.4), in order to obtain a Hamiltonian formulation in
the deviation vector to be terms of 3D dynamical variables with canonical Poisson
Sµνp brackets.
ξµ =− ν. (3.13) This involves a choice of an arbitrary fixed orthonor-
−p2
mal frame or tetrad e µ on the background spacetime,
a
Wethenseethatp willbe(covariantly)conjugatetozµ satisfying e µe = η , where η is the Minkowski
µ a bµ ab ab
if we choose metric and is used to raise and lower the frame in-
dices. We write e µ = (e µ,e µ), where the frame in-
g µ˜p˜ =p +δp (3.14) a 0 i
µ µ˜ µ µ dices a,b,c,... take values 0 for the temporal compo-
=p − 1R αβS ξν − 1R ν p ξαξβ, nent and i,j,k,... = 1,2,3 for the spatial components.
µ 2 µν αβ 2 µα β ν We also use A = (0,i) for the body-fixed frame indices
which is the inverse of (2.17). The complete transfor- on ΛAµ. We continue using Greek letters µ,ν,α,β,...
mation from generic- to covariant-SSC variables is then forcoordinate-basisindices(thoughtheycouldalsohave
given by (3.14) and been interpreted as abstract indices up to now). The
Greek coordinate-basis indices take values t for the time
S˜µ˜ν˜ =gµ˜ gν˜ (Sµν +2p[µξν]), (3.15) coordinate and i,j,k,... (= r,θ,φ, say) for the spatial
µ ν
Λ˜ µ˜ =L µ˜gν˜ Λ ν, (3.16) coordinates, with the unitalicized font distinguishing the
A ν˜ ν A latter from spatial frame indices i,j,k. We use frame
alongwiththeworldlineshiftdefinedby(3.1)and(3.13). components of tensors such as p = (p ,p ) = e µp , to
a 0 i a µ
In the end, (3.12) has become be distinguished from the coordinate-basis components
p =(p ,p).
1 1 µ t i
p˜ z˜˙µ˜+ S˜ Ω˜µ˜ν˜ =p z˙µ+ S Ωµν +O(S3), (3.17) With this notation in order, we consider the generic-
µ˜ 2 µ˜ν˜ µ 2 µν
SSC action (2.18):
and inserting this into the covariant-SSC action (2.5)
(cid:90) (cid:20) 1 (cid:21)
yields the generic-SSC action (2.18), with appropriately S = ds p z˙µ+ S Ωµν −H , (4.1)
modified Lagrange multiplier terms. The expression µ 2 µν D
(2.19) for the effective squared dynamical mass M2 = λ(cid:16) (cid:17)
H =χaC + p2+M2(p,S,z) .
−p2 follows from the transformation (3.14) of pµ and D a 2
from M˜2 =−p˜2.
As discussed in [69], the spin gauge constraint,
(cid:32) (cid:33)
pb
IV. CANONICAL HAMILTONIAN Ca =Sab (cid:112) +Λ0b =0, (4.2)
−p2
We now take the final form (2.18) of the action for a is not itself a SSC, but it becomes a specific SSC with
generic SSC and specialize to the Newton-Wigner SSC a specific choice of the “gauge field” Λ a. The following
0
8
choices for the gauge field Λ a turn (4.2) into various Using (4.6) and (4.8) in (4.1), we see that the action
0
familiar SSCs: in the NW SSC has the form
pa (cid:90) (cid:20) 1 (cid:21)
Λ0a = (cid:112)−p2 ⇒ Sabpb =0, (4.3) S = ds Pµz˙µ+ 2SijΛkiΛ˙kj −HD , (4.10)
2p0δa−pa
Λ a = 0 ⇒ S =0, (4.4) where we have defined a new momentum,
0 (cid:112) a0
−p2
(cid:112) 1
Λ0a =δ0a ⇒ Sab(pb+ −p2δ0b)=0. (4.5) Pµ =pµ+ 2ωµabSab, (4.11)
The first choice represents the covariant Tulczyjew SSC
whose coordinate-basis components P = (P ,P ) are
[61,62]andthesecondyieldstheCorinaldesi-Papapetrou µ t i
canonically conjugate to the worldline coordinates zµ =
SSC[34,75,76]. Thethirdcondition(4.5),leadingtothe
(t,zi). WerefertoP asthecanonicalmomentumandto
Newton-Wigner (NW) SSC [33–35], will be the one used µ
p as the covariant momentum, and we work with both
here. The NW SSC allows one to formulate a canonical µ
below.
