A General Definition of “Conserved Quantities” in General Relativity and Other Theories of Gravity
In general relativity, the notion of mass and other conserved quantities at spatial infinity can be defined in a natural way via the Hamiltonian framework: Each conserved quantity is associated with an asymptotic symmetry and the value of the conserved quantity is defined to be the value of the Hamiltonian which generates the canonical transformation on phase space corresponding to this symmetry. However, such an approach cannot be employed to define “conserved quantities” in a situation where symplectic current can be radiated away (such as occurs at null infinity in general relativity) because there does not, in general, exist a Hamiltonian which generates the given asymptotic symmetry. (This fact is closely related to the fact that the desired “conserved quantities” are not, in general, conserved!) In this paper we give a prescription for defining “conserved quantities” by proposing a modification of the equation that must be satisfied by a Hamiltonian. Our prescription is a very general one, and is applicable to a very general class of asymptotic conditions in arbitrary diffeomorphism covariant theories of gravity derivable from a Lagrangian, although we have not investigated existence and uniqueness issues in the most general contexts. In the case of general relativity with the standard asymptotic conditions at null infinity, our prescription agrees with the one proposed by Dray and Streubel from entirely different considerations.
Notions of energy and angular momentum have played a key role in analyzing the behavior of physical theories. For theories of fields in a fixed, background spacetime, a locally conserved stress-energy-momentum tensor, , normally can be defined. If the background spacetime has a Killing field , then is a locally conserved current. If is a Cauchy surface, then defines a conserved quantity associated with ; if is a timelike or null surface, then has the interpretion of the flux of this quantity through .
However, in diffeomorphism covariant theories such as general relativity, there is no notion of the local stress-energy tensor of the gravitational field, so conserved quantities (which clearly must include gravitational contributions) and their fluxes cannot be defined by the above procedures, even when Killing fields are present. Nevertheless, in general relativity, for asymptotically flat spacetimes, conserved quantities associated with asymptotic symmetries have been defined at spatial and null infinity.
A definition of mass-energy and radiated energy at null infinity, , was first given about 40 years ago by Trautman  and Bondi et al . This definition was arrived at via a detailed study of the asymptotic behavior of the metric, and the main justification advanced for this definition has been its agreement with other notions of mass in some simple cases as well as the fact that the radiated energy is always positive (see, e.g., ,  for further discussion of the justification for this definition). A number of inequivalent definitions of quantities associated with general (BMS) asymptotic symmetries at null infinity have been proposed over the years, but it was not until the mid-1980’s that Dray and Streubel  gave a general definition that appears to have fully satisfactory properties . This definition generalized a definition of angular momentum given by Penrose  that was motivated by twistor theory.
In much of the body of work on defining “conserved quantities” at null infinity, little contact has been made with the Hamiltonian formulation of general relativity. An important exception is the work of Ashtekar and Streubel  (see also ), who noted that BMS transformations correspond to canonical transformations on the radiative phase space at . They identified the Hamiltonian generating these canonical transformations as representing the net flux of the “conserved quantity” through . They then also obtained a local flux formula under some additional assumptions not related to the canonical framework (in particular, by their choice of topology they, in effect, imposed the condition that the local flux formula contain no “second derivative terms”). However, they did not attempt to derive a local expression for the “conserved quantity” itself within the Hamiltonian framework, and, indeed, until the work of  and , it was far from clear that, for arbitrary BMS generators, their flux formula corresponds to a quantity that could be locally defined on cross-sections of .
The status of the definition of “conserved quantities” at null infinity contrasts sharply with the situation at spatial infinity, where formulas for conserved quantities have been derived in a clear and straightforward manner from the Hamiltonian formulation of general relativity , . As will be reviewed in sections 2 and 3 below, for a diffeomorphism covariant theory derived from a Lagrangian, if one is given a spacelike slice and a vector field representing “time evolution”, then the Hamiltonian generating this time evolution—if it exists—must be purely a “surface term” when evaluated on solutions (i.e., “on shell”). It can be shown that if extends to spatial infinity in a suitable manner and if is a suitable infinitesimal asymptotic symmetry, then a Hamiltonian does exist (see “case I” of section 4 below). The value of this Hamiltonian “on shell” then can be interpreted as being the conserved quantity conjugate to . One thereby directly obtains formulas for the ADM mass, momentum, and angular momentum as limits as one approaches spatial infinity of surface integrals over two-spheres.
