Observer dependence of angular momentum in general relativity and its relationship to the gravitational-wave memory effect

Eanna´ E.´ Flanagan and David A. Nichols Cornell Center for Astrophysics and Planetary Science (CCAPS), Cornell University, Ithaca, New York 14853 USA

We define a procedure by which observers can measure a type of special-relativistic linear and angular momentum (P a,J ab) at a point in a curved spacetime using only the spacetime geometry in a neighborhood of that point. The method is chosen to yield the conventional results in stationary spacetimes near future null infinity. We also explore the extent to which spatially separated observers can compare the values of angu- lar momentum that they measure and find consistent results. We define a generalization of parallel transport along curves which gives a prescription for transporting values of angular momentum along curves that yields the correct result in special relativity. If observers use this prescription, then they will find that the angular momenta they measure are observer dependent, because of the effects of spacetime curvature. The observer dependence can be quantified by a kind of general- ized holonomy. We show that bursts of gravitational waves with memory generically give rise to a nontrivial generalized holonomy: there is, in this context, a close relation between the observer dependence of angular momentum and the gravitational-wave memory effect.

PACS numbers: 04.20.Ha, 04.25.Nx

I. INTRODUCTION AND SUMMARY pick out a preferred Poincar´egroup with which to define a special-relativistic angular momentum. Instead of special-relativistic linear and angular mo- A. Angular momentum in general relativity mentum, one has an infinite set of conserved charges, one associated with each generator of the BMS group An interesting nonlinear feature of dynamical asymp- [7,8]. These charges can be computed from a surface totically flat solutions in general relativity is that there integral over any cross section of future null infinity, and is no canonical way to define a special-relativistic angu- the difference between the values of a charge at two dif- lar momentum at future null infinity. This result fol- ferent cross sections is given by the integral of a 3-form lows from the work of Bondi, van der Burg, and Met- (or “flux”) over the region of future null infinity between zner [1] and Sachs [2,3]. They showed that the group of the two cross sections. These BMS charges transform asymptotic symmetries of asymptotically flat spacetimes covariantly under BMS transformations, just as special- at future null infinity is not the Poincar´egroup but is relativistic linear and angular momentum transform un- instead an infinite-dimensional group now known as the der Poincar´etransformations, and they include the Bondi Bondi-Metzner-Sachs (BMS) group. Its structure is sim- 4-momentum. ilar to that of the Poincar´egroup: rather than being Some researchers have argued that it is necessary or de- a semidirect product of the Lorentz group with a four- sirable to give a definition of a preferred finite set of con- parameter Abelian group of spacetime translations, it is served charges, that would be more similar to the familiar a similar product of the conformal group on a 2-sphere conserved charges of special relativity [9–13]. An alterna- (which is isomorphic to the universal covering group of tive philosophy, which we espouse, is that all of the BMS arXiv:1411.4599v4 [gr-qc] 1 Feb 2016 the Lorentz group) with an infinite-dimensional commu- charges are physically relevant and that one should try tative group called the supertranslations [4]. The transla- to understand more deeply their physical nature, start- tions are a four-parameter normal subgroup of the larger ing with operational prescriptions by which they can be group of supertranslations, from which the Bondi energy- measured by asymptotic observers. momentum [5] is defined. The supertranslations, how- The purpose of this paper is an initial attempt to un- ever, make relativistic angular momentum (the charge derstand how BMS charges can be measured. For sim- associated with the Lorentz symmetries) behave differ- plicity, we take a “bottom-up” approach: we suppose ently in asymptotically flat spacetimes than in Minkowski that observers who are unaware of the BMS group at- space. In the latter, angular momentum depends only tempt to measure conserved charges and ask how the upon a choice of origin, which is a consequence of the charges they measure are related. (This is analogous to four spacetime translations in the Poincar´egroup; in the Newtonian observers who are ignorant of special relativ- former, angular momentum depends upon a smooth func- ity making measurements in Minkowski spacetime; their tion on the 2-sphere that parametrizes the supertrans- observations of Newtonian quantities are inconsistent be- lations in the BMS group. This property is typically cause of Lorentz contraction, etc.) Similarly, here we called the supertranslation ambiguity of angular momen- find that the charges measured by different observers us- tum (e.g., [6]). It arises because there is no unique way to ing special-relativistic methods are inconsistent due to 2 spacetime curvature. One of our goals is to character- text. In addition, there is a close relation between ize or interpret the inconsistencies between different ob- gravitational-wave memory and the BMS super- servers’ measurements, and relate the inconsistencies to translation that relates the shear-free cuts of a sta- other measurable quantities. tionary spacetime before a burst of gravitational For simplicity, we will restrict attention to measure- waves to those after the burst, as noted by, for ments made in stationary regions, and focus on how mea- example, Strominger and Zhiboedov [17]. There- surements made in two successive stationary regions can fore, our examples in Secs. III and IV also highlight be compared to one another. the role of BMS supertranslations in the observer More specifically, our main results are as follows: dependence of angular momentum in these simple contexts. • We give a local, operational definition of an “an- gular momentum” that can be measured by an ob- server at a point in a curved spacetime, using only B. Universality of observer dependence of angular information contained in the geometry in a neigh- momentum borhood of that point. The result is a pair of ten- sors (P a,J ab) at that spacetime point. This pre- As discussed above, if different observers attempt to scription is chosen to give the expected result in measure angular momentum in general relativity using stationary spacetime regions near future null infin- special-relativistic methods, they will disagree on the re- ity. See Sec.II for details. sults. In other words, angular momentum becomes ob- • We define a method by which two observers at two server dependent. In this section, we will show that this different points in a curved spacetime can compare observer dependence is a universal and local feature of the values of angular momenta that they measure. general relativity, independent of the choice of asymp- The philosophy we adopt is to imagine observers totic boundary conditions. (However, this observer de- who assume the validity of special relativity and pendence does not necessarily imply the existence of am- who make measurements based on this assump- biguities of the BMS type, as we discuss in Sec.V). We tion. We devise a method of comparison based will compute the observer dependence explicitly in a spe- on a generalization of parallel transport, which re- cific simple case, and show that it is closely related to duces to the correct method in flat spacetimes. In gravitational-wave memory. curved spacetimes, the method of comparison will Consider two observers A and B (Alice and Bob) in be curve dependent, and, in general, inconsistencies a flat region of spacetime, who are at rest with respect will arise when observers attempt to compare val- to one another and share a common inertial frame (t, x). ues of angular momenta. Therefore, from this point Suppose that they both measure the angular momentum of view, angular momentum inevitably becomes ob- of a nearby particle. The observer A will obtain the result server dependent in curved spacetimes. See Sec.III for details. JA = S + (xp − xA) × p, (1.1) • We identify a simple physical mechanism that ac- where S is the intrinsic angular momentum of the parti- counts for and explains the observer dependence in cle, p is its momentum in the observers’ common inertial simple cases. Specifically, two observers who mea- frame, xp is the location of the particle, and xA is A’s sure the change in angular momentum of a given location. Here we assume that A measures the angular source can disagree on that change, since they dis- momentum about her own location. Similarly, observer agree on where they believe the source used to be. B will measure an angular momentum about his own lo- They disagree on the source’s original location be- cation, and obtain the result cause of the gravitational-wave memory effect, the permanent relative displacement of observers due JB = S + (xp − xB) × p. (1.2) to the passage of a burst of waves [14–16]. The memory effect has, unbeknownst to the observers, If A and B compare their measurements, they will find displaced them by different amounts. This argu- a difference given by ment is given in more detail in Sec.IB below. JA − JB = −(xA − xB) × p, (1.3) • We argue that the close relation between gravitational-wave memory and the observer de- which is consistent with their measured relative displace- pendence of angular momentum is in fact very gen- ment xA − xB. eral, by using covariant methods and looking at Now suppose that a gravitational-wave burst of finite a number of examples (Secs.III andIV). While duration is incident on the observers. Thus the spacetime the connection between gravitional-wave memory consists of a flat region, followed by a gravitational-wave and angular-momentum ambiguity has often been pulse, followed by a subsequent flat region. We can adopt noted, our analysis shows the explicit and pre- transverse-traceless (TT) coordinates (T,Xi) to describe cise form of the relationship in a general con- the entire relevant spacetime region—before, during, and 3 after the burst of waves—which we chose to coincide with far as observer A is concerned, δxA vanishes, since she the inertial-frame coordinates (t, xi) before the burst. In is an inertial observer sitting at the origin of her iner- the TT coordinates and in the linearized approximation, tial frame. In particular, she is unaware of the effects the metric is of the gravitational-wave burst. (More generally, if the observer were accelerated by nongravitational forces, she 2 2 i j ds = −dT + [δij + hij(T − Z)]dX dX , (1.4) could measure δxA using an accelerometer carried with her. In the present context, the accelerometer reading where for simplicity we have specialized to a burst prop- would be zero.) agating in the +Z direction. At late times, the met- Inserting the assumption δx = 0 into Eq. (1.9) and ric perturbation becomes constant, h (T − Z) → h∞ = A ij ij subtracting a similar equation for B finally yields (constant), while at early times hij vanishes, and T = t, i i X = x . δJ − δJ = (x − x ) × p − (x0 − x0 ) × p . (1.10) We now extend the definition of the coordinates (t, xi) B A B A B A to the region after the burst by defining Using the gravitational-wave-memory formula (1.6) sim-   plifies this to i 1 ∞ j t = T, x = δij + hij X . (1.5) 2 1 ∞ δJB − δJA = − [h · (xB − xA)] × p . (1.11) These coordinates are then inertial coordinates after the 2 burst. Now the observers A and B are freely falling, Thus, A and B disagree on the change in angular mo- i which implies that their TT coordinate locations XA and mentum, by an amount which is proportional to the i XB are conserved. Hence their relative displacement in gravitational-wave memory. Essentially what has hap- the (t, xi) inertial frame after the burst is pened is that the two observers disagree on where the   particle used to be, because they have been displaced rel- 0 0 1 ∞ ative to one another by the gravitational-wave memory xA − xB = 1 + h · (xA − xB). (1.6) 2 effect, and they assume there is no such relative displace- ment. This is the standard formula for gravitational-wave mem- The result (1.11) will be rederived by a more formal ory. Here x0 and x0 are the locations of A and B in the A B and covariant computation in Sec.IVA below. inertial-frame coordinates after the burst. In the spacetime region after the burst has passed, the observers A and B can again measure the angular mo- C. Covariant description of angular momentum’s mentum of the particle, in the new inertial frame (t, xi). observer dependence: Methods of this paper The observer A obtains the result