phase space algebra for the reduced degrees of freedom
The form (4.10) of the action still has unphysical de-
on the constraint surface, as we shall see below. In gen-
grees of freedom associated with reparametrization in-
eral relativity, this SSC saw a widespread use only more
variance. We can fix these by choosing the worldline pa-
recently. It was employed for post-Newtonian calcula-
rameter to be the time coordinate, s = t, so that t˙= 1,
tions in [30, 77–79], where [30, 78] apply it in the Feyn-
and thus,
man rules, in the ADM canonical formulation of spin
[79, 80], for the test-spin Hamiltonian in [24], and at
P z˙µ =P +P z˙i. (4.12)
the level of the MPD equations in [81]. However, while µ t i
Refs. [24, 77, 78] use the condition on the spin in (4.5),
We can then solve the mass-shell constraint p2 = −M2
their condition on Λ a differs from (4.5).
0 for P , using (4.11). This is most easily accomplished
It is useful to write the rotational kinematic term in t
order by order in the spin, and we will discuss the so-
the action in the local frame,
lution to linear order in the following subsection and to
quadratic order in Sec. VII.
D(ΛAbe ν)
S Ωµν =S Λ ae µ b Having solved both constraints, H vanishes, and we
µν µν A a ds D
obtain from (4.10) and (4.12) the final canonical form of
(cid:16) (cid:17)
=S Λ aΛ˙Ab+ω abz˙µ , (4.6) the action,
ab A µ
(cid:90) (cid:20) 1 (cid:21)
where dots denote the ordinary derivative d/ds, and S = dt P z˙i+ S ΛkiΛ˙kj −H , (4.13)
i 2 ij
ω ab =eb ∇ eaν (4.7)
µ ν µ
where
aretheRiccirotationcoefficients. ChoosingtheNWSSC
(4.5) removes all temporal components from the SΛΛ˙ H(zi,Pi,Sij)=−Pt. (4.14)
term (notice that also Λ 0 = δ0), leaving only spatial
A A
components: A variation of the action with respect to the dynamical
variables zi, P , Λij, and S leads to the equations of
i ij
S Λ aΛ˙Ab =S ΛkiΛ˙kj, (4.8) motion
ab A ij
∂H ∂H ∂H
where we understand that the first index of Λki refers z˙i = , P˙ =− , S˙ =(cid:15) S , (4.15)
∂P i ∂zi i ijk∂S k
to the body-fixed frame and the second one to the lo- i j
cal frame. Thus, the RHS of (4.8) provides a canonical
where
kinematic term for the physical degrees of freedom Λij
and Sij, and the dependent degrees of freedom Λ0µ and 1
S0i have no kinematic terms. The latter are fixed by the Si = 2(cid:15)ijkSjk. (4.16)
gauge choice Λ a =δa and the resultant NW SSC (4.5),
0 0
which can be solved to yield These have the form of Hamilton’s canonical equations
with H being the Hamiltonian. The canonical Poisson
S = Sijpj , (4.9) brackets for the dynamical variables zi, Pi, and Sij can
0i p0+M be “read off” from these equations of motion as
having used p2 = −M2. These arguments allow us to {zi,P }=δi, {S ,S }=(cid:15) S , (4.17)
j j i j ijk k
avoid the Dirac brackets for handling the constraints,
which would be considerably more complicated [24]. with all others vanishing.