It might seem natural to try a similar approach at null infinity: Let be a spacelike slice which is asymptotically null in the sense that in the unphysical spacetime its boundary is a cross-section, , of . Let the vector field be an infinitesimal BMS asymptotic symmetry. Then, when evaluated on solutions, the Hamiltonian generating this time evolution—if it exists—must again be purely a “surface term” on , i.e., it must be expressible as an integral of a local expression over the cross-section . This expression would then provide a natural candidate for the value of the “conserved quantity” conjugate to at “time” .
As we shall see in section 3 below, the above proposal works if is everywhere tangent to . However, if fails to be everywhere tangent to , then it is easy to show that no Hamiltonian generating the time evolution exists. The obstruction to defining a Hamiltonian arises directly from the possibility that symplectic current can escape through .
The main purpose of this paper is to propose a general prescription for defining “conserved quantities” in situations where a Hamiltonian does not exist. This proposal consists of modifying the equation that a Hamiltonian must satisfy via the addition of a “correction term” involving a symplectic potential that is required to vanish whenever the background spacetime is stationary. If such a symplectic potential exists and is unique—and if a suitable “reference solution” can be chosen to fix the arbitrary constant in the definition of the “conserved quantity”—we obtain a unique prescription for defining a “conserved quantity” associated with any infinitesimal asymptotic symmetry. In the case of asymptotically flat spacetimes at null infinity in vacuum general relativity, we show in section 5 that existence and uniqueness does hold, and that this prescription yields the quantities previously obtained in .
In section 2, we review some preliminary material on the diffeomorphism covariant theories derived from a Lagrangian. In section 3, we investigate the conditions under which a Hamiltonian exists. In section 4, we present, in a very general setting, our general proposal for the definition of “conserved quantities” associated with infinitesimal asymptotic symmetries. This general proposal is then considered in the case of asymptotically flat spacetimes at null infinity in general relativity in section 5, where it is shown to yield the results of . Some further applications are briefly discussed in section 6.
In this paper, we will follow closely both the conceptual framework and the notational conventions of  and . Further details of most of what is discussed in this section can be found in those references.
On an -dimensional manifold, , we consider a theory of dynamical fields, collectively denoted , which consist of a Lorentzian metric, , together with other tensor fields, collectively denoted as . To proceed, we must define a space, , of “kinematically allowed” field configurations, on . A precise definition of would involve the specification of smoothness properties of , as well as possible additional restrictions on (such as global hyperbolicity or the requirement that a given foliation of by hypersurfaces be spacelike) and asymptotic conditions on (such as the usual asymptotic flatness conditions on fields at spatial and/or null infinity in general relativity). The precise choice of that would be most suitable for one’s purposes would depend upon the specific theory and issues being considered. In this section and the next section, we will merely assume that a suitable has been defined in such a way that the integrals occurring in the various formulas below converge. In section 4, we will impose a general set of conditions on that will ensure convergence of all relevant integrals. In section 5, we will verify that asymptotically flat spacetimes at null infinity in vacuum general relativity satisfy these conditions.
We assume that the equations of motion of the theory arise from a diffeomorphism covariant -form Lagrangian density 
where denotes the derivative operator associated with , denotes the Riemann curvature tensor of . (An arbitrary (but finite) number of derivatives of and are permitted to appear in .) Here and below we use boldface letters to denote differential forms on spacetime and, when we do so, we will suppress the spacetime indices of these forms. Variation of yields
where no derivatives of appear in the first term on the right side. The Euler-Lagrange equations of motion of the theory are then simply . Note that—when the variation is performed under an integral sign—the term corresponds to the boundary term that arises from the integrations by parts needed to remove derivatives from . We require that be locally constructed out of and in a covariant manner. This restricts the freedom in the choice of to111If we change the Lagrangian by , the equations of motion are unaffected. Under such change in the Lagrangian, we have . Thus, if such changes in the Lagrangian are admitted, we will have this additional ambiguity in . However, this ambiguity does not affect the definition of the presymplectic current form (see eq.(4) below) and will not affect our analysis.
where is locally constructed out of and in a covariant manner.
