Gravitational Multipoles in General Stationary Spacetimes Daniel R. Mayerson

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Gravitational Multipoles in General

Stationary Spacetimes

Daniel R. Mayerson

Institute for Theoretical Physics, KU Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium

daniel.mayerson @ kuleuven.be

Abstract

The Geroch-Hansen and Thorne (ACMC) formalisms give rigorous and equivalent deﬁni-

tions for gravitational multipoles in stationary vacuum spacetimes. However, despite their

ubiquitous use in gravitational physics, it has not been shown that these formalisms can

be generalized to non-vacuum stationary solutions, except in a few special cases. This pa-

per shows how the Geroch-Hansen formalism can be generalized to arbitrary non-vacuum

stationary spacetimes for metrics that are suﬃciently smooth at inﬁnity. The key is the

construction of an improved twist vector, which is well-deﬁned under a mild topological

condition on the spacetime (which is automatically satisﬁed for black holes). Ambiguities

in the construction of this improved twist vector are discussed and ﬁxed by imposing natu-

ral “gauge ﬁxing” conditions, which also immediately lead to the equivalence between the

Geroch-Hansen and Thorne formalisms for such arbitrary stationary spacetimes.

Contents

1 Introduction and Summary 2

1.1 Summary .................................... 3

2 Geroch-Hansen and Thorne Formalisms in Vacuum 4

2.1 Geroch-Hansen multipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Thorne ACMC-coordinates and multipoles . . . . . . . . . . . . . . . . . . 7

2.3 Equivalence of Geroch-Hansen and Thorne formalisms . . . . . . . . . . . . 7

3 Generalization to Arbitrary Spacetimes 8

3.1 Construction of the improved twist vector . . . . . . . . . . . . . . . . . . 9

3.2 Conditions on exactness of W(2) ........................ 13

3.3 The improvement form “gauge” choice . . . . . . . . . . . . . . . . . . . . 14

3.4 The existence of the ACMC expansion and smoothness of the metric . . . 18

A STF Tensors 22

B Improvement Twist Vector for N= 2 Supergravity 23

B.1 Gauge potentials, ﬁeld strengths, and symmetries . . . . . . . . . . . . . . 24

B.2 Uniqueness of electrostatic potentials . . . . . . . . . . . . . . . . . . . . . 24

arXiv:2210.05687v2 [gr-qc] 12 Oct 2023

1 Introduction and Summary

Multipole moments of a ﬁeld encode the angular structure of the ﬁeld as determined by

its sources; successive multipole moments can typically be read oﬀ from the angular de-

pendence of terms in an asymptotic radial expansion. This makes it a tricky business to

deﬁne gravitational multipoles in general relativity due to general coordinate invariance.

Nevertheless, Geroch [1, 2] and Hansen [3] managed to deﬁne and calculate gravitational

multipoles of stationary, asymptotically ﬂat vacuum spacetimes in an elegant, manifestly

coordinate-invariant formalism. Thorne [4] developed a separate formalism for extract-

ing coordinate-independent multipoles from a stationary, vacuum metric, which relies on

properties of a preferred family of coordinate systems called asymptotically Cartesian and

mass-centered (ACMC) coordinates. These two formalisms were shown to give equivalent

deﬁnitions of multipoles by G¨ursel [5].

There are two families of gravitational multipole tensors in a stationary, vacuum space-

time: the mass multipoles Ma1···aℓand the current multipoles (or angular momentum multi-

poles) Sa1···aℓ. For example, for Kerr, the multipole tensors reduce to single numbers Mℓ, Sℓ

at each order due to axisymmetry, and are given by Mℓ=M(−a2)ℓand Sℓ=Ma(−a2)ℓ.

The most familiar multipoles are the mass M0=Mand angular momentum S1=Ma.

The original Geroch-Hansen and Thorne formalisms, as well as the proof of their equiv-

alence, are all stated in asymptotically ﬂat vacuum spacetimes.1The Geroch-Hansen for-

malism was generalized to electrovacuum solutions (i.e. solutions to the Einstein-Maxwell

equations) [7, 8, 9], and scalar-tensor theories [10]. However, for a generic matter content,

it has never been shown how — or if — the Geroch-Hansen formalism can be generalized.