0 0 0 0 JA = S + (xp − xA) × p , (1.7) While the example of the previous section was intu- itively useful and suggestive of the generality of the phe- where primes denote quantities as measured after the nomenon, it is important to have a covariant method for burst. We imagine that the particle’s spin and location comparing angular momenta at different times and as may have changed in the intervening period, but for sim- measured by different observers. There are, however, sev- plicity we assume that its momentum p has not. A sim- eral subtle aspects of how to define angular momentum ilar formula applies to the observer B, and once again if and how to compare values between different observers A and B compare their measurements, they will find a in curved spacetime. The remainder of this paper is de- difference given by voted to articulating a procedure that treats these issues

0 0 0 0 and allows observers to compare angular momentum co- JA − JB = −(xA − xB) × p, (1.8) variantly. We now give a brief sketch of the approach taken in this paper and summarize the organization of which is consistent with their measured relative displace- this paper’s sections. ment after the burst x0 −x0 . So far, there is no observer A B SectionII contains a local operational definition of a dependence. a ab Next, we assume that observer A wishes to compute linear and angular momentum (P ,J ) that can be mea- the change in angular momentum of the particle between sured by individual observers. For simplicity we will re- early and late times. This is given by fer to this pair simply as “angular momentum”. Section IIA defines the mathematical space in which the local 0 angular momentum lives: the dual space of the space δJA = JA − JA + δxA × p. (1.9) of Poincar´etransformations from the tangent space at Here δxA is the change in A’s location between early a point in spacetime to itself. The next part, Sec.IIB, and late times, and the third term is necessary to trans- defines a prescription whereby an angular momentum (a form the original angular-momentum measurement to particular element of this dual space) can be obtained her new location, so that she is subtracting angular mo- from local measurements of the Riemann tensor and its menta as measured about the same point. However, as derivatives. SectionIIC shows that the prescription for 4 measuring angular momentum yields the expected value tempt to measure the angular momentum of that source, in stationary spacetimes near future null infinity, Sec. by using only information about the geometry of space- IID describes the accuracy and errors of the algorithm time in the observer’s vicinity. Specifically, we describe in more general spacetimes, Sec.IIE notes that the algo- an algorithm by which an angular momentum can be rithm is not unique, and Sec.IIF focuses on the accuracy constructed from the Riemann tensor and its gradients with which the center-of-mass worldline can be measured. at the observer’s location. The algorithm we propose, SectionIII describes how to compare angular momen- moreover, is not unique, and the angular momenta ob- tum at different spacetime points. It defines a trans- tained will differ from one observer to another. However, port law in Sec.IIIA—which will be called the affine in a certain limit (Sec.IIC below), the angular momenta transport—that can be used for comparing angular mo- will become observer independent and characterize the mentum at two different spacetime points. SectionIIIB source. In more general situations, the nonuniqueness of explains in detail how to compare angular momentum at the algorithm will be unimportant, and the angular mo- two points using the affine transport. When the curve is a menta will be observer dependent. In these situations, closed loop, the transport law defines a generalized holon- the nature of this observer dependence will be physically omy operation, the basic properties of which are given in interesting, as discussed in the remaining sections of this Sec.IIIC. When the generalized holonomy reduces to the paper. identity, it indicates that there is a consistent (observer- independent) notion of angular momentum for different While the literature on angular momentum in general observers along the curve; when it does not, it provides a relativity is extensive and well developed (see, e.g., [19]), notion of the size of the observer dependence in angular our approach here introduces a new perspective, in that momentum between different observers along the curve. it focuses on a local and operational definition of a quan- SectionIIID shows that the generalized holonomy con- tity that observers can measure. Our procedure can be tains four independent pieces. When the closed curve is applied at any point in any spacetime (subject to a small generated by portions of worldlines of two freely falling number of local assumptions) and yields the expected re- observers, connected by spatial geodesics, each of the four sult in the limit of large distances from a source in an pieces can be interpreted as a kind of gravitational-wave isolated, linearized, stationary, vacuum spacetime. memory. In particular, for nearby geodesics, the gen- A measurement of the general type considered here, eralized holonomy contains the usual gravitational-wave where the angular momentum of a source is extracted memory. from measurements of the geometry of spacetime, has SectionIV gives two examples of the generalized holon- been carried out once in the history of physics: the mea- omy for an idealized spacetime consisting of a region of surement of the spin of the Earth to ∼ 20% by Gravity flat Minkowski space followed by a burst of linearized Probe B [20].1 gravitational waves with memory that propagates away leaving a second flat Minkowski spacetime region. The We start in Sec.IIA by defining a vector space that can first half of the section, Sec.IVA, reproduces the nearly be interpreted as the space of angular momenta for an ob- Newtonian argument of Sec.IB using the language of the server at a given point P in a curved spacetime. We give generalized holonomy. The next half of the section, Sec. the general algorithm for measuring angular momentum IV B, examines the more general example of a gravita- in Sec.IIB. In Sec.IIC we explain the motivation for tional wave expanded in symmetric trace-free multipoles this algorithm: namely, that it gives the expected result that is emitted radially outward from a pointlike source. in stationary linearized spacetimes near future null infin- The paper concludes in Sec.V. ity. We discuss the physical interpretation of the mea- Throughout this paper, we use units in which G = sured angular momentum in general spacetimes in Sec. c = 1, and we use the conventions of Misner, Thorne, IID, the nonuniqueness of the algorithm in Sec.IIE, and and Wheeler [18] for the metric and curvature tensors. the accuracy of the center-of-mass measurement in Sec. We use Latin letters from the beginning of the alphabet IIF. for general spacetime indices and Greek letters for those associated with specific coordinate systems. Latin letters from the middle of the alphabet (starting at i) will be reserved for spatial indices, and a 0 will denote a time index in the latter context.

1 Our measurement procedure is not quite the same as that used II. OPERATIONAL DEFINITION OF THE by Gravity Probe B. While both extract angular-momentum in- ANGULAR MOMENTUM OF A SOURCE AS formation from the spacetime geometry, our procedure uses the MEASURED BY A LOCAL OBSERVER curvature tensors in an infinitesimal region about a spacetime point, while Gravity Probe B uses information about the geom- etry in the vicinity of an entire orbit in addition to information In this section, we describe a method by which an ob- about asymptotic inertial frames provided by the direction to a server in the vicinity of some source of gravity can at- guide star. 5

A. Definition of a linear space of angular momenta geodesic deviation equation. Similarly, the mag- at a given point in spacetime netic pieces can be measured by monitoring the relative angular velocity of gyroscopes induced by

At a point P in a spacetime (M, gab), let TP (M) denote frame dragging [21]. By repeating these measure- the tangent space. Let GP be the Poincar´egroup that ments at nearby spacetime points, the observer can acts on TP (M), that is, the space of affine maps from in principle also measure the components of the TP (M) to itself that preserve the metric. Since GP is gradient ∇aRbcde. a Lie group, it has an associated Lie algebra GP that 2. Compute the curvature invariants consists of infinitesimal Poincar´etransformations. The ∗ abcd corresponding dual space GP , the space of linear maps K1 ≡ RabcdR , (2.2a) from G to the real numbers, is the space of linear and P K ≡ ∇ R ∇aRbcde. (2.2b) angular momenta at the event P. 1 a bcde

To see this explicitly, consider an affine coordinate sys- We assume that K1 > 0 and K1 > 0. Then, com- a tem x on TP (M). Such a coordinate system is associ- pute quantities M and r using ated with a choice of basis vectors ~ea and a fixed vector √ a 2 ~x0 such that the coordinates x of a vector ~x are given 15 5K1 a M = , (2.3a) by ~x = ~x0 + x ~ea. In this coordinate system, the maps 3/2 4K1 in GP have the usual form of a Poincar´etransformation: a a b a a r x → Λ bx + κ . Here Λ b is a Lorentz transforma- 15K1 a r = . (2.3b) tion and κ is a translation. The infinitesimal versions K1 of these maps in GP have the same form, but with in- a a a a finitesimal κ and with Λ b = δ b + ω b, where ωab is an infinitesimal antisymmetric tensor. Now consider the 3. Repeat the above measurements and computations ∗ at nearby2 spacetime points, thus measuring the dual space, GP . A general linear map from GP to real numbers can be written as gradient ∇ar of the quantity r.