The presympletic current -form, —which is a local function of a field configuration, , and two linearized perturbations, and off of —is obtained by taking an antisymmetrized variation of
On account of the ambiguity (3) in the choice of , we have the ambiguity
in the choice of .
Now let be a closed, embedded -dimensional submanifold without boundary; we will refer to as a slice. The presymplectic form, , associated with is a map taking field configurations, , together with a pairs of linearized perturbations off of , into the real numbers—i.e., it is a two-form on ---defined by integrating222The orientation of relative to the spacetime orientation is chosen to be where is a future-directed timelike vector. over ,
Although this definition depends, in general, upon the choice of , if and satisfy the linearized field equations and is required to be a Cauchy surface, then does not depend upon the choice of , provided that is compact or suitable asymptotic conditions are imposed on the dynamical fields . The ambiguity (5) in the choice of gives rise to the ambiguity
in the presymplectic form . In this equation, by the integral over , we mean a limiting process in which the integral is first taken over the boundary, , of a compact region, , of (so that Stokes’ theorem can be applied333We choose the orientation of to be the one specified by Stokes’ theorem, i.e., we dot the first index of the orientation form on into an outward pointing vector.), and then approaches all of in a suitably specified manner. (Note that since is a slice, by definition it does not have an actual boundary in the spacetime.) Thus, for example, if is an asymptotically flat spacelike slice in an asymptotically flat spacetime, the integral on the right side of eq.(7) would correspond to the integral over a two-sphere on in the asymptotically flat region in the limit as the radius of the two-sphere approaches infinity. Of course, the right side of eq.(7) will be well defined only if this limit exists and is independent of any of the unspecified details of how the compact region, , approaches . In section 4 below, we will make some additional assumptions that will ensure that integrals over “” of certain quantities that we will consider are well defined.
Given the presymplectic form, , we can factor by the orbits of the degeneracy subspaces of to construct a phase space, , in the manner described in . This phase space acquires directly from the presymplectic form on a nondegenerate symplectic form, . One also obtains by this construction a natural projection from to . Now, a complete vector field on naturally induces the field variation on fields . If is such that corresponds to a tangent field on (i.e., if the diffeomorphisms generated by map into itself), then we may view as the dynamical evolution vector field corresponding to the notion of “time translations” defined by . If, when restricted to the solution submanifold444The solution submanifold, , is sometimes referred to as the “covariant phase space” ., , of , this time evolution vector field on consistently projects to phase space, then one has a notion of time evolution associated with on the “constraint submanifold”, , of , where is defined to be the image of under the projection of to . If this time evolution vector field on preserves the pullback to of , it will be generated by a Hamiltonian, . (As argued in the Appendix of , this will be the case when is compact; see section 3 below for some general results in the noncompact case.) Thus, this construction provides us with the notion of a Hamiltonian, , conjugate to a vector field on .
However, a number of complications arise in the above construction. In particular, in order to obtain a consistent projection of from to , it is necessary to choose to be “field dependent”, i.e., to depend upon . As explained in , this fact accounts for why, in a diffeomorphism covariant theory, the Poisson bracket algebra of constraints does not naturally correspond to the Lie algebra of infinitesimal diffeomorphisms. However, these complications are not relevant to our present concerns. To avoid dealing with them, we prefer to work on the original field configuration space with its (degenerate) presymplectic form rather than on the phase space . The notion of a Hamiltonian, , on can be defined as follows:
Definition: Consider a diffeomorphism covariant theory within the above framework, with field configuration space and solution submanifold . Let be a vector field on the spacetime manifold, , let be a slice of , and let denote the presymplectic form (6). (If the ambiguity (5) in the choice of gives rise to an ambiguity in (see eq.(7)), then we assume that a particular choice of has been made.) Suppose that , , and have been chosen so that the integral converges for all and all tangent vectors to at . Then a function is said to be a Hamiltonian conjugate to on slice if for all and all field variations tangent to (but not necessarily tangent to ) we have
Note that if a Hamiltonian conjugate to on slice exists, then—assuming that is connected—its value on is uniquely determined by eq.(8) up to the addition of an arbitrary constant. In many situations, this constant can be fixed in a natural way by requiring to vanish for a natural reference solution, such as Minkowski spacetime. On the other hand, the value of off of is essentially arbitrary, since eq.(8) fixes only the “field space gradient” of in directions off of at points of .
