# Local symmetries as constraints on the motion of freely-falling extended
bodies

Abraham I. Harte  David Dwyer  Centre for Astrophysics and Relativity
School of Mathematical Sciences
Dublin City University, Glasnevin, Dublin 9, Ireland

###### Abstract

Different extended objects can fall in different ways, depending on their
internal structures. Some motions are nevertheless impossible, regardless of
internal structure. This paper derives universal constraints on extended-body
motion, both in Newtonian gravity and in general relativity. In both theories,
we identify a weak notion of “local symmetry” which precludes certain force
and torque combinations. Local symmetries imply that certain components of a
body’s quadrupole moment cannot affect its motion. They also imply that some
forces can arise only in combination with appropriate torques. Many of these
symmetries are shown to be determined by the algebraic structure of the tidal
tensor. In general relativity, we thus relate qualitative features of
extended-body motion to the Petrov type of the spacetime. Doing so shows that
local symmetries are in fact ubiquitous. In general relativity, there are at
least two such symmetries in all algebraically-special spacetimes. Some of
these are generated by Killing vectors and some by conformal Killing-Yano
tensors. However, many local symmetries do not fall into either of these
classes.

##  I Introduction

One of the most fundamental results in general relativity is that freely-
falling objects move on geodesics Ehlers and Geroch (2004); Yang (2013);
Geroch and Weatherall (2018); Gralla and Wald (2008). However, geodesic motion
is only an approximation. Real objects have finite mass, and are therefore
affected by self-interaction Poisson  _et al._ (2011); Harte (2015); Barack
and Pound (2018). Real objects also have finite size, and are thus affected by
internal structure Dixon (1979); Thorne and Hartle (1985). Regardless,
geodesic motion has the property that all future trajectories are uniquely
determined by initial positions and initial velocities. To the extent that
objects do move along geodesics, free-fall is therefore “universal:” Every
object with the same initial conditions falls on the same trajectory.

Expanding in powers of an object’s size, the first extended-body correction to
geodesic motion involves an object’s angular momentum. However, taking this
into account does not spoil the universality of free-fall. Once an appropriate
center-of-mass definition has been fixed, a unique trajectory follows from a
given position, a given linear momentum, and a given angular momentum at some
initial time. While more initial data is required than in the geodesic case,
all bodies with the same initial conditions still follow the same
trajectories. We may thus view this as a refinement rather than a failure of
universal free-fall. Failure occurs when expanding through one higher order in
an object’s size.

More precisely, the universality of free-fall breaks down once quadrupole
moments are taken into account. Although the trajectories of some objects can
be uniquely predicted given, e.g., their initial positions, initial linear
momenta, initial angular momenta, and initial quadrupole moments, equations of
motion can still differ from one object to another. Moreover, many bodies can
be said only to satisfy laws of motion rather than equations of motion:
Although trajectories are constrained by the laws of motion, they may fail to
be uniquely determined from any reasonable initial data set111One generally
expects that unique predictions are always possible given a sufficiently
detailed model and sufficiently detailed initial data. However, the question
addressed here is whether or not the future can be uniquely predicted, to
adequate precision, using only the small amount of initial data which might be
ascribed to an “effective point particle” (possibly with structure)..
Regardless, different bodies can fall in different ways, depending on the
evolution of their quadrupole and higher-order moments. Our goal here is to
constrain these differences, both in general relativity and in Newtonian
gravity. For simplicity, we focus only on quadrupolar effects.

Although extended-body effects are instantaneously small in most astrophysical
systems, their effects can grow over time. Perhaps the best-known Newtonian
example is the phenomenon of tidal locking Hut (1981); Tremaine (2023);
Poisson and Will (2014); Ferroglia and Fiolhais (2020); another is the chaotic
tumbling of Saturn’s moon Hyperion Tremaine (2023); Wisdom  _et al._ (1984);
Wisdom (1987). Indeed, tidal effects are known to produce secular changes in,
e.g., orbital eccentricities, inclinations, and radii Hut (1981); Tremaine
(2023); Ogilvie (2014). In general relativity, tidal effects are often studied
as something which affects the late-stage evolution of inspiraling binaries,
potentially allowing gravitational wave observations to probe the composition
of neutron star interiors Flanagan and Hinderer (2008); Chatziioannou  _et
al._ (2018); Baiotti and Rezzolla (2017).

Regardless, the primary goal here is not to model any particular astrophysical
system. Instead, we identify fundamental, model-independent limitations on
extended-body motion: What is possible and what is not? From this perspective,
it is useful to imagine a hypothetical spacecraft which has been designed to
be able to control its internal mass distribution. Changes in that
distribution can then be used to modulate extended-body forces, and thus to
control a spacecraft’s motion. Such systems have been analyzed before, both in
Newtonian gravity Murakami (1981); Martinez-Sanchez and Gavit (1987); Beletsky
(2001); Gratus and Tucker (2003); Harte and Gaffney (2021) and in general
relativity Wisdom (2003); Harte (2007); Avron and Kenneth (2006); Bergamin
_et al._ (2009); Silva  _et al._ (2016); Mendes and Poisson (2017); Veselý and
Žofka (2019); Harte (2020); Silva (2022). In both contexts, shape-changing
spacecraft have been found to be able to produce large orbital changes simply
by modulating extended-body forces over many orbits. Roughly speaking, this is
accomplished by exchanging internal energy with orbital energy. It is also
possible (and in fact simpler) for a spacecraft to control its rotation by
similarly manipulating its internal mass distribution.

There are limitations, however. Certain course corrections cannot be produced,
no matter how cleverly a spacecraft has been engineered. This can be seen most
easily in a uniform Newtonian gravitational field, where internal structure
has no effect whatsoever; all objects fall identically. Relativistically, the
same is true in all maximally-symmetric spacetimes Dixon (1970, 1979).
Different objects can fall differently only when there is some inhomogeneity
to “grab onto.” This can be made more precise in general relativity by noting
that for each Killing field which may exist, certain force and torque
combinations are impossible, regardless of an object’s internal structure
Dixon (1970, 1974); Ehlers and Rudolph (1977); Dixon (1979). Analogous results
are also known in Newtonian gravity Harte and Gaffney (2021). However, it is
natural to ask if these are the only fundamental constraints on extended-body
forces and torques.

They are not. At least in vacuum spacetimes which are of Petrov type D,
certain torques are known to be impossible even when there are no Killing
fields which exclude them Harte (2020). Such constraints can in fact be
related to the presence of conformal Killing-Yano tensors, which describe a
different kind of symmetry. Indeed, we show below that any conformal Killing-
Yano tensor in a (not necessarily type D) vacuum spacetime precludes certain
torque components. However, even this does not exhaust all fundamental
restrictions on extended-body motion.

We find that constraints due to Killing vectors and constraints due to
conformal Killing-Yano tensors are both special cases of a certain type of
“local symmetry.” Crucially, these symmetries are very common. In general
relativity, we show that local symmetries exist in all algebraically-special
spacetimes, and in many algebraically-general ones as well. We also find that
every Newtonian gravitational field admits at least one local symmetry.

A related theme in this paper is to describe how extended-body effects depend
qualitatively on the algebraic structure of the relevant tidal tensor. In
general relativity, we derive local symmetries, and discuss quadrupolar forces
and torques, in each of the Petrov types which can be associated with four-
dimensional vacuum spacetimes. There are essentially three types of result:
First, what is the space of possible torques which can arise due to extended-
body effects? This is either four or six-dimensional, depending on the Petrov
type (where the six-dimensional case allows any torque whatsoever). Our second
type of result asks for the space of possible forces which can be varied
without simultaneously varying the torque. This lies between zero and four
dimensions. Our third type of result asks how many of a body’s ten quadrupole
components can affect its motion. The answer here lies between four and ten.
In some Petrov type I spacetimes, an appropriately-engineered spacecraft could
vary its ten quadrupole components in order to arbitrarily control all four
force components and all six torque components. In other spacetimes,
considerably less is possible.

Although our main motivation here is to understand motion in general
relativity, the Newtonian case is already rich and largely unexplored. In
fact, all of the concepts which appear in general relativity are already
present in Newtonian gravity. We thus begin in Sect. II by describing
Newtonian extended bodies. Newtonian tidal tensors are classified in terms of
their algebraic structure, and the corresponding constraints on extended-body
motion are derived. A concept of local symmetry is introduced as well. Sect.
III performs the same analysis for extended bodies in general relativity,
introducing local symmetries in that context and explaining how extended-body
motion depends on the Petrov type. Appendix A summarizes our notational
conventions and provides a table of symbols. Appendix B reviews some material
on principal null directions and the Petrov classification. Appendix C
describes objects with tidally-induced quadrupole moments.

##  II Extended bodies in Newtonian gravity

Before analyzing extended-body motion in general relativity, we first discuss
motion in Newtonian gravity. This is partially because model-independent
features of Newtonian extended-body motion do not appear to have been explored
before, and are interesting in their own right. However, a thorough
understanding of the Newtonian problem also allows us to better understand
which effects are “fundamentally” relativistic and which are not.

Regardless, Sect. II.1 reviews the Newtonian theory of extended-body motion in
a form which emphasizes the role of symmetry and which easily carries over
into general relativity. Sect. II.2 considers the effects of symmetry on
objects with arbitrary quadrupole moments. Sect. II.3 then classifies
different tidal tensors according to their eigenvalues and discusses how
extended-body motion differs in each case. Lastly, Sect. II.4 applies our
formalism in order to describe motion in certain example gravitational fields.

###  II.1 Generalized momentum and generalized force

We begin by reviewing a perspective on Newtonian motion which was developed in
Harte (2008a, 2015); Harte and Gaffney (2021), and which grew out of Dixon’s
formulation of extended-body motion in general relativity Dixon (1970, 1974,
1979, 2015). The central object of study is the “generalized momentum,” which
unifies a body’s linear momentum and angular momentum into a single object. If
an extended Newtonian body has momentum density $\rho_{a}$, its generalized
momentum at time $t$ is defined to be

$\mathcal{P}_{\xi}(t)\equiv\int\rho_{a}(x,t)\xi^{a}(x)dV,$ (1)

where $\xi^{a}(x)$ is any Euclidean Killing field. At fixed $t$, the
generalized momentum may be viewed as a linear map from the space of Killing
fields into $\mathbb{R}$, and may therefore be interpreted as a vector in the
six-dimensional space which is dual to the space of Euclidean Killing fields.
Three of those six dimensions describe a body’s linear momentum; the remaining
three describe its angular momentum.

Extracting linear and angular momenta from the generalized momentum requires a
choice of origin which is not required for $\mathcal{P}_{\xi}$ itself. Letting
$\gamma_{t}$ be such an origin at time $t$, the associated linear momentum
$p_{a}(t;\gamma_{t})$ and angular momentum $S^{ab}=S^{[ab]}(t;\gamma_{t})$ are
implicitly defined by

$\mathcal{P}_{\xi}(t)=p_{a}(t;\gamma_{t})\xi^{a}(\gamma_{t})+\frac{1}{2}S^{ab}(t;\gamma_{t})\nabla_{a}\xi_{b}(\gamma_{t}).$
(2)

In this Euclidean context, $\nabla_{a}\nabla_{b}\xi_{c}=0$ so
$\nabla_{a}p_{b}=0$. The angular momentum does however depend on the choice of
origin, as is familiar even from elementary discussions of Newtonian
mechanics. In fact, the linear and the angular momenta defined by (1) and (2)
are essentially222The only difference is that it is more conventional to
consider the angular momentum vector
$S^{a}\equiv\tfrac{1}{2}\epsilon^{abc}S_{bc}$ in place of the bivector
$S^{ab}$. Both $S^{a}$ and $S^{ab}$ nevertheless encode the same information
in Newtonian mechanics. equivalent to elementary textbook definitions: Using
Cartesian coordinates $x^{i}$,

$\displaystyle p_{i}(t;\gamma_{t})=\int\rho_{i}(x,t)d^{3}x,$ (3a)
$\displaystyle
S^{ij}(t;\gamma_{t})=2\int(x-\gamma_{t})^{[i}\rho^{j]}(x,t)d^{3}x.$ (3b)

The generalized momentum may be viewed as describing a body’s “bulk” state.
Mass and momentum conservation constrain the evolution of that state, and
therefore the evolution of the generalized momentum: Differentiating (1) may
be shown to yield the “generalized force” Harte (2008a, 2015)

$\mathcal{F}_{\xi}(t)\equiv\frac{d}{dt}\mathcal{P}_{\xi}(t)=-\int\rho(x,t)\mathcal{L}_{\xi}\Phi(x,t)d^{3}x,$
(4)

where $\rho$ denotes the body’s mass density and $\Phi$ the Newtonian
gravitational potential. Like the generalized momentum, the generalized force
is, at fixed $t$, a six-dimensional vector in the space which is dual to the
space of Euclidean Killing fields.

In the same way that the generalized momentum can be decomposed into a linear
momentum and an angular momentum, the generalized force can be decomposed into
an ordinary force $F_{a}(t;\gamma_{t})$ and a torque
$N^{ab}=N^{[ab]}(t;\gamma_{t})$, both of which satisfy

$\displaystyle\mathcal{F}_{\xi}(t)=F_{a}(t;\gamma_{t})\xi^{a}(\gamma_{t})+\frac{1}{2}N^{ab}(t;\gamma_{t})\nabla_{a}\xi_{b}(\gamma_{t}).$
(5)

Comparing this expression to the time derivative of (2) recovers the laws of
motion

$\frac{D}{dt}p_{a}=F_{a},\qquad\frac{D}{dt}S^{ab}=2p^{[a}\dot{\gamma}_{t}^{b]}+N^{ab}.$
(6)

The time derivatives here act on both arguments of $p_{a}(t;\gamma_{t})$ and
$S^{ab}(t;\gamma_{t})$. Also note that the $p^{[a}\dot{\gamma}_{t}^{b]}$ term
which affects the angular momentum is purely kinematic, and is therefore
natural to separate from the “dynamical” torque $N^{ab}$. It vanishes when,
e.g., $\gamma_{t}$ is placed at an object’s center of mass. Although the force
and the torque which appear here are equivalent to elementary expressions, we
find it convenient to work with the more-abstract concepts of generalized
momentum and generalized force. One reason for this is that doing so allows
forces and torques to be considered simultaneously in a single calculation.
Another advantage is that the Lie derivative in (4) provides an immediate
connection with symmetries and conservation laws.

Regardless, the gravitational potential $\Phi$ which appears in the
generalized force is, a priori, the sum of an external field and a self-field.
The gravitational self-field may nevertheless be shown not to contribute to
the generalized force Harte (2008a, 2015). The $\Phi$ which appears there can
thus be reinterpreted as a purely-external potential. Doing so while further
assuming that all lengthscales associated with the external field are large
compared with the size of the body, it becomes useful to Taylor expand
$\mathcal{L}_{\xi}\Phi$ in the generalized force (4). Doing so around
$\gamma_{t}$ results in

$\displaystyle\mathcal{F}_{\xi}(t)=-M\mathcal{L}_{\xi}\Phi(\gamma_{t},t)-D^{a}(t;\gamma_{t})\mathcal{L}_{\xi}\nabla_{a}\Phi(\gamma_{t},t)$
$\displaystyle~{}+\tfrac{1}{2}\tilde{Q}^{ab}(t;\gamma_{t})\mathcal{L}_{\xi}\mathcal{E}_{ab}(\gamma_{t},t)+\ldots,$
(7)

where $M$ is the body’s mass,

$D^{i}(t;\gamma_{t})\equiv\int(x-\gamma_{t})^{i}\rho(x,t)d^{3}x$ (8)

is its mass dipole moment, and

$\tilde{Q}^{ij}(t;\gamma_{t})\equiv\int(x-\gamma_{t})^{i}(x-\gamma_{t})^{j}\rho(x,t)d^{3}x$
(9)

is its “full” (not necessarily trace-free) quadrupole moment. We have also
used

$\mathcal{E}_{ab}(x,t)\equiv-\nabla_{a}\nabla_{b}\Phi(x,t)$ (10)

to denote the Newtonian tidal tensor. This tensor is always symmetric and
trace-free, where the latter property follows from the vacuum field equation
$\nabla^{2}\Phi=0$. The definition (10) also implies that

$\nabla_{[a}\mathcal{E}_{b]c}=0,$ (11)

which may be viewed as a Newtonian analog of the Bianchi identity.

Our focus here is on the quadrupolar contribution to the force and torque,
which is given by the second line of (7). That term can, however, be
simplified by noting that since $\mathcal{E}_{ab}$ is trace-free and
$\mathcal{L}_{\xi}g_{ab}=0$, where $g_{ab}$ denotes the Euclidean metric,
arbitrary multiples of $g^{ab}$ can be added to $\tilde{Q}^{ab}$ without
affecting $\tilde{Q}^{ab}\mathcal{L}_{\xi}\mathcal{E}_{ab}$. The quadrupole
moment in that expression may therefore be replaced by its trace-free
counterpart

$Q^{ab}\equiv(\delta^{a}_{c}\delta^{b}_{d}-\tfrac{1}{3}g^{ab}g_{cd})\tilde{Q}^{cd}.$
(12)

From now on, we refer to $Q^{ab}$ (and not $\tilde{Q}^{ab}$) as “the”
quadrupole moment. Like $\mathcal{E}_{ab}$, this moment is symmetric and
trace-free. In terms of it, the quadrupolar contribution to the generalized
force is

$\mathcal{F}_{\xi}^{(q)}=\frac{1}{2}Q^{ab}\mathcal{L}_{\xi}\mathcal{E}_{ab}.$
(13)

Quadrupolar forces and torques therefore arise only when the tidal field fails
to share the same symmetries as the background Euclidean space. Combining (5)
and (13), the ordinary force and torque are given by

$F_{a}^{(q)}=\frac{1}{2}Q^{bc}\nabla_{a}\mathcal{E}_{bc},\qquad
N_{ab}^{(q)}=2Q^{c}{}_{[a}\mathcal{E}_{b]c}$ (14)

at quadrupolar order.