1 4. Assuming that the vector ∇ar is spacelike, de- (κ , ω ) → P aκ − J abω (2.1) a a ab a ab fine the unit vector n in the√ direction of ∇ar by 2 a −1 a a n = N ∇ r where N = ∇ r∇ar. Compute for some vector P a and some antisymmetric tensor J ab. the quantity ∗ Therefore, elements of GP can be parametrized in terms ya = −rna (2.4) of pairs of tensors (P a,J ab), a linear momentum and an angular momentum. The angular momentum J ab trans- which the observer interprets as the displacement forms under changes of origin in TP (M) as angular mo- vector from her own location to the center-of-mass ~ ab ab [a b] mentum should: for ~x0 → ~x0 +δx, J → J +2P δx . worldline of the source. ab The angular momentum J would be interpreted by an 5. Compute the symmetric tensor H from observer at P as angular momentum about a point which ab c d is “displaced from P by an amount ~x0,” even though such Hab = Racbdn n . (2.5) a displacement is ambiguous in general relativity. Compute the eigenvectors ζa and eigenvalues λ of b this matrix from Habζ = λζa. From the definition (2.5), one of the eigendirections will be ζa = na B. Definition of the general prescription for with corresponding eigenvalue λ = 0. We as- measuring an angular momentum sume that there is at least one eigenvector with a strictly positive eigenvalue, and we denote the In this section, we define a prescription for how an ob- eigendirection corresponding to the largest eigen- server at an event P can measure an element of the dual value by ta. It follows that this vector is orthogonal space G∗ of linearized Poincar´etransformations on the a P to na, t na = 0. tangent space at P. The prescription requires several as- sumptions about the geometry near P, as discussed fur- 6. Assuming that the vector ta is timelike, define a ther below, and therefore it is applicable only in certain unit, future-directed timelike vector ua by ua = −1 a 2 a situations. N t where N = −tat and the sign of N is The steps of the prescription are as follows: chosen so that ua is future directed. The linear momentum is then given by P a = Mua. 1. Measure all the components of the Riemann tensor Rabcd and of its gradient ∇aRbcde at the event P. The electric pieces of the Riemann tensor in the 2 Equivalently, measure the Riemann tensor and its first two observer’s frame can be measured by monitoring derivatives at P; the quantity ∇ar can then be expressed in the relative acceleration of test masses using the terms of these using Eqs. (2.2) and (2.3). 6

2 α α β β 7. Compute the curvature invariant and define the distancer ˆ byr ˆ =p ˆαβ(x − zˆ )(x − zˆ ). Finally, we define the unit vectorn ˆα by 1 ab cdef K2 ≡ abcdR ef R . (2.6) 2 1 β β nˆα = ∇αrˆ = pˆαβ(x − zˆ ). (2.10) From this compute a spin vector Sa by rˆ

r7K 1 In terms of these quantities, the total angular momen- a 2 a 4 abcd e αβ α S = n + r  ubncHdeu . (2.7) tum Jˆ about the point x is 288M 2 3 Jˆαβ = αβγδuˆ Sˆ +y ˆαPˆβ − yˆβPˆα, (2.11) 8. Compute the angular momentum J ab by γ δ

ab abcd a b b a wherey ˆα =p ˆα (ˆzβ −xβ) = −rˆnˆα is a vector which points J =  ucSd + y P − y P . (2.8) β from the field point xα to the center-of-mass worldline. a ab ∗ Finally from (P ,J ) compute an element of GP The metric perturbation is using the definition (2.1) specialized to ~x0 = 0. 2Mˆ 4 h (xα) = (η + 2ˆu uˆ ) − uˆ  Sˆγ nˆδuˆ , Although the procedure is somewhat lengthy, these αβ rˆ αβ α β rˆ2 (α β)γδ eight steps define a method for computing an element (2.12) ∗ of GP from the Riemann tensor and its derivatives at a which is equivalent to the stationary limit of the lin- point P. earized metric perturbation in [22], after one makes the i i substitution that uα = (dt)α + viδ α, where (t, x ) are harmonic coordinates. Finally, the Riemann tensor is C. Motivation for the prescription: Stationary linearized spacetimes near future null infinity 1 R = (h + h − h − h ) , (2.13) αβγδ 2 αδ,βγ βγ,αδ αγ,βδ βδ,αγ We now explain the motivation for the choice of pre- scription described in the last subsection: it is designed to where give the expected answer in a certain limit. Specifically, 2Mˆ we consider spacetimes that are stationary and free of hαδ,βγ = (ηαδ + 2ˆuαuˆδ)(3ˆnβnˆγ − pˆβγ ) (2.14) matter in the neighborhood of an observer and for which rˆ3 the sources are sufficiently distant from the observer that 12 ˆλ ν µ µ − uˆ(αδ)λµν S uˆ [(5ˆnβnˆγ − pˆβγ )ˆn − 2ˆn(βpˆγ) ] . the metric can be described by a linearized multipolar ex- rˆ4 pansion. For these distant sources, the dominant terms in We now compute the angular momentum that an ob- the multipolar expansion will be the mass monopole and α the current dipole or spin, with the remaining multipoles server at x would measure in this spacetime, using the being negligible. In this situation, the measured P a and algorithm described in the last subsection. The curvature J ab coincide with the conserved charges of the spacetime invariants (2.2) are given by to a good approximation, as we now show. This require- ment does not fix the prescription uniquely, but we shall 48Mˆ 2 argue in Sec.IIE below that the nonuniqueness is not K1 = 6 [1 + O()] , (2.15a) significant. rˆ We start by writing down a Poincar´ecovariant expres- 720Mˆ 2 K = [1 + O()] , (2.15b) sion for the metric for stationary linearized spacetimes, 1 rˆ8 keeping only the first two multipoles. This metric can 2 α β where for ease of notation we have defined be written as ds = (ηαβ + hαβ)dx dx , where we have specialized to Lorentzian coordinates xα for the back- ! ! Sˆ2 Mˆ ground metric, and indices are raised and lowered with O() ≡ O + O . (2.16) α α 2 2 ηαβ. Let the 4-momentum of the source be Pˆ = Mˆ uˆ , Mˆ rˆ rˆ whereu ˆa is the 4-velocity and Mˆ is the rest mass. (We use a hatted notation for these quantities to distinguish Note that correction terms linear in the spin are forbid- them from the quantities, defined in the previous sub- den by parity considerations. Computing M and r using section, that the observer measures.) Let the intrinsic Eqs. (2.3) yields α α angular momentum of the source be Sˆ with Sˆ uˆα = 0, and letz ˆα be a point on the center-of-mass worldline M = Mˆ [1 + O()], r =r ˆ[1 + O()]. (2.17) of the source. Let xα be the point at which we want to evaluate the metric perturbation hαβ. We define the Similarly by evaluating the gradient of r according to projection tensor steps 3 and 4, we find

α α α α pˆαβ = ηαβ +u ˆαuˆβ (2.9) n =n ˆ [1 + O()] , y =y ˆ [1 + O()] . (2.18) 7

Next, we evaluate the symmetric tensor (2.5) using the larger ˆ. By continuity, therefore, they will also be satis- expression (2.13) for the Riemann tensor. The result is fied in regions of spacetimes that are sufficiently close to this case. We now discuss in more detail how the mea- Mˆ surement procedure applies to these more general situa- H = − (η + 3ˆu uˆ − nˆ nˆ ) αβ rˆ3 αβ α β α β tions and spacetimes. 6 There are a number of physical effects that can make + uˆ  Sˆγ nˆδuˆ . (2.19) rˆ4 (α β)γδ the spacetime geometry measured by observers differ from the idealized case discussed above of asymptotic re- Because of the symmetries of the Riemann tensor, the gions in linearized stationary spacetimes with two multi- tensor H is symmetric and has nα as an eigenvector αβ poles. The effects that we consider include nonlinearities, with its corresponding eigenvalue being identically zero. higher-order multipoles, nonisolated systems, and non- The three remaining eigenvectors at leading order in an stationarity. We now estimate the size of these effects in expansion in 1/rˆ areu ˆα,  Sˆβnˆγ uˆδ, and a third vec- αβγδ more general contexts and thereby determine both when tor that is orthogonal to those two as well asn ˆα. The we might expect the assumptions listed above to break eigenvalues associated with these eigenvectors are (again down and also when the algorithm yields physically sen- ˆ 3 at leading order in an expansion in 1/rˆ) 2M/rˆ and a sible results. The various effects are: repeated eigenvalue equal to −M/ˆ rˆ3 for the latter two, respectively. Therefore, if we follow step 5 and choose • Nonlinearities: Our analysis above assumed that the normalized eigenvector corresponding to the largest the spacetime could be described as a linear per- eigenvalue, we obtain uα =u ˆα[1 + O()]. It follows that turbation about Minkowski spacetime. For an iso- lated, stationary source in an asymptotically flat P α = Pˆα[1 + O()]. (2.20) spacetime, there will be corrections to the met- ric arising from nonlinearities. These nonlinearities Next, from Eqs. (2.13) and (2.14), the curvature in- will give corrections to the metric perturbation hαβ variant (2.6) is given by that scale3 as

2 ! ! 2 ! ˆ 2 Mˆ Mˆ Sˆ Sˆ 288M α O ,O ,O . (2.23) K = (Sˆ nˆ )[1 + O()] . (2.21) 2 3 4 2 rˆ7 α rˆ rˆ rˆ

Inserting this equation and the expression (2.19) for Hαβ The form of these corrections can be found, for into the formula (2.7) for the intrinsic angular momen- example, from the leading nonlinear terms in the tum, we determine post-Newtonian expansion of the metric (given in, e.g., [23]). These corrections will be small4 com- α α S = Sˆ [1 + O()]. (2.22) pared to the leading-order terms ∼ M/ˆ rˆ and ∼ S/ˆ rˆ2 in the metric perturbation (2.12), as long asr ˆ Thus, the algorithm successfully recovers the linear mo- ˆ p ˆ ˆ2/3 ˆ −1/3 mentum and intrinsic angular momentum of the space- is large compared to M, S, and S M . For time. Also, from Eqs. (2.8), (2.11), and (2.18), we find sufficiently larger ˆ, therefore, the effects of nonlin- that J αβ = Jˆαβ[1 + O()], so that the algorithm yields earities can be neglected. the total angular momentum of the source about the ob- • Higher-order multipoles: Our analysis in Sec.IIC α server’s location x . above included only the mass and spin and ne- glected higher-order mass and current multipoles. However, as is well known, the effect of these mul- D. Physical interpretation of the measured linear tipoles will be small at sufficiently larger ˆ. The and angular momenta in more general contexts

