Partial Differential Equations of Physics

Transcription

1 Partial Differential Equations of Physics arxiv:gr-qc/ v1 27 Feb 1996 Robert Geroch Enrico Feri Institute, 5640 Ellis Ave, Chicago, Il Introduction February 3, 2008 The physical world is traditionally organized into various systes: electroagnetis, perfect fluids, Klein-Gordon fields, elastic edia, gravitation, etc. Our descriptions of these individual systes have certain features in coon: Use of fields on a fixed space-tie anifold M, a geoetrical interpretation of the fields in ters of M, partial differential equations on these fields, an initial-value forulation for these equations. Yet beyond these coon features there are nuerous differences of detail: Soe systes of equations are linear, and soe are not; soe have constraints, and soe do not; soe arise fro Lagrangians, and soe do not; soe are firstorder, and soe higher-order. Systes also differ in other respects, e.g., as to what fields they need as background, what interactions they perit (or require). It alost sees as though, in the end, every physical syste has its own special character. It ight be useful to have a systeatic treatent of the fields and equations that arise in the description of physical systes. Thus, there would be a general definition of a field, and a general for for a syste of partial differential equations on such fields. The treatent would consist of a fraework sufficiently broad to encopass the systes found in nature, but no broader. One would, for exaple, treat the initial-value forulation once and for all within this broad fraework, with the forulations for individual physical systes eerging as special cases. In a siilar way, one would treat within a quite general context constraints, the geoetrical character of physical fields, how soe systes require other fields as a background, 1

2 how interactions operate, etc. The goal of such a treatent would be to get a better grip on the structural features of the partial differential equations of physics. Here are two exaples of issues on which such a treatent ight shed light. What, if any, is the physical basis on which the various fields on the anifold M are grouped into separate physical systes? Thus, for instance, the fields (n, F ab, ρ, u a ) are grouped into (n, ρ, u a ) (a perfect fluid), and (F ab ) (electroagnetic field). By physical basis, we ean in ters of the dynaical equations on these fields. A second issue is this: How does it coe about that the fields of general relativity are singled out as those for which diffeoorphiss on M are gauge? On its face, this singling out sees surprising, for the diffeoorphiss act equally well on all the physical fields on M. We shall here discuss, in a general, systeatic way, the structure of the partial differential equations describing physical systes. We take it as given that there is a fixed, four-diensional anifold M of space-tie events, on which all the action takes place. Thus, for instance, we are not considering discrete odels. Further, physical systes are to be described by certain fields on M. These ay be ore general than ere tensor fields: Our fraework will adit spinors, derivative operators, and perhaps other field-types not yet thought of. But we shall insist largely for atheatical convenience that the set of field-values at each space-tie point be finitediensional. We shall further assue that these physical fields are subject to systes of partial differential equations. That is, we assue, aong other things, that physics is local in M. Finally, we shall deand that these partial differential equations be first-order (i.e., involving no higher than first space-tie derivatives of the fields), and quasilinear (i.e., linear in those first derivatives). That the equations be first-order is no real assuption: Higherorder equations can and will be cast into first-order for by introducing new auxiliary fields. (Thus, to treat the Klein-Gordon equation on scalar field ψ, we introduce an auxiliary vector field playing the role of ψ.) It is y sense that this is ore than a ere atheatical device: The auxiliary fields tend to have direct physical significance. It is not so clear what we are actually assuing when we deand quasilinearity. It is certainly possible to write down a first-order syste of partial differential equations that is not even close to being quasilinear (e.g., ( ψ/ x) 2 + ( ψ/ t) 2 = ψ 2 ). But all known physical systes see to be described by quasilinear equations, and it is anyway hard to proceed without this deand. In any case, first-order, 2