A body which does not eject or absorb mass has no control over the monopolar
generalized force $-M\mathcal{L}_{\xi}\Phi$. Moreover, the dipolar force
$-D^{a}\mathcal{L}_{\xi}\nabla_{a}\Phi$ can always be set to zero by placing
$\gamma_{t}$ at the center of mass. The first nontrivial contribution to “non-
universal” free-fall therefore arises at quadrupolar order, which is our
focus. In astrophysical contexts, it is often assumed that all quadrupole
moments are induced by the tidal field. Such cases are discussed briefly in
Appendix C, where it is shown that introducing an effective potential which
depends on $\mathcal{E}^{ab}\mathcal{E}_{ab}$ can allow the quadrupolar force
to be absorbed into the monopole. However, our goal here is not to model any
particular system: Unless otherwise noted, we allow for arbitrary quadrupole
moments below.

###  II.2 Constraints from symmetry

Intuitively, extended-body effects arise from inhomogeneities in the
gravitational field. Depending on a body’s internal mass distribution,
different parts of it may interact with slightly different gravitational
fields, resulting in different net effects. Indeed, no extended-body effects
are possible in a uniform gravitational field where
$\nabla_{a}\Phi=\mathrm{constant}$. This suggests that extended-body effects
should be constrained by any symmetries which may exist.

The simplest such constraints arise from symmetries of the potential. It is
immediately clear from (2) and (4) that if there exists a Killing field
$\Xi^{a}(x)$ such that333We use $\xi^{a}$ here to denote a generic Killing
field, but $\Xi^{a}$ to denote a specific Killing field which also generates a
symmetry of $\Phi$.

$\mathcal{L}_{\Xi}\Phi(x,t)=0$ (15)

throughout the body of interest, one component of the generalized momentum
must be conserved:

$\mathcal{P}_{\Xi}=p_{a}\Xi^{a}+\frac{1}{2}S^{ab}\nabla_{a}\Xi_{b}=\mathrm{constant}.$
(16)

This is in fact not restricted to the quadrupole approximation. It is exact.
As a consequence,

$\mathcal{F}_{\Xi}=F_{a}\Xi^{a}+\frac{1}{2}N^{ab}\nabla_{a}\Xi_{b}=0.$ (17)

This too is exact. It implies that when $\Phi$ shares a symmetry with the
background Euclidean space, certain force and torque combinations are
impossible, regardless of a body’s internal structure. Such constraints hold
regardless of whether or not $\gamma_{t}$ lies at the center-of-mass.

One simple example concerns the motion of an object in a spherically-symmetric
gravitational field. In that case, three generalized momentum components are
conserved, one for each of the three rotational symmetries. Similarly, three
generalized force components vanish. In more elementary language, the angular
momentum 3-vector which is associated with motion around the origin is
conserved. As a consequence, non-radial forces—which affect an object’s
orbital angular momentum—can arise only in combination with torques which
produce compensating changes in the spin angular momentum. Certain linear
combinations of force and torque components therefore vanish, and these are
precisely the generalized force components which are associated with the
rotational symmetries.

Returning to our discussion of generic gravitational fields (which are not
necessarily spherically-symmetric), symmetry in the sense of (15) is a fairly
strong requirement. It is therefore interesting to ask if that requirement can
be weakened while still retaining interesting physical consequences. Can
constraints such as (17) continue to hold even when $\mathcal{L}_{\Xi}\Phi\neq
0$? Indeed they can. If there is a 1-parameter family of Killing fields
$\Xi^{a}_{t}(x)$ such that

$\mathcal{L}_{\Xi_{t}}\mathcal{E}_{ab}(\gamma_{t},t)=-\nabla_{a}\nabla_{b}\mathcal{L}_{\Xi_{t}}\Phi(\gamma_{t},t)=0,$
(18)

inspection of (13) shows that at least the quadrupolar contribution to the
generalized force must vanish:

$\mathcal{F}^{(q)}_{\Xi_{t}}=F^{(q)}_{a}\Xi_{t}^{a}+\frac{1}{2}N^{(q)}_{ab}\nabla^{a}\Xi^{b}_{t}=0.$
(19)

The quadrupolar component of the constraint (17) therefore generalizes in
three ways: First, we may consider Killing fields which are symmetries of the
tidal field but not of the potential. Second, we may consider Killing fields
which preserve the tidal field only at $\gamma_{t}$. Third, we may consider
different Killing fields at different times. Although these generalizations
are straightforward, they considerably weaken our notion of symmetry while
still implying that certain force and torque combinations are impossible. We
describe a 1-parameter family of Killing fields $\Xi^{a}_{t}$ which satisfy
(18) as the generators of a local symmetry. Somewhat more precisely, these are
local symmetries of the tidal field. The “ordinary” symmetries which satisfy
(15) are special cases. We refer to local symmetries which do not preserve
$\Phi$ as “proper.”

Unlike ordinary symmetries of the potential, proper local symmetries are not
necessarily associated with conservation laws. A natural candidate for a
potentially-conserved conserved quantity in this context is
$\mathcal{P}_{\Xi_{t}}$. However, the rate of change of this quantity is not
quite given by the generalized force $\mathcal{F}_{\Xi_{t}}$, since now the
Killing fields may depend on time. Instead,

$\frac{d}{dt}\mathcal{P}_{\Xi_{t}}=\mathcal{F}_{\Xi_{t}}+\mathcal{P}_{\dot{\Xi}_{t}}.$
(20)

The first term on the right-hand simplifies due to (19), but does not
necessarily disappear. In some cases, both terms simplify when $\gamma_{t}$ is
placed at an object’s center of mass; one such example is given in Sect.
II.4.2 below.

What is interesting here is not so much that local symmetries imply force and
torque constraints; that much is obvious from (13). What is more important is
that proper local symmetries are ubiquitous. We show below that at least one
(not necessarily proper) local symmetry exists in every Newtonian
gravitational field, and in many cases there are more. Local symmetries
therefore play an important role in constraining extended-body motion. We now
identify these symmetries and their consequences in different types of tidal
fields.

###  II.3 Constraints from algebraic structure

Any nonzero Newtonian tidal tensor can be classified, at each point, in terms
of its eigenvalues. These tensors must be real, symmetric, and trace-free, and
therefore admit three real eigenvalues (counting multiplicity) which sum to
zero. Either,

  1. 1.

$\mathcal{E}_{ab}$ has three distinct and nonzero eigenvalues.

  2. 2.

$\mathcal{E}_{ab}$ has two distinct nonzero eigenvalues and one vanishing
eigenvalue.

  3. 3.

$\mathcal{E}_{ab}$ has one doubly-degenerate nonzero eigenvalue and one non-
degenerate nonzero eigenvalue.

Depending on which of these descriptions hold, we describe $\mathcal{E}_{ab}$
as being of algebraic type 1, 2, or 3. Type 3 tidal tensors can be described
as “algebraically special.” Type 1 and type 2 tidal tensors are instead
“algebraically general.” It is shown in Appendix B that the algebraically-
special Newtonian tidal tensors may be viewed as approximations to Petrov type
D spacetimes in general relativity. Type 1 and 2 tidal fields instead
correspond to Petrov type I spacetimes, which are conventionally described as
algebraically general.

Regardless, the three eigenvalues $\mathcal{E}_{+}$, $\mathcal{E}_{-}$, and
$-\mathcal{E}_{+}-\mathcal{E}_{-}$ of the tidal tensor can all be encoded in
the complex “tidal scalar”

$\displaystyle\mathcal{E}\equiv(\mathcal{E}_{+}+\mathcal{E}_{-})+i(\mathcal{E}_{+}-\mathcal{E}_{-}),$
(21)

which is analogous to the Weyl scalars used in general relativity444A four-
dimensional Weyl tensor is associated, in general, with five complex Weyl
scalars. Without aligning the triad, a Newtonian tidal tensor would be
associated with two complex scalars and one real scalar. In both cases,
however, certain scalars can be made to vanish by appropriately aligning the
basis vectors. In the Newtonian case, doing so leaves only $\mathcal{E}$. In
the relativistic case, simplifications which arise when aligning the tetrad
are discussed in Sect. III.3.1 below.. However, $\mathcal{E}$ depends on the
ordering of the eigenvalues. If $\mathcal{E}_{+}$ and $\mathcal{E}_{-}$ are
swapped, $\mathcal{E}\mapsto\bar{\mathcal{E}}$; if $\mathcal{E}_{+}$ and
$-\mathcal{E}_{+}-\mathcal{E}_{-}$ are swapped,
$\mathcal{E}\mapsto-\frac{1}{2}i[\mathcal{E}+(2-i)\bar{\mathcal{E}}]$; if
$\mathcal{E}_{-}$ and $-\mathcal{E}_{+}-\mathcal{E}_{-}$ are swapped,
$\mathcal{E}\mapsto\frac{1}{2}i[\mathcal{E}+(2+i)\bar{\mathcal{E}}]$. The
eigenvalues of a type 3 tidal tensor may nevertheless be ordered such that
$\mathcal{E}$ real. For type 2 tidal tensors, the eigenvalues may be ordered
such that $\mathcal{E}$ is imaginary. In the generic type 1 case,
$\mathcal{E}$ must have both real and imaginary components.

One order-independent way to determine the algebraic type of the tidal tensor
is to compute the dimensionless ratio

$\displaystyle\frac{4(\det\mathcal{E}_{ab})^{2}}{(\mathcal{E}_{cd}\mathcal{E}^{cd})^{3}}$
$\displaystyle=\frac{[(\mathcal{E}+\bar{\mathcal{E}})(\mathcal{E}^{2}+\bar{\mathcal{E}}^{2})]^{2}}{(\mathcal{E}^{2}+4|\mathcal{E}|^{2}+\bar{\mathcal{E}}^{2})^{3}}.$
(22)

If this vanishes, the tidal tensor is of type 2; if it is equal to $2/27$, the
tidal tensor is of type 3; in all other cases, the tidal tensor is of type 1.

Forces and torques which arise in gravitational fields with each of the three
algebraic types may be understood by diagonalizing $\mathcal{E}_{ab}$. If
$e^{a}_{+}$ and $e^{a}_{-}$ are real orthonormal eigenvectors associated with
the eigenvalues $\mathcal{E}_{+}$ and $\mathcal{E}_{-}$, it will be useful to
define the complex null vector

$m^{a}\equiv\frac{1}{\sqrt{2}}(e^{a}_{+}+ie^{a}_{-}).$ (23)

Also defining $\ell_{a}\equiv
i\epsilon_{abc}m^{b}\bar{m}^{c}=\epsilon_{abc}e_{+}^{b}e_{-}^{c}$, which is an
eigenvector of $\mathcal{E}_{ab}$ with eigenvalue
$-\mathcal{E}_{+}-\mathcal{E}_{-}$, the triad $(\ell^{a},m^{a},\bar{m}^{a})$
forms a convenient basis with inner products

$m^{a}m_{a}=m^{a}\ell_{a}=0,\qquad m^{a}\bar{m}_{a}=\ell^{a}\ell_{a}=1.$ (24)

Using it, the tidal tensor can be written as

$\mathcal{E}_{ab}=\frac{1}{2}(g_{ab}-3\ell_{a}\ell_{b})\operatorname{Re}\mathcal{E}+\operatorname{Re}(m_{a}m_{b})\operatorname{Im}\mathcal{E}.$
(25)

The triad here is adapted to the tidal tensor, not the quadrupole moment, so
the latter can look more complicated when written in an analogous form:
Introducing the three “quadrupole scalars,” $Q_{\ell m}\equiv
Q_{ab}\ell^{a}m^{b}$, $Q_{mm}\equiv Q_{ab}m^{a}m^{b}$, and $Q_{\ell\ell}\equiv
Q_{ab}\ell^{a}\ell^{b}$,

$\displaystyle
Q_{ab}=\frac{1}{2}Q_{\ell\ell}\big{(}3\ell_{a}\ell_{b}-g_{ab}\big{)}+2\operatorname{Re}\big{[}\big{(}Q_{mm}\bar{m}_{(a}$
$\displaystyle~{}+2Q_{\ell m}\ell_{(a}\big{)}\bar{m}_{b)}\big{]}.$ (26)

While $Q_{\ell\ell}$ is real, both $Q_{\ell m}$ and $Q_{mm}$ can be complex.
Together, these scalars encode all five real components of $Q_{ab}$.

Eqs. (25) and (26) can now be substituted into (13) in order to show that the
quadrupolar generalized force is

$\displaystyle\mathcal{F}_{\xi}^{(q)}=(\operatorname{Im}Q_{mm})(\operatorname{Im}\mathcal{E})i\bar{m}^{a}\mathcal{L}_{\xi}m_{a}-\operatorname{Re}[Q_{\ell
m}(3\bar{m}^{a}\operatorname{Re}\mathcal{E}$
$\displaystyle~{}+m^{a}\operatorname{Im}\mathcal{E})]\mathcal{L}_{\xi}\ell_{a}-\frac{1}{4}\operatorname{Re}[(3Q_{\ell\ell}+2i\operatorname{Re}Q_{mm})\mathcal{L}_{\xi}\mathcal{E}].$
(27)

This holds for all tidal tensors and for all quadrupole moments. Using it and
(5) shows that

$\displaystyle
F_{a}^{(q)}=(\operatorname{Im}Q_{mm})(\operatorname{Im}\mathcal{E})i\bar{m}^{b}\nabla_{a}m_{b}-\operatorname{Re}[Q_{\ell
m}(3\bar{m}^{b}\operatorname{Re}\mathcal{E}$
$\displaystyle~{}+m^{b}\operatorname{Im}\mathcal{E})]\nabla_{a}\ell_{b}-\frac{1}{4}\operatorname{Re}[(3Q_{\ell\ell}+2i\operatorname{Re}Q_{mm})\nabla_{a}\mathcal{E}],$
(28)

and

$\displaystyle N^{ab}_{(q)}=2\operatorname{Re}[Q_{\ell
m}\ell^{[a}(3\bar{m}^{b]}\operatorname{Re}\mathcal{E}+m^{b]}\operatorname{Im}\mathcal{E})]$
$\displaystyle~{}+2(\operatorname{Im}Q_{mm})(\operatorname{Im}\mathcal{E})i\bar{m}^{[a}m^{b]}.$
(29)

Although the quadrupole components $Q_{\ell\ell}$ and
$\operatorname{Re}Q_{mm}$ can (at least sometimes) affect the force, these
expressions show that they can never affect the torque. By contrast, both the
force and the torque can depend on $Q_{\ell m}$ and on
$\operatorname{Im}Q_{mm}$.

This shared dependence on $Q_{\ell m}$ and on $\operatorname{Im}Q_{mm}$ can be
used to write the force partially in terms of the torque. From (29), first
note that

$\displaystyle\operatorname{Re}[Q_{\ell
m}(3\bar{m}^{a}\operatorname{Re}\mathcal{E}+m^{a}\operatorname{Im}\mathcal{E})]=-N^{ab}_{(q)}\ell_{b},$
(30a)
$\displaystyle(\operatorname{Im}Q_{mm})\operatorname{Im}\mathcal{E}=-iN^{ab}_{(q)}m_{a}\bar{m}_{b}.$
(30b)

Substituting these expressions into (27) then results in

$\displaystyle
F_{a}^{(q)}=(\ell^{d}\nabla_{a}\ell^{c}+\bar{m}^{b}m^{c}\bar{m}^{d}\nabla_{a}m_{b})N_{cd}^{(q)}+\operatorname{Re}[\mathcal{Q}\nabla_{a}\mathcal{E}],$
(31)

where

$\mathcal{Q}\equiv-\frac{1}{4}(3Q_{\ell\ell}+2i\operatorname{Re}Q_{mm})$ (32)

is a complex quadrupole component which does not affect the torque. The
quadrupolar force is therefore an affine function of the quadrupolar torque.
Furthermore, the space of forces which can be varied independently of the
torque is spanned by the real and the imaginary components of
$\nabla_{a}\mathcal{E}$. These forces are all that can be produced if, e.g.,
the torque vanishes.

One interesting implication of this is that if the torque vanishes, and if
$\gamma_{t}$ is chosen such that $D^{a}=0$, the total force, up to quadrupolar
order, may be viewed as a purely-monopolar force in the effective potential

$\Phi_{\mathrm{eff}}=\Phi-\operatorname{Re}[(\mathcal{Q}/M)\mathcal{E}].$ (33)

Even in this restricted regime, a shape-changing spacecraft can exert
considerable control over its motion simply by modulating $\mathcal{Q}$ at
appropriate points in its orbit Harte and Gaffney (2021). It may also be noted
that this $\Phi_{\mathrm{eff}}$ is “physically equivalent” (but not equal) to
the effective potential (155) when the quadrupole moment is tidally induced.

####  II.3.1 Type 3 tidal tensors

We have now determined the quadrupolar contributions to the generalized force
(27), the torque (29), and the ordinary force (31). These expressions hold for
any extended body in any gravitational field, but can now be specialized to
discuss extended-body motion in each of the three types of tidal field
discussed above. Type 3 tidal tensors are the simplest, so we begin with them.

Every type 3 tidal tensor admits a degenerate eigenvalue, and the orthonormal
eigenvectors $e^{a}_{\pm}$ may be chosen to span the associated eigenspace.
Then $\mathcal{E}_{+}=\mathcal{E}_{-}$ and $\mathcal{E}=2\mathcal{E}_{+}$. The
eigenvector $\ell^{a}$ is associated with the non-degenerate eigenvalue
$-2\mathcal{E}_{+}$, and the tidal tensor (25) reduces to

$\mathcal{E}_{ab}=(g_{ab}-3\ell_{a}\ell_{b})\mathcal{E}_{+}.$ (34)

It follows from (31) that at fixed torque, the quadrupolar force in a type 3
field can be modulated only in the direction parallel to
$\nabla_{a}\mathcal{E}_{+}$.