9
A. Hamiltonian to linear order in spin by solving the mass-shall constraint p2 = −M2, where
P = p + 1ω abS as in (4.11). But we must now
µ µ 2 µ ab
We can find the explicit Hamiltonian H =−P to lin- also take into account the spin-squared contributions to
t
ear order in the spin by solving the mass shell constraint the effective dynamical mass M, which arise both from
p2 = −M2 for P in terms of zi, P , and S , using intrinsic couplings in the covariant-SSC dynamical mass
P =p + 1ω abSt as in (4.11), and uising theijsolution M˜ andfromwhatonemightcallthekinematiccouplings
foµrS µgiven2bµy(4.9ab). DefiningthelapseN,shiftNi,and of (2.19),
0i
inverse spatial metric γij of the background spacetime, M2 =M˜2−R paξb(Scd+pcξd)+O(S3), (5.1)
abcd
1
N = , where
(cid:112)
−gtt
Sabp
gti ξa =− b. (5.2)
Ni =N2gti =− , (4.18) −p2
gtt
NiNj gtigtj Theformofthecovariant-SSCdynamicalmassM˜ which
γij =gij+ =gij− ,
N2 gtt encodes a spin-induced quadrupole moment is given by
[38, 48, 80]
we find
p˜a˜p˜c˜
H =−Pt(zi,Pi,Sij) (4.19) M˜2 =m2+CRa˜˜bc˜d˜−p˜2S˜˜be˜S˜d˜e˜+O(S3) (5.3)
(cid:32) (cid:33)
=HNS− NQPˆµ ωµ2ij + Pωˆ0µ0+iPˆmj Sij +O(S2), =m2+CRabcdp−app2cS˜beS˜de+O(S3), (5.4)
where m and C are constants, and
where
S˜ab =PaPbScd =Sab+2p[aξb] (5.5)
H =NQ−NiP , (4.20) c d
NS i
(cid:113) istheprojectionofthespintensororthogonaltothemo-
Q= m2+γijP P ,
i j mentum (which coincides with the covariant-SSC spin
Pˆ =(−H ,P ), tensorS˜a˜˜b,uptoparalleltransport,attheconsideredor-
µ NS i
der). The constant C measures the body’s spin-induced
Pˆa =eaµPˆ =(Pˆ0,Pˆi)=(e0µPˆ ,eiµPˆ ),
µ µ µ quadrupolar deformation response. It is equal to 1 when
the body is a black hole [48, 83], and we will see that
and where we have taken M2 = m2 + O(S2) with m
special simplifications occur in this case. For material
being a constant.
bodies such as neutron stars, C depends on the equation
The Hamiltonian becomes somewhat simpler if we
of state [83, 84]. The constant mass m(S) is a function
adoptthe“timegauge”[82],i.e.ifwespecializethelocal
(cid:112)
Lorentzframeeaµ sothatitstimelikevectorpointsalong of the likewise constant spin length S = 21 S˜abS˜ab and
the direction of the time coordinate, so that e0 = Nδt encodes the moment of inertia [38]. Notice that the spin
µ µ
and also e t = δ0/N. We will refer to this choice as a length is defined with the projected spin tensor (or with
a a
the covariant-SSC spin tensor).
time-aligned tetrad from now on. This choice also im-
plies that P0 = NPt and thus, from (4.18) and (4.20), The couplings to the Riemann tensor—–the kine-
that Pˆ0 =Q. It also implies that Pˆi =eijP =Pi =P , matic couplings of (5.1) and the intrinsic spin-induced
j i quadrupolecouplingof(5.3)—–canbebetterunderstood
which is then independent of P . We can then write the
t byusingtheelectric/magneticdecompositionoftheWeyl
Hamiltonian (4.19) as
tensor. This goes hand-in-hand with the decomposition
N (cid:18)ω ω P (cid:19) of the spin tensor Sab in terms of a Pauli-Lubanski spin
H =HNS−QPˆa a2ij − Qa+0imj Sij+O(S2), (4.21) vector sa (5.14) and the vector (cid:112)−p2ξa which encodes
the mass dipole.
where Pˆa = (Q,Pi), with H and Q still given by We restrict attention to vacuum spacetimes in four di-
NS
(4.20). This agrees with Eqs. (4.41-45) of [24] if we note mensions. ThentheRiemanntensorequalstheWeylten-
ω =2E and mind some raised and lowered indices sor,anditcanbedecomposedintocontributionsfroman
µab µab
and changes of bases. electric part E(p) and a magnetic part B(p) with respect
ab µν
to a timelike vector pµ [85–87]. In a compact complex
notation this reads
V. CURVATURE COUPLINGS AT QUADRATIC
1 pcpd
ORDER IN SPIN E(p)+iB(p) = G efR (5.6)
ab ab 2 ac efbd−p2
At quadratic order in the spin, the action is still given pcpd
=(R +i∗R ) , (5.7)
by (4.13), with the Hamiltonian H = −Pt determined acbd acbd −p2
10
where The R SabScd coupling was also considered e.g. in
abcd
[88, 89], but therein the prefactor is an arbitrary con-
Gabcd =gacgbd−gadgbc+iηabcd (5.8) stant, analogous to C here. However, the present deriva-
tion shows that this prefactor is actually fixed (by kine-
is the tensor which projects a 2-form onto (four times)
matics). Aswasarguedin[30],theonlynonminimalcou-
its anti-self-dual part, where the volume form is η =
√ µναβ plings which carry arbitrary coefficients should be con-
−g(cid:15)µναβ or ηabcd = (cid:15)abcd with (cid:15)0123 = 1, and where structed from the projected spin S˜ab (or the vector sa).