3 quasilinear allows us to cast all the partial differential equations into a convenient coon for, and it is on this coon for that the progra is based. A case could be ade that, at least on a fundaental level, all the partial differential equations of physics are hyperbolic that, e.g., elliptic and parabolic systes arise in all cases as ere approxiations of hyperbolic systes. Thus, Poisson s equation for the electric potential is just a facet of a hyperbolic syste, Maxwell s equations. In Sect. 2, we introduce our general fraework for systes of first-order, quasilinear partial differential equations for the description of physical systes. The physical fields becoe cross-sections of an appropriate fibre bundle, and it is on these cross-sections that the differential equations are written. So, for instance, the coefficients in these equations becoe certain tensor fields on the bundle space. This fraework, while broad in its reach, is not particularly useful for explicit calculations. The reaining sections describe various structural features of these syste of partial differential equations. A hyperbolization (Sect. 3) is a casting of the syste of equations (or, coonly, a subsyste of that syste) into what is called syetric, hyperbolic for. To such a for there is applicable a general theore on existence and uniqueness of solutions. This is the initial-value forulation. The constraints (Sect. 4) represent a certain subsyste of the full syste, the equations of which play a dual role: providing conditions that ust be satisfied by initial data, and leading to differential identities on the equations theselves. The constraints are integrable if these differential identities really are identities; and coplete if the constraint subsyste, together with the subsyste involved in the hyperbolization, exhausts the full syste of equations. The geoetrical character of the physical fields has to do with how they transfor, i.e., with lifting diffeoorphiss on M to diffeoorphiss on the bundle space (Sect. 5). Cobining all the systes of physics into one aster bundle B, then the full set of equations on this bundle will be M-diffeoorphis invariant. This diffeoorphis invariance requires an appropriate adjustent in the initial-value forulation for this cobined syste. Finally, we turn (Sect. 6) to the relationships between the various physical fields, as reflected in their differential equations. Physical fields on space-tie can interact on two broad levels: dynaically (through their derivative-ters), and kineatically (through ters algebraic in the fields). Roughly speaking, two fields are part of the sae physical syste if their derivative-ters cannot be separated into individual equations; and 3

4 one field is a background for another if the forer appears algebraically in the derivative-ters of the latter. The kineatical (algebraic) interactions are the ore failiar couplings between physical systes. It is the exaples that give life to this general theory. We have assebled, in Appendix A, a variety of standard exaples of physical systes: the fields, the equations, the hyperbolizations, the constraints, the background fields, the interactions, etc. We shall refer to this aterial frequently as we go along. Thus, this is not the standard type of appendix (to be read later, if at all, by those interested in technical details), but rather is an integral part of the general theory. Indeed, it ight be well to review this aterial first as a kind of introduction. Appendix B contains a stateent and an outline of the proof of the theore on existence and uniqueness of solutions of syetric, hyperbolic systes of partial differential equations. All in all, this subject fors a pleasant coingling of analysis, geoetry, and physics. 2 Preliinaries Fix, once and for all, a sooth, four-diensional anifold 1 M. The points of M will be interpreted as the events of space-tie, and, thus, M itself will be interpreted as the space-tie anifold. We do not, as yet, have a etric, or any other geoetrical structure, on M. We next wish to introduce various types of fields on M. To this end, let b π M be a sooth fibre bundle 2 over M. That is, b is soe sooth anifold (called the bundle anifold) and π is soe sooth apping (called the projection apping); and these are such that locally (in M) b can be written as a product in such a way that π is the projection onto one factor 3. An exaple is the tangent bundle of M: Here, b is the eight-diensional anifold of all tangent vectors at all points of M, and π is the apping that extracts, fro a tangent vector at a point of M, the point of M. That the 1 We take M to be connected, paracopact, and Hausdorff. 2 See, e.g., Steenrod, The Topology of Fibre Bundles (Princeton University Press, Princeton, l954). Note that, in contrast to what is done in this reference, we introduce no Lie group acting on b. 3 This eans, in ore detail, that, given any point x M, there exists an open neighborhood U of x, a anifold F, and a diffeoorphis ζ fro U F to π 1 [U] such that π ζ is the projection of U F to its first factor. 4