Assuming that the tidal tensor remains type 3 in a neighborhood of the
relevant point, (27) reduces to

$\displaystyle\mathcal{F}_{\xi}^{(q)}=-\frac{3}{2}Q_{\ell\ell}\mathcal{L}_{\xi}\mathcal{E}_{+}-6\mathcal{E}_{+}\operatorname{Re}(Q_{\ell
m}\bar{m}^{a})\mathcal{L}_{\xi}\ell_{a}.$ (35)

The motion is therefore unaffected by $Q_{mm}$; at least two of the five
(real) quadrupole components are irrelevant in type 3 tidal fields. It can
also be observed that the torque (29) reduces to

$N_{(q)}^{ab}=-12\mathcal{E}_{+}\operatorname{Re}\big{(}Q_{\ell
m}\bar{m}^{[a}\big{)}\ell^{b]},$ (36)

which is controlled only by $Q_{\ell m}$.

As $Q_{\ell m}$ encodes only two real control parameters, it is not possible
for an extended body to use its quadrupole moment in order to control all
three torque components: For any such moment,

$N^{ab}_{(q)}m_{[a}\bar{m}_{b]}=0.$ (37)

In terms of a vector torque $N_{c}$ which satisfies
$N^{ab}=\epsilon^{abc}N_{c}$, this is equivalent to

$N^{a}_{(q)}\ell_{a}=0.$ (38)

Regardless, quadrupole moments cannot affect a body’s torque within the
degenerate eigenplane of the tidal tensor. They do however affect the other
two torque components.

One way to understand this torque constraint, and also the fact that forces
depend in part on torques, is via local symmetries. Type 3 tidal fields admit
at least three local symmetries with the properties discussed in Sect. II.2
above. To find them, first use (5), (31), and (37) to note that

$\displaystyle\mathcal{F}^{(q)}_{\xi}=\frac{1}{2}\big{[}(2\xi^{a}\ell^{c}\nabla_{a}\ell^{b}+\nabla^{b}\xi^{c})N_{bc}^{(q)}-3Q_{\ell\ell}\mathcal{L}_{\xi}\mathcal{E}_{+}\big{]}.$
(39)

Eq. (19) implies that we would like to find a 1-parameter family of Killing
fields $\Xi^{a}_{t}(x)$ such that $\mathcal{F}_{\Xi_{t}}(t)=0$ for all
possible quadrupole moments. Varying $Q_{\ell\ell}$ while noting that that
quadrupole component cannot affect the torque, one necessary condition is
clearly

$\Xi_{t}^{a}(\gamma_{t})\nabla_{a}\mathcal{E}_{+}(\gamma_{t})=0.$ (40)

Moreover, since the torque can be varied throughout the two-dimensional space
which is not excluded by (37), the other necessary condition is that

$\nabla^{a}\Xi^{b}_{t}(\gamma_{t})=2\Xi^{c}_{t}(\gamma_{t})\ell^{[a}\nabla_{c}\ell^{b]}+i\lambda_{t}m^{[a}\bar{m}^{b]},$
(41)

where $\lambda_{t}$ is real but otherwise arbitrary. Each Killing field which
satisfies these constraints preserves the tidal tensor at $\gamma_{t}$, and
therefore generates a local symmetry. Choosing $\Xi^{a}_{t}(\gamma_{t})=0$
while varying $\lambda_{t}$ produces a family of pure rotations about
$\gamma_{t}$; these imply the torque constraint (37). Setting $\lambda_{t}=0$
while varying $\Xi^{a}_{t}$ throughout the space which is consistent with (40)
generates at least two more local symmetries; these imply that there are at
least two force components which cannot arise without accompanying torques.

Our discussion thus far has assumed only that $\mathcal{E}_{ab}$ is given by
(34). However, somewhat more can be said by recalling that a tidal tensor must
arise as two derivatives of a scalar field. That implies the “Bianchi
identity” (11), which may be used to show that for type 3 tidal fields,

$\displaystyle\nabla_{a}\mathcal{E}_{+}$
$\displaystyle=-\frac{3}{2}\mathcal{E}_{+}(\nabla\cdot\ell)\ell_{a},$ (42a)
$\displaystyle\nabla_{a}\ell_{b}$
$\displaystyle=(\nabla\cdot\ell)m_{(a}\bar{m}_{b)}.$ (42b)

The non-degenerate eigenvector is therefore geodesic, shear-free, and twist-
free, which is reminiscent of the Goldberg-Sachs theorem for repeated
principal null directions in a four-dimensional spacetime. Regardless,
applying these expressions to (31) shows that

$F^{(q)}_{a}=\frac{1}{4}(\nabla\cdot\ell)[9Q_{\ell\ell}\mathcal{E}_{+}\ell_{a}+2N_{ab}^{(q)}\ell^{b}].$
(43)

The force component which is independent of the torque must therefore be
parallel to the non-degenerate eigenvector $\ell^{a}$. Although force
components which are orthogonal to $\ell^{a}$ can be controlled as well (at
least when $\nabla\cdot\ell\neq 0$), they can arise only with accompanying
torques.

####  II.3.2 Type 2 tidal tensors

By definition, one of the eigenvalues of a type 2 tidal tensor must vanish.
Identifying $\ell^{a}$ with the unit eigenvector which corresponds to that
eigenvalue, $\mathcal{E}_{+}=-\mathcal{E}_{-}$ and
$\mathcal{E}=2i\mathcal{E}_{+}$. Eq. (25) then reduces to

$\mathcal{E}_{ab}=2\mathcal{E}_{+}\operatorname{Re}(m_{a}m_{b}).$ (44)

Inspection of (27) shows that in a type 2 field, $Q_{\ell\ell}$ cannot affect
the motion; up to four (out of five) real quadrupole components matter in
these cases. It also follows from (29) that the quadrupolar torque can be
controlled arbitrarily in type 2 fields. However, the force can be varied in
only one direction without also varying the torque.

These are consequences of the fact that there are at least two local
symmetries in each type 2 tidal field. To identify them, note from (31) that

$\displaystyle\mathcal{F}_{\xi}^{(q)}=\frac{1}{2}[\xi^{a}(2\ell^{d}\nabla_{a}\ell^{c}+\bar{m}^{b}m^{c}\bar{m}^{d}\nabla_{a}m_{b})+\nabla^{c}\xi^{d}]N_{cd}^{(q)}$
$\displaystyle~{}+\operatorname{Re}Q_{mm}\mathcal{L}_{\xi}\mathcal{E}_{+}.$
(45)

As $\operatorname{Re}Q_{mm}$ cannot affect the torque, ensuring that this
vanishes for all possible quadrupole moments implies that the local symmetries
are given by all Killing fields which satisfy

$\displaystyle\Xi^{a}_{t}(\gamma_{t})\nabla_{a}\mathcal{E}_{+}(\gamma_{t})=0,$
(46a)
$\displaystyle\nabla^{a}\Xi^{b}_{t}(\gamma_{t})=2\Xi^{c}_{t}(\gamma_{t})(\ell^{[a}\nabla_{c}\ell^{b]}+m^{[a}\bar{m}^{b]}m^{d}\nabla_{c}\bar{m}_{d}).$
(46b)

This space is at least two dimensional.

####  II.3.3 Type 1 tidal tensors

Type 1 tidal tensors admit three distinct and nonzero eigenvalues. In these
cases, the quadrupolar torque can be varied arbitrarily. It also follows from
(31) that at fixed torque, forces can be varied through the space spanned by
$\nabla_{a}\mathcal{E}_{+}$ and by $\nabla_{a}\mathcal{E}_{-}$. That space is
at most two dimensional, so quadrupolar effects can never be used to fully
control all force and torque components. At least one force component cannot
be varied without an accompanying torque. This is a consequence of the fact
that type 1 tidal fields admit at least one local symmetry. Indeed, all
Newtonian tidal tensors admit at least one local symmetry. This symmetry can
be found by determining all Killing fields which satisfy

$\displaystyle\Xi^{a}_{t}(\gamma_{t})\nabla_{a}\mathcal{E}_{+}(\gamma_{t})=\Xi^{a}_{t}(\gamma_{t})\nabla_{a}\mathcal{E}_{-}(\gamma_{t})=0,$
(47a)
$\displaystyle\nabla^{a}\Xi^{b}_{t}(\gamma_{t})=2\Xi^{c}_{t}(\gamma_{t})(\ell^{[a}\nabla_{c}\ell^{b]}+m^{[a}\bar{m}^{b]}m^{d}\nabla_{c}\bar{m}_{d}).$
(47b)

The impossibility of completely controlling all forces and torques could have
been anticipated by a counting argument: The five components of the quadrupole
moment are not sufficient to independently control all six generalized force
components. Nevertheless, there are cases in which all force and torque
components can be independently controlled using octupole and higher-order
moments. The situation is different in general relativity, where there are ten
quadrupole components and also ten generalized force components. We shall see
in Sect. III.3 below that complete control of all relativistic forces and
torques is possible in the quadrupole approximation, at least in some
spacetimes. This suggests that there may be a sense in which, at quadrupolar
order, some “essentially Newtonian” force or torque components can be
controlled only relativistically, via a body’s current (rather than mass)
quadrupole.

####  II.3.4 Summarizing the Newtonian constraints

We may now summarize our Newtonian results by describing how much control
would be available to a spacecraft which has been engineered to arbitrarily
control its quadrupole moment. First, we have found that in a type $n$ tidal
field, there are at least $n$ local symmetries (with $n=1,2,3$). In
algebraically general tidal fields, which are of types 1 or 2, appropriate
spacecraft have complete control over the quadrupolar torques which are
exerted upon them. In the algebraically-special type 3 case, torque vectors
can instead be controlled only within the 2-plane which is orthogonal to the
non-degenerate eigenvector of $\mathcal{E}_{ab}$. How such a spacecraft can
control the quadrupolar forces which act upon it is more complicated. However,
one general statement is that without changing the torque, suitable spacecraft
can arbitrarily control forces only throughout the space which is spanned by
the gradients of the eigenvalues $\mathcal{E}_{\pm}$. For tidal tensors of
types 2 and 3, this space is at most one-dimensional. For tidal tensors of
type 1, it is at most two-dimensional. These and related results are collected
in Table 1.

Alg. type | $\big{\\{}\Xi^{a}_{t}\big{\\}}$ | $\big{\\{}Q_{ab}\big{\\}}$ | $\big{\\{}N_{ab}^{(q)}\big{\\}}$ | $\big{\\{}F_{a}^{(q)}\big{|}N_{bc}^{(q)}\big{\\}}$
---|---|---|---|---
1 | 1–3 | 3–$5^{*}$ | 3* | 0–2
2 | 2–3 | 3–4 | 3* | 0–1
3 | 3–4 | 2–3 | 2 | 0–1
Table 1: Forces, torques, and local symmetries in Newtonian tidal fields with
different algebraic types. The second column displays the number of local
symmetries $\Xi^{a}_{t}$. The third column specifies the number of real
quadrupole components which can affect the force or torque. The fourth column
specifies the dimension of the space of possible quadrupolar torques. The
rightmost column displays the dimension of the space of quadrupolar forces
which can be varied at fixed torque. Starred numbers are used to indicate that
there is no constraint. All ranges which appear here depend on the dimension
of the space which is spanned $\nabla_{a}\mathcal{E}_{+}$ and
$\nabla_{a}\mathcal{E}_{-}$.

###  II.4 Examples of Newtonian tidal fields

In order to illustrate our results, we now discuss some simple examples of
Newtonian tidal fields, including their local symmetries.

####  II.4.1 Type 3 examples

The prototypical example of a type 3 tidal tensor is generated by the
spherically-symmetric potential $\Phi=-k/r$, where $k$ is a constant and $r$
is a radial coordinate. Then,

$\mathcal{E}_{ab}=\frac{k}{r^{3}}(3\hat{r}_{a}\hat{r}_{b}-g_{ab}),$ (48)

where $\hat{r}_{a}\equiv\nabla_{a}r$ denotes the radial unit vector and
$g_{ab}$ is again the Euclidean metric. The vector $\hat{r}_{a}$ is a non-
degenerate eigenvector of $\mathcal{E}_{ab}$, so we may identify it with
$\ell_{a}$. Doing so, the tidal scalar (21) reduces to
$\mathcal{E}=-2k/r^{3}$. At fixed torque, only the radial force can thus be
controlled using extended-body effects. Additionally, the torque constraint
(38) implies that there can be no quadrupolar torque along the radial
direction. These constraints are intuitively clear given the conservation of
angular momentum. What is perhaps less clear is that even with these
conservation laws, an object which controls its quadrupole moment can still
exert considerable control over its orbit Harte and Gaffney (2021). In this
case, the three local symmetries determined by (40) and (41) are in fact the
ordinary rotational symmetries of $\Phi$. They are not “proper” local
symmetries.

Another type 3 example is provided by $\Phi=k(r^{2}-3z^{2})$, where $z$ is a
Cartesian coordinate and $k$ is again a constant. This describes a constant
tidal field, and is essentially the $r\to\infty$ limit of the spherically-
symmetric example above. Although quadrupolar forces vanish in this case,
torque vectors can be arbitrarily controlled in all directions which are
orthogonal to the $z$ axis. Also, since $\nabla_{a}\mathcal{E}=0$, there are
four local symmetries rather than three—three translations and a rotation
around the $z$ axis. Only the rotation is however a symmetry of $\Phi$. Each
translation $\partial_{i}$ is a proper local symmetry, and the corresponding
generalized momentum varies according to

$\frac{d}{dt}\mathcal{P}_{\partial_{i}}=\frac{dp_{i}}{dt}=-M\partial_{i}\Phi-D^{j}\partial_{i}\partial_{j}\Phi,$
(49)

at least through quadrupolar order. The linear momentum $p_{a}$ is therefore
unaffected by a body’s quadrupole moment; its behavior is (at least
instantaneously) “universal.”

####  II.4.2 Type 2 examples

The simplest example of a non-constant type 2 tidal tensor is generated by the
cylindrically-symmetric potential $\Phi=k\ln r$, where $k$ is another constant
and $r$ is now the distance away from the axis of symmetry. Let $\ell^{a}$ be
a unit eigenvector of $\mathcal{E}_{ab}$ with eigenvalue 0, which must be
parallel to the symmetry axis. Also defining $\hat{r}_{a}\equiv\nabla_{a}r$,

$\mathcal{E}_{ab}=\frac{k}{r^{2}}(2\hat{r}_{a}\hat{r}_{b}+\ell_{a}\ell_{b}-\delta_{ab})$
(50)

and $\mathcal{E}=4ik/r^{2}$. The two local symmetries which can be found by
solving (46) are in fact the ordinary translational and azimuthal symmetries
of $\Phi$. The translational symmetry along $\ell^{a}$ clearly precludes any
force in that direction. The azimuthal symmetry instead requires that any
azimuthal force be accompanied by a torque along the symmetry axis; one cannot
exist without the other. Moreover, because $\nabla_{a}\mathcal{E}$ is radial,
only the radial force can be controlled independently of the torque.

More interesting type 2 examples can be found by superposing the potentials
from multiple long, parallel cylinders. Although doing so breaks the azimuthal
symmetry of $\Phi$, a proper local symmetry takes its place. The geometric
significance of this symmetry is, however, difficult to visualize.

We therefore consider a simpler example instead: Letting $y$ and $z$ denote
Cartesian coordinates, and letting $k$ and $a$ be constants, suppose that

$\Phi=ke^{z/a}\sin(y/a).$ (51)

The corresponding tidal tensor is then

$\displaystyle\mathcal{E}_{ab}=\frac{k}{a^{2}}e^{z/a}[(\nabla_{a}y\nabla_{b}y-\nabla_{a}z\nabla_{b}z)\sin(y/a)$
$\displaystyle~{}-2\nabla_{(a}y\nabla_{b)}z\cos(y/a)],$ (52)

which clearly admits $\ell^{a}=\epsilon^{abc}\nabla_{b}y\nabla_{c}z$ as an
eigenvector with vanishing eigenvalue. That eigenvector generates an ordinary
translational symmetry of $\Phi$, which implies that it is not possible to
produce a quadrupolar force orthogonal to the $yz$ plane. The other two
eigenvectors of $\mathcal{E}_{ab}$ may be arranged such that
$\mathcal{E}=2ike^{z/a}/a^{2}$, which implies that the force can be controlled
independently of the torque only in the $z$ direction. Quadrupolar forces in
the $y$ direction can be controlled as well, but only at the cost of
accompanying torques.

This last statement is a consequence of the proper local symmetry

$\displaystyle\Xi^{i}_{t}(x)=\epsilon^{ijk}(x-\gamma_{t})_{j}\ell_{k}+2a\nabla^{i}y,$
(53)

which may be found by applying (46) to (52). It may be verified that although
$\mathcal{L}_{\Xi_{t}}\mathcal{E}_{ab}(x)$ vanishes when $x=\gamma_{t}$, it
does not vanish more generally. Geometrically, $\Xi^{a}_{t}$ corresponds to a
rotation in the $yz$ plane, with origin $\gamma_{t}$, together with a
translation in the $y$ direction. Alternatively, it may be interpreted as a
pure rotation around a point which is displaced from $\gamma_{t}$ by a
distance $2a$ in the $z$ direction. Regardless, the combination of
translational and rotational components here is what links forces to torques.

While proper local symmetries are not necessarily associated with conservation
laws, we can again see how close they can get. Using the local symmetry (53)
together with (7), (19), and (20),

$\frac{d}{dt}\mathcal{P}_{\Xi_{t}}(t)=-M\mathcal{L}_{\Xi_{t}}\Phi-D^{a}\mathcal{L}_{\Xi_{t}}\nabla_{a}\Phi-\epsilon_{abc}p^{a}\dot{\gamma}_{t}^{b}\ell^{c}$
(54)

through quadrupolar order. If $\gamma_{t}$ is chosen to lie at the body’s
center of mass, the last two terms here vanish, leaving only the monopolar
contribution $-M\mathcal{L}_{\Xi_{t}}\Phi$. While $\mathcal{P}_{\Xi_{t}}$ can
change, it does so only in the same way as for a monopolar particle. In this
sense, its behavior is universal.