∗R = 1η efR is the dual of the Riemann ten-
acbd 2 ac efbd The coupling terms agree with [48] in the case of the
sor. The tensors E(p) and B(p) are orthogonal to pa, covariant SSC.
ab ab
and thus effectively three-dimensional, and are symmet- Using(5.16),andusingP =p +1ω bcS asin(4.11),
a a 2 a bc
ric and trace-free, making them easier to handle than we can rewrite the mass-shell constraint p2 =−M2 as
R . The following useful relations hold,
abcd
µ2 ≡−P2 (5.17)
G efG =4G , (5.9)
ab efcd abcd 1
p p =m2−Paω bcS + ω bcωadeS S
G eG gf e f =−G , (5.10) a bc 4 a bc de
abg cd −p2 abcd
1
R +i∗R = 1G efR + 4RabcdSabScd−(C−1)Ea(pb)sasb+O(S3).
abcd abcd 2 ab efcd
1 We can then give a formal solution for the Hamiltonian
= 2RabghGcdgh as
1 (cid:113)
= 8GabefRefghGcdgh. (5.11) H =−Pt =N µ2+γijPiPj−NiPi. (5.18)
Notethataproofof(5.10)canrequireusingη p =0. This is only a formal solution because µ2 depends on P .
[abcd e] t
In (5.11), the equality of the left and right duals of the But because this dependence starts only at O(S), this
Riemann tensor was used. From these relations together equation can be relatively easily solved for P order by
t
with (5.6), the Riemann tensor can be recovered as the order in the spin. We saw the fully expanded solution
real part of to linear order in spin, for the general case in (4.19),
and with a time-aligned tetrad, e0 = Nδt, in (4.21).
p p (cid:16) (cid:17) µ µ
R +i∗R =G efG gh e g E(p)+iB(p) . We give the solution to quadratic order, in the case of a
abcd abcd ab cd −p2 fh fh
time-aligned tetrad, in (7.31).
(5.12)
We conclude this section with a remark on the con-
Using (5.7) along with (5.5) allows us to express the
served spin length. The action (4.1) has a symmetry un-
curvature couplings in (5.1) and (5.3) as
der spatial rotations of the body-fixed frame. The corre-
sponding Noether conserved quantity is the spatial spin
R papcS˜b S˜de =p2E(p)sasb, (5.13)
abcd e ab inthebody-fixedframeΛ aΛ bS . Contractingthisten-
i j ab
R paξbpcξd =−p2E(p)ξaξb, sor with itself, we obtain the conserved scalar
abcd ab
RabcdpaξbScd =2(cid:112)−p2Ba(pb)saξb+2p2Ea(pb)ξaξb, S˜µνS˜µν =2sasa ≡2S2, (5.19)
where the Pauli-Lubanski spin vector sa is defined as where S is the conserved spin length. It should be noted
that the scalar S Sab is in general not conserved. The
1 p ab
sa =− ηabcd b S˜ constant mass m(S) is actually a function of S and en-
2 (cid:112)−p2 cd codes the moment of inertia [38].
1 p
=− ηabcd b S . (5.14)
2 (cid:112)−p2 cd
VI. THE KERR SPACETIME AND ITS
One further useful identity, which follows from (5.12), is RIEMANN TENSOR
1
R SabScd (5.15) Wenowspecializetothecasewherethebackgroundis
4 abcd the Kerr geometry, giving the vacuum spacetime around
=−E(p)sasb−2(cid:112)−p2B(p)saξb−p2E(p)ξaξb. a spinning black hole with mass M and angular mo-
ab ab ab
mentumMa. ThemetricinBoyer-Lindquistcoordinates
By combining (5.1), (5.3), (5.13), and (5.15), we can ex- (t,r,θ,φ) reads
press the total effective dynamical mass as
(cid:18) (cid:19)
2Mr Σ
M2 =m2−CE(p)sasb−2(cid:112)−p2B(p)saξb−p2E(p)ξaξb ds2 =− 1− Σ dt2+ ∆dr2+Σdθ2
ab ab ab (6.1)
1 Λ 4Mar
=m2+ R SabScd−(C−1)E(p)sasb. (5.16) + sin2θdφ2− sin2θdtdφ,
4 abcd ab Σ Σ