If a Hamiltonian conjugate to on slice exists, then its value provides a natural definition of a conserved quantity associated with at “time” . However, in many cases of interest—such as occurs in general relativity when, say, is an asymptotic time translation and the slice goes to null infinity—no Hamiltonian exists. In the next section, we shall analyze the conditions under which a Hamiltonian exists. In section 4, we shall propose a definition of the “conserved quantity” conjugate to on a slice when no Hamiltonian exists.
3 Existence of a Hamiltonian
When does a Hamiltonian conjugate to on slice exist? To analyze this issue, it is very useful to introduce the Noether current -form associated with , defined by
where the “” denotes the contraction of the vector field into the first index of the differential form . One can show (see the appendix of ) that for a diffeomorphism covariant theory, always can be written in the form
where when the equations of motion hold, i.e., corresponds to “constraints” of the theory. Equation (10) defines the Noether charge -form, . It was shown in  that the Noether charge always takes the form
Thus, we obtain
Consequently, if there exists a Hamiltonian, , conjugate to on , then for all and all it must satisfy the equation
where the integral over has the meaning explained below eq.(7). Note that for field variations which are “on shell”, i.e., such that satisfies the linearized equations of motion, we have
Consequently, if exists, it is given purely as a “surface term” (i.e., an integral over ) when evaluated on .
Equation (14) gives rise to an obvious necessary condition for the existence of : Let (i.e., is a solution to the field equations) and let and be tangent to (i.e., and satisfy the linearized field equations). Let be a two-parameter family with , , and . Then if eq.(14) holds, by equality of mixed partial derivatives, we must have
Conversely, if eq.(16) holds, then—assuming that is simply connected (and has suitable differentiable properties)—it will be possible to define on so that eq.(14) holds whenever is tangent to . (Proof: On each connected component of choose a “reference solution” and define at . Let and let for be a smooth, one-parameter family of solutions that connects to . Define
This definition will be independent of the choice of path when eq.(16) holds since, by simple-connectedness, any other path will be homotopic to and one can apply Stokes’ theorem to the two-dimensional submanifold spanned by this homotopy.) However, if is defined on , there is no obstruction to extending to so that eq.(14) holds on for all tangent to (i.e., including that are not tangent to ), since the additional content of that equation merely fixes the first derivative of in the “off shell” directions of field space.
Therefore, the necessary and sufficient condition for the existence of a Hamiltonian conjugate to on is that for all solutions and all pairs of linearized solutions tangent to , we have
Note that since this condition refers only to the “covariant phase space” , we shall in the following restrict attention to entirely and use eq.(15) for (even though the “off shell” volume integral in eq.(14) is crucial to justifying the interpretation of as the generator of dynamics conjugate to ).
Note that there are two situations where eq.(18) will automatically hold: (i) if the asymptotic conditions on are such that goes to zero sufficiently rapidly that the integral of over vanishes in the limit as approaches ; (ii) if is such that can always be chosen so that is tangent to , since then the pullback of to vanishes. In two these cases, a Hamiltonian conjugate to will exist on . However, if these conditions do not hold, then in general no Hamiltonian will exist.
We turn, now, to giving a general prescription for defining “conserved quantities”, even when no Hamiltonian exists.
4 General Definition of “Conserved Quantities”
In this section, we will propose a definition of conserved quantities under very general assumptions about asymptotic conditions “at infinity”. We begin by specifying these assumptions.
We shall assume that the desired asymptotic conditions in the given diffeomorphism covariant theory under consideration are specified by attaching a boundary, , to the spacetime manifold, , and requiring certain limiting behavior of the dynamical fields, , as one approaches . We shall assume that is an -dimensional manifold, so that is an -dimensional manifold with boundary555The assumption that is an -dimensional manifold structure is not essential in cases where vanishes at (see “Case I” below). In particular, there should be no difficulty in extending our framework to definitions of asymptotic flatness at spatial infinity in which is comprised by a single point .. In cases of interest, will be equipped with additional nondynamical structure (such as a conformal factor on or certain tensor fields on ) that will enter into the specification of the limiting behavior of and thereby be part of the specification of the field configuration space, , and the covariant phase space, . We will refer to such fixed, non-dynamical structure as the “universal background structure” of .