5 local-product condition holds, in this exaple, is seen by expressing tangent vectors in ters of their coponents with respect to a local basis in M. Returning to the general case, for any point x of M, the fibre over x is the set of points π 1 (x), i.e., the set of points κ b such that π(κ) = x. It follows fro the local-product condition that each fibre is a sooth subanifold of b, and that all the fibres are diffeoorphic with each other. In the exaple of the tangent bundle, for instance, the fibre over point x M is the set of all tangent vectors at x. Next, let A be any sooth subanifold of M. A cross-section over A is a sooth apping A φ b such that π φ is the identity apping on A. Thus, a cross-section assigns, to each point x of A, a point of the fibre over x. Typically, A will be of diension four (i.e., an open subset of M), or three. We interpret the fibre over x as the space of allowed physical states at the space-tie point x, i.e., as the space of possible field-values at x. Then the bundle anifold b is interpreted as the space of all field-values at all points of M. A cross-section over subanifold A becoes a field, defined at the points of A. In ost, but not all, exaples (Appendix A) b will be a tensor bundle. Thus, for electroagnetis b is the ten-diensional anifold of all antisyetric, second-rank tensors at all points of M. For general relativity, by contrast, b is the fifty-four-diensional anifold a point of which is coprised of a point of M, a Lorentz-signature etric at that point, and a torsion-free derivative operator at that point. In both of these exaples, the projection π erely extracts the point of M. It is convenient to introduce the following notation. Denote tensors in M by lower-case Latin indices; and tensors in b by lower-case Greek indices. Then, at any point κ b, we ay introduce ixed tensors, where Latin indices indicate tensor character in M at π(κ), and Greek indices tensor character in b at κ. For exaple, the derivative of the projection ap is written ( π) α a, i.e., it sends tangent vectors in b at point κ to tangent vectors in M at π(κ). The derivative of a cross-section, φ, over a four-diensional region of M is written ( φ) a α ; and we have, fro the defining property of a cross-section, ( φ) a α ( π) α b = δ a b. (1) A vector λ α at κ b is called vertical if it is tangent to the fibre through κ, i.e., if it satisfies λ α ( π) b α = 0. Eleents of the space of vertical vectors at κ will be denoted by pried Greek superscripts. Thus, λ α eans λ is 5

6 a tangent vector in b, a vector which, by the way, is vertical. Eleents of the space dual to that of the vertical vectors will be denoted by pried Greek subscripts. Thus, µ α eans µ is a linear apping fro vertical vectors in b to the reals. More generally, these pried indices ay appear in ixed tensors. Note that we ay freely reove pries fro superscripts (i.e., ignore the verticality of an index), and add pries to subscripts (i.e., restrict the apping fro all tangent vectors to just vertical ones), but not the reverses. As an exaple of this notation, we have: ( π) a α = 0. To illustrate these ideas, consider electroagnetis. Then a typical point of the bundle anifold b is κ = (x, F ab ), where x is a point of M and F ab is an antisyetric tensor at x. A tangent vector λ α in b at κ can be represented as an infinitesial change 4 in both the point x of M and the antisyetric tensor F ab. Given such a λ α, the cobination λ α ( π) a α is that tangent vector in M at x represented by just the change in x -part of λ α (ignoring the change in F ab -part). Such a λ α is vertical provided its change in x vanishes so, a vertical vector is represented siply as an infinitesial change in the antisyetric tensor F ab, with x fixed. In this exaple, we ight introduce the field µ α ab = µ α [ab] on b, which takes any such vertical vector, λ α, and returns, as λ α µ α ab, the change in the tensor F ab at x. A connection on fibre bundle b π M is a sooth field γ α a on b satisfying γ α a ( π) b α = δ b a. Given a connection γ α a, those vectors at κ b that can be written in the for ξ a γ α a for soe ξ a are called horizontal. Of course, there exist any possible connections, and so any such notions of horizontality. It follows directly fro these definitions that, fixing a connection, every tangent vector in b at κ can be written, uniquely, as the su of a horizontal and a vertical vector, i.e., that every vector can be split into its horizontal and vertical parts. We ay incorporate this observation into the notation by allowing ourselves the operations with pries that were previously prohibited: In the presence of a fixed connection, γ α a, we ay affix a prie to a Greek superscript (by taking the vertical projection); and, in a siilar way, we ay reove a prie fro a Greek subscript. For exaple, we have α γ a = 0. Note that in every case the reoval and subsequent affixing of a prie leaves a tensor unchanged (but not so for affixing a prie and its subsequent reoval.) Again consider, as an exaple, the case of electroagnetis. (Any other 4 By infinitesial change in the point of..., we ean tangent vector to a curve in.... 6