In the previous subsections, we showed that an ob- 3 server that is sufficiently distant from a stationary source This will be true in suitable coordinates, for which the limit Mˆ → 0 of the metric at fixed S/ˆ Mˆ is the Minkowski metric of gravity can measure that source’s linear and an- in Minkowski coordinates (for example, Cartesian Kerr-Schild gular momentum to a good approximation, using just coordinates in the Kerr spacetime). For more general coordi- the spacetime geometry in the vicinity of the observer. nates (such as Boyer-Lindquist coordinates), other terms can oc- The measurement procedure required several assump- cur that are larger than some of the terms in Eq. (2.23) [e.g., 2 2 2 tions about that spacetime geometry: (i) the curvature Sˆ /(Mˆ rˆ )]. These larger terms are gauge effects, and they can invariants (2.2) needed to be positive, (ii) the vector ∇ r be ignored for the argument given here. a 4 An exception is the term ∼ Mˆ 2/rˆ2 which will be comparable to needed to be spacelike, (iii) the tensor Hab needed to the ∼ S/ˆ rˆ2 term in the metric (2.12) when Sˆ ∼ Mˆ 2. One might have at least one strictly positive eigenvalue, and (iv) the expect that this term would give rise to fractional corrections of corresponding eigenvector needed to be timelike. These order unity to the measured momentum and angular momentum; assumptions are satisfied for linearized stationary space- the corrections, however, are suppressed, because the S/ˆ rˆ2 term times described by just two multipoles at sufficiently is parity odd while the Mˆ 2/rˆ2 term is parity even. 8

dominant correction to the metric perturbation in the angular dependence to disentangle the effects the parity-even sector will be of the locally produced curvature ∼ M/ˆ rˆ3 from the curvature Eij produced by distant sources. Q δh ∼ , (2.24) When nonlinearities are included, however, there αβ rˆ3 is an unavoidable ambiguity: the linear and angu- where Q is the mass quadrupole. Using the esti- lar momenta of individual objects cannot be de- mate Q ∼ Mˆ L2, where L is the size of the source, fined in general. We can estimate the ambigui- we see that this correction will be small compared ties from nonlinearities using the fact that different to M/ˆ rˆ in the regime definitions of the “mass of an object” in post-1- Newtonian theory differ by a quantity of order the 2 rˆ  L. (2.25) tidal-interaction energy, QijEij, where Qij ∼ Mˆ L is a mass quadrupole. Therefore, objects of mass Similarly, in the parity-odd sector, the dominant Mˆ , size L and separated by distances ∼ D have an correction will be uncertainty or ambiguity in their masses of order5 S δh ∼ , (2.26) ∆M/ˆ Mˆ ∼ Mˆ L2/D3 . (2.27) αβ rˆ3 The measurement method discussed in Sec.IIB ˆ where S ∼ SL is the current quadrupole. This above will be subject to this ambiguity; however, correction will be small compared to the spin term in many situations the ambiguity will be negligible. in Eq. (2.12) wheneverr ˆ  L. Therefore, in the regime (2.25), corrections to the measured linear • Nonstationary systems: For dynamical, radiating and angular momentum P α and J αβ will be small. sources, it is immediately clear that our measure- We note that in the context of linearized gravity, ment procedure will not be applicable in general. it is possible in principle to measure P α and J αβ The reason is that the Weyl tensor for radiated accurately even in the regimer ˆ ∼ L, by using mea- gravitational waves falls off at larger ˆ as 1/rˆ, whereas the static piece of the Weyl tensor associ- surement procedures more sophisticated than those 3 envisaged in this paper. As is well known, in lin- ated with the mass and spin falls off as 1/rˆ . There- earized gravity the charges P α and J αβ can be ex- fore, at sufficiently larger ˆ, if an observer measures tracted unambiguously from the metric perturba- the Riemann tensor and its derivatives at her loca- tion using surface integrals [18]. Therefore, a family tion, her result will be dominated by the radiative of observers distributed over the surface of a sphere, pieces of the metric, and the measurement method who make measurements of the spacetime geome- of Sec.IIB above will fail. try in their vicinity and compare notes in a suitable As discussed in the Introduction, however, the mea- way, can measure P α and J αβ with high accuracy. surement method can still yield interesting infor- In this paper, we will not need to consider such mation about dynamical systems, for an intermit- nonlocal measurement procedures, because the is- tently stationary spacetime (by which we mean a sues we want to explore are all present in the regime spacetime which is stationary at early times and (2.25) in which our local measurement procedure is again at late times). Observers can apply the mea- sufficient. surement procedure at early and at late times and then attempt to compare their results. This sce- • Nonisolated systems: So far we have considered nario is discussed in detail in the remaining sections observers near isolated sources in asymptotically of the paper. flat spacetimes. Suppose, however, that there are As an aside, we note that we can classify nonsta- also distant sources, or that the spacetime is not tionary systems into two types. The first is what we asymptotically flat. In linearized gravity, the ef- will call asymptotically linear systems, that is, sys- fect of distant sources can be quantified in terms tems for which the linear approximation is valid6 at of the tidal tensor Eij (the electric components of the associated Riemann tensor). The correspond- ing fractional corrections to the linear momentum measured by observers using the procedure of Sec. 5 This estimate is valid for generic sources which have a nonvanish- IIB will be of order ∼ Erˆ3/M.ˆ Similarly the frac- ing intrinsic quadrupole moment. It is not valid for spherically tional corrections to the angular momentum will be symmetric sources whose intrinsic quadrupole vanishes. For such of order ∼ Brˆ4/S,ˆ where B is the magnetic tidal sources, the scaling of the mass ambiguity can be estimated from ij ˆ 5 3 tensor. These effects limit the accuracy and util- the quadrupoles Q ∼ ML /D induced by tidal interactions; it is of order ∆M/ˆ Mˆ ∼ Mˆ L5/D6. ity of our measurement method of Sec.IIB above. 6 Here the assumption is that the linear approximation is valid in Within the context of linearized gravity, it is possi- a neighborhood of some two-sphere which encloses the source, ble to circumvent this difficulty using the nonlocal not the weaker assumption the linear approximation is valid in measurement method discussed above, which uses a neighborhood of some observer. 9

sufficiently larger ˆ. For these systems, one can de- In particular, the displacement vector yα from the ob- fine unambiguous linear and angular momenta us- server to the center-of-mass worldline [cf. Eq. (2.8) above] ing surface integrals, and they can be measured us- will be measured with this accuracy: ing the nonlocal measurement procedure discussed " !# above. Our local measurement procedure can work Mˆ yα =y ˆα 1 + O . (2.29) for such systems, but only if L  rˆ  λ, where rˆ λ is the wavelength of the radiation. The second

type of system, asymptotically nonlinear systems, α are those for which the linear approximation is not However,y ˆ is of orderr ˆ, and, therefore, the error in the valid at larger ˆ. These are the systems for which measurement is of order the BMS asymptotic symmetry group is most rele- α ˆ vant. Neither our local measurement procedure nor δy ∼ M. (2.30) the nonlocal measurement procedure based on sur- face integrals of linearized theory apply to systems This error is large; it is of the same order as the maximum in this regime.7 displacements caused by gravitational-wave memory ef- fects.8

E. Nonuniqueness of the measurement algorithm III. AFFINE TRANSPORT AND The algorithm discussed above is not uniquely deter- GENERALIZED HOLONOMY: PROPERTIES mined by the requirement that it give the correct answer AND APPLICATION TO ANGULAR in linearized stationary spacetimes with two multipoles, MOMENTUM because the information about the linear and angular mo- mentum of the spacetime is encoded redundantly in the We now turn to the question of how two observers at Riemann tensor and its first two derivatives at any point. different locations in a curved spacetime can compare Therefore, there are several methods that can be used to values of linear and angular momentum. The philosophy extract these momenta. For example, Eq. (2.4) could be we adopt is to imagine that the observers attempt to replaced by ya = −∇ar2/2, which would give the same compare values using the same methods they would use result to leading order. in special relativity (i.e., in the absence of gravity). In stationary linearized spacetimes with two multi- The first part of this section introduces a curve- poles, there is a unique and accepted definition of the dependent transport law, which we call affine transport linear and angular momentum of the spacetime; there- and which serves as the basis for our method of compar- fore, any nonuniqueness or ambiguities in the measure- ing angular momentum. The next subsection describes ment procedure must vanish in this limit as the measure- how the affine transport can be used to compare values ment is taken at large distances from the source. More of the angular momentum defined at different spacetime specifically, this implies that the effects of these ambi- points. The final subsection describes the affine trans- guities all scale as 1/r as r → ∞ (or as 1/v, where v port around a closed curve, which we call the generalized is a null coordinate with goes to infinity at future null holonomy, and it explains its relation to the inevitable ob- infinity). Most importantly, they are small compared to server dependence of angular momentum in curved space- the observer dependence of angular momentum that we times. discuss in the remainder of the paper (that characterized by generalized holonomies, which we show gives rise to finite effects in the limit v → ∞). 8 It is possible, however, to modify the measurement method to increase the accuracy as follows. Modify the definitions of M F. Accuracy of measurement of the center-of-mass and r in Eqs. (2.3) to worldline √ " √ # 15 5K2 15 3(K )3/2 M = 1 1 − 1 , (2.31a) 3/2 4K1 The procedure discussed above allows an observer to 4K1 measure the angular momentum of the spacetime about s " √ # 15K 5 3(K )3/2 his own location to an accuracy of  = M/ˆ rˆ: r = 1 1 − 1 , (2.31b) K1 4K1 αβ ˆαβ J = J [1 + O()] . (2.28) and leave the rest of the measurement algorithm unaltered. Then the fractional error in measurement of yα is decreased to O(Mˆ 2/rˆ2), and the errors in yα vanish as the observers approach future null infinity. This modified algorithm is derived from the 7 Except to the extent that measurements before and after the expressions for the curvature invariants of the Kerr spacetime nonstationarity can probe effects of the nonstationarity, as we in Boyer-Lindquist coordinates and therefore yields the Boyer- discuss in the remainder of this paper. Lindquist radial-coordinate values at the observers’ locations. 10