##  III Extended-body effects in general relativity

We now move on to discussing extended-body constraints in general relativity.
Following essentially the same steps as in the Newtonian theory, we begin in
Sect. III.1 by reviewing the generalized momentum and the generalized force in
a relativistic context. Sect. III.2 applies these concepts to determine how
symmetries constrain extended-body motion. It focuses on local symmetries in
general relativity and shows that some of these are generated by conformal
Killing-Yano tensors. Sect. III.3 then analyzes quadrupolar forces and torques
in vacuum spacetimes, deriving local symmetries and their physical
consequences for each of the possible Petrov types. Lastly, Sect. III.4 uses
pp-wave and Kasner spacetimes as examples with which to illustrate our
results.

###  III.1 Generalized momentum and generalized force

As in Newtonian theory, the bulk state of an extended body in general
relativity can be described in terms of a generalized momentum
$\mathcal{P}_{\xi}(s)$ Harte (2008b, a, 2012, 2015). At least for a test body
with stress-energy tensor $T^{ab}$, it is useful to define this as

$\mathcal{P}_{\xi}(s)\equiv\int_{\Sigma_{s}}T^{a}{}_{b}(x)\xi^{b}(x)dS_{a}$
(55)

at “time” $s$, where the hypersurfaces $\Sigma_{s}$ are chosen to foliate the
body’s worldtube. Since there may not be any Killing fields here, the vector
fields $\xi^{a}$ must be chosen more broadly than in the Newtonian setting:
They are “generalized Killing fields,” or “GKFs.” A complete definition for
the GKFs may be found in Harte (2008b, 2015), but for our purposes, it
suffices to note that they require for their specification the aforementioned
hypersurfaces $\Sigma_{s}$, as well as a reference worldline which we
parameterize by $\gamma_{s}$. Both of these structures can, e.g., be fixed
using center-of-mass conditions Ehlers and Rudolph (1977); Dixon (1979);
Schattner (1979a, b). In that case, $\gamma_{s}$ would be regarded as a point
on the body’s center-of-mass worldline. Regardless, any GKF is uniquely
determined by its value and that of its first derivative anywhere on the
reference worldline—both of which can be chosen arbitrarily so long as
$\nabla_{(a}\xi_{b)}(\gamma_{s})=0$. The space of possible choices for
$\xi^{a}(\gamma_{s})$ and $\nabla_{[a}\xi_{b]}(\gamma_{s})$ is ten
dimensional, so the space of GKFs is also ten dimensional. Noting that
$\mathcal{P}_{\xi}$ is linear in the GKFs, the generalized momentum may be
viewed as an $s$-dependent vector in the ten-dimensional space which is dual
to the space of generalized Killing fields.

These ten dimensions encode the four components of a body’s linear momentum
$p_{a}$ and the six components of its angular momentum $S^{ab}=S^{[ab]}$, both
of which are tensors on the reference worldline. As in the Newtonian setting,
the linear and the angular momenta can be defined implicitly by using (2) to
relate them to $\mathcal{P}_{\xi}$. For extended test bodies, doing so results
in the same momenta as those found by Dixon Dixon (1970, 1974, 1979, 2015).
When self-fields are significant, those fields finitely renormalize the
generalized momentum (55), and therefore $p_{a}$ and $S^{ab}$ as well Harte
(2012, 2015).

Whether or not self-interaction is significant, we may again introduce the
generalized force $\mathcal{F}_{\xi}\equiv d\mathcal{P}_{\xi}/ds$ in order to
describe changes in a body’s generalized momentum. Both an ordinary force
$F_{a}$ and a torque $N^{ab}=N^{[ab]}$ can be extracted from
$\mathcal{F}_{\xi}$ using (4). However, the force and the torque are not
simply the rates of change of $p_{a}$ and $S^{ab}$. To see this, first note
that all GKFs satisfy Killing’s equation at least through first order on the
reference worldline Harte (2008b, 2015):

$\mathcal{L}_{\xi}g_{ab}(\gamma_{s})=\nabla_{c}\mathcal{L}_{\xi}g_{ab}(\gamma_{s})=0.$
(56)

Applying this and (4) while differentiating (2) results in the Mathisson-
Papapetrou-Dixon equations

$\displaystyle\frac{D}{ds}p_{a}$
$\displaystyle=-\frac{1}{2}R_{abcd}\dot{\gamma}_{s}^{b}S^{cd}+F_{a},$ (57a)
$\displaystyle\frac{D}{ds}S^{ab}$
$\displaystyle=2p^{[a}\dot{\gamma}_{s}^{b]}+N^{ab}.$ (57b)

The force and the torque, or equivalently $\mathcal{F}_{\xi}$, encode only
dynamical contributions to the evolution. The terms
$-\frac{1}{2}R_{abcd}\dot{\gamma}_{s}^{b}S^{cd}$ and
$2p^{[a}\dot{\gamma}_{s}^{b]}$ which appear in (57) are instead kinematical.
They are related to the fact that the approximate Poincaré symmetry which is
encoded in (56) mixes translations, rotations, and boosts from one moment in
time to the next. This mixing of approximate symmetries physically manifests
as a mixing of linear and angular momenta over time. However, our focus here
is only on understanding dynamical contributions which can differ from one
extended body to another, all of which are encoded in $F_{a}$ and in $N^{ab}$.

Assuming that $\nabla_{a}T^{ab}=0$, differentiation of (55) shows that at
least for test bodies,

$\mathcal{F}_{\xi}=\frac{1}{2}\int_{\Sigma_{s}}T^{ab}\mathcal{L}_{\xi}g_{ab}w^{c}dS_{c},$
(58)

where $w^{c}$ is a time evolution vector field for the foliation $\Sigma_{s}$.
The generalized force component which is associated with a particular GKF
$\xi^{a}$ therefore measures the degree by which that GKF fails to be a
genuine Killing field. However, this “measurement” is weighted by $T^{ab}$,
which can vary from one extended body to another. Different weightings allows
different objects to experience different forces and different torques in the
same spacetime.

It is inconvenient to analyze these differences using an integral expression
for $\mathcal{F}_{\xi}$. We instead assume that all bodies we consider are
sufficiently small that multipolar expansions can be employed. Since
$\mathcal{L}_{\xi}g_{ab}$ vanishes through first order around $\gamma_{s}$,
the first nontrivial contribution in such an expansion arises at second—i.e.,
quadrupolar—order. A calculation shows that in fact Dixon (1970, 1974); Harte
(2010, 2015)

$\mathcal{F}_{\xi}^{(q)}=-\frac{1}{6}\tilde{J}^{abcd}\mathcal{L}_{\xi}R_{abcd},$
(59)

where $\tilde{J}^{abcd}$ denotes the body’s full (not necessarily trace-free)
quadrupole moment. This moment has all of the same algebraic properties as a
Riemann tensor. For a test body, it is the quadrupole moment derived by Dixon;
see Eq. (9.12) of Dixon (1974) or Eq. (2.8) of Harte (2020). If self-
interaction is significant, the relevant $\tilde{J}^{abcd}$ is finitely
renormalized with respect to Dixon’s definition Harte (2012, 2015). Also, the
$R_{abcd}$ which appears in the generalized force must then be understood as
the Riemann tensor which is associated with a certain effective metric.
Although the details are not important here, we assume that the metric which
appears in all of our equations is this effective one. It reduces to the
physical metric when considering test bodies, but more generally includes both
“external” and “self-field” contributions. In a Newtonian limit, it
incorporates only the external gravitational field.

Regardless, all of our discussion is confined to spacetimes which satisfy the
vacuum Einstein equation, perhaps with a cosmological constant555The
cosmological constant has no direct influence on extended-body effects, so
there is no downside to including it. $\Lambda$. The Ricci tensor is therefore

$R_{ab}=\Lambda g_{ab},$ (60)

and the Weyl tensor is related to the Riemann tensor by

$\displaystyle C_{abcd}=R_{abcd}+\frac{1}{6}\Lambda g_{a[d}g_{c]b}.$ (61)

Substituting this into (59) while using (56) shows that
$\mathcal{L}_{\xi}R_{abcd}(\gamma_{t})=\mathcal{L}_{\xi}C_{abcd}(\gamma_{t})$
for every GKF $\xi^{a}$. Similarly, $\tilde{J}^{abcd}$ may be replaced in (59)
by its trace-free counterpart $J^{abcd}\equiv(\tilde{J}^{abcd})_{\mathrm{TF}}$
without affecting $\mathcal{F}_{\xi}^{(q)}$ Harte (2010); Bini and Geralico
(2014); Harte (2020). The quadrupolar generalized force in any vacuum
spacetime may therefore be written as

$\mathcal{F}^{(q)}_{\xi}=-\frac{1}{6}J^{abcd}\mathcal{L}_{\xi}C_{abcd}.$ (62)

Using (4), the corresponding force and torque are

$\displaystyle F^{(q)}_{a}$
$\displaystyle=-\frac{1}{6}J^{bcde}\nabla_{a}C_{bcde},$ (63a) $\displaystyle
N_{ab}^{(q)}$ $\displaystyle=-\frac{4}{3}J_{[a}{}^{cde}C_{b]cde}.$ (63b)

Much of our discussion below is focused on these expressions.

In astrophysical contexts, it is often assumed that $J^{abcd}$ vanishes for
isolated and non-spinning bodies, but that nonzero quadrupole moments can be
induced either by external tidal fields or by rotation. The former case is
briefly discussed in Appendix C, where the simplest models are shown to result
in very simple forces and torques. Although the mass

$M\equiv\sqrt{-p_{a}p^{a}}$ (64)

can vary in these (and other) models, a certain effective mass is conserved at
least for some bodies with tidally-induced quadrupole moments; see (160). That
is in turn analogous to the existence of the Newtonian effective potential
(155). Nevertheless, our main goal here is to understand model-independent
features of the extended-body problem. We therefore make no assumptions below
regarding the specific form of the quadrupole moment.

One model-independent feature which can already be deduced is that in a vacuum
spacetime, many objects with differing internal structures can experience
identical forces and torques. To see this, first note that $J^{abcd}$ has all
of the same algebraic properties as a Weyl tensor, and therefore has ten
independent components. This contrasts with the twenty independent components
of $\tilde{J}^{abcd}$. Einstein’s equation thus implies that at least ten
components of the full quadrupole moment cannot affect an object’s motion666As
noted in Sect. II.1 above, the Newtonian analog of this statement is that the
vacuum field equation implies that one of the six components of
$\tilde{Q}^{ab}$ cannot affect the motion.. Depending on the Petrov type of
the spacetime, we shall see below that even more quadrupole components can
fail to affect the motion. As our focus is only on vacuum spacetimes, we now
refer only to $J^{abcd}$ (and not to $\tilde{J}^{abcd}$) as “the” quadrupole
moment in relativistic contexts.

###  III.2 Constraints from symmetry

If there exists a Killing field $\Xi^{a}$, it is clear from (58) that
regardless of an object’s internal structure, $\mathcal{P}_{\Xi}$ is conserved
and $\mathcal{F}_{\Xi}=0$. Killing fields therefore place universal
constraints on extended-body effects. In fact, these constraints hold not only
for the full generalized force, but also for each term in its multipole
expansion. This much has been known since at least the 1970s Dixon (1970,
1974); Ehlers and Rudolph (1977); Dixon (1979). What is new here is the
concept of a local symmetry.

####  III.2.1 Local symmetries

A Killing field in a curved spacetime is analogous, in Newtonian gravity, to a
vector field which preserves not only the Euclidean metric, but also the
potential $\Phi$. However, we found in Sect. II.2 above that it was useful to
generalize this by considering symmetries of the tidal tensor
$\mathcal{E}_{ab}$ which are not necessarily symmetries of $\Phi$. We also
found it useful to allow the tidal tensor to perhaps be preserved at only a
single point.

Applying these ideas in a relativistic context, we now define the generator of
a local symmetry to be a 1-parameter family of generalized Killing fields
$\Xi^{a}_{s}$ which locally preserve the curvature:

$\mathcal{L}_{\Xi_{s}}R_{abcd}(\gamma_{s})=0.$ (65)

If such a family exists, it follows from (59) that

$\mathcal{F}_{\Xi_{s}}^{(q)}=F_{a}^{(q)}\Xi^{a}_{s}+\frac{1}{2}N_{ab}^{(q)}\nabla^{a}\Xi^{b}_{s}=0$
(66)

at time $s$. Conversely, a family of GKFs generates a local symmetry whenever
$\mathcal{F}_{\Xi_{s}}^{(q)}=0$ for all possible quadrupole moments.

As noted in Sect. III.1 above, GKFs always satisfy Killing’s equation through
first order along the reference worldline. Eq. (65) asks if it is possible to
extend this through one higher order, in which case (56) is supplemented by

$\nabla_{a}\nabla_{b}\mathcal{L}_{\Xi_{s}}g_{cd}(\gamma_{s})=0.$ (67)

This is not possible in general. However, cases where it is possible are not
uncommon and are physically interesting.

It is clear that every Killing field which may exist generates a local
symmetry, and each of these is associated with a conservation law. A “proper”
local symmetry, which is a local symmetry which is not generated by a Killing
field, might fail to be associated with any conservation law.

####  III.2.2 Conformal Killing-Yano tensors as local symmetries

We now show that some proper local symmetries are generated by conformal
Killing-Yano (CKY) tensors. By definition, a CKY tensor $f_{ab}=f_{[ab]}(x)$
must satisfy

$\nabla_{(a}f_{b)c}=g_{ab}f_{c}-f_{(a}g_{b)c},$ (68)

where $f_{a}\equiv\frac{1}{3}\nabla^{b}f_{ba}$. A CKY tensor for which
$f_{a}=0$ is called a Killing-Yano tensor. Regardless, the vacuum Einstein
equation implies that $\nabla_{(a}f_{b)}=0$ Jezierski and Łukasik (2006).
Using this as well as (68) and the Ricci identity, every CKY tensor may be
shown to satisfy

$\displaystyle\nabla_{c}\nabla_{d}f_{ab}=-\nabla_{a}\nabla_{b}f_{cd}+2R_{ca[d}{}^{e}f_{b]e}+\nabla_{a}(2g_{bc}f_{d}$
$\displaystyle~{}-g_{cd}f_{b})+\nabla_{c}(2g_{ad}f_{b}-g_{ab}f_{d}).$ (69)

Antisymmetrizing over the index pairs $ab$ and $cd$, it then follows that

$R_{ab[c}{}^{e}f_{d]e}=-R_{cd[a}{}^{e}f_{b]e}.$ (70)

This can be viewed as an integrability condition for the existence of a CKY
tensor. Introducing the 1-parameter family of GKFs which are determined by

$\Xi^{a}_{s}(\gamma_{s})=0,\qquad\nabla^{a}\Xi^{b}_{s}(\gamma_{s})=f^{ab}(\gamma_{s}),$
(71)

it implies that $\mathcal{L}_{\Xi_{s}}R_{abcd}(\gamma_{s})=0$. Every CKY
tensor therefore generates a local symmetry. Since
$\Xi^{a}_{s}(\gamma_{s})=0$, these symmetries have no translational
components. They may be viewed as generating a family of curvature-preserving
Lorentz transformations on the reference worldline.

It follows from (66) and (71) that the physical consequence of such a symmetry
is that one component of the quadrupolar torque must vanish. In fact, since
the Hodge dual $f^{*}_{ab}\equiv\frac{1}{2}\epsilon_{ab}{}^{cd}f_{cd}$ of any
CKY tensor is also a CKY tensor Jezierski and Łukasik (2006), two real torque
components must vanish

$N^{ab}_{(q)}f_{ab}=N^{ab}_{(q)}f_{ab}^{*}=0.$ (72)

These constraints have previously been derived in Petrov type D spacetimes
Harte (2020), although it was not clear then whether or not the connection
with CKY tensors was anything more than coincidence. Our derivation here shows
that it was not a coincidence, that these constraints are not restricted only
to type D spacetimes, and also that they can be related to curvature-
preserving vector fields.

CKY tensors have seen a number of other applications in the literature,
perhaps most prominently in the Kerr spacetime. There, the square of a
Killing-Yano tensor can be used to construct a quadratic conserved quantity
for geodesics: the Carter constant. Nontrivial Killing-Yano tensors
nevertheless exist in spacetimes which are not Kerr, and Carter-like constants
may be constructed in those cases as well. Killing-Yano tensors can also be
used in order to construct conserved quantities—or at least conserved
currents—for various field equations Carter (1977); Carter and McLenaghan
(1979); Kastor and Traschen (2004); Andersson  _et al._ (2015); Grant and
Flanagan (2020a, b); Jezierski (2002); Jezierski and Łukasik (2006). However,
there are reasons to expect that exact generalizations of the Carter constant
do not exist for generic compact objects Grant and Flanagan (2015).

In the context of this paper, it is interesting to ask if the torque
constraint (72) can nevertheless be used to construct a quantity which is at
least approximately conserved. In the pole-dipole approximation where
quadrupole and higher-order moments are neglected, the existence of conserved
quantities which are either linear or quadratic in the momenta have been
investigated Rüdiger (1981, 1983); Compère and Druart (2022) at least in
combination with the Tulczyjew spin supplementary (or center-of-mass)
condition

$p_{a}S^{ab}=0.$ (73)

In that context, an approximate conservation law was found which directly
generalized the Carter constant. Additionally, the quantity

$\mathcal{P}_{\Xi_{s}}=\frac{1}{2}S^{ab}f_{ab}$ (74)

was found to be conserved when $f_{ab}^{*}$ is Killing-Yano and when it
satisfies certain other conditions as well. These extra conditions are not
satisfied in Kerr Rüdiger (1981); Santos and Batista (2020), although their
failure there merely reduces $\mathcal{P}_{\Xi_{s}}$ to a quantity which is
approximately conserved within the pole-dipole approximation: In powers of the
spin, $d\mathcal{P}_{\Xi_{s}}/ds=\mathcal{O}(S^{2})$. If quadrupole moments
are included, but are assumed to be spin-induced and with the deformability
which is expected for black holes, this quantity is instead conserved up to
terms of order $S^{3}$ Compère  _et al._ (2023). However,
$\mathcal{P}_{\Xi_{s}}$ is not so well preserved for objects with any other
deformabilities.