7 (nonscalar) tensor bundle would be siilar.) Fix any sooth derivative operator a on the anifold M. Then this a gives rise to a connection γ a α on b, in the following anner. For κ = (x, F ab ) any point of b, and ξ a any tangent vector in M at x, let λ α = ξ a γ a α be that tangent vector in b at κ represented as follows: The infinitesial change in x is that dictated by ξ a, while the infinitesial change in F ab is that resulting fro parallel transport, via a, of F ab fro x along ξ a. We thus specify the cobination ξ a γ a α for every ξ a, and so the tensor γ a α itself. Note that we have λ α ( π) α a = ξ a, which shows that the γ a α so defined is indeed a connection. So, the horizontal vectors at κ in this exaple are those for which the infiniestial change in F ab is exactly that resulting fro parallel transport. Clearly, every tangent vector in b can be written, uniquely, as the su of a vertical vector and such a horizontal vector. While every derivative operator on M gives rise, as above, to a connection on b, there are any other connections on b (corresponding roughly to non-linear parallel transport ). We shall not routinely ake use of a connection in what follows, for two reasons. First, for soe fields, such as the derivative operator of general relativity, we have no natural connection. Second, even when there is a natural connection (e.g., for electroagnetis), that connection will itself be a dynaical variable. It is awkward having one dynaical field playing a crucial role in the kineatics of another. We now wish to write down a certain class of partial differential equations on cross-sections. To this end, let k A α and j A be sooth fields on b. Here, the index A lives in soe, as yet unspecified, vector space. Norally, this vector space will be soe tensor product involving tensors in M and in b, i.e., A will erely stand for soe cobination of Latin and Greek indices. But, at least in principle, this could be soe newly constructed vector space attached to each point of b, in which case we would have to introduce a new fibre bundle, with base space b, to house it. Consider now the partial differential equation k A α ( φ) α + j A = 0, (2) where U φ b is a sooth cross-section over soe open subset U of M. This equation is to hold at every point x U, where k and j are evaluated on the cross-section, i.e., at φ(x). Note that this is a first-order equation on the cross-section, linear in its first derivative. The nuber of unknowns at each point is the diension of the fibre; the nuber of equations the diension 7