~ A. Definition of an affine transport law 6. Finally, for curves in a flat spacetime, ∆ξPQ is just the vectorial displacement Q~ − P~ in any inertial In this section, we define a transport law that can be coordinate system. In particular, it vanishes for used to transport vectors along curves and which is a gen- closed curves in a flat spacetime, because, as we eralization of parallel transport. Let C be a curve between show later, it is only nontrivial in the presence of the spacetime points P and Q, and let the curve have spacetime curvature. ~ tangent vector k. Next, define a map χC from TP (M) to We will use these properties frequently in the calcula- TQ(M) through the solution of the differential equation tions in the remainder of this paper. ~ ~ ∇~kξ = αk . (3.1) Here α is a dimensionless constant. Namely, starting B. Application to a curve-dependent definition of ~ angular-momentum transport from an initial condition ξP in TP (M), we solve the dif- ~ ~ ferential equation to obtain the value ξQ of ξ at Q. The Using the affine transport law, we can define a method image of ξ~ under the map χ is then defined to be ξ~ . P C Q of comparing the local values of angular momentum at Since we are not aware of a name for this specific trans- two different spacetime points, by transporting angu- port law, we will call it the affine transport of the vector lar momenta from one spacetime point to another, in ξ~ along the curve C with tangent ~k. This map satisfies a curve-dependent manner. six important properties that are listed below: ∗ ∗ We define a map from GP to GQ that depends on a choice of curve C that joins these two points. Because the 1. It is independent of the choice of parametrization elements of G∗ act on maps in G [those from the tangent along the curve (which follows because both sides of P P space T (M) to itself], a natural map is one based on the equation are linear in the tangent to the curve P the affine transport of elements in G to elements of G . ~k). P Q Namely, for hP ∈ GP , the corresponding hQ ∈ GQ is 2. When two curves are composed, the composition defined by of maps is equivalent to the map on the composed −1 hP = χC ◦ hQ ◦ χC , (3.3) curve (i.e., if C = C1 ∪ C2 then χC = χC1 ◦ χC2 ). 3. For a fixed curve, C, Eq. (3.1) is a linear differential where χC is the affine transport along C. For an element q ∈ G∗ , therefore we define the corresponding element equation in ξ~. The solution for given initial data, P P q ∈ G∗ by therefore, can be expressed as the sum of two terms: Q Q the first term is the solution of the homogeneous q (h ) = q (χ−1 ◦ h ◦ χ ) = q (h ) . (3.4) differential equation (parallel transport) with the Q Q P C Q C P P same initial data, and the second is the solution of To recover the correct transformation properties of angu- the inhomogeneous differential equation with zero lar momentum under displacements with this definition, initial data. The complete solution is it is necessary to choose the value ξa¯ = Λ a¯ ξa + α∆ξa¯ , (3.2) Q PQ a P PQ α = −1 (3.5) where Λ a¯ denotes the parallel transport opera- PQ a of the parameter in the definition (3.1) of the function tion from P to Q and ∆ξa¯ is the inhomogeneous PQ χ , which we now show by writing this mapping from the solution for α = 1. The notation here is that over- C angular-momentum space G∗ to the angular-momentum lined indices are associated with the point Q and in- P space G∗ in a more explicit notation. dices without extra adornment are associated with Q To do so, let us represent elements of the algebra h P. P as pairs

4. It follows that (unlike parallel transport) affine P P transport does not preserve the norm of the trans- hP ↔ (κa , ωab) (3.6) ported vector.

5. For geodesic curves, one can show that the inho- ~ affine parameter along a geodesic curve with λ = 0 corresponding mogeneous part of the solution ∆ξPQ is just the ~ 9 to P and λ = 1 be the point Q, and denote the directional deriva- tangent to the curve at the point Q (i.e., αkQ). 0 tive along the geodesic by ∇~k = D/Dλ. For a point P between a¯ a¯ a a¯ P and Q, one can confirm that ξP0 = ΛPP0 aξP + αλkP0 is the a¯ a solution along the geodesic curve, because ∇~kΛPP0 aξP = 0 ~ and ∇~kk = 0. Evaluating the expression at λ = 1, one finds that 9 ~ The calculation which shows this is short: Let λ ∈ [0, 1] be an the inhomogeneous part of the solution is αkQ. 11 and the map between algebras at different points P and C. Generalized holonomy: A measure of observer Q, χC, as dependence of angular momentum χ ↔ (α∆ξa¯ , Λ a¯ ) , (3.7) C PQ PQ a For closed curves starting from a point P, the affine where the quantities in this equation are exactly those ap- transport around the curve defines a generalized holon- pearing in Eq. (3.2). The transformation rule for the al- omy, a map from the tangent space at P to itself. For flat gebra elements (3.3) then has the representation in terms spacetimes, the generalized holonomy is always the iden- of these pairs as tity map. Specializing the result (3.12) to closed curves P P (when Q is the same point as P), yields the mapping (κa , ωab) = a¯ Q a¯ ¯b Q b a¯ ¯b Q ab a b cd [c d] (ΛPQ aκa¯ + αΛQP aΛQP bωa¯¯b∆ξPQ, ΛQP aΛQP bωa¯¯b) , J → Λ cΛ d(J − 2∆ξ P ) . (3.16) (3.8) Thus, if there is a nontrivial holonomy of parallel trans- which respects the multiplication rule for a semidirect- port or a nonzero inhomogeneous solution, then observers product structure. In the above expression, we have used along the curve will find that angular momentum is ob- a a a¯ the notation ∆ξPQ = ΛPQa¯ ∆ξPQ and the fact that server dependent. The extent to which a generalized a¯ a¯ ΛQP a and ΛPQ a are related in accord with the typ- holonomy is nontrivial is a measure of how much space- ical notation for the inverse of the parallel propagator. time curvature is an obstruction to separated observers Keeping in mind the representation of the maps qP as arriving at a consistent definition of angular momentum. 1 As a simple example of this generalized holonomy, con- q (h ) = P a κP − J abωP (3.9) P P P a 2 P ab sider an infinitesimal quadrilateral starting from a point P with legs given by ua and va where  is small. The given in (2.1), we can then take the definition of the trans- quadrilateral is traversed first in the direction of ua, then formation property of angular momentum in Eq. (3.4) va, then −ua, then −va. If we start at P with some initial above and substitute in the result of (3.8); by equating vector ξa and solve the transport equation (3.1) around the coefficients of ωQ and κQ, we find that the angular a¯¯b a¯ the loop, a relatively straightforward calculation shows momenta at the two points are related by that the homogeneous part of the solution is a¯¯b a¯ ¯b  ab [a b] JQ = ΛQP aΛQP b JP + 2α∆ξPQPP , (3.10) a 2 a b c d 3 ξ +  R bcdξ v u + O( ). (3.17) and the corresponding momenta are related by This is the usual expression for the holonomy around a a¯ a¯ a PQ = ΛPQ aPP . (3.11) small loop. The inhomogeneous part of the solution is We, therefore, see that to have the usual transformation ∆ξa = 1 3Ra vcud(ub + vb) + O(4). (3.18) law for angular momentum 2 bcd

¯ ¯  [a b] A more detailed calculation is given in Ref. [24]. Thus, J a¯b = Λ a¯ Λ b J ab − 2∆ξ P , (3.12) Q QP a QP b P PQ P while the holonomy of parallel transport is proportional we must choose α = −1. An alternative and simpler to the Riemann tensor contracted with the area of the formulation of the transport law given by Eqs. (3.12) and quadrilateral, the generalized holonomy contains an ad- (3.11) is discussed in AppendixA. ditional term proportional to the Riemann tensor con- If we decompose the angular momentum J ab into an tracted with both the area and the perimeter of the re- intrinsic spin Sa and a displacement vector ya using the gion. definitions 1 a ab D. Relation between generalized holonomy and y = − 2 J Pb , (3.13a) M gravitational-wave memory a 1 a b cd S = bcd P J , (3.13b) 2M In this section, we give a precise and covariant def- a then from Eqs. (3.11) and (3.12), the fact that P Pa = inition of an observable that can be interpreted as a 2 a −M , and that y Pa = 0, we can show after some algebra “gravitational-wave memory” generalized to an arbitrary that the spin is parallel transported just like the linear spacetime. We then show that the generalized holonomy momentum, around a suitably constructed loop contains information Sa¯ = Λ a¯ Sa , (3.14) about this covariant gravitational-wave memory, show- Q PQ a P ing a very general relationship between these two observ- while the displacement vector transforms as ables. However, we also show that the generalized holonomy ya¯ = Λ a¯ yb − ∆ξa
. (3.15) Q QP a P PQ contains additional information and specifically contains Additional properties of the affine transport for closed three other independent pieces, each of which could arise curves are discussed next. as a kind of “memory” effect due to the passage of a 12