We now state our two main assumptions concerning the asymptotic conditions on the dynamical fields, , and the asymptotic behavior of the allowed hypersurfaces, : (1) We assume that has been defined so that for all and for all tangent to , the -form defined on extends continuously666It should be emphasized that we require that the full extend continuously to —not merely its pullback to hypersurfaces that approach . to . (2) We restrict consideration to slices, , in the “physical spacetime”, , that extend smoothly to in the “unphysical spacetime”, , such that this extended hypersurface intersects in a smooth -dimensional submanifold, denoted . Following terminology commonly used for null infinity, we shall refer to as a “cross-section” of . We also shall assume that is compact—although it would be straightforward to weaken this assumption considerably, since only the behavior of near is relevant to our considerations.
An important immediate consequence of the above two assumptions is that the integral (6) defining always converges, since it can be expressed as the integral of a continuous -form over the compact -dimensional hypersurface .
We turn, now, to the definition of infinitesimal asymptotic symmetries. Let be a complete vector field on (so that, in particular, is tangent to on ). We say that is a representative of an infinitesimal asymptotic symmetry if its associated one-parameter group of diffeomorphisms maps into , i.e., if it preserves the asymptotic conditions specified in the definition of . Equivalently, is a representative of an infinitesimal asymptotic symmetry if (which automatically satisfies the linearized field equations ) satisfies all of the asymptotic conditions on linearized solutions arising from the asymptotic conditions imposed upon , i.e., if corresponds to a vector tangent to .
If is a representative of an infinitesimal asymptotic symmetry, then the integral appearing on the right side of eq.(15), namely
always is well defined via the limiting proceedure described below eq.(7), and, indeed, depends only on the cross-section of , not on . To see this777A similar argument has previously been given to show that the “linkage formulas” are well defined (see, , )., let be a nested sequence of compact subsets of such that approaches , and let
Then, since “on shell” we have
(see eq.(13) above) we have by Stokes’ theorem for ,
where denotes , i.e., the portion of lying between and . As a direct consequence of our assumptions that extends continuously to and that is compact, it follows that is a Cauchy sequence, and hence it has a well defined limit, , as . Note that this limit always exists despite the fact that there is no guarantee that the differential forms or themselves extend continuously to . A similar argument establishes that this limit is independent of , i.e., for a slice such that = , a similarly defined sequence of integrals on will also converge to .
Let and be representatives of infinitesimal asymptotic symmetries. We say that is equivalent to if they coincide on and if, for all , tangent to , and for all on , we have , where is given by eq.(19) and is given by the same expression with replaced by . The infinitesimal asymptotic symmetries of the theory are then comprised by the equivalence classes of the representatives of the infinitesimal asymptotic symmetries.
Now consider an infinitesimal asymptotic symmetry, represented by the vector field , and let be a slice in the spacetime with boundary on . We would like to define a conserved quantity associated with at “time” via eq.(15). As we have seen above, the right side of eq.(15) is well defined under our asymptotic assumptions, but, as discussed in the previous section, in general, there does not exist an which satisfies this equation. The analysis naturally breaks up into the following two cases:
Case I: Suppose that the continuous extension of to has vanishing pullback to . Then by eq.(18), exists for all infinitesimal asymptotic symmetries (assuming that is simply connected and has suitable differentiable properties) and is independent of the choice of representative . Furthermore, if and are cross-sections of that bound a region , we have888We define the orientation of to be that obtained by dotting the first index of the orientation of into an outward pointing vector. The orientation of was previously specified in footnotes 2 and 3. The signs in eq.(24) to correspond to the case where lies to the future of . by eqs.(15) and (21)
Thus, is independent of choice of cross-section within the same homology class. If the arbitrary constant (for each cross-section) in is fixed in such a way that there is a “reference solution” for which on all cross-sections (see below), then on all solutions will be independent of choice of cross-section within the same homology class. Thus, in this case, not only does exist, but it truly corresponds to a conserved quantity, i.e., its value is independent of “time”, .
Case II: Suppose that the continuous extension of to does not, in general, have vanishing pullback to . Then, in general, there does not exist an satisfying eq.(15). One exception is the case where and are such that is everywhere tangent to . In this case, if is tangent to cross-sections and , we have
Since the right side of this equation is nonvanishing in general, we see that even when is tangent to cross-sections so that exists, will not be conserved.