8 of whatever is the vector space in which the index A lives. The coefficients in this equation, k A α and j A, are functions on the bundle anifold b, i.e., these coefficients ay depend on both the point of M and the field-value φ. Apparently, every syste of partial differential equations describing a physical syste in space-tie can be cast into the for of Eqn. (2). Various exaples are given in Appendix A. Many, such as those for a perfect fluid, the electroagnetic field, or the charged Dirac particle, are already packaged in the appropriate for. Others ust be brought into this for by introducing auxiliary fields. In the Klein-Gordon case, for exaple, we ust augent the scalar field ψ by its space-tie derivative, ψ a, resulting in a bundle space with five-diensional fibres. We then obtain, on (ψ, ψ a ), a first-order syste of equations of the for (2). For general relativity, the fibre over x M consists of pairs (g ab, a ), where g ab is a Lorentz-signature etric and a a torsion-free derivative operator at x. The curvature tensor arises in (2) as the derivative of the derivative operator. Let us agree that all first-order equations on the fields (even those that follow fro differentiating other equations) are to be included in (2). Thus, for exaple, Eqn. (35) is included for the Klein-Gordon syste. Note that the only structure we are iposing on the physical fields at this stage is a differentiable structure, as carried by the anifold b. If you wish to utilize any additional features on these fields e.g., the ability to add fields, to ultiply the by nubers, to ultiply the by each other, etc. then this ust be introduced, separately and explicitly, as additional structure on the bundle space b. For exaple, for electroagnetis, but not for a perfect fluid, each fibre has the additional structure of a vector space The present forulation of partial differential equations carries with it a certain gauge freedo. Let λ A b be any sooth field on b. Then Eqn. (2) reains invariant under adding to k A α the expression λ A b ( π) α b, and at the sae tie to j A the expression λ A. That is, the solutions φ of (2) before these changes in k and j are precisely the sae as the solutions after. Note that k A α, (i.e., what results fro contracting k A α only with vertical v α ) is gauge-invariant. Furtherore, this tensor exhausts the gaugeinvariant inforation, in the following sense: Given any field ˆk A α satisfying ˆk A α = k A α, then there exists one and only one gauge transforation that sends k A α to ˆk A α. This gauge freedo reflects the idea that the horizontal part of the α-contraction in (2) does not really involve the derivative of the 8

9 cross-section, by virtue of the identity (1). Thus, the coponents of k A α that participate in this part of the α-contraction are not contributing to the dynaics. It would be ost convenient if we could soehow circuvent this gauge freedo, e.g., by rewriting Eqn. (2) to involve only the gauge-invariant part, k A α, of k A α. Unfortunately, this cannot be done in any natural way in general. But it can be done in the presence of soe fixed connection, γ a α, on the bundle b. In fact, given a connection, we ay always achieve through gauge a k A α that is vertical in α in the sense that it annihilates every horizontal vector h α. Furtherore, this requireent on k A α copletely exhausts the gauge freedo. Indeed, given k A α and connection γ a α, then the gauge transforation with λ A b = γ b α k A α, uniquely, does the job. It will soeties be convenient, when a connection is available, to exploit this gauge-choice. 3 Hyperbolizations A key feature of the partial differential equations of physics is their initialvalue forulation, i.e., their forulation in ters of initial data and tie - evolution. It turns out that this forulation can be carried out in a rather general setting. This is the subject of the present, and uch of the following, section. Fix a partial differential equation of the for (2), so we have in particular fixed sooth fields k A α and j A on b. By a hyperbolization of Eqn. (2), we ean a sooth field h A α on b such that i) the field ha α k A β on b is syetric in α, β ; and ii) for each point κ b, there exists a covector n in M at π(κ) such that the tensor n h A α k A β at κ is positive-definite. Note that the definition involves only k A α, and neither j A nor the rest of k. Thus, in particular, the definition is gauge-invariant. Note also that the hyperbolizations at a point κ b for an open subset of a vector space. For h A α any hyperbolization, and v α any nonzero (vertical) vector at a point, the cobination v α h A α at that point ust be nonzero. (This follows, contracting the positive-definite tensor in ii) with v α v β.) But this iplies, in turn, that the diension of the space of equations in (2) (that of the index A ) ust be greater than or equal to the diension of the space of unknowns (that of the index α ). So, if this diensionality criterion fails, then there can be no hyperbolization. But suppose this criterion is satisfied: Can we then 9