burst of gravitational waves. The first is a difference in The inhomogeneous part of the generalized holonomy proper time measured by two observers (a gravitational about the loop C is given by redshift effect). The second is a relative boost of two ~ h ˆi ˆi i initially comoving observers. The third is a relative ro- ∆ξ = ξB(τ1) − ξB(τ2) ~eˆi + (∆τB − ∆τA)~wB tation of the inertial frames of two observers. In the ˆi limit of nearby geodesics at large distances from a source + ξB(τ1)(Λ · ~eˆi − ~eˆi) . (3.20) emitting a burst of gravitational waves with memory, Here ∆τA = τ2 − τ1 is the interval of A’s proper time these effects reduce to a combination of more familiar between P and R, and ∆τB is the interval of B’s proper notions of gravitational-wave memory arising from solu- time between Q and S. The quantity Λa is the usual tions of the equation of geodesic deviation and of differ- b holonomy around the loop C. Finally ~wB is the 4-vector ential frame dragging and the difference in proper time at R obtained by parallel transporting B’s 4-velocity of nearby geodesics [25]. (We are also investigating the ~uB(τ2) at S along the spatial geodesic to R. Equiva- relationship between the new gravitational-wave memory lently, it can be obtained by acting with the holonomy of Pasterski et al., [26] and the memory effects quantified a a b around the loop on A’s 4-velocity at R, wB = Λ buA(τ2). by the generalized holonomy in [25].) We now describe a The first term in the generalized holonomy (3.20) can calculation that elucidates the relationship between the be interpreted as (a generalization of) the gravitational- generalized holonomy and the ordinary memory plus dif- wave memory effect. It is the change in the relative dis- ferences in proper time, relative rotations, and relative placement of the observers A and B, as seen by A in her boosts. Fermi normal coordinates, when A and B are initially Consider two freely falling observers A and B in an comoving. The second term depends on the difference in arbitrary spacetime. We fix attention on an interval of the proper times measured by A and B along the cor- A’s worldline between two events P and R, where A’s responding segments of their worldlines. It also depends proper time τ varies between τ1 and τ2, as illustrated in on the boost that relates the final velocity of B to that Fig.1. We denote by ~uA(τ) the 4-velocity of A along of A. Finally, the third term depends on the holonomy her worldline. We also introduce an orthonormal tetrad a Λ b around the loop, which in general will consist of a ~eαˆ(τ) which is parallel transported along A’s worldline, spatial rotation together with the aforementioned boost. where ~e0ˆ = ~uA. We now turn to the derivation of the formula (3.20). At each point on A’s worldline, there is a unique spatial The inhomogeneous part of the generalized holonomy can vector be obtained by solving the differential equation (3.1) with ~ ξˆi (τ)~e (τ) (3.19) α = 1 around the loop C starting with ξR = 0 at the B ˆi initial point R. The solution at the next point P can be such that the exponential map evaluated on this vector obtained from the fifth property listed in Sec.IIIA above, 10 is a point zB(τ) on B’s worldline. Or, equivalently, that the inhomogeneous term for a geodesic is just the ˆi tangent to the geodesic. It is given by (τ, ξB(τ)) gives the location of B’s worldline in Fermi nor- mal coordinates centered on A’s worldline. We denote by ~ ξP = −∆τA~uA(τ1). (3.21) ~uB(τ) the 4-velocity of observer B at the point zB(τ). We denote by Q and S the initial and final points zB(τ1) and We now solve the differential equation along the leg PQ zB(τ2) on B’s worldline (see Fig.1 below). Finally, we of the loop. The solution at Q will be the sum of the ~ let fαˆ(τ) be the orthonormal tetrad at zB(τ) obtained by parallel transport of the initial condition (3.21), together parallel transporting ~eαˆ(τ) from the corresponding point with an inhomogeneous term that is the tangent to the on A’s worldline along the spatial geodesic with initial spatial geodesic. The result is tangent (3.19).11 ~ ˆi ~ We assume that the observers A and B are initially ξQ = −∆τA~uB(τ1) + ξB(τ1)fˆi(τ1). (3.22) ~ comoving, in the sense that f0ˆ(τ1) is B’s 4-velocity at Q. We define a closed loop C by starting at R, traveling Here the first term is the parallel transport term, and we along A’s worldline back to P, traveling along the spatial have used the fact that the parallel transport of A’s initial geodesic with initial tangent (3.19) to Q, traveling along 4-velocity is B’s initial 4-velocity. The second term is the B’s worldline to S, and then back to R along the spatial tangent to the spatial geodesic at Q, from the definitions ˆi ~ geodesic whose final tangent at R is the vector (3.19) at (3.19) of ξB and of fαˆ. Next, we solve the differential equation along the seg- τ = τ2. ment QS of B’s worldline. Since B parallel transports his own 4-velocity, the result is

~ ˆi ~ 10 Uniqueness requires that B is sufficiently close to A to be inside ξS = −∆τA~uB(τ2)+ξB(τ1)Γ·fˆi(τ1)+∆τB~uB(τ2). (3.23) a convex normal neighborhood. 11 a Note that the parameter τ need not be the proper time along B’s Here Γ b is the parallel transport operator from Q to S, ~ worldline, and the orthonormal tetrad fαˆ need not be parallel and the last term is the inhomogeneous term, the tan- transported along B’s worldline. gent to B’s worldline at S. Finally we transport this 13

to a different Minkowski region. The first example treats R S a linearized plane wave, which reproduces the result of Sec.IB in a covariant language. The second example deals with a linearized pulse of waves heading radially outward from a pointlike source. This more general ex- ample gives an indication of the magnitude and the form of the disagreement that observers will have when mea- suring angular momentum. A schematic spacetime diagram with the curve used to compute the generalized holonomy is depicted in Fig. 1. As discussed in the previous section, there are two freely falling observers A and B, and we consider a closed curve consisting of segments of their worldlines together with spatial geodesics that join the two worldlines. In this section, we additionally assume that spacetime is flat initially and at late times (the unshaded portions of the diagram), and that at intermediate times there is a burst of gravitational waves present (the gray shaded region). The two spacelike curves—the one, before the burst, with endpoints at P and Q and the other, after the burst, with endpoints at R and S—are geodesics of Minkowski space (i.e., straight lines in some surface of P Q constant time). The figure describes both the plane wave (Sec.IVA) and a local region of the radially propagating FIG. 1. Spacetime diagram of a burst of gravitational waves gravitational wave (Sec.IVB). and the curve used to compute the generalized holonomy. The gray region represents the spacetime location of the gravita- tional waves, while the unshaded regions are Minkowski space- A. Generalized holonomy for a gravitational plane times before and after the burst. The curve bounded by P wave with memory and R is the worldline of observer A, and that bordered by Q and S is that of B. The curves with endpoints (P, Q) and (R, S) are spacelike geodesics before and after the burst, re- We consider a spacetime which is flat at early and at spectively, which are just straight lines in the flat spacetime late times and which contains a linearized plane wave at regions. intermediate times. We use the conventions described in Sec.IB above: the metric is given by the expression (1.4) in global TT coordinates (T,Xi), with the metric pertur- result along the leg SR of the loop. When we parallel bation being hij(T − Z). This metric perturbation van- transport B’s 4-velocity ~uB, the result is the vector ~wB ishes at early times but at late times asymptotes to the ∞ defined above. Similarly, when we parallel transport the constant value hij . We also introduce a coordinate sys- ~ a i vector Γ · fˆi, the result is the holonomy operator Λ b of tem (t, x ) which at early times is an inertial coordinate i the loop acting on the basis vector ~eˆi at R. This is be- system and coincides with the TT coordinates (T,X ) cause ~eˆi is parallel transported along RP, and because and which at late times is again inertial and related to ~ the TT coordinates by Eq. (1.5). fˆi is obtained from ~eˆi by parallel transporting along PQ. Thus, we obtain The observers A and B are freely falling and are there- fore stationary with respect to the TT coordinates, with ~ ˆi ˆi i i i i ξR = (∆τB − ∆τA)~wB + ξB(τ1)Λ · ~eˆi − ξB(τ2)~eˆi, X = XA = constant for A, and X = XB = constant for (3.24) B. The inertial-frame locations of the observers at early times are where the last term is the inhomogeneous term. This is i i i i equivalent to the formula (3.20). xA = XA, xB = XB, (4.1) while at late times they are

IV. GENERALIZED HOLONOMY IN x0i = (δi + 1 h∞ i )Xj , x0i = (δi + 1 h∞ i )Xj , (4.2) LINEARIZED GRAVITY A j 2 j A B j 2 j B as discussed in Sec.IB above. This section provides two related examples of the gen- To compute the generalized holonomy, it will be useful eralized holonomy. Both spacetimes consist of a flat to recall results from Sec.IIIA. First, recall that the gen- Minkowski region followed by a burst of gravitational eralized holonomy along a curve composed of several seg- waves with memory, after which the spacetimes settle ments is just the composition of the individual solutions 14 to Eq. (3.1). Second, remember that the general solution 2. Homogeneous solution and the generalized holonomy can be written as the sum of a homogeneous solution (i.e., the usual holonomy) and an inhomogeneous solution (the It is not too difficult to see that the holonomy of par- part that is independent of the initial data), which allows allel transport is the identity map, the two solutions to be computed independently. Third, note that for geodesic curves the inhomogeneous part of a a Λ b = δ b (4.4) the solution is proportional to the tangent to the curve at the endpoint. Thus, the generalized holonomy can be for the curve shown in Fig.1 even though the space- found by computing the affine transport in four steps (P time has nontrivial curvature. It follows from the fact to R to S to Q to P), while computing the inhomoge- that the parallel transport is trivial in the flat regions of neous and homogeneous parts separately. spacetime and that it is identical on the two worldlines of the two different observers. Consequently, the inhomoge- neous solution is the only relevant part of the generalized 1. Calculation of the inhomogeneous solution holonomy.