Case I arises in general relativity for spacetimes which are asymptotically flat at spatial infinity as defined in , and our prescription for defining corresponds to that given in  and ; see  for the explicit details of how eq.(15) gives rise to the usual expression for ADM mass when is an asymptotic time translation. As we shall discuss in detail in the next section, case II arises in general relativity for spacetimes which are asymptotically flat at null infinity.
The main purpose of this paper is to provide a general definition of a “conserved quantity” conjugate to an arbitrary infinitesimal asymptotic symmetry in case II. In the following, we will restrict attention to this case, and we will denote the quantity we seek as to distinguish it from a true Hamiltonian . As we have seen, in this case an attempt to define by eq.(15) fails the consistency check (16) and thus does not define any quantity. However, consider the following simple modification of eq.(15): On , let be a symplectic potential for the pullback, , of the (extension of the) symplectic current form to , so that on we have for all and , tangent to
We require that be locally constructed999More precisely, by “locally constructed” we mean the following: Suppose that is a diffeomorphism which preserves the universal background structure. Suppose and are such that there exists an open (in ) neighborhood, , of such that for all we have and , where denotes the pullback map on tensor fields associated with the diffeomorphism . Then we require that at we have . out of the dynamical fields, , and their derivatives (or limits of such quantities to ) as well as any fields present in the “universal background structure”. In the case where (and, hence ) is an analytic function101010The condition that be an analytic function of its variables (as occurs in essentially all theories ever seriously considered) has nothing to do with any smoothness or analyticity conditions concerning the behavior of the dynamical fields themselves on . We do not impose any analyticity conditions on the dynamical fields. of its variables (see eq.(1)), we also require that depend analytically on the dynamical fields; more precisely, if is a one-parameter family of fields on that depends analytically on and satisfies suitable uniformity conditions111111For the case of asymptotically flat spacetimes at null infinity, , a suitable uniformity condition would be to require the unphysical fields to vary analytically with at . near , we require that the corresponding also depends analytically on . If any arbitrary choices are made in the specification of the background structure (such as a choice of conformal factor in the definition of null infinity in general relativity), then we demand that be independent of such choices (so, in particular, in the case of null infinity, is required to be conformally invariant). Our proposal is the following: Let satisfy121212Here it should be noted that the new term on the right side of this equation is an ordinary integral over the surface of , whereas, as explained above, the first term in general is defined only as an asymptotic limit.
Then it is easily seen that this formula satisfies the consistency check (16) and, thus, defines a “conserved quantity” up to an arbitrary constant. Finally, let this arbitrary constant be fixed by requiring that vanish (for all infinitesimal asymptotic symmetries and all cross-sections ) on a suitably chosen “reference solution” . We will specify below the necessary conditions that must be satisfied by .
However, the above proposal fails to define a unique prescription because the choice of symplectic potential is ambiguous up to131313Note that the ambiguity in is of an entirely different nature than the ambiguity (3) in . The quantity is defined from the Lagrangian (before has been defined) and its ambiguity arises from eq.(2). The quantity is defined from and its ambiguity arises from eq.(25).
where is an -form on locally constructed out of the dynamical fields as well as the universal background structure defined on , with independent of any arbitrary choices made in the specification of the background structure. Thus, in order to obtain a prescription which defines , we must specify an additional condition or conditions which uniquely select .
An additional requirement on can be motivated as follows. We have already seen from eq.(24) above that cannot, in general, be conserved, i.e., there must be a nonzero flux, , on associated with this “conserved quantity”. This is to be expected on account of the possible presence of radiation at . However, it seems natural to demand that vanish (and, thus, that be conserved) in the case where no radiation is present at . Such a case should occur when is a stationary solution, i.e., when there exists a nonzero infinitesimal asymptotic symmetry represented by an exact symmetry —so that in —and is timelike in in a neighborhood of . Hence, we wish to require that vanish on for all for stationary solutions.