10 guarantee a hyperbolization? The answer is no. In fact, there is no known, practical procedure, given a general partial differential equation (2), for finding its hyperbolizations, or, indeed, for even deterining whether or not one exists. (This is essentially a little algebra proble: Given a tensor k A α at a point, what are the tensors h A α at that point with ha α k A β syetric?) In practice, hyperbolizations are found, in sufficiently low diensions, by solving explicitly the algebraic equations inherent in i) and ii) ; and, in higher diensions, by guessing. Physical considerations frequently suggest candidates. Consider again the exaple of electroagnetis (Appendix A). We have already rearked that, at point κ = (x, F ab ) of the bundle space b, a typical vertical vector, which we now write δφ α, is represented by an infiniesial change, δf ab, in the electroagnetic field at x. Since the left sides of Maxwell s equations, (29) and (30), consist of a vector and a third-rank, antisyetric tensor, the index A lies in the eight-diensional vector space of such objects. That is, a typical vector in this space can be written σ A = (s a, s abc ), with s abc = s [abc]. (Note that, since di A = 8 > 6 = di α, our diensionality criterion above is satisfied.) The fields k A α and j A are to be read off by coparing Maxwell s equations, (29) and (30), with the general partial differential equation (2). We thus obtain k A β σ A n δˆφ β = s a (n b δ ˆF ab ) + s abc (n [a δ ˆF bc] ). (3) Here, we have represented k A β by giving the scalar that results fro contracting away its indices, on vectors σ A, n, and δˆφ β. The field j A of (2), on the other hand, depends on gauge. If we choose for the gauge that deterined by the derivative operator a on M used in Maxwell s equations (29), (30), then we have j A = 0. Now fix any vector t a at x, and consider the tensor h A α given, in Eqn. (31), as the A-index vector that results fro the contraction h A α δφα. Substituting this vector for σ A in (3), we obtain h A α δφα k A β n δˆφ β = δf a t (n b δ ˆF ab ) 3 2 t[a δf ab] (n [a δ ˆF bc] ) = 2t a n b [δf (a δ ˆF b) 1/4g ab δf n δ ˆF n ]. (4) It now follows, provided only that the vector t a is chosen tielike, that the h A α of (31) is a hyperbolization. Indeed, condition i) follows fro the fact that the last expression in (4) is syetric under interchange of δf ab and 10

11 δ ˆF ab ; and condition ii) follows fro the fact that, whenever n is tielike with t n < 0, the last expression in (4) defines a positive-definite quadratic for in δf ab. Thus, every tielike vector field t a on M gives rise to a hyperbolization of Maxwell s equations. In fact, this faily exhausts the hyperbolizations in the Maxwell case. The situation is siilar for any other physical exaples. (See Appendix A.) Thus, the hyperbolizations of the Klein-Gordon equation are characterized by two vector fields on M; and those for the perfect-fluid equation by two scalar fields. Even dissipative fluids 5 can be described by equations aditting a hyperbolization. There are only two physical exaples, as far as I a aware, for which there exist no hyperbolization. One is Einstein s equation, for which the lack of a hyperbolization is related to the diffeoorphisinvariance of the theory; and the other is dust. We shall return to each of these exaples later. Fix a hyperbolization, h A α, of Eqn. (2). For each point κ b, denote by s κ the collection of all covectors n in M at π(κ) such that the tensor n h A α k A β is positive-definite. Then s κ is a nonepty (by condition ii)), open, convex cone. The physical interpretation of these cones will turn out to be the following. The tangent vectors p a in M at π(κ) such that p a n a > 0 for all n a s κ represent the signal-propagation directions of the physical field. Note that these p a for a closed, nonepty, convex cone at each point, the dual cone of s κ. These cones depend not only on the point x of M, but also in general on the value of the field at x, i.e., on where we are in the fibre over x. In cases in which there is ore than one hyperbolization, these cones could also depend on which hyperbolization has been selected. But it turns out that, for ost physical exaples, these cones are essentially independent of hyperbolization. Thus, in the case of electroagnetis, the signal propagation directions p a consist of all tielike and null vectors lying in one of the two halves of the light cone. In the case of a perfect fluid, the p a for the sound cone. Is it possible to isolate, via a definition, the crucial algebraic feature of k A α in such physical exaples that is responsible for hyperbolization-independent cones? Suppose that, included aong the various physical fields on M is a spacetie etric, g ab. In that case, we say that the syste (2) is causal if all the 5 See, e.g., I.S. Liu, I. Muller, T. Ruggeri, Ann Phys (NY) 169, 191 (1986); R. Geroch, L. Lindblo, Ann Phys (NY) 207, 394 (1991); R. Geroch, J. Math. Phys. 36, 4226 (1995). 11