Our comment on this is that the situation may differ with different spin
supplementary conditions. Relaxing the Tulczyjew condition (73) while also
allowing for an arbitrary angular momentum and an arbitrary (not necessarily
spin-induced) quadrupole moment, the torque constraint (72) implies that for
an arbitrary CKY tensor $f_{ab}$,

$\displaystyle\frac{d}{ds}\mathcal{P}_{\Xi_{s}}$
$\displaystyle=\mathcal{P}_{\dot{\Xi}_{s}}=\dot{\gamma}_{s}^{b}(p^{a}f_{ab}+\tfrac{1}{2}S^{ac}\nabla_{b}f_{ac}),$
(75)

at least if the octupole and higher-order moments are neglected. It follows
that $\mathcal{P}_{\Xi_{s}}$ is conserved whenever $\dot{\gamma}_{s}^{b}$ is
orthogonal to $p^{a}f_{ab}+\tfrac{1}{2}S^{ac}\nabla_{b}f_{ac}$. It may be
possible to construct spin supplementary conditions in which this is
guaranteed to occur. If so, we would have a new conserved quantity for objects
with arbitrary quadrupole moments.

Another point to note is that although $\mathcal{P}_{\Xi_{s}}$ is the most
obvious guess for a conserved quantity associated with the local symmetry
$\Xi_{s}^{a}$, it is not necessarily optimal. In the Schwarzschild spacetime,
one of the torque constraints which is associated with a conformal Killing-
Yano tensor can in fact be derived from ordinary Killing symmetries Harte
(2020). That implies that there is an associated conservation law. However,
the quantity which is conserved in that case does not coincide with
$\mathcal{P}_{\Xi_{s}}$, except at one moment in time. We find a similar
result when discussing motion in plane wave spacetimes in Sect. III.4.1 below.

###  III.3 Constraints from algebraic structure

The majority of local symmetries we consider are not related either to Killing
vector fields or to conformal Killing-Yano tensors. We now use the algebraic
structure of the Weyl tensor to identify these symmetries and to determine
their physical consequences. We consider general vacuum spacetimes and
determine how extended-body motion depends on the Petrov type of the spacetime
in which it moves. No spin supplementary condition is assumed, and there might
not be any Killing vectors or CKY tensors.

####  III.3.1 A convenient basis

Simple expressions for forces and torques require decompositions which are
adapted to the spacetime geometry, and not, e.g., to an object’s rest frame.
Mathematically, this corresponds to expressing forces and torques in terms of
a basis which is adapted to the principal null directions (PNDs) of the
spacetime’s Weyl tensor. As reviewed in Appendix B, Weyl tensors can be
classified according to their Petrov type, which is determined by the numbers
of PNDs with different multiplicities; see the second column in Table 2.
Although it is possible for the Petrov type to vary from point to point, we
assume below that it does not.

In order to introduce an appropriate basis, first let
$(\ell^{a},n^{a},m^{a},\bar{m}^{a})$ be a null tetrad where $\ell^{a}$ and
$n^{a}$ are real, $\bar{m}^{a}$ is the complex conjugate of $m^{a}$, and the
only non-vanishing scalar products are

$m^{a}\bar{m}_{a}=-\ell^{a}n_{a}=1.$ (76)

Many such tetrads exist. However, the space of possibilities can be reduced by
aligning $\ell^{a}$ and $n^{a}$ with PNDs of the Weyl tensor. More precisely,
we choose $\ell^{a}$ and $n^{a}$ to be parallel to “the” PNDs with the largest
and the second-largest multiplicities, respectively777Even if rescalings of
$\ell^{a}$ and $n^{a}$ are ignored, this prescription is not necessarily
unique. In the type N case, $\ell^{a}$ is aligned with the sole PND while
$n^{a}$ is essentially free. In type I spacetimes where there are four PNDs
with equal multiplicity, twelve alignments are possible. Two possible
alignments are possible in type D and in type II spacetimes.; see the third
column of Table 2.

Petrov type | PNDs | Multiplicities | Vanishing $\Psi_{I}$
---|---|---|---
I | 1, 1, 1, 1 | 1, 1 | $\Psi_{0},\Psi_{4}$
II | 2, 1, 1 | 2, 1 | $\Psi_{0},\Psi_{1},\Psi_{4}$
D | 2, 2 | 2, 2 | $\Psi_{0},\Psi_{1},\Psi_{3},\Psi_{4}$
III | 3, 1 | 3, 1 | $\Psi_{0},\Psi_{1},\Psi_{2},\Psi_{4}$
N | 4 | 4, - | $\Psi_{0},\Psi_{1},\Psi_{2},\Psi_{3}$
Table 2: The Petrov classification and the Weyl scalars which vanish with an
appropriately-aligned tetrad. The second column summarizes the multiplicities
of all principal null directions which are associated with a given Weyl
tensor. The third column gives the respective multiplicities of the PNDs which
are tangent to $\ell^{a}$ and $n^{a}$ when these vectors are chosen as
described in the text. The fourth column lists all Weyl scalars which
necessarily vanish with this alignment.

Given an arbitrary null tetrad, the Weyl tensor can be described in terms of
the five Weyl scalars $\Psi_{0},\ldots,\Psi_{4}$ which are defined by (144) in
Appendix B. However, some of these scalars vanish when $\ell^{a}$ and $n^{a}$
are aligned as described in the previous paragraph. In type D spacetimes, only
$\Psi_{2}$ can be nonzero; in type III spacetimes, it is only $\Psi_{3}$ which
fails to vanish; in type N spacetimes, it is only $\Psi_{4}$; in type II
spacetimes, both $\Psi_{2}$ and $\Psi_{3}$ are nonzero; in type I spacetimes,
$\Psi_{1}$, $\Psi_{2}$, and $\Psi_{3}$ can all fail to vanish. These
simplifications justify our choices for $\ell^{a}$ and $n^{a}$, and are
summarized in the final column of Table 2.

Further simplifications can sometimes be performed by employing the type III
tetrad transformations which are described in Appendix B. These rescale
$\ell^{a}$ and $n^{a}$ while rotating $m^{a}$ and $\bar{m}^{a}$. Referring to
(143) and (144), a type III transformation which is generated by the complex
scalar $c$ transforms the Weyl scalars via

$\Psi_{I}\mapsto c^{2-I}\Psi_{I},\qquad I=0,\ldots,4.$ (77)

In the Petrov type D case, where aligning $\ell^{a}$ and $n^{a}$ with the two
PNDs leaves only $\Psi_{2}$ nonzero, type III tetrad transformations have no
effect. In the Petrov type III case, a type III tetrad transformation can be
used to set $\Psi_{3}$ equal to any nonzero constant. Similarly, $\Psi_{4}$
can be made constant in any Petrov type N spacetime. In the Petrov type I
case, a type III tetrad transformation can be used to ensure that
$\Psi_{1}=\Psi_{3}$. Although the simplifications afforded by type III
transformations are occasionally useful, we employ them below only in type I
spacetimes.

However a particular tetrad has been fixed, it is convenient to define from it
the complex bivectors

$\begin{gathered}X^{ab}\equiv 2\ell^{[a}m^{b]},\qquad Y^{ab}\equiv
2n^{[a}\bar{m}^{b]},\\\ Z^{ab}\equiv
2(\ell^{[a}n^{b]}-m^{[a}\bar{m}^{b]}).\end{gathered}$ (78)

These and and their complex conjugates form a basis for all bivectors. The
basis elements $X^{ab}$, $Y^{ab}$, and $Z^{ab}$ are self-dual, meaning that,
e.g., $X^{*ab}=iX^{ab}$. Their complex conjugates are instead anti self-dual.
It can also be shown, using (76), that the non-vanishing antisymmetrized
products between members of this basis are

$X^{[a}{}_{c}Y^{b]c}=\tfrac{1}{2}Z^{ab},\quad
X^{[a}{}_{c}Z^{b]c}=-X^{ab},\quad Y^{[a}{}_{c}Z^{b]c}=Y^{ab}.$ (79)

If both pairs of indices are contracted, the only non-vanishing inner products
are

$Z_{ab}Z^{ab}=2X_{ab}Y^{ab}=-4.$ (80)

The main motivation for introducing this bivector basis is that it allows the
curvature and the quadrupole moment to be written down and manipulated without
having to perform coordinate computations; see, e.g., (145). These bivectors
also provide a convenient basis for the torque which is experienced by an
extended body.

####  III.3.2 Quadrupolar forces and torques in general vacuum spacetimes

The vector and bivector bases which have just been described can now be used
to compute forces and torques. We allow for general vacuum spacetimes which
are not conformally flat, and also for arbitrary quadrupole moments (but no
octupole or higher moments). As observed in Harte (2020), the trace-free
quadrupole moment $J_{abcd}$ has the same algebraic properties as a Weyl
tensor, and may therefore be described in terms of five complex scalars
$J_{0},\ldots,J_{4}$ which are analogous to the five Weyl scalars. Comparing
with (144) and (145), any quadrupole moment can thus be written as

$\displaystyle
J_{abcd}=2\operatorname{Re}\big{[}J_{0}Y_{ab}Y_{cd}+J_{1}(Y_{ab}Z_{cd}+Z_{ab}Y_{cd})$
$\displaystyle~{}+J_{2}(Z_{ab}Z_{cd}-X_{ab}Y_{cd}-Y_{ab}X_{cd})$
$\displaystyle~{}-J_{3}(X_{ab}Z_{cd}+Z_{ab}X_{cd})+J_{4}X_{ab}X_{cd}\big{]},$
(81)

where

$\displaystyle J_{0}\equiv\tfrac{1}{4}J_{abcd}X^{ab}X^{cd},\quad
J_{1}\equiv\tfrac{1}{8}J_{abcd}X^{ab}Z^{cd},$ (82a) $\displaystyle
J_{2}\equiv-\tfrac{1}{4}J_{abcd}X^{ab}Y^{cd}=\tfrac{1}{16}J_{abcd}Z^{ab}Z^{cd},$
(82b) $\displaystyle J_{3}\equiv-\tfrac{1}{8}J_{abcd}Y^{ab}Z^{cd},\quad
J_{4}=\tfrac{1}{4}J_{abcd}Y^{ab}Y^{cd}.$ (82c)

At quadrupolar order, the generalized force may now be computed by
substituting these equations and the Weyl expansion (145) into (62). As
$\ell^{a}$ has already been assumed to have been aligned with one of the PNDs,
$\Psi_{0}$ will always vanish. Contributions from the other Weyl scalars are

$\displaystyle\mathcal{F}^{(q)}_{\xi}=\frac{4}{3}\operatorname{Re}\big{\\{}\big{[}\Psi_{1}(J_{4}X^{ab}-3J_{2}Y^{ab})\mathcal{L}_{\xi}Z_{ab}-2J_{3}X^{ab}\mathcal{L}_{\xi}(\Psi_{1}Y_{ab})\big{]}+3\big{[}\Psi_{2}(J_{1}Y^{ab}-J_{3}X^{ab})\mathcal{L}_{\xi}Z_{ab}-2J_{2}\mathcal{L}_{\xi}\Psi_{2}\big{]}$
$\displaystyle~{}+\big{[}\Psi_{3}(3J_{2}X^{ab}-J_{0}Y^{ab})\mathcal{L}_{\xi}Z_{ab}-2J_{1}Y^{ab}\mathcal{L}_{\xi}(\Psi_{3}X_{ab})\big{]}+\big{[}\Psi_{4}(J_{0}Y^{ab}+J_{1}Z^{ab})\mathcal{L}_{\xi}X_{ab}-J_{0}\mathcal{L}_{\xi}\Psi_{4}\big{]}\big{\\}}.$
(83)

The terms here which involve $\Psi_{2}$ and $\Psi_{4}$ were already obtained
in Harte (2020); the others are new. Much of the remainder of this paper
analyses the implications of this expression. One consequence which is already
apparent is that each Weyl scalar can couple only to certain quadrupole
scalars, as summarized in Table 3.

Weyl scalar | Quadrupole scalars
---|---
$\Psi_{1}$ |  $J_{2}$, $J_{3}$, $J_{4}$
$\Psi_{2}$ |  $J_{1}$, $J_{2}$, $J_{3}$
$\Psi_{3}$ |  $J_{0}$, $J_{1}$, $J_{2}$
$\Psi_{4}$ |  $J_{0}$, $J_{1}$
Table 3: Summary of which quadrupole scalars can couple to which Weyl scalars
in the generalized force (83). The scalar $\Psi_{0}$ has been omitted as it
always vanishes with our tetrad choice.

####  III.3.3 Type N spacetimes

The simplest nontrivial Weyl tensors are of Petrov type N. In that case,
aligning $\ell^{a}$ with the sole PND results in only $\Psi_{4}$ being
nonzero. Eq. (83) then reduces to

$\displaystyle\mathcal{F}^{(q)}_{\xi}=\frac{4}{3}\operatorname{Re}\left[\Psi_{4}(J_{0}Y^{ab}+J_{1}Z^{ab})\mathcal{L}_{\xi}X_{ab}-J_{0}\mathcal{L}_{\xi}\Psi_{4}\right].$
(84)

Equivalently, (4) and (79) can be used to convert this into the force

$F^{(q)}_{a}=\frac{4}{3}\operatorname{Re}\big{[}\Psi_{4}(J_{0}Y^{bc}+J_{1}Z^{bc})\nabla_{a}X_{bc}-J_{0}\nabla_{a}\Psi_{4}\big{]},$
(85)

and the torque

$N^{(q)}_{ab}=\frac{8}{3}\operatorname{Re}\big{[}\Psi_{4}(2J_{1}X_{ab}-J_{0}Z_{ab})\big{]}.$
(86)

At most $J_{0}$ and $J_{1}$ can thus affect the motion, meaning that there are
only four real quadrupole components which must be considered in type N
spacetimes (rather than the ten components which might affect motion in a
general vacuum spacetime).

It also follows from (80) and (86) that regardless of the quadrupole moment,

$N_{ab}^{(q)}X^{ab}=0.$ (87)

The real and the imaginary components of this constraint imply that there is a
two-dimensional space of real torques which cannot be produced by any
quadrupole moment in a type N spacetime. Torques can, however, be controlled
throughout the four-dimensional space which is spanned by real combinations of
$X_{ab}$, $Z_{ab}$, and their complex conjugates. Moreover, given any torque
within this space, the quadrupole scalars $J_{0}$ and $J_{1}$ which produce it
are uniquely determined. This can be used to show that the force is a linear
function of the torque:

$\displaystyle
F_{a}^{(q)}=\frac{3}{4}\operatorname{Re}\big{[}(Z^{bc}Y^{df}-Y^{bc}Z^{df})\nabla_{a}X_{df}$
$\displaystyle~{}-Z^{bc}\nabla_{a}\ln\Psi_{4}\big{]}N_{bc}^{(q)}.$ (88)

It is therefore impossible to vary the force without simultaneously varying
the torque.

Both (87) and (88) are consequences of local symmetries. One way to identify
these symmetries is to note that the generalized force can be written as

$\displaystyle\mathcal{F}^{(q)}_{\xi}=\frac{1}{4}\big{\\{}3\xi^{a}\operatorname{Re}\big{[}(Z^{bc}Y^{df}-Y^{bc}Z^{df})\nabla_{a}X_{df}$
$\displaystyle~{}-Z^{bc}\nabla_{a}\ln\Psi_{4}\big{]}+2\nabla^{b}\xi^{c}\big{\\}}N_{bc}^{(q)},$
(89)

where $\xi^{a}$ is any GKF. Recalling that local symmetries are generated by
families of GKFs $\Xi^{a}_{s}$ whose associated generalized forces vanish for
all possible quadrupole moments, it follows that

$\displaystyle\nabla^{a}\Xi^{b}_{s}=\frac{3}{2}\Xi^{f}_{s}\operatorname{Re}\big{[}(Y^{ab}Z^{cd}-Z^{ab}Y^{cd})\nabla_{f}X_{cd}$
$\displaystyle~{}+Z^{ab}\nabla_{f}\ln\Psi_{4}\big{]}+\bar{\lambda}_{s}X^{ab}+\lambda_{s}\bar{X}^{ab}$
(90)

at $\gamma_{s}$, where $\lambda_{s}$ is any family of complex scalars and
$\Xi^{a}_{s}(\gamma_{s})$ is arbitrary. Setting $\Xi^{a}_{s}(\gamma_{s})=0$
while varying $\lambda_{s}$ recovers the two real symmetries which generate
the torque constraint (87). Setting $\lambda_{s}=0$ while varying
$\Xi^{a}_{s}(\gamma_{s})$ results in four more local symmetries; these imply
that forces and torques must be linked via (88). In total, there are six local
symmetries in each type N spacetime.

####  III.3.4 Type III spacetimes

In Petrov type III spacetimes, we align the tetrad such that $\Psi_{3}$ is the
only non-vanishing Weyl scalar. The generalized force (83) then reduces to

$\displaystyle\mathcal{F}^{(q)}_{\xi}=\frac{4}{3}\operatorname{Re}\big{[}\Psi_{3}(3J_{2}X^{ab}-J_{0}Y^{ab})\mathcal{L}_{\xi}Z_{ab}$
$\displaystyle~{}-2J_{1}Y^{ab}\mathcal{L}_{\xi}(\Psi_{3}X_{ab})\big{]},$ (91)

which depends only on $J_{0}$, $J_{1}$, and $J_{2}$. Six real quadrupole
components can therefore affect motion in type III spacetimes. Converting the
generalized force into an ordinary force and a torque,

$\displaystyle
F^{(q)}_{a}=\frac{4}{3}\operatorname{Re}\big{[}\Psi_{3}(3J_{2}X^{bc}-J_{0}Y^{bc})\nabla_{a}Z_{bc}$
$\displaystyle~{}-2J_{1}Y^{bc}\nabla_{a}(\Psi_{3}X_{bc})\big{]},$ (92)

and

$N_{ab}^{(q)}=\frac{16}{3}\operatorname{Re}\big{[}\Psi_{3}(J_{1}Z_{ab}-J_{0}Y_{ab}-3J_{2}X_{ab})\big{]}.$
(93)

Any torque whatsoever can thus be generated by an appropriately-structured
object.