~ •P to R: We transport the initial vector ξP = 0 along the geodesic from P to R using the affine 3. Relation to the memory effect and the observer transport law (3.1) with α = 1. The result is dependence of angular momentum ~ ξR = (δt)∂T = (δt)∂t, In this example, the generalized holonomy is directly where δt is the interval of A’s proper time between related to the change in proper distance between the P and R. In the second equation, we have trans- two observers that arises from the solution to the equa- formed from TT coordinates to the inertial coordi- tion of geodesic deviation (the usual physical effect of nates. the gravitational-wave memory). This leads to an ob- server dependence in angular momentum which is given ~ αβ [α β] •R to S: Next, we use the vector ξR as an initial by δJ = 2∆ξ P , from Eqs. (3.12) and (4.4). Using condition for the affine transport along the straight the result (4.3) this corresponds to an observer depen- line extending from R to S in the flat spacetime dence of the spatial angular momentum of the spatial 1 ∞ j l k k region after the burst. It is easiest to perform this angular momentum of − 2 εijkh lδx p , where p is the i computation in the inertial coordinates (t, x ). The spatial momentum and δx = xB − xA. This is precisely result is the result (1.11) found in Sec.IB. ∂ ξ~ = (δt)∂ + (x0i − x0i ) . S t B A ∂xi B. Generalized holonomy for a gravitational wave Next we use Eqs. (4.1) and (4.2) and transform at large radius back to the TT coordinates, giving ∂ For a gravitational wave propagating radially outward ξ~ = (δt)∂ + (xi − xi ) . S T B A ∂Xi from a pointlike source, the computation of the general- ized holonomy is very similar to that of the plane wave, (Here in the spatial components, there was a can- but the expressions are somewhat lengthier. The lin- 1 ∞ cellation between a factor of 1 + 2 h and its in- earized metric of this spacetime has the same form as verse.) that of Eq. (1.4), but the function hαβ(t − z) gets re- •S to Q: This part of the affine transport removes placed by an outgoing wave solution in spherical polar the timelike component of the vector, and it trans- coordinates. The most common form of this metric is a given in Lorentz gauge—see, e.g., Eqs. (8.13a)–(8.13c) forms the spatial part of ξS because the spatial vec- tors change under parallel transport. As a result, of [22]—which is often expressed as a sum of terms pro- the outcome of the transport is portional to mass and current multipoles and the time derivatives of the multipoles. To compute the general- ∂ ξ~ = (δi + 1 h∞ i )(xj − xj ) . ized holonomy, only the leading-order terms in a series Q j 2 j B A ∂Xi in 1/r will be needed. In addition, it will be most useful to express the metric perturbation in a TT gauge rather •Q to P: The affine transport takes place along a than Lorentz gauge. straight line in a flat spacetime region, and so its net effect is to add the corresponding displacement vector along the line. The final result at P gives 1. Transverse-traceless metric perturbation the inhomogeneous piece of the general solution ∂ ∆~ξ = ξ~ = 1 h∞ i (xj − xj ) . (4.3) The quickest way to compute the TT metric perturba- P 2 j B A ∂Xi tion is to compute the Riemann tensor and use the fact 15 that the TT metric perturbation is related to the gauge- burst of waves: invariant Riemann tensor (at linear order in the metric ∞ X ` + 2 (`) perturbation) via the relation Ξ = I nA` , (4.7a) 0 `(`!) A` `=2 ¨TT ∞  R0i0j = hij , (4.5) X 1 1 (`) 1 (`) Ξ = − I nA`−1 + I nA` n i `! ` − 1 iA`−1 2 A` i `=2 where the pair of dots over hTT indicates taking two time ij (` + 2) (`−1) (`−1)  A`−1 A` derivatives. The metric perturbation can be found by − ( I iA`−1 n − I A` n ni) . integrating Eq. (4.5) twice with respect to time. In co- 2r ordinates (u, r, θ, ϕ) where u = t − r, and starting from (4.7b) Eqs. (8.13a)–(8.13c) of [22], the result is For the current multipoles, the only linearized gauge gen- erator that can be constructed from the `th time deriva- ∞ 1  1 (`) (`) tive of S , the radial vectors ni, and the antisymmetric TT X A`−1 A`−2 A` h = [4n(i I j)A n − 2 I ijA n ij r `! `−1 `−2 tensor εipq would be proportional to the following: `=2 (`) ∞ A 4` q (`) ` (S) X A`−1 q − (δij + ninj) I A` n ] + n × Ξ = ε n n . (4.8) (` + 1)! i ipq S pA`−1 `=2 (`) (`)  [n ε nA`−1 − ε nA`−2 ] (i j)pq S pA`−1 pq(i S j)pA`−2 A second quick calculation will show that this transfor- 2 mation does not make the spacetime flat. As a result, we + O(1/r ) . (4.6) will require that the multipoles satisfy the conditions

(`) (`) Here IA` = IA` (u) is an `-pole mass moment and SA` = I A` = const. and S A` = 0 , (4.9) SA` (u) is an `-pole current moment, which are symmetric trace-free (STF) tensors with ` indices (the subscript A ` when u > uf . is one notation used to represent ` spatial indices). The With the metric determined by Eq. (4.6), subject to notation (`) above the symbols for the moments means the condition (4.9), the gauge transformation (4.7) is suf- th i the ` derivative with respect to u. The vector n is ficient to define Minkowski coordinates after the pulse of i A a unit radial vector (i.e., x /r) and n ` is the tensor waves via the relation product of ` radial unit vectors. yα = xα + Ξα. (4.10)

This also provides the necessary information to compute 2. Multipoles and coordinate change after the burst the generalized holonomy. As in the previous plane-wave example, we will split the calculation into the inhomoge- neous and homogeneous parts, which are treated in the As in the example of the plane wave, it will be as- next subparts, respectively. sumed that before a retarded time u ≡ t − r = 0 all the multipoles vanish; they are dynamical between 0 and uf ; and after the retarded time uf , the spacetime 3. Calculation of the inhomogeneous solution is again Minkowski, but some of the multipoles and their time derivatives can have nonzero constant values which •P to R: This is identical to the equivalent calcu- correspond to the gravitational-wave memory. Interest- lation involving the gravitational plane wave: the ingly, only certain multipoles can go to constant values α α vector after affine transport is ξR = (δt)u , where and still have the spacetime be Minkowski (and, hence, ~u = ∂ . As before, it is helpful to transform to the th t stationary). Specifically, the ` time derivative of the flat coordinates yα after the pulse, defined by Eqs. mass-multipole STF tensors I can take nonzero values A` (4.10) and (4.7). This introduces two new terms whereas the equivalent time derivatives of the current α0 00 α0 α0 into the result: ξ = (δt − Ξ )u − Ξ 0 δt. Here multipoles S cannot asymptote to a nonzero value and R R R ,0 A` we use primes to denote tensor components in the 0 still be Minkowski space. This seems to be closely related Minkowski coordinates, xα = yα. to the fact that there is no magnetic-type memory from physically realistic sources [27]. •R to S: In the flat Minkowski space after the burst, α0 00 α0 First, consider just the mass multipoles, and assume the affine transport gives ξS = (δt − ΞR)u − th α0 α0 i0 α that the ` time derivatives go to constant values. A ΞR ,00 δt + δ i0 δy . Changing back to the x coor- short calculation can show that the generator of lin- dinates alters the spatial part of the vector so that α α 0 α α i α i earized gauge transformations below can remove the con- ξS = u (δt + δΞ ) + δΞ ,0δt + ΞS ,iδx + δ i(δx + i α α α stant time derivatives of the mass multipoles after the δΞ ). Here δΞ = ΞS − ΞR has been defined. 16