12 signal-propagation directions are tielike or null. This is equivalent to the condition that each s κ includes all tielike vectors lying within one of the two halves of the light cone. A perfect fluid, for exaple, is causal provided its sound speed, p + n p, does not exceed the speed of light. ρ ρ+p n Fix a hyperbolization h A α of Eqn. (2). This hyperbolization leads, as we now explain, to an initial-value forulation. By initial data we ean a sooth, three-diensional subanifold S of M, together with a sooth cross-section, S φ 0 b, over S, such that, for every point x S, a noral n to S at x lies in the cone s φ0 (x). In other words, we ust specify the physical state of the syste at each point of the three-diensional anifold S, in such a way that, at every point of S, all signal-propagation directions are transverse to S. Note that the role of the cross-section, φ 0, in this definition is to deterine the cone within which the noral to S ust lie. Thus, a change of cross-section, keeping S fixed, could destroy the initial-data character of (S, φ 0 ). (Changing the hyperbolization could, in principle, also change the initial-data character, but, as we rearked earlier, it generally does not.) As an exaple of these definitions we have: If we have on M a spacetie etric g ab with respect to which (2) is causal, then every (S, φ 0 ), with S spacelike, is initial data. We ay now suarize the fundaental existence-uniqueness theore as follows. Given initial data (S, φ 0 ), there exists, in a suitable neighborhood U of S, one and only one cross-section, U φ b, such that i) φ S = φ 0 and ii) h A β [k A α( φ) α + j A ] = 0. (5) Condition i) ensures that the field φ, specified over the neighborhood U of S, agree, on S itself, with the given initial conditions, φ 0. Condition ii) ensures that the field φ satisfy a certain partial differential equation derived fro (2) (specifically, by contracting it with h A β ). In short, the theore states that we can solve the partial differential equation (5), uniquely, subject to any given initial conditions. There is given in Appendix B a ore detailed version of this theore (including ore inforation regarding the neighborhood U), and a sketch of the proof. This version, in particular, supports our interpretation of the cones s κ in ters of signal-propagation. Since every solution of Eqn. (2) is autoatically a solution of (5), the theore above guarantees local uniqueness of the solutions of any syste, (2), of partial differential equations aditting a hyperbolization. Thus, for ost 12

13 systes of interest in physics, initial data lead to a unique local solution. Furtherore, if the hyperbolization h A α is invertible (which holds, by the way, if and only if di A = di α ), then Eqn. (5) is equivalent to Eqn. (2). In this case, e.g., for a perfect fluid, the theore also guarantees local existence of solutions of (2). But in any physical exaples electroagnetis included h A α is not invertible so part of Eqn. (2) is lost in the passage to (5). In these cases, we cannot guarantee, directly fro the theore, local existence of solutions of Eqn. (2). The fate of these lost equations is the subject of the following section. Let us now return briefly to the exaple of dust. (See Appendix A.) With the traditional choice of fields ρ (ass density) and u a (unit, tielike fourvelocity) the dust equations, (67) and (68), adit no hyperbolization. This is perhaps surprising, for this syste obviously has an initial-value forulation. It turns out that, if we introduce the auxiliary field w b a = a u b, then the corresponding syste of equations on this new set of fields, (ρ, u a, w b a ), does adit a hyperbolization. It is not clear what, if any, is the physical eaning of this odification. Furtherore, the hyperbolization it produces is apparently lost on coupling the dust with gravitation via Einstein s equation. (This behavior is a consequence of the appearance of a Rieann tensor in the equations on (ρ, u a, w b a ).) What is going on physically in this exaple? 4 Constraints Fix a partial differential equation of the for (2), so we have in particular fixed sooth fields k A α and j A on b. While uch of the aterial of this section finds application to the initial-value forulation, we require at this stage no specific hyperbolization nor, indeed, even the existence of one. A constraint at point κ b is a tensor c An at κ such that c A(n k A ) α = 0. (6) Note that the definition is gauge-invariant, and that the constraints at κ for a vector space. A nuber of exaples, for various physical systes, is given in Appendix A. For instance, the equations for a perfect fluid adit only the zero constraint; those for Klein-Gordon, a ten-diensional vector space of constraints; and those for general relativity an eighty-four diensional vector space. Maxwell s equations, on the other hand, adit a two-diensional 13