In addition, the equivalence between the Geroch-Hansen and Thorne formalisms has not

been generalized beyond vacuum spacetimes.

Since the extension of Geroch-Hansen to stationary spacetimes with arbitrary matter

content was uncertain, it is also not clear if the entire concept of the two families of

multipole tensors Ma1···aℓand Sa1···aℓstill applies in such theories. For example, it was

suggested that there could exist theories in which stationary solutions admit a third family

of multipole tensors [11].2

Nevertheless, the usage of the multipoles Ma1···aℓand Sa1···aℓare ubiquitous in modern

gravitational physics, and especially in phenomenology. For example, multipoles have

been discussed for solutions of N= 2 supergravity theories (with many scalars and gauge

ﬁelds) [15, 16, 17, 18, 19], or solutions of higher-derivative gravity theories [20, 21, 22].

Multipoles have been argued to be important, theory-independent characteristics of the

metric in gravitational wave phenomenology, as pioneered by Ryan [23, 24] and developed

and generalized in other works [25, 26, 27, 28]. Measuring the mass quadrupole is already

an important facet of tests of deviations from general relativity with current gravitational

wave detections, see e.g. [29].

It is clear that there is a need to expand our formal deﬁnitions and understanding

of gravitational multipoles beyond (electro)vacuum spacetimes, to include solutions in

theories with arbitrary matter content. That is precisely the goal of this paper.

1The condition of asymptotic ﬂatness can be relaxed to include a NUT parameter [6].

2Of course, it is clear that in the presence of matter, more information is needed besides these two

metric multipole families for the full metric reconstruction; a simple illustration of this is the Kerr and

Kerr-Newman black holes which both have identical gravitational multipoles Ma1···aℓand Sa1···aℓ. Only

in vacuum are the two multipole families suﬃcient information to reconstruct the metric unambiguously

[12, 13, 14].

1.1 Summary

This paper discusses how the Geroch-Hansen formalism can be generalized to stationary

spacetimes with arbitrary matter Lagrangians, in order to give a rigorous deﬁnition of

gravitational multipoles for any stationary spacetime. In addition, I show that the equiv-

alence between the Geroch-Hansen and Thorne multipole formalisms similarly generalizes

to arbitrary stationary spacetimes. The key to these proofs is the deﬁnition and properties

of an “improved” twist vector ωI

µin the Geroch-Hansen formalism (Section 3.1), which can

be formulated in terms of the energy-momentum tensor featuring in Einstein’s equations.

For generic matter content, this paper shows (Section 3.2) that this improved twist

vector can be mathematically deﬁned as long as a certain closed two-form constructed from

the energy-momentum tensor on constant-time slices of the spacetime is also an exact form.

Physically, this condition is that there are no spatial two-cycles over which there is a net ﬂux

of perpendicular (radial) momentum. This is a rather reasonable assumption for stationary

spacetimes. For example, for a single, stationary black hole with a spherical horizon,

asymptotic ﬂatness ensures this condition is always satisﬁed so that the construction of ωI

is always guaranteed to hold.

The construction of the improved twist vector further contains an ambiguity or “gauge

dependence” which aﬀects the (current) multipole moments. I discuss (Section 3.3) the

“gauge ﬁxing” conditions that must be imposed upon the improved twist vector in order

for the multipole moments to be unambiguously deﬁned. With the proposed “gauge ﬁxing”

conditions, the equivalence of the Geroch-Hansen and Thorne multipoles also immediately

follows for such arbitrary spacetimes.

Along the way, I also exhibit (Section 3.1.1) the explicit form of the improved twist

vector ωI

µfor N= 2 supergravity with an arbitrary number of vector multiplets, which

has not been reported previously.

An assumption that must be made on the metric, beyond asymptotic ﬂatness, is suf-

ﬁcient smoothness at inﬁnity (see Sections 2.1 and 2.3 for a more precise deﬁnition and

discussion) and the existence of an ACMC coordinate frame (see Section 2.2). Whereas

these assumptions seem rather intuitively mild, they do leave open the possibility that the

Thorne ACMC formalism is slightly more general, in the sense that metrics could exist

where the Thorne ACMC formalism applies but the Geroch-Hansen formalism does not.