In fact, a given torque uniquely determines $J_{0}$, $J_{1}$ and $J_{2}$. That
may be used to show that the force is again a linear function of the torque:

$\displaystyle
F_{a}^{(q)}=\frac{1}{4}\operatorname{Re}\big{[}(X^{bc}Y^{df}-Y^{bc}X^{df})\nabla_{a}Z_{bc}+Z^{df}$
$\displaystyle~{}\times(Y^{bc}\nabla_{a}X_{bc}-2\nabla_{a}\ln\Psi_{3})\big{]}N_{df}^{(q)}.$
(94)

It is therefore impossible to control the force independently of the torque in
type III spacetimes.

Local symmetries may be found in type III spacetimes by first writing the
generalized force which is associated with a generic GKF $\xi^{a}$ as

$\displaystyle\mathcal{F}_{\xi}^{(q)}=\frac{1}{4}\operatorname{Re}\big{\\{}\xi^{a}\big{[}(X^{bc}Y^{df}-Y^{bc}X^{df})\nabla_{a}Z_{bc}+Z^{df}$
$\displaystyle~{}\times(Y^{bc}\nabla_{a}X_{bc}-2\nabla_{a}\ln\Psi_{3})\big{]}+\tfrac{1}{2}\nabla^{d}\xi^{f}\big{\\}}N_{df}^{(q)}.$
(95)

This vanishes for all quadrupole moments when we choose a 1-parameter family
of GKFs which satisfy

$\displaystyle\nabla^{a}\Xi^{b}_{s}=2\Xi^{f}_{s}\big{[}(X^{ab}Y^{cd}-Y^{ab}X^{cd})\nabla_{f}Z_{cd}+Z^{ab}$
$\displaystyle~{}\times(2\nabla_{f}\ln\Psi_{3}-Y^{cd}\nabla_{f}X_{cd})\big{]}$
(96)

at $\gamma_{s}$, where $\Xi^{a}_{s}$ is arbitrary. It follows that there are
four local symmetries in type III spacetimes. Physically, these imply that the
force and the torque must be linked by (94).

####  III.3.5 Type D spacetimes

In Petrov type D spacetimes, we align the tetrad such that $\Psi_{2}$ is the
only non-vanishing Weyl scalar. The generalized force then reduces to

$\displaystyle\mathcal{F}^{(q)}_{\xi}=4\operatorname{Re}\big{[}\Psi_{2}(J_{1}Y^{ab}-J_{3}X^{ab})\mathcal{L}_{\xi}Z_{ab}-2J_{2}\mathcal{L}_{\xi}\Psi_{2}\big{]},$
(97)

which depends only on $J_{1}$, $J_{2}$, and $J_{3}$. Up to888In the type N and
type III cases discussed above, we were able to say exactly how many
quadrupole components contribute to the motion. In the type D case, the answer
varies depending on the properties of $\nabla_{a}\Psi_{2}$. six real
quadrupole components therefore contribute to the motion in type D spacetimes.

This statement can be refined by first using (5) to extract the force

$F^{(q)}_{a}=4\operatorname{Re}\big{[}\Psi_{2}(J_{1}Y^{bc}-J_{3}X^{bc})\nabla_{a}Z_{bc}-2J_{2}\nabla_{a}\Psi_{2}\big{]},$
(98)

and the torque

$N_{ab}^{(q)}=16\operatorname{Re}\big{[}\Psi_{2}(J_{3}X_{ab}+J_{1}Y_{ab})\big{]}.$
(99)

The torque therefore depends on $J_{1}$ and $J_{3}$, but not $J_{2}$. It is
also apparent that regardless of the quadrupole moment,

$N_{ab}^{(q)}Z^{ab}=0.$ (100)

Quadrupolar torques can therefore be varied only within the four-dimensional
space which is spanned by real combinations of $X_{ab}$, $Y_{ab}$, and their
complex conjugates.

Unlike in type N or type III spacetimes, the force in a type D spacetime is
not necessarily a linear function of the torque. Instead,

$\displaystyle
F_{a}^{(q)}=\frac{1}{4}\operatorname{Re}\big{[}(X^{bc}Y^{df}-Y^{bc}X^{df})\nabla_{a}Z_{bc}\big{]}N_{df}^{(q)}$
$\displaystyle~{}-8\operatorname{Re}\big{[}J_{2}\nabla_{a}\Psi_{2}\big{]}.$
(101)

Unless $\Psi_{2}$ is constant, this implies that the force can be varied
independently of the torque. In particular, the space of forces which can be
controlled at fixed torque is spanned by the real and the imaginary components
of $\nabla_{a}\Psi_{2}$.

If $\Psi_{2}$ is constant, the force is instead linear in the torque and
$J_{2}$ disappears from the laws of motion. Such an example (necessarily with
a nonzero cosmological constant $\Lambda$) is provided by the Nariai or anti-
Nariai spacetimes999This is described as a Bertotti-Robinson spacetime in
Barnes (2015) but as a Nariai or anti-Nariai spacetime (depending on the sign
of $\Lambda$) in Sect. 18.6 of Griffiths and Podolský (2010). Barnes (2015)
with line elements

$ds^{2}=-2dudv+\Lambda
v^{2}du^{2}+\frac{dy^{2}+dz^{2}}{[1+\tfrac{1}{4}\Lambda(y^{2}+z^{2})]^{2}}.$
(102)

In fact, $F_{a}^{(q)}=0$ in these spacetimes; extended-body effects can
influence only the torque, at least at quadrupolar order.

In more complicated type D spacetimes where $\Psi_{2}$ is not constant, the
space of forces which can be produced at fixed torque is either one- or two-
dimensional. In Kerr spacetimes with nonzero angular momentum, forces may be
varied throughout a two-dimensional space without also varying the torque. In
the Schwarzschild limit where the angular momentum goes to zero, forces can
instead be varied in only one direction without also varying the torque Harte
(2020).

Local symmetries may be derived in an arbitrary type D spacetimes by first
writing the generalized force as

$\displaystyle\mathcal{F}^{(q)}_{\xi}=\frac{1}{4}\operatorname{Re}\big{\\{}\xi^{a}\big{[}(X^{bc}Y^{df}-Y^{bc}X^{df})\nabla_{a}Z_{bc}\big{]}$
$\displaystyle~{}+2\nabla^{d}\xi^{f}\big{\\}}N_{df}^{(q)}-8\operatorname{Re}[J_{2}\mathcal{L}_{\xi}\Psi_{2}].$
(103)

This vanishes for all quadrupole moments when we construct a 1-parameter
family of GKFs which satisfy

$\displaystyle\nabla^{a}\Xi^{b}_{s}=\frac{1}{2}\Xi^{f}_{s}\operatorname{Re}\big{[}(X^{ab}Y^{cd}-Y^{ab}X^{cd})\nabla_{f}Z_{cd}\big{]}$
$\displaystyle~{}+\bar{\lambda}_{s}Z^{ab}+\lambda_{s}\bar{Z}^{ab}$ (104)

at $\gamma_{s}$. Here, $\lambda_{s}$ is arbitrary and $\Xi^{a}_{s}$ must be
orthogonal to $\nabla_{a}\Psi_{2}$ at $\gamma_{s}$. Setting
$\Xi^{a}_{s}(\gamma_{s})=0$ while varying $\lambda_{s}$ recovers the two real
symmetries which imply the torque constraint (100). These are related to the
fact that the real and the imaginary components of $Z_{ab}$ are proportional
to conformal Killing-Yano tensors. If we instead set $\lambda_{s}=0$ and vary
$\Xi^{a}_{s}(\gamma_{s})$, our prescription results in two to four additional
symmetries. These imply that the force components which are orthogonal to
$\nabla_{a}\Psi_{2}$ and its complex conjugate must be linked to torques via
(101). In total, there are between four and six local symmetries in type D
spacetimes. There are, e.g., four such symmetries in Kerr (with nonzero
angular momentum), five in Schwarzschild, and six in the Nariai and anti-
Nariai spacetimes.

####  III.3.6 Type II spacetimes

In Petrov type II spacetimes, we choose the tetrad such that only $\Psi_{2}$
and $\Psi_{3}$ are nonzero. The generalized force (83) then reduces to

$\displaystyle\mathcal{F}^{(q)}_{\xi}$
$\displaystyle=\frac{4}{3}\operatorname{Re}\big{\\{}4J_{1}\mathcal{L}_{\xi}\Psi_{3}-6J_{2}\mathcal{L}_{\xi}\Psi_{2}-2\Psi_{3}J_{1}Y^{ab}\mathcal{L}_{\xi}X_{ab}$
$\displaystyle~{}+[3(\Psi_{3}J_{2}-\Psi_{2}J_{3})X^{ab}+(3\Psi_{2}J_{1}-\Psi_{3}J_{0})Y^{ab}]\mathcal{L}_{\xi}Z_{ab}\big{\\}},$
(105)

the ordinary force is given by

$\displaystyle F_{a}^{(q)}$
$\displaystyle=\frac{4}{3}\operatorname{Re}\big{\\{}4J_{1}\nabla_{a}\Psi_{3}-6J_{2}\nabla_{a}\Psi_{2}-2\Psi_{3}J_{1}Y^{bc}\nabla_{a}X_{bc}$
$\displaystyle~{}+[3(\Psi_{3}J_{2}-\Psi_{2}J_{3})X^{bc}+(3\Psi_{2}J_{1}-\Psi_{3}J_{0})Y^{bc}]\nabla_{a}Z_{bc}\big{\\}},$
(106)

and the torque by

$\displaystyle
N_{ab}^{(q)}=\frac{16}{3}\operatorname{Re}\big{[}\Psi_{3}J_{1}Z_{ab}+(3\Psi_{2}J_{1}-\Psi_{3}J_{0})Y_{ab}$
$\displaystyle~{}+3(\Psi_{2}J_{3}-\Psi_{3}J_{2})X_{ab}\big{]}.$ (107)

These expressions depend on all quadrupole scalars except for $J_{4}$, so up
to eight quadrupole components can affect motion in type II spacetimes.

It also follows that any torque whatsoever can be produced by appropriately
varying the quadrupole moment. However, unlike in the type N, type III, and
type D cases discussed above, a given torque does not uniquely determine the
relevant quadrupole scalars. Instead, fixing $N_{ab}^{(q)}$ fixes only
$J_{0}$, $J_{1}$, and $\Psi_{2}J_{3}-\Psi_{3}J_{2}$. Using this, the force can
nevertheless be shown to be affine function of the torque:

$\displaystyle
F_{a}^{(q)}=\frac{1}{4}\operatorname{Re}\big{[}Y^{bc}Z^{df}\nabla_{a}X_{bc}-2Z^{df}\nabla_{a}\ln\Psi_{3}$
$\displaystyle~{}(X^{bc}Y^{df}-Y^{bc}X^{df})\nabla_{a}Z_{bc}\big{]}N_{df}^{(q)}$
$\displaystyle~{}-8\operatorname{Re}[J_{2}\nabla_{a}\Psi_{2}].$ (108)

By varying $J_{2}$ and $J_{3}$ at fixed $\Psi_{2}J_{3}-\Psi_{3}J_{2}$, the
quadrupolar force can thus be controlled, at fixed torque, throughout the
space which is spanned by the real and the imaginary components of
$\nabla_{a}\Psi_{2}$. That space has at most two dimensions. If $\Psi_{2}$ is
constant, which occurs only in certain Kundt spacetimes Barnes (2015), the
force cannot be controlled independently of the torque and only six quadrupole
components affect the motion. If $\nabla_{a}\Psi_{2}$ is nonzero and linearly
independent of its complex conjugate, there are eight quadrupole components
which affect the motion.

All local symmetries may be found in type II spacetimes by first writing the
generalized force as

$\displaystyle\mathcal{F}_{\xi}=\frac{1}{4}\operatorname{Re}\big{\\{}\xi^{a}\big{[}Y^{bc}Z^{df}\nabla_{a}X_{bc}-2Z^{df}\nabla_{a}\ln\Psi_{3}+(X^{bc}Y^{df}$
$\displaystyle~{}-Y^{bc}X^{df})\nabla_{a}Z_{bc}\big{]}+2\nabla^{d}\xi^{f}\big{\\}}N_{df}^{(q)}-8\operatorname{Re}[J_{2}\mathcal{L}_{\xi}\Psi_{2}].$
(109)

Ensuring that this vanishes for all possible quadrupole moments, local
symmetries are seen to be generated by the 1-parameter family of GKFs whose
gradients satisfy

$\displaystyle\nabla^{a}\Xi^{b}_{s}=\frac{1}{2}\Xi_{s}^{f}\operatorname{Re}\big{[}2Z^{ab}\nabla_{f}\ln\Psi_{3}-Y^{cd}Z^{ab}\nabla_{f}X_{cd}$
$\displaystyle~{}+(X^{ab}Y^{cd}-Y^{ab}X^{cd})\nabla_{f}Z_{cd}\big{]}$ (110)

at $\gamma_{s}$, where $\Xi^{a}_{s}(\gamma_{s})$ is constrained only to be
orthogonal to $\nabla_{a}\Psi_{2}$. This implies that there are between two
and four local symmetries. Their presence requires forces which are orthogonal
to $\nabla_{a}\Psi_{2}$ and its complex conjugate to be linked to torques via
(108).

####  III.3.7 Type I spacetimes

In type I spacetimes, only $\Psi_{0}$ and $\Psi_{4}$ can necessarily be made
to vanish by choosing an appropriate tetrad. However, as noted in Sect.
III.3.1 above, a type III tetrad transformation can always be used to ensure
that $\Psi_{1}=\Psi_{3}$. Applying such a transformation for simplicity, the
generalized force (83) reduces to

$\displaystyle\mathcal{F}^{(q)}_{\xi}=\frac{8}{3}\operatorname{Re}\big{\\{}2(J_{1}+J_{3})\mathcal{L}_{\xi}\Psi_{1}+\Psi_{1}(J_{3}-J_{1})Y^{ab}\mathcal{L}_{\xi}X_{ab}$
$\displaystyle~{}-3J_{2}\mathcal{L}_{\xi}\Psi_{2}+\tfrac{1}{2}\big{[}\big{(}\Psi_{1}(3J_{2}+J_{4})-3\Psi_{2}J_{3}\big{)}X^{ab}$
$\displaystyle~{}+\big{(}3\Psi_{2}J_{1}-\Psi_{1}(3J_{2}+J_{0})\big{)}Y^{ab}\big{]}\mathcal{L}_{\xi}Z_{ab}\big{\\}}.$
(111)

All five quadrupole scalars $J_{0},\ldots,J_{4}$ appear here, so all of a
body’s quadrupole moment can affect its motion in at least some type I
spacetimes.

Using (5) to extract the torque from the generalized force,

$\displaystyle
N^{(q)}_{ab}=\frac{16}{3}\operatorname{Re}\big{\\{}[3\Psi_{2}J_{3}-\Psi_{1}(3J_{2}+J_{4})]X_{ab}+[3\Psi_{2}J_{1}$
$\displaystyle~{}-\Psi_{1}(3J_{2}+J_{0})]Y_{ab}+\Psi_{1}(J_{1}-J_{3})Z_{ab}\big{\\}}.$
(112)

All torques are therefore possible in type I spacetimes. However, although all
five quadrupole scalars appear here, fixing the torque fixes only the three
combinations

$\begin{gathered}J_{1}-J_{3},\qquad J_{0}-J_{4},\\\
3\Psi_{2}(J_{1}+J_{3})-\Psi_{1}(6J_{2}+J_{0}+J_{4})\end{gathered}$ (113)

of quadrupole scalars. This observation allows the force to again be written
as an affine function of the torque:

$\displaystyle
F_{a}^{(q)}=\frac{1}{4}\operatorname{Re}\big{[}Y^{bc}Z^{df}\nabla_{a}X_{bc}+(X^{bc}Y^{df}-Y^{bc}X^{df})\nabla_{a}Z_{bc}\big{]}$
$\displaystyle~{}\times
N_{df}^{(q)}+\frac{8}{3}\operatorname{Re}\big{[}2(J_{1}+J_{3})\nabla_{a}\Psi_{1}-3J_{2}\nabla_{a}\Psi_{2}\big{]}.$
(114)

Noting that $J_{1}+J_{3}$ and $J_{2}$ can be varied arbitrarily without
affecting the quadrupole components (113), forces can thus be varied, at fixed
torque, throughout the space which is spanned by the real and imaginary
components of $\nabla_{a}\Psi_{1}$ and $\nabla_{a}\Psi_{2}$. If these
gradients are all linearly independent, that space is four-dimensional. All
ten force and torque components can then be controlled independently. At the
opposite extreme, $\Psi_{1}$ and $\Psi_{2}$ may both be constant Barnes
(2015), in which case the force is entirely determined by the torque.

Unlike in the algebraically-special spacetimes discussed above, there might
not be any local symmetries in type I spacetimes. Any local symmetries which
do exist nevertheless satisfy

$\displaystyle\nabla^{a}\Xi^{b}_{s}=\frac{1}{2}\Xi^{f}_{s}\operatorname{Re}\big{[}(X^{ab}Y^{cd}-Y^{ab}X^{cd})\nabla_{f}Z_{cd}$
$\displaystyle~{}-Z^{ab}Y^{cd}\nabla_{f}X_{cd}\big{]}$ (115)

at $\gamma_{s}$, where $\Xi^{a}_{s}$ can be varied arbitrarily at $\gamma_{s}$
as long as it is orthogonal to the real and the imaginary components of both
$\nabla_{a}\Psi_{1}$ and $\nabla_{a}\Psi_{2}$. This results in between zero
and four local symmetries, depending on which particular type I spacetime is
considered.