•S to Q: Transporting back through the burst terms are α changes the spatial part of the vector to ξQ = ∞ 1 ` + 2 (`) 0 α α α S ,α i α i i 0 X A` δΞ u +δΞ ,0δt+ 2 (ΞS ,i−Ξi )δx +δ i(δx +δΞ ), ∆ξ = ( I n ) , (4.12a) (1) `(`!) A` SR where the change occurred from the parallel trans- `=2 ∞ port of the affine frame back to the original point  (`) α α ,α (1) X 1 1 A and where the fact that h i = Ξ ,i + Ξi was used ∆ξ = − ( I n `−1 ) i `! ` − 1 iA`−1 SR to simplify the change in the spatial part of the `=2 vector. 1 (`)  + ( I nA` n ) , (4.12b) 2 A` i SR ∞ (`) (δt/r) X ` + 2 •Q to P: Along this flat geodesic in the Minkowski ∆ξ = − δt [( I nA`−1 /r) i 2(`!) iA`−1 SR space prior to the burst, the affine transport adds `=2 i α the displacement vector δx to the result of ξQ. (`) A` Thus, the complete inhomogeneous solution is − ( I A` n ni/r)SR] , (4.12c) ∞ δxi ` + 2 (`) 0 X A`−1 α 0 α α 1 α S ,α i α i ∆ξ(δx/r) = − [( I iA`−1 n )S ∆ξ = δΞ u + δΞ ,0δt + (Ξ ,i − Ξ )δx + δ iδΞ . rS `! 2 S i `=2 (4.11) (`) A` − ( I A` n ni)S ] , (4.12d) ∞ 2δxj 1 (`) (δx/r) X A`−1 We will discuss the relationship between the terms that ∆ξi = (n[i I j]A`−1 n )S . (4.12e) rS `! a `=2 appear in ∆ξP and the gravitational-wave memory in more detail below. In the expression above, the subscript SR means to take the difference of the quantity within parentheses evalu- ated at the values of the coordinate points S and R. The local infinitesimal Lorentz transformation, which will be denoted as ω = ω is strictly of order 1/r 4. Homogeneous solution and the generalized holonomy αβ [αβ] and can be written as ∞ X ` + 2 (`) (`) The calculation of the homogeneous part of the solu- ω = [( I nA`−1 /r) − ( I nA` n /r) ] i0 `! iA`−1 SR A` i SR tion is simpler than that of the inhomogeneous portion `=2 above. The first set of nontrivial terms comes from the + O(1/r2) , (4.13a) parallel transport along the worldline extending from P ∞ to R, and from the coordinate change at R. For an ar- X 2 (`) ω = (n I nA`−1 /r) + O(1/r2) . bitrary initial condition ξα , this vector will be modified ij `! [i j]A`−1 SR (0) `=2 1 α R,α β by an amount − 2 (ΞR,β −Ξβ )ξ(0). There will be a sim- (4.13b) ilar contribution with the opposite sign involving quan- tities at the point S from the parallel transport along the worldline from S to Q and the coordinate change at 5. Relation to the memory effect and the observer S. Thus, the part of the holonomy that differs from the dependence of angular momentum 1 α ,α β identity is given by 2 (Ξ ,β −Ξβ )SRξ(0), where the sub- script SR implies it is the difference of the values at the The relation between the generalized holonomy and the quantities at the coordinate points S and R, transported physical effects associated with the gravitational-wave back to P. memory is somewhat more involved than it was for a gravitational plane wave. The term δΞi is a measure of α From the expression for Ξ in the gauge transformation the change in distance between the observers that oc- (4.7), it is possible to show that the generalized holonomy curs from the memory. In addition, the part δΞ0 gives has a homogeneous piece in the form of a local infinites- information about the difference in proper time mea- imal Lorentz transformation that scales as 1/r, and an sured by the two observers that is a result of the mem- inhomogeneous part that contains terms independent of ory of the gravitational-wave burst. The other term δx and δt that are zeroth order in 1/r (and also terms 1 α S ,α i 2 (ΞS ,i − Ξi )δx takes into account a boosting and ro- that go as 1/r, which we will not show), in addition to tation of the spatial displacement vector along the other terms that scale as δx/r, and δt/r. For the inhomoge- observer’s worldline from the wave’s memory, and the neous part, these terms will be labeled by ∆ξα , ∆ξα , α (1) (δx/r) part δΞ ,0δt represents a relative change in the tangent α α and ∆ξ(δt/r). The zeroth-order terms come from −δΞ , to the observers’ worldlines from the memory. whereas the terms of order δt/r and δx/r come from the Because the inhomogeneous solution has a zeroth-order α 1 α S ,α i terms δΞ ,0δt and 2 (ΞS ,i −Ξi )δx , respectively. These piece in 1/r, the center of mass and the angular mo- 17 mentum will have an observer dependence with a mag- dependent observables that can be nontrivial when the 0 i i 0 j k nitude of order P ∆ξ(1) + P ∆ξ(1) and ijk∆ξ(1)P , re- burst has departed and that can be considered to be types spectively, where P a is the 4-momentum of the source. of “gravitational-wave memory.” There is the usual dis- For separations for which δx is of order r, then the terms placement memory, a residual relative boost, a relative a rotation, and a difference in elapsed proper time between ∆ξ(δx/r) will also have leading-order contributions to the observer dependence of the center of mass and of the an- the two observers. These four observables are all encoded gular momentum of the form P 0∆ξi + P i∆ξ0 in the generalized holonomy around a suitably defined (δx/r) (δx/r) closed loop in spacetime. Thus, we clarified and general- and  ∆ξj P k, respectively. Similarly, for times δt ijk (δx/r) ized the often-noted close relation between gravitational- of order the light-travel time to the source (i.e., of or- wave memory and observer dependence of angular mo- der r), then there will be additional observer dependence mentum. 0 i j k from terms of the form P ∆ξ(δt/r) and ijk∆ξ(δt/r)P . Finally, we performed explicit computations in two Equations (2.4) and (2.8) imply that the angular mo- different specific contexts that illustrate the relation- mentum tensor J ab will have terms proportional to r ships between generalized holonomy, observer depen- at large radii: specifically, it is the orbital-like part of dence of angular momentum, and gravitational-wave the angular momentum 2y[aP b] = −2rn[aP b] that has memory. The first context was a plane gravitational this scaling. When the angular momentum transforms wave with memory passing through flat spacetime, and by Eq. (3.16), the 1/r parts of the holonomy will in- the second was an outgoing linearized gravitational wave duce a change in the angular momentum that is of order near future null infinity. The plane wave only showed the unity in a series in 1/r. These terms will have the form displacement memory effect, but the multipolar gravita- ab a [c b] a [b c] δJ = 2(ω cy P − ω cy P ). The lowest-order part tional wave displayed all four of the physical observables of the 4-momentum will still be unambiguous, and any associated with the memory. observer dependence will be a relative 1/r effect. Although our goal was to provide physical insight into the nature of the BMS group, the generalized holonomy tool does not quite achieve this goal: in AppendixB, V. CONCLUSIONS we show that the generalized holonomy can be nontrivial for certain spacelike curves in the Schwarzschild space- In this paper, we noted that bursts of gravitational time, even as the curves tend to spatial infinity. Hence, waves cause spatially separated observers to disagree on observers along this curve would find their measured an- their changes in displacement and therefore to disagree gular momentum to be observer dependent, even though on their measured special-relativistic angular momenta angular momentum is well defined in Schwarzschild (as of a source. This observer dependence of angular mo- the BMS group has a preferred Poincar´esubgroup in sta- mentum is related to the gravitational-wave memory of tionary spacetimes). Thus, our prescription for assessing the pulse of waves. We derived this phenomenon first in observer dependence in angular momentum not only cap- a simple context of linearized plane waves, and later in a tures BMS/memory ambiguities in angular momentum more systematic and covariant framework. but also reflects other, more trivial effects of spacetime We defined a procedure by which observers could mea- curvature on angular momentum measurements. Finding sure a type of special-relativistic linear and angular mo- a method to isolate just the BMS ambiguities is a topic mentum at their locations, from the spacetime geometry we will investigate in future work. in their vicinity. The procedure gives the correct result Because the affine transport law defines a way to com- when the spacetime is linear and stationary, and the mea- pare other vectors in addition to the angular momentum surement takes place near future null infinity. We esti- at different spacetime points, it could find application to mated the errors in the procedure when the spacetime is other problems. For example, if a burst of gravitational nonlinear, dynamical, or the source is not isolated. waves passes through a post-Newtonian spacetime, the To compare angular momentum at different spacetime momenta and angular momenta of the particles that en- points, we defined a transport equation, the affine trans- ter into the post-Newtonian equations of motion could port, which is a slight generalization of parallel trans- differ before and after the burst. The affine transport port. The transport around a closed curve, the gener- may be useful for deriving a prescription for matching the alized holonomy, consists of a Poincar´etransformation, post-Newtonian spacetimes before and after the bursts, rather than a Lorentz transformation as for a normal in a manner that allows one to compute the motion of an holonomy. The generalized holonomy contains an inho- N-body system. mogeneous displacement term. The extent to which the generalized holonomy is nontrivial is a measure of how much spacetime curvature prevents different observers ACKNOWLEDGMENTS from arriving at a consistent definition of linear and an- gular momentum. We thank Justin Vines for reading an earlier draft of For two freely falling observers, who encounter a burst this paper and for providing helpful comments and also of gravitational waves, we showed that there are four in- Abhay Ashtekar, Mike Boyle, and Leo Stein for helpful 18 discussions. This work was supported in part by NSF grants No. PHY-1404105 and No. PHY-1068541. r M δr2 Appendix A: Angular Momentum Transport Laws δx1

In the body of this paper, we introduced a method of transporting a pair of tensors (P a,J ab) along a curve δr1 δx2 from one point to another in a curved spacetime. In this Appendix, we show that the transport method is equiv- FIG. 2. Curve used to compute the generalized holonomy in alent to solving the following simple set of differential the Schwarzschild spacetime of mass M at large radii. The lengths of the four segments that compose the curve, δr , δr , equations along the curve: 1 2 δx1, and δx2 are all of order r, where r  M is the closest distance to the source along the segment labeled by δx1. This a b a bc [b c] k ∇aP = 0, k ∇aJ = 2P k . (A1) curve has a nontrivial generalized holonomy as r goes to spa- tial infinity, even though the Schwarzschild spacetime has a The equivalence between the two methods was pointed well-defined angular momentum. out to us by Justin Vines [28]. Suppose we have a curve xα = xα(λ) that joins at point P at λ = 0 to another point Q at λ = 1. We introduce Appendix B: Generalized holonomy of a spacelike an orthonormal basis of vectors ~eαˆ at Q, and extend it curve in the Schwarzschild spacetime along the curve by parallel transport. We decompose the 4-momentum and angular momentum on this basis as In this Appendix, we compute the generalized holon- omy of certain curves in the Schwarzschild spacetime and ˆ P a = P αˆea ,J ab = J αˆβea eb . (A2) show that the generalized holonomy does not become the αˆ αˆ βˆ identity in the limit when the curves asymptote to fu- The transport equations (A1), when written in terms of ture null infinity. As discussed in the body of the paper, this basis, become this property implies that we cannot use the generalized holonomy as a tool to diagnose whether a given space- d time admits a well-defined angular momentum (since in P αˆ = 0, (A3a) dλ stationary spacetimes the BMS group has a preferred Poincar´esubgroup and so normal angular momentum is d ˆ ˆ ˆ J αˆβ = P αˆkβ − P βkαˆ. (A3b) well defined). dλ The specific curve we consider is shown in Fig.2. Let- αˆ αˆ ting M denote the mass of the spacetime and r  M be The first of these gives P (λ) = P0 = constant. We now make the ansatz for the angular momentum solution the distance to the closest point along the segment la- beled by δx1, we will assume that the lengths of the four αˆβˆ αˆβˆ αˆ βˆ βˆ αˆ sides of the curve, δx1, δr1, δx2, and δr2 are all of order J (λ) = J0 + P0 χ (λ) − P0 χ (λ), (A4) r (or equivalently, the area enclosed by the curve is of order r2). The curvature at the loop will scale as M/r3. αˆ αˆβˆ for some vector χ (λ), where J0 is the initial value of Because the holonomy associated with parallel transport J αˆβˆ at λ = 0. Using this ansatz we see that the differen- scales as the curvature times the area, when the vector tial equation (A3b) will be satisfied if χαˆ vanishes at P transported has magnitude of order r (such as the dis- a and satisfies placement vector y ), the vector will undergo changes of order M. Similarly, because the inhomogeneous part of d the generalized holonomy scales as curvature times the χαˆ = kαˆ. (A5) dλ area to the three-halves power, the vector ∆ξa will also be of order M. Even as r approaches spatial infinity, This differential equation coincides with the differential this estimate suggests that there will be nontrivial gen- equation (3.1) that defines the generalized parallel trans- eralized holonomy and observer dependence in angular port, for the case α = +1. By comparing with Eq. (3.2) momentum. we find We did, in fact, compute the exact generalized holon- omy in the case in which δr1 and δr2 are radial curves, αˆ αˆ χ (Q) = ∆ξPQ. (A6) and δx1 and δx2 are two coordinate lines between the endpoints of these two curves, respectively. The pre- Substituting this result into the ansatz (A4) gives an ex- cise answer, while not particularly insightful, does in- pression for the angular momentum at Q which agrees deed scale as M as r goes to spatial infinity. We con- with Eq. (3.12), establishing the result. clude that the generalized holonomy is not specifically 19 linked to BMS ambiguities in angular momentum, and is prevents observers from consistently measuring and com- more generally a diagnostic of when spacetime curvature paring a type of special-relativistic angular momentum.

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