(This is discussed further in Sections 3.4.3 and 3.4.4.)

The generalization presented here of Geroch-Hansen and Thorne to non-vacuum sta-

tionary spacetimes puts the deﬁnition of multipole moments and their usage in character-

izing arbitrary stationary spacetimes on ﬁrmer mathematical ground. These multipoles

being well-deﬁned has been implicitly assumed in many (recent) works, but this paper is

the ﬁrst to show that gravitational multipoles should indeed be unambiguous observables

in the presence of arbitrary matter ﬁelds.

An earlier partial discussion of generalizing the Geroch-Hansen formalism beyond vac-

uum spacetimes was given in [30] in the context of so-called bumpy black holes. There —

as is also of the essence in this paper — the key fact allowing the generalization of the

multipole formalism was the suﬃcient fall-oﬀ of the energy-momentum tensor; however,

for many such bumpy black holes, only the ﬁrst few multipole moments can be deﬁned as

the (improved) twist vector cannot be deﬁned at higher orders in 1/r. Correspondingly,

the metric cannot be brought into ACMC form beyond a certain order.3Other attempts to

3This was not shown in [30] but can easily be seen to be the case for the metrics discussed in [30] for

which only a few multipoles are found to be well-deﬁned. The reason the ACMC expansion fails in these

cases is essentially because the energy-momentum tensor has an angular dependence that is “too strong”

for its 1/r fall-oﬀ — e.g. Tθϕ ∼cos3θsin3θ/r5(eq. (4.4) in [30]).

generalize the notion of multipoles include generalizing multipoles to de Sitter spacetimes

[31] using the Noether charge formalism of [11] which allows generalization of multipoles

to non-stationary spacetimes. Perhaps the most obvious direction for future work is to un-

derstand how also the Noether charge formalism of [11] generalizes to general non-vacuum

spacetimes, and its relation with the the (extended) Geroch-Hansen and Thorne formalisms

discussed here.

Section 2 reviews the Geroch-Hansen and Thorne formalisms and the proof of their

equivalence, all for vacuum stationary spacetimes. In Section 3, I discuss the generaliza-

tion to non-vacuum spacetimes of the Geroch-Hansen formalism and multipole deﬁnitions

by constructing the improved twist vector and discussing its properties, and I discuss

the necessity of the assumptions of the existence of an ACMC coordinate frame and the

smoothness of the metric at inﬁnity.

2 Geroch-Hansen and Thorne Formalisms in Vacuum

This Section is a review of the formalisms of Geroch-Hansen [2, 3] and Thorne [4] which

both deﬁne the gravitational multipoles Ma1···aℓand Sa1···aℓof a stationary, vacuum space-

time, as well as the proof of their equivalence by G¨ursel [5].

2.1 Geroch-Hansen multipoles

An elegant and manifestly coordinate-invariant formalism for deﬁning multipoles of a vac-

uum spacetime was developed by Geroch [2] for static spacetimes, and later expanded to

stationary spacetimes by Hansen [3]. (See also [32].)

Let ξbe the (asymptotically) timelike Killing vector of the four-dimensional stationary,

asymptotically ﬂat spacetime with metric gµν ; the scalar ﬁeld λis given by its norm:

λ=ξ2,(1)

We can deﬁne a manifold Mas the collection of the orbits of ξ— this is indeed a (three-

dimensional, Riemannian) manifold [33, 34] and a natural metric on it is:

hαβ =gαβ −λ−1ξαξβ.(2)

The metric hαβ can also be used to project tensors from the four-dimensional spacetime to

M; note that ξµhα

µ= 0. The natural covariant derivative Don Mis simply [33, 34]:

Dα=hµ

α∇µ.(3)

Note that the correct way to project a tensor covariant derivative is DαTβγ =hµ

αhν

βhρ

γ∇µTνρ.