To see what condition on will ensure that this holds, we note that from eq.(26), it follows immediately that
where the variation of the flux -form, , on is given by
Here the first term in this equation arises from taking ‘’ of the integrand of the first term in eq.(26) (using eq.(21) above), whereas the second term is just the ‘’ of the integrand of the second term in eq.(26). However, we have
Thus, we obtain
We now impose the requirement that vanish whenever is stationary (even when is non-stationary). We also explicitly assume that the reference solution, , (on which vanishes for all cross-sections and hence ) is stationary. Since both and vanish on , we obtain from eq.(31) the remarkably simple formula
It then follows immediately (as a consequence of our choice of ) that vanishes (for all ) on stationary solutions, as we desired. Equation (32) also implies an additional desirable property of : We have whenever is an exact symmetry—i.e., whenever —regardless of whether radiation may be present.
If a symplectic potential satisfying our above condition exists and is unique, then eq.(26) together with the requirement that vanish (for all cross-sections and all ) on a particular, specified solution, , uniquely determines . However, there remains a potential difficulty in specifying : If then we also have , where is any diffeomorphism generated by a representative of an infinitesimal asymptotic symmetry. Since we have no meaningful way of distinguishing between and , if we demand that vanish on we must also demand that it vanish on . However, this overdetermines (so that no solution exists) unless the following consistency condition holds: Let be a representative of an infinitesimal asymptotic symmetry and consider the field variation about given by . Since this corresponds to the action of an infinitesimal asymptotic symmetry on , under this field variation we must have . On the other hand, is specified by eq.(26). Since under this field variation we have
and since, by assumption, vanishes at , we find that the consistency requirement on is that for all representatives and of infinitesimal asymptotic symmetries and for all cross-sections , we must have
From eqs.(21) and (25) together with the vanishing of at , it follows that the right side of eq.(34) is independent of cross-section and thus need only be checked at one cross-section. In addition, eq.(34) manifestly holds when is an exact symmetry of —i.e., when —since in that case. Using
(Here, the integral over is to be interpreted as an asymptotic limit, with the limit guaranteed to exist by the argument given above. If extends continuously to then the term makes no contribution to the integral since both and are tangent to .) Since eq.(36) is manifestly antisymmetric in and , it follows that the consistency condition also is automatically satisfied whenever is an exact symmetry of . However, if both and are asymptotic symmetries that are not exact symmetries of , then eq.(34) (or, equivalently, eq.(36)) yields a nontrivial condition that must be satisfied by .
To summarize, we propose the following prescription for defining “conserved quantities” in case II: Let be a symplectic potential on (see eq.(25) above) which is locally constructed out of the dynamical fields and background structure (and is an analytic function of the dynamical fields when is analytic), is independent of any arbitrary choices made in specifying the background structure, and is such that vanishes for all tangent to whenever is stationary. (If it exists, such a is unique up to addition of a term where is locally constructed out of the dynamical fields and background structure (and is analytic in the dynamical fields when is analytic), is independent of any arbitrary choices made in specifying the background structure, and is such that vanishes for all whenever is stationary.) Let be a stationary solution that satisfies eq.(34) (or, equivalently, eq.(36)) for all infinitesimal asymptotic symmetries and . Then we define by eq.(26) together with the requirement that vanish on . To the extent that a satisfying the above requirements exists and is unique, and to the extent that a stationary satisfying (34) exists, this defines a prescription for defining “conserved quantities” associated with asymptotic symmetries. This prescription automatically gives rise to the flux formula (32), so that the flux vanishes whenever is stationary or is an exact symmetry.
In the next section, we analyze what this general prescription yields for the case of asymptotically flat spacetimes at null infinity in vacuum general relativity.