####  III.3.8 Summarizing the relativistic constraints

We have now derived quadrupolar forces and torques in vacuum spacetimes and
discussed qualitative differences which depend on the Petrov type of the
relevant spacetime. As summarized in Table 4, our focus has been on three
characteristics of extended-body motion: the number of torque components which
can be affected by internal structure, the number of force components which
can be controlled independently of the torque, and the number of quadrupole
components which affect the motion. These characteristics have also been
related to the presence of local symmetries. Roughly speaking, there are fewer
local symmetries in spacetimes which are “more” algebraically general, and in
those cases, extended bodies can exert more control over their motion.

Petrov type | $\big{\\{}\Xi^{a}_{s}\big{\\}}$ | $\big{\\{}J_{abcd}\big{\\}}$ | $\big{\\{}N_{ab}^{(q)}\big{\\}}$ | $\big{\\{}F_{a}^{(q)}\big{|}N_{bc}^{(q)}\big{\\}}$
---|---|---|---|---
I | 0–4 | 6-10* | 6* |  $\leq 4$*
II | 2–4 | 6-8 | 6* | $\leq 2$
D | 4–6 | 4-6 | 4 | $\leq 2$
III | 4 | 6 | 6* | 0
N | 6 | 4 | 4 | 0
Table 4: Qualitative features of quadrupolar forces and torques in spacetimes
with different Petrov types. Column 2 lists the number of local symmetries.
The upper bounds there also provide an upper bound for the number of Killing
vectors which can exist. Column 3 lists the number of real quadrupole
components which affect the motion. The number of controllable torque
components is provided in column 4. Column 5 lists the number of force
components which can be controlled at fixed torque. Stars are used to indicate
that a given number is unconstrained.

Any torque whatsoever can be produced in type I, type II, and type III
spacetimes, but not in type N or type D spacetimes. In the latter cases, a
two-dimensional space of torques is inaccessible, regardless of the quadrupole
moment. It is interesting in this context to recall that the relativistic
torque is qualitatively different from its Newtonian countepart. The
relativistic torque includes three components which are physically similar to
the Newtonian torque, but it also involves three components which are
fundamentally non-Newtonian. These additional components may be viewed as
controlling the misalignment between the 4-velocity and the 4-momentum: the
“hidden momentum” Harte (2007); Gralla  _et al._ (2010); Harte (2020).
Constraints on the relativistic torque therefore affect an object’s ability
not only to control its spin, but also to directly control its velocity. One
example of this is given in Sect. III.4.2 below, where an object in a Kasner
spacetime is shown to be able to move itself arbitrarily simply by controlling
its torque. More generally, since at least four torque components can be
controlled in every nontrivial vacuum spacetime, the torque can always be used
to control at least some of an object’s velocity.

Another of our results is that the quadrupolar force can always be written as
an affine function of the quadrupolar torque: Eqs. (88), (94), (101), (108),
and (114) all have the form

$F_{a}^{(q)}=\chi_{a}{}^{bc}N_{bc}^{(q)}+\operatorname{Re}\sum_{I}\alpha_{I}J_{4-I}\nabla_{a}\Psi_{I},$
(116)

where the $\alpha_{I}$ are coefficients and $\chi_{a}{}^{bc}$ depends only on
the geometry (but not on an object’s internal structure). The first term here
describes that portion of the force which is universally tied to the torque.
The second term provides all portions of the force which can be varied
independently of the torque, and is also the only force which remains when
$N_{ab}^{(q)}=0$. Interestingly, this latter term is simply a linear
combination of gradients of the Weyl scalars. It does not depend on, e.g., any
gradients of the tetrad. Recalling (31), a similar result holds also for
quadrupolar forces in Newtonian gravity.

One generic feature of extended-body motion is that the mass (64) is not
necessarily constant at quadrupolar and higher orders. Our result (116)
nevertheless implies that at least for torque-free, spin-free bodies with
constant quadrupole scalars, there exists an “effective mass” which is
conserved. As we have discussed already, it is always possible to arrange for
the torque to vanish. Adopting the Tulczyjew spin supplementary condition
(73), doing so implies that if $S^{ab}$ is initially zero, it remains so. Then
$p_{a}=M\dot{\gamma}_{s}^{a}$ Ehlers and Rudolph (1977), and it follows from
(57) that

$\frac{DM}{ds}=-\operatorname{Re}\sum_{I}\alpha_{I}J_{4-I}\dot{\gamma}^{a}_{s}\nabla_{a}\Psi_{I}.$
(117)

If all of the relevant quadrupole scalars (i.e., the ones for which
$\alpha_{I}J_{4-I}\neq 0$) are constant, the effective mass

$M_{\mathrm{eff}}\equiv M+\operatorname{Re}\sum_{I}\alpha_{I}J_{4-I}\Psi_{I}$
(118)

is therefore conserved. A special case of this was used in Harte (2020) in
order to understand how extended-body effects can be used to alter orbits in
the Schwarzschild spacetime. Appendix C identifies a somewhat different
effective mass (160) which is conserved when quadrupole moments are tidally
induced with constant deformabilities. Using different assumptions on the
nature of the quadrupole moment, certain other effective masses can be found
as well Dixon (1970, 1979).

###  III.4 Example spacetimes

We now apply the general results derived above to two specific examples:
extended-body motion in pp-wave spacetimes and extended-body motion in Kasner
spacetimes. Kasner spacetimes are type I while pp-waves are type N, so the
Weyl tensors in these examples lie at the two extremes of algebraic
speciality.

####  III.4.1 Motion in pp-wave spacetimes

A pp-wave spacetime describes a plane-fronted gravitational wave with parallel
rays Stephani  _et al._ (2009); Griffiths and Podolský (2010); Jordan  _et
al._ (2009). Besides their interpretation as idealized gravitational waves,
some pp-waves also arise as ultrarelativistic limits of other (not necessarily
radiative) spacetimes. This can occur both via “global” boosts Aichelburg and
Sexl (1971); Podolský and Griffiths (1998), or via the Penrose limit, which
locally describes the geometry near arbitrary null geodesics as effective
plane waves Penrose (1976); Blau  _et al._ (2006).

Regardless of interpretation, any pp-wave spacetime can be described by the
line element

$ds^{2}=-2dudv+H(u,y,z)du^{2}+dy^{2}+dz^{2},$ (119)

where $H(u,x,y)$ is a dimensionless “waveform,” $y$ and $z$ are transverse
coordinates, and $u$ is a null “phase” coordinate. Imposing the $\Lambda=0$
vacuum Einstein equation shows that

$(\partial_{y}^{2}+\partial_{z}^{2})H(u,y,z)=0,$ (120)

so the waveform here must be harmonic101010That the nonlinearity of Einstein’s
equation disappears in this class of spacetimes is due to the fact that pp-
waves are Kerr-Schild transformations of Minkowski spacetime Xanthopoulos
(1978); Harte and Vines (2016). on each $u=\mathrm{constant}$ hypersurface.
Harmonic functions in two real dimensions can be related to complex analytic
functions of one variable, so the waveform of an arbitrary vacuum pp-wave can
be written as

$H(u,y,z)=\operatorname{Re}\,\mathcal{H}(u,\zeta(y,z)),$ (121)

where $\mathcal{H}$ is complex and analytic in the complexified transverse
coordinate $\zeta\equiv(y+iz)/\sqrt{2}$. A pp-wave is said to be linearly
polarized when $\arg\mathcal{H}=\mathrm{constant}$, and other properties of
$\mathcal{H}$ can be used to classify pp-waves as described in Jordan  _et
al._ (2009); see also Table 24.2 of Stephani  _et al._ (2009).

The most important category in this classification are the vacuum “plane
waves,” which satisfy

$\partial_{\zeta}^{3}\mathcal{H}=0.$ (122)

In any plane wave spacetime, there exist coordinates in which

$\mathcal{H}(u,\zeta)=\mathcal{h}(u)\zeta^{2},$ (123)

where the complex function $\mathcal{h}$ determines the curvature as a
function of phase; see $\Psi_{4}$ in (131) below. Einstein’s equation does not
constrain $\mathcal{h}$.

All vacuum pp-waves, whether plane waves or not, are type N wherever they are
not flat. The lone principal null direction is parallel to

$\ell_{a}=-\nabla_{a}u,$ (124)

which physically describes the direction along which the gravitational wave
propagates. Its integral curves are the “rays” of that wave. A calculation
shows that

$\nabla_{a}\ell_{b}=0,$ (125)

so these rays are geodesic, non-expanding, shear-free, and twist-free. It also
follows that $\ell^{a}$ is Killing. For some pp-waves, $\ell^{a}$ is the only
Killing field. In special cases, there can be up to five more111111If the
vacuum restriction is relaxed, there can be up to seven Killing vectors in
total Sippel and Goenner (1986). Jordan  _et al._ (2009). All plane waves
admit at least five Killing fields in total, although some admit six. In this
latter case, all of the local symmetries described by (90) are ordinary
Killing fields. In other pp-wave spacetimes, some local symmetries are Killing
while some are not. Regardless, $\mathcal{P}_{\ell}=p_{a}\ell^{a}$ is
conserved for any extended body moving in any pp-wave spacetime. Furthermore,

$\mathcal{F}_{\ell}=F_{a}\ell^{a}=0.$ (126)

This constraint holds not only for the full force, but also for its
quadrupolar contribution.

In any pp-wave spacetime, it is convenient to use $\ell_{a}$ as one element of
the null tetrad $(\ell_{a},n_{a},m_{a},\bar{m}_{a})$, where
$n_{a}\equiv-(\nabla_{a}v+\tfrac{1}{2}H\ell_{a})$ and
$m_{a}\equiv\nabla_{a}\zeta$. Employing this to construct the $X_{ab}$ defined
by (78), a calculation shows that

$\nabla_{c}X_{ab}=0.$ (127)

The real and imaginary components of $X_{ab}$ are therefore Killing-Yano
tensors. As described in Sect. III.2.2 above, each such tensor generates a
local symmetry. For generic pp-waves, these symmetries are truly local and are
not necessarily associated with any conservation law. However, there are
special pp-wave spacetimes where the local symmetries associated with Killing-
Yano tensors are related to ordinary Killing symmetries. In those cases, there
are genuine conservation laws which can be associated with Killing-Yano
tensors (regardless of, e.g., spin supplementary conditions).

To see this, consider the special case of a plane wave spacetime. The waveform
$\mathcal{H}$ is then given by (123). Letting $\lambda(u)$ be any possibly-
complex solution to the differential equation
$\chi^{\prime\prime}(u)=\frac{1}{2}\mathcal{h}(u)\bar{\chi}(u)$, and letting
$\zeta_{\tau}$ and $u_{\tau}$ be any families of constants, the vector fields

$\displaystyle\Xi^{a}_{\tau}(x)=\operatorname{Re}\big{\\{}[\zeta_{\tau}\chi^{\prime}(u_{\tau})-\zeta(x)\chi^{\prime}(u(x))]\ell^{a}(x)$
$\displaystyle~{}-\chi(u(x))m^{a}(x)\big{\\}}$ (128)

are Killing. At fixed $\tau$, varying over all possible $\chi(u)$ results in
four real Killing fields with this form. In a flat limit, two of these Killing
fields describe translations transverse to the rays of the gravitational wave.
The remaining two describe mixed boosts together with rotations, and are a
consequence of the fact that moving transverse to a plane wave appears only to
rotate it. For all four of these Killing fields,

$\nabla^{a}\Xi^{b}_{\tau}=\operatorname{Re}[\chi^{\prime}(u)X^{ab}].$ (129)

If $u_{\tau}=u(\gamma_{\tau})$ and $\zeta_{\tau}=\zeta(\gamma_{\tau})$ now
denote an object’s phase and transverse coordinates at some time $\tau$, and
if $\chi(u_{\tau})=0$, it follows that $\Xi^{a}_{\tau}(\gamma_{\tau})=0$. Also
choosing $\chi^{\prime}(u_{\tau})$ to be equal to $1$ or to $-i$ reproduces
the generalized Killing fields determined by (71), when $s=\tau$, where the
CKY tensor $f_{ab}$ which appears there is understood to be either the real or
the imaginary component of $X_{ab}$. Since $\Xi^{a}_{\tau}$ is genuinely
Killing, the generalized momentum component

$\displaystyle\mathcal{P}_{\Xi_{\tau}}=\operatorname{Re}\big{\\{}\big{[}\zeta_{\tau}\chi^{\prime}(u_{\tau})-\zeta_{s}\chi^{\prime}(u_{s})\big{]}\mathcal{P}_{\ell}-\chi(u_{s})p_{a}m^{a}$
$\displaystyle~{}+\tfrac{1}{2}\lambda^{\prime}(u_{s})S_{ab}X^{ab}\big{\\}}$
(130)

is conserved for any fixed $\tau$; it is independent of $s$. Given the
aforementioned initial conditions for $\chi(u)$, the first line here
necessarily vanishes when $s=\tau$, implying that
$\mathcal{P}_{\Xi_{\tau}}=\frac{1}{2}\operatorname{Re}[\chi^{\prime}(u_{\tau})X^{ab}]S_{ab}(\tau)$.
However, the first line in (130) must be retained when $s\neq\tau$. It is
interesting to note that in this case, where the symmetries associated with
Killing-Yano tensors can definitively be associated with conservation laws,
the quantities which are conserved do not coincide with the real and the
imaginary components of $S_{ab}X^{ab}$, except at one moment in time. The
situation here is similar to the one in the Schwarzschild spacetime, where the
local symmetry associated with one (but not both) of the conformal Killing-
Yano tensors can be derived from ordinary Killing symmetries Harte (2020). In
that case as well, the associated conservation law is not trivial.

Returning to the case of a general pp-wave spacetime, use of (144) shows that
with the above tetrad, the only non-vanishing Weyl scalar is

$\Psi_{4}=\frac{1}{4}(\partial_{z}^{2}-\partial_{y}^{2}+2i\partial_{y}\partial_{z})H=-\frac{1}{4}\partial_{\zeta}^{2}\mathcal{H}.$
(131)

Substituting this and (127) into the generic type N force (85) shows that the
quadrupolar force which acts on an arbitrary extended body is

$\displaystyle F^{(q)}_{a}$
$\displaystyle=-\frac{4}{3}\operatorname{Re}[J_{0}\nabla_{a}\Psi_{4}],$
$\displaystyle=\frac{1}{3}\operatorname{Re}\big{[}J_{0}(m_{a}\partial_{\zeta}-\ell_{a}\partial_{u})\partial_{\zeta}^{2}\mathcal{H}\big{]}.$
(132)

This is clearly consistent with the Killing constraint (126). It may also be
seen that although the torque (86) depends on the quadrupole components
$J_{0}$ and $J_{1}$, the force here depends only on $J_{0}$. This suggests
that there are two real control parameters with which to control the force (at
least if the torque is allowed to vary as well). However, these two parameters
do not necessarily have independent effects. If a pp-wave is linearly
polarized, for example, the quadrupolar force can be varied only in the one
direction parallel to $\nabla_{a}|\Psi_{4}|$. The force can also be varied in
only one direction in a plane wave spacetime—whether it is linearly polarized
or not. In fact, the force is always proportional to $\ell_{a}$ plane wave
spacetimes; it is longitudinal.

####  III.4.2 Motion in Kasner spacetimes

The vacuum Kasner spacetimes may be viewed as describing homogeneous but
anisotropic (and empty) universes Griffiths and Podolský (2010); Stephani  _et
al._ (2009); Harvey (1990). They have the line elements

$ds^{2}=-dt^{2}+\sum_{i=1}^{3}t^{2u_{i}}(dx^{i})^{2},$ (133)

where $u_{1}$, $u_{2}$, and $u_{3}$ are constants121212These are more commonly
denoted by $p_{1}$, $p_{2}$, and $p_{3}$. We use a different notation in order
to avoid confusion with the momentum.. Applying the $\Lambda=0$ vacuum
Einstein equation results in

$\sum_{i=1}^{3}u_{i}=\sum_{i=1}^{3}u_{i}^{2}=1,$ (134)

which imples that the space of vacuum Kasner spacetimes may be viewed, in
$\mathbb{R}^{3}$, as the intersection of a unit sphere with a plane. This
leaves a 1-parameter family of solutions.

Except in special cases which we do not consider, the Kasner spacetimes are of
Petrov type I. Being spatially homogeneous, they admit the three translational
Killing fields $\partial_{i}$. Since the Weyl scalars in an appropriately-
adapted tetrad can at most depend on $t$, there is a three-dimensional space
of local symmetries described by (115). However, these three symmetries are
simply the three translational Killing fields. There are no proper local
symmetries.

As in all type I spacetimes, the torque which acts on an extended body in a
Kasner spacetime is unconstrained. However, the force can be varied only along
$t_{a}\equiv-\nabla_{a}t$ without simultaneously varying the torque.
Homogeneity implies that the three spatial components

$\mathcal{P}_{\partial_{i}}=p_{i}+u_{i}t^{2u_{i}-1}S^{ti}$ (135)

of the generalized momentum are conserved, where no sum over $i$ is implied.
Similarly, the force and the torque are related via
$\mathcal{F}_{\partial_{i}}=F_{i}+u_{i}t^{2u_{i}-1}N^{ti}=0$, where again, no
sum is implied.

One interesting feature of motion in Kasner spacetimes is that despite their
spatial homogeneity, it is still possible for an extended body to exert
essentially arbitrary control over its trajectory. To see this in a special
case, first fix a centroid using the spin supplementary condition

$S_{ab}t^{b}=0,$ (136)

which demands that $\gamma_{s}$ be chosen such that the mass dipole moment
vanishes in the frame which is associated with the background homogeneity.
Given (135), this spin supplementary condition implies that the three momentum
components $p_{i}$ must be conserved. However, the velocity is not necessarily
proportional to the momentum and is not necessarily conserved: Differentiating
(136) while using (57) instead shows that

$(-p\cdot t)\dot{\gamma}^{a}_{s}=(-\dot{\gamma}_{s}\cdot
t)p^{a}-N^{a}{}_{b}t^{b}-S^{ab}\dot{\gamma}^{c}_{s}\nabla_{c}t_{b}.$ (137)

Suppose for simplicity that $p_{i}=0$. Choosing the body’s quadrupole moment
such that $N_{(q)}^{ij}=0$, it is then possible to arrange for the angular
momentum to vanish for all time, at least through quadrupolar order. Doing so,
the spatial velocity becomes proportional to $N^{it}_{(q)}$, which can be
controlled arbitrarily. An extended body with vanishing spatial momentum and
vanishing angular momentum may therefore translate itself arbitrarily, simply
by controlling its quadrupole moment. This takes advantage of the fact that
although Kasner spacetimes are spatially homogeneous, they are not boost-
invariant. A similar phenomenon has been discussed before in flat Friedmann-
Robertson-Walker spacetimes Harte (2007), which are both homogeneous and
isotropic (though not vacuum). Torques may be used to control translations in
other spacetimes as well, but then changes in the momentum can complicate the
interpretation.