We can also introduce a rescaled version of the three-dimensional metric, hab:

hab =−ξ2gab +ξaξb.(4)

When we consider the metric hab, we will denote the corresponding three-dimensional

manifold M, so that there is no possible confusion with Mwhich is endowed with the

metric h. The metric hab can also be understood as the three-dimensional metric coming

from a (timelike) Kaluza-Klein reduction over ξ. This amounts to introducing coordinates

(t, xi) such that locally ξ=∂tand the metric takes the standard Kaluza-Klein form:

gµν dxµdxν=λ(dt +A)2−λ−1hijdxidxj,(5)

In such coordinates, we have:

hij =−g00gij +g0ig0j.(6)

We will always denote µ, ν, ··· for four-dimensional indices; a, b, ··· for the induced

three-dimensional indices on M;α, β, ··· for the three-dimensional indices on M, and

ﬁnally i, j, ··· as well as a1, a2,··· for the three-dimensional indices when we are using

coordinates where ξ=∂t. Four-dimensional covariant derivatives will always be denoted

with ∇µand three-dimensional ones by D(or D, ˜

D). Unfortunately, there are many kinds

of indices to keep track of, but fortunately it is usually clear from the context in what

space (and with what metric) we are working in.

We can introduce the twist vector ωµ, deﬁned by:

ωµ=ϵµνρσξν∇ρξσ.(7)

Equivalently in form notation, ω=∗(ξ∧dξ). The curl of this vector is given by:

∂[µων]=−ϵµνρσξρRσλξλ.(8)

Since the spacetime is a solution to the vacuum Einstein equations, Rµν = 0, this curl

vanishes. It follows that the twist vector is derivable from a potential ω:

∇µω=ωµ,(9)

which deﬁnes the scalar ﬁeld ω. From the Kaluza-Klein point of view (5), the Kaluza-Klein

vector Awith ﬁeld strength F=dAis related to the twist vector ωµin (7) as ωt= 0 and:

ωi=−λ2(∗3F)i,(10)

where ∗3is the Hodge dual with respect to hij . The vanishing of the curl of ωµis then

simply the statement that the three-dimensional equation of motion for the gauge ﬁeld F

is sourceless, d(∗3λ2F) = 0, and the scalar ωis the three-dimensional dual gauge potential,

−dω =˜

F=λ2∗3F. The Bianchi identity for Ftranslates to an “equation of motion”

for ωi:

Di(λ−2ωi)=0.(11)

The manifold Mcan be conformally transformed into a new three-manifold ˜

Mwith

metric ˜

hab by: ˜

hab = Ω2hab,(12)

where Ω ∼1/r2at spatial inﬁnity (with rthe distance from the object). In this way,

spatial inﬁnity of Mis brought to a single point Λ, which is a regular point on the compact

manifold ˜

The conformal factor Ω, together with (local) coordinates at Λ on ˜

M, must be chosen

such that the metric coeﬃcients ˜

hab and Ω are smooth (inﬁnitely diﬀerentiable) functions

of the coordinates at Λ. For stationary, vacuum spacetimes it is known that this is always

possible, and that moreover one can choose coordinates such that ˜

hab and Ω are analytic

at Λ [14, 12, 13, 35, 36]. (The demand of smoothness of ˜

hab and Ω at Λ may seem like a

trivial demand, but it is not — see further in Section 3.4.)

Indeed, Geroch and Hansen used the existence of such a conformal factor Ω to provide

a deﬁnition of asymptotic ﬂatness. The manifold Mwith metric hab (and thus the full

stationary four-dimensional spacetime gab) is asympotically ﬂat if there exists a manifold

Mwith metric ˜

hab such that ˜

M=M ∪ Λ where Λ is a single point; ˜

hab = Ω2hab is a

smooth metric on ˜

M; and Ω|Λ=˜

DaΩ|Λ= 0, ˜

Da˜

DbΩ|Λ= 2˜

hab|Λ[3]. We will also always

assume this particular deﬁnition of asymptotic ﬂatness.

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GravitationalMultipolesinGeneralStationarySpacetimesDanielR.MayersonInstituteforTheoreticalPhysics,KULeuven,Celestijnenlaan200D,B-3001Leuven,Belgiumdaniel.mayerson@kuleuven.beAbstractTheGeroch-HansenandThorne(ACMC)formalismsgiverigorousandequivalentdefini-tionsforgravitationalmultipolesinstationaryv...

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