5 “Conserved Quantities” at Null Infinity in General Relativity
In vacuum general relativity, the manifold is taken to be -dimensional and the only dynamical field, , is the spacetime metric, . We shall write the varied field as
The Einstein-Hilbert Lagrangian of general relativity is
where denotes the scalar curvature of and is the spacetime volume form associated with . The presymplectic potential -form is given by
where is the derivative operator associated with . The corresponding presymplectic current -form is 
Finally, the Noether charge -form associated with a vector field is given by 
We wish to consider spacetimes that are asymptotically flat at future and/or past null infinity. For definiteness, we will consider future null infinity. (Sign changes would occur in several formulas when we consider past null infinity on account of our orientation convention on (see footnote 8).) We denote future null infinity by and adopt the standard definition of asymptotic flatness there (see, e.g., ). The key ingredient of this definition is that there exist a smooth141414The requirement of smoothness could be weakened considerably without affecting our analysis. metric, , on and a smooth function, , on such that on , on , and is null151515For solutions to the vacuum field equations, it follows from the fact that on that is null on in the metric . and nonvanishing everywhere on , and such that throughout we have
We also assume that has topology . In the following all indices will be raised and lowered using the “unphysical metric”, . We write
(Here denotes the derivative operator associated with , although, of course, since is a scalar, is independent of the choice of derivative operator.) We may use the freedom with a smooth, strictly positive function on to assume, without loss of generality, that the Bondi condition
holds. An immediate consequence of eq.(47) is that on we have , so in the Bondi gauge
Without loss of generality (see, e.g., ), we also may assume that the conformal factor, , on and the unphysical metric, , on are universal quantities, i.e., they may be assumed to be independent of the physical metric, , on . Without loss of generality, we may (by use of freedom remaining in the choice of ) take the universal unphysical metric , on to be such that the induced spatial metric on all cross-sections of is that of a round two-sphere of scalar curvature . In the following, we will fix an allowed choice of on and a choice of . We will then take161616Note that our imposition of this rather rigid structure on as a result of our gauge fixing is not done merely for convenience, but is necessary in order that extend to . to consist of metrics, , on such that extends smoothly to and equals there, and such that the Bondi condition (47) holds on . It may then be checked that the general notion of infinitesimal asymptotic symmetries given in the previous section corresponds to the usual notion of infinitesimal BMS symmetries; indeed, our general definition of infinitesimal asymptotic symmetries corresponds closely to the definition of infinitesimal BMS symmetries171717The only difference between our definition and the definition given in  concerns the notion of the equivalence of two representatives, and . In addition to requiring agreement of and at , we impose the extra requirement that they give rise to the same asymptotic integral (19). However, it is not difficult to show that if and agree at then they automatically give rise to the same asymptotic integral (19). given in .
It follows immediately from our conditions on that the unphysical perturbed metric
extends smoothly to and vanishes there, so it can be written in the form
where extends smoothly to and, in general, is nonvanishing there. Furthermore, since , we have
This, in turn, implies that vanishes on , so we may write
where is smooth (and, in general, nonvanishing) at . This implies that
The crucial issue with regard to the applicability of the ideas of the previous section is whether the presymplectic current -form181818As noted in section 2, has the ambiguity (5). However, Iyer  has shown that if is such that maintains the general form given by eq.(23) of  with the coefficients in that formula being regular, analytic functions of the fields, then must vanish on . Consequently, in vacuum general relativity, the limit to of is, in fact, unique. extends continuously to . To investigate this, we express the quantities appearing in eq.(41) in terms of and variables that extend smoothly to . Clearly, the unphysical volume element
extend smoothly to and are nonvanishing there. We eliminate the the action of the physical derivative operator, , on in terms of the unphysical derivative operator, , via
where (see, e.g., )
Finally, we substitute
The terms appearing in the resulting expression for may now be classified as follows: (i) Terms in which acts on or . For these terms, the powers of resulting from eqs.(55), (56), and (59) cancel, so these terms extend smoothly to and are, in general, nonvanishing there. (ii) Terms in which does not act on or and is proportional to . These terms cancel due to the antisymmetry in and . (iii) Terms in which does not act on or but is not proportional to . These terms necessarily contain a contraction of with or , and eq.(53) can then be used. The extra power of picked up by the use of this equation ensures that these terms extend smoothly to , where they are, in general, nonvanishing. The upshot is that extends smoothly to and is, in general, nonvanishing there. Thus, with our definition of , asymptotically flat spacetimes at null infinity in general relativity do indeed fall into the category of “case II” of the previous section.
To apply the proposed prescription of the previous section to define a “conserved quantity”, , for each BMS generator, , and each cross-section, , of , we need an explicit formula for the pullback, , of the extension of to . To do so, we define by
so that the pullback, , of to defines a positively oriented volume element191919For past null infinity, this volume element would be negatively oriented, resulting in sign changes in some of the formulas below. on (see footnote 8). We have
where we have written and “” denotes the same terms as in the preceding expression with and interchanged.
The above formula can be rewritten in a more useful form as follows. By a direct computation using eq.(7.5.14) of , the variation of the unphysical Ricci tensor at is given by
Hence, defining by