##  IV Conclusion

We have derived universal constraints on the gravitational forces and torques
which can be produced by an object’s quadrupole moment, both in Newtonian
gravity and in general relativity. Depending on the algebraic structure of the
relevant tidal tensor, certain quadrupole moments can be irrelevant, certain
torques can be impossible, and only certain forces can be produced without an
accompanying torque. These results are summarized at the ends of Sects. II.3
and III.3, and particularly by Tables 1 and 4. We have also found that the
quadrupolar force can be viewed as an affine function of the quadrupolar
torque; see (31) and (116). These results are independent of any spin
supplementary conditions.

Fundamentally, our results are explained by the existence of “local
symmetries.” In the Newtonian case, local symmetries correspond to Euclidean
Killing fields which locally preserve the tidal tensor. In general relativity,
local symmetries are generalized Killing fields which locally preserve the
Riemann tensor. Regardless, each local symmetry precludes certain force and
torque combinations. This generalizes the well-known fact that Killing fields
constrain motion in general relativity. In fact, no further generalization is
possible: Every universal constraint on extended-body motion is associated
with a local symmetry, at least at quadrupolar order. Any generalized force
which is not forbidden by local symmetries may be experienced by a suitably-
structured object.

The local symmetries we have introduced are an essentially geometric concept,
and may thus be of interest not only in the theory of motion. Roughly
speaking, the generalized Killing fields introduced in Harte (2008b) provide a
sense in which full Poincaré symmetry can exist around a given worldline in a
curved spacetime. This results in Killing’s equation being satisfied at least
through first order on the reference worldline. Additionally, certain
geometric structures (though not the metric) are preserved even away from that
worldline. However, it is natural to ask if Killing’s equation can be made to
hold through one higher order, at least at one point along the reference
worldline. When this occurs, we have a local symmetry. Perhaps surprisingly,
examples are common and have the physically-interesting consequences described
above. We have shown explicitly how to construct all local symmetries, both in
general relativity and in Newtonian gravity. Their number depends, in part, on
the algebraic structure of the tidal tensor, and is summarized in the second
columns of Tables 1 and 4. All ordinary Killing fields generate a local
symmetry in a curved spacetime, and we have shown that conformal Killing-Yano
tensors do so as well. Many local symmetries are, however, unrelated either to
Killing vectors or to conformal Killing-Yano tensors.

##  Appendix A Notation

We use the same sign conventions as Wald Wald (1984), so, e.g., the Riemann
tensor satisfies $R_{abc}{}^{d}\omega_{d}=2\nabla_{[a}\nabla_{b]}\omega_{c}$
for any covector $\omega_{a}$. The letters $a,b,\ldots$ are used to denote
abstract indices in both three and four dimensions, $i,j,\ldots$ are used to
denote three-dimensional coordinate components, $\alpha,\beta,\ldots$ are used
to denote four-dimensional coordinate components, and $I,J,\ldots$ are used
for numerical indices which are not associated with any coordinates. Hodge
duals are indicated by $*$ and overbars are used to denote complex conjugates.

Symbols which are commonly used in the text are summarized in Table 5.
Generalized forces, as well as ordinary forces and torques, are often
supplemented with a “$(q)$” superscript to refer only to quadrupolar
contributions. We use the abbreviations “GKF” (generalized Killing field),
“PND” (principal null direction), and “CKY” (conformal Killing-Yano). Three
classification schemes are also used: The algebraic structure of Newtonian
tidal tensors is summarized in II.3, the analogous Petrov classification for
four-dimensional Weyl tensors is summarized in Appendix B, and that Appendix
also specifies the three types of tetrad transformations which may be
performed in four spacetime dimensions.

Symbol |  Description |  Reference
---|---|---
$x$ |  Generic point |  -
$x^{i}$ |  Spatial (usually Cartesian) coordinates |  -
$\gamma_{t}$, $\gamma_{s}$ |  Reference point for object’s location at time $t$ (or $s$) |  -
$g_{ab}$ |  Metric |  -
$\xi^{a}$ |  Generalized or ordinary Killing vector |  (56)
$\Xi_{t}^{a}$, $\Xi_{s}^{a}$ |  Local symmetry generator |  (18), (65), (67)
$M$ |  Mass |  (64)
$\mathcal{P}_{\xi}$ |  Generalized momentum associated with $\xi^{a}$ |  (1), (2), (55)
$\mathcal{F}_{\xi}$ |  Generalized force associated with $\xi^{a}$ |  (4), (6), (57), (58)
$p_{a}$, $S^{ab}$ |  Linear and angular momenta |  (2), (3)
$F_{a}$, $N^{ab}$ |  Force and torque |  (5), (6), (57)
$\ell^{a}$, $m^{a}$, $\bar{m}^{a}$ |  Newtonian semi-null triad |  (24)
$\Phi$, $\mathcal{E}_{ab}$ |  Newtonian potential and tidal tensor |  (10), (25)
$\mathcal{E}_{\pm}$, $\mathcal{E}$ |  Real eigenvalues of $\mathcal{E}_{ab}$ and complex tidal scalar |  (21), (25)
$\tilde{Q}_{ab}$, $Q_{ab}$ |  Full and trace-free Newtonian quadrupole moments |  (9), (12), (26)
$Q_{\ell\ell}$, $Q_{\ell m}$, $Q_{mm}$ |  Quadrupole scalars |  (26)
$\ell^{a}$, $n^{a}$, $m^{a}$, $\bar{m}^{a}$ |  Relativistic null tetrad |  (76)
$X^{ab}$, $Y^{ab}$, $Z^{ab}$ |  Bivector basis elements |  (78)
$\Lambda$ |  Cosmological constant |  (60)
$R_{abcd}$, $C_{abcd}$ |  Riemann and Weyl tensors |  (61), (145)
$\Psi_{0},\ldots,\Psi_{4}$ |  Weyl scalars |  (144), (145)
$\tilde{J}_{abcd}$, $J_{abcd}$ |  Full and trace-free relativistic quadrupole moments |  (81)
$J_{0},\ldots,J_{4}$ |  Quadrupole scalars |  (81), (82)
$f_{ab}$ |  Conformal Killing-Yano tensor |  (68)
Table 5:  Table of symbols. The first group of symbols are used in both
Newtonian and relativistic contexts. The second group lists Newtonian symbols
while the third lists relativistic ones.

##  Appendix B Tetrad transformations, principal null directions, and the
Petrov classification

It is convenient in a four-dimensional spacetime to introduce a complex null
tetrad $(\ell^{a},n^{a},m^{a},\bar{m}^{a})$, and using (78), any such tetrad
can be used to construct the bivector basis
$(X^{ab},Y^{ab},Z^{ab},\bar{X}^{ab},\bar{Y}^{ab},\bar{Z}^{ab})$. However,
different tetrads are possible, and different choices result in different
bivectors. This appendix reviews some facts regarding the admissible tetrad
transformations, as well as their application to the construction of principal
null directions and to the Petrov classification. We also comment on relations
between the algebraic classifications of relativistic and Newtonian tidal
tensors.

It is explained in, e.g., Sect. 1.8(g) of Chandrasekhar (1998) that all null
tetrads which are normalized according to (76) can be generated from a single
example using three types of transformation. Type I transformations preserve
$\ell^{a}$, and in terms of an arbitrary complex scalar $a$, these are given
by

$\displaystyle\ell^{a}\mapsto\ell^{a},\qquad m^{a}\mapsto m^{a}+a\ell^{a},$
(138a) $\displaystyle n^{a}\mapsto
n^{a}+\bar{a}m^{a}+a\bar{m}^{a}+|a|^{2}\ell^{a}.$ (138b)

Type II transformations instead preserve $n^{a}$, and in terms of an arbitrary
complex scalar $b$, they are given by

$\displaystyle n^{a}\mapsto n^{a},\qquad m^{a}\mapsto m^{a}+bn^{a},$ (139a)
$\displaystyle\ell^{a}\mapsto\ell^{a}+\bar{b}m^{a}+b\bar{m}^{a}+|b|^{2}n^{a}.$
(139b)

Lastly, the type III transformations

$\ell^{a}\mapsto|c|\ell^{a},\quad n^{a}\mapsto\frac{1}{|c|}n^{a},\quad
m^{a}\mapsto\frac{c}{|c|}m^{a}$ (140)

preserve the directions (though not the scales) of both $\ell^{a}$ and
$n^{a}$, and can be applied for any nonzero complex scalar $c$. All three
types of tetrad transformation affect the bivector basis defined by (78). Type
I transformations do so via

$\displaystyle X^{ab}\mapsto X^{ab},\quad Y^{ab}\mapsto
Y^{ab}-\bar{a}Z^{ab}-\bar{a}^{2}X^{ab},$ (141a) $\displaystyle Z^{ab}\mapsto
Z^{ab}+2\bar{a}X^{ab},$ (141b)

type II transformations via

$\displaystyle X^{ab}\mapsto X^{ab}+bZ^{ab}-b^{2}Y^{ab},\quad Y^{ab}\mapsto
Y^{ab},$ (142a) $\displaystyle Z^{ab}\mapsto Z^{ab}-2bY^{ab},$ (142b)

and type III transformations via

$\displaystyle X^{ab}\mapsto cX^{ab},\quad Y^{ab}\mapsto c^{-1}Y^{ab},\quad
Z^{ab}\mapsto Z^{ab}.$ (143)

The bivector basis may be used to decompose a Weyl tensor $C_{abcd}$ into the
five Weyl scalars Stephani  _et al._ (2009); Chandrasekhar (1998)

$\displaystyle\Psi_{0}\equiv\tfrac{1}{4}C_{abcd}X^{ab}X^{cd},\quad\Psi_{1}\equiv\tfrac{1}{8}C_{abcd}X^{ab}Z^{cd},$
(144a)
$\displaystyle\Psi_{2}\equiv\tfrac{1}{16}C_{abcd}Z^{ab}Z^{cd}=-\tfrac{1}{4}C_{abcd}X^{ab}Y^{cd},$
(144b)
$\displaystyle\Psi_{3}\equiv-\tfrac{1}{8}C_{abcd}Y^{ab}Z^{cd},\quad\Psi_{4}\equiv\tfrac{1}{4}C_{abcd}Y^{ab}Y^{cd},$
(144c)

which are in general complex. Working in the opposite direction, one can
instead write the Weyl tensor in terms of the Weyl scalars and the given
bivectors:

$\displaystyle
C_{abcd}=2\operatorname{Re}\big{[}\Psi_{0}Y_{ab}Y_{cd}+\Psi_{1}(Y_{ab}Z_{cd}+Z_{ab}Y_{cd})$
$\displaystyle~{}+\Psi_{2}(Z_{ab}Z_{cd}-X_{ab}Y_{cd}-Y_{ab}X_{cd})$
$\displaystyle~{}-\Psi_{3}(X_{ab}Z_{cd}+Z_{ab}X_{cd})+\Psi_{4}X_{ab}X_{cd}\big{]}.$
(145)

Regardless, it follows from (141), (142), and (143) that type I tetrad
transformations preserve $\Psi_{0}$, type II transformations preserve
$\Psi_{4}$, and type III transformations preserve $\Psi_{2}$.

The algebraic structure of a Weyl tensor largely depends on its principal null
directions (PNDs). Recall that each PND may be defined as parallel to a
nonzero real null vector field $k^{a}$ which satisfies Wald (1984); Hall
(2004); Stephani  _et al._ (2009)

$k_{[a}C_{b]cd[e}k_{f]}k^{c}k^{d}=0.$ (146)

Noting that $\ell^{a}$ is tangent to a PND iff $\Psi_{0}=0$, PNDs may be
generated by using type II tetrad transformations to rotate $\ell^{a}$ until
the zeroth Weyl scalar vanishes. If $n^{a}$ is not already aligned with a PND,
which would occur only when $\Psi_{4}=0$, this method can in fact be used to
identify all PNDs. Applying it, (139), (142), and (144) show that any $b$
satisfying

$\Psi_{0}+4b\Psi_{1}+6b^{2}\Psi_{2}+4b^{3}\Psi_{3}+b^{4}\Psi_{4}=0$ (147)

is associated with a PND which is tangent to

$k^{a}=\ell^{a}+\bar{b}m^{a}+b\bar{m}^{a}+|b|^{2}n^{a}.$ (148)

Assuming that $\Psi_{4}\neq 0$, (147) is a quartic polynomial in $b$. This
implies that there are at most four distinct PNDs. The multiplicity of each
PND is defined to be equal to the algebraic multiplicity of the relevant root.
Equivalently, multiplicities can be determined by checking whether or not
(146) can be strengthened according to the Bel criteria Hall (2004); Griffiths
and Podolský (2010) which are listed in the second column of Table 6. The
final column of that table describes how multiple Weyl scalars must vanish
when $\ell^{a}$ is aligned with a degenerate PND; we take advantage of this in
Sect. III.3 in order to eliminate as many Weyl scalars as possible.

The Petrov type of the Weyl tensor is determined by the multiplicities of its
PNDs. Assuming that $C_{abcd}\neq 0$, there are five possibilities
Chandrasekhar (1998); Hall (2004); Griffiths and Podolský (2010); Stephani
_et al._ (2009), described as Petrov types I, II, D, III, and N:

  1. I.

Four multiplicity-1 PNDs.

  2. II.

One multiplicity-2 and two multiplicity-1 PNDs.

  3. D.

Two multiplicity-2 PNDs.

  4. III.

One multiplicity-3 and one multiplicity-1 PND.

  5. N.

One multiplicity-4 PND.

These cases are summarized in the second column of Table 2 on page 2.
“Generic” (or “algebraically general”) spacetimes are of Petrov type I; all
other possibilities are referred to as “algebraically special.”

Multiplicity | Weyl constraint | Vanishing $\Psi_{I}$
---|---|---
1 | $\ell_{[a}C_{b]cd[e}\ell_{f]}\ell^{c}\ell^{d}=0$ | $\Psi_{0}$
2 | $\ell_{[a}C_{b]cde}\ell^{c}\ell^{d}=0$ | $\Psi_{0},\Psi_{1}$
3 | $C_{bcde}\ell^{c}\ell^{d}=0$ | $\Psi_{0},\Psi_{1},\Psi_{2}$
4 | $C_{bcde}\ell^{c}=0$ | $\Psi_{0},\Psi_{1},\Psi_{2},\Psi_{3}$
Table 6: Summary of Bel criteria and vanishing Weyl scalars when $\ell^{a}$ is
aligned with PNDs of differing multiplicities.

The Kerr family of spacetimes are all type D, and the two PNDs which appear
there are associated with shear-free families of ingoing and outgoing null
geodesics. In the Schwarzschild case, these geodesics are purely radial and
twist-free; more generally, they are twisted by the rotation of the black
hole. Type N solutions include, e.g., gravitational plane waves, where the
lone PND is parallel to the rays of the gravitational wave. Although many
solutions are known with Petrov types II and III Griffiths and Podolský
(2010); Stephani  _et al._ (2009), most of their interpretations are
physically obscure. Nevertheless, there is a sense in which all Petrov types
appear generically when expanding the Weyl tensor at large distances in an
asymptotically-flat spacetime: The peeling property states that as one
approaches future null infinity along an outgoing null geodesic with
increasing affine parameter $r$,

$C_{abcd}=\frac{N_{abcd}}{r}+\frac{III_{abcd}}{r^{2}}+\frac{II_{abcd}}{r^{3}}+\frac{I_{abcd}}{r^{4}}+\mathcal{O}(r^{-5}),$
(149)

where $N_{abcd}$ is of Petrov type N, $III_{abcd}$ is of Petrov type III,
$II_{abcd}$ is either of Petrov type II or Petrov type D, and $I_{abcd}$ is of
Petrov type I; see Sect. 11.1 of Wald (1984).

The Petrov classification may be related to the classification of Newtonian
tidal tensors which is presented in Sect. II.3 above. First recall that a
Newtonian gravitational potential $\Phi$ can be associated with the
approximate line element Wald (1984); Poisson and Will (2014)

$ds^{2}=-(1+2\Phi)dt^{2}+(1-2\Phi)(dx^{2}+dy^{2}+dz^{2}).$ (150)

If $\nabla^{2}\Phi=0$, the corresponding Weyl tensor is

$C_{abcd}=2[\mathcal{E}_{d[a}(\eta_{b]c}+2t_{b]}t_{c})-\mathcal{E}_{c[a}(\eta_{b]d}+2t_{b]}t_{d})]$
(151)

through first order in $\Phi$, where $t_{a}\equiv-\nabla_{a}t$ and
$\mathcal{E}_{ab}\equiv-\nabla_{a}\nabla_{b}\Phi$ again denotes the Newtonian
tidal tensor. If that tidal tensor is of type 3, meaning that it admits a
doubly-degenerate eigenvalue, and if $\ell^{a}$ corresponds to the unit
spacelike eigenvector of $\mathcal{E}_{ab}$ which is associated with the non-
degenerate eigenvalue, the two null vectors $k_{\pm}^{a}\equiv
t^{a}\pm\ell^{a}$ both satisfy

$k^{a}_{\pm}k^{c}_{\pm}C_{abc}{}^{[d}k^{e]}_{\pm}=0.$ (152)

Referring to Table 6, this is the Bel criterion for a multiplicity-2 PND. Type