14 vector space of constraints: The general c An is given, in this case, by Eqn. (32), where x and y are arbitrary nubers. To check that this c An does indeed satisfy (6), cobine it with the k A α for Maxwell s equations given by (3), to obtain c An k A α δˆφ α = xδ ˆF n + yǫ nbc δ ˆF bc. (7) Now syetrize both sides over n,. Each constraint, as we shall see, plays two distinct roles: It signals a differential condition that ust be iposed on initial data for Eqn. (2), as well as a differential identity involving Eqn. (2). In the case of Maxwell s equations, for exaple, the first role is reflected in the failiar spatial constraint equations, E = 0, B = 0. The second role is reflected in the fact that identities result fro taking the divergence and curl, respectively, of Maxwell s equations, (29) and (30). We begin with the first role. Fix constraint c An. Let U φ b be any solution of Eqn. (2), defined in open U M, and let S be any threediensional subanifold of U. Consider now the equation n a c Aa [k A α ( φ) α + j A ] = 0, (8) at points x of S, where n a is a noral to S at x and the coefficients are evaluated at κ = φ(x). We first note that Eqn. (8) holds on S, for it is a consequence of (2). We next clai that the left side of Eqn. (8) involves only φ 0 = φ S, i.e., only φ restricted to S. To see this, first note that φ 0 alone deterines ( φ) α at points of S up to addition of a ter of the for n v α. But such a ter annihilates n a c Aa k A α, by the defining equation, (6), for a constraint, and so does not contribute to the left side of Eqn. (8). What we have shown, then, is that Eqn. (8) is a constraint equation : It is a differential equation on cross-sections over S that ust be satisfied by every restriction to S of a solution of Eqn. (2). In the Maxwell case, for exaple, the two independent constraints give rise, via (8), to the vanishing of the divergence of the electric and agnetic fields. Note that (S, φ 0 ) above need not be initial data: We have as yet introduced no hyperbolization. We next introduce a notion of sufficiently any constraints. We say the constraints are coplete if, for any point κ b and any nonzero covector n n at π(κ) M, we have di(c An n n ) + di(v α ) = di(σ A ). (9) 14

15 The first ter 6 is the diension of the space of all vectors of the indicated for, as c An runs over all constraints at κ. The second ter is the diension of the space of vertical vectors, i.e., the diension of the fibres. The last ter is the diension of the space of equations represented by (2). Eqn. (9) eans, roughly speaking, that there are at least as any equations as unknowns in Eqn. (2), and that any excess is taken up entirely by constraint equations, (8). This interpretation will be ade ore precise shortly. The constraints are coplete for the vast ajority of physical exaples. (See Appendix A.) Thus, Eqn. (9) reads, for the perfect-fluid equations, = 5 ; for Maxwell s equations, = 8 ; for the Klein-Gordon equations, = 11. For Einstein s equation, the constraints are not coplete: Eqn. (9) reads = 110. This, as we shall see later, is related to the diffeoorphis-invariance of the theory 7. Is there soe siple characterization of those tensors k A α that yield coplete constraints? The second role of a constraint is in signaling a differential identity involving Eqn. (2). The idea here is very siple. Eqn. (2) is to hold at every point x of soe open subset U of M. Taking the x-derivative, n, of this equation, and contracting with any constraint c An, we obtain an equation involving the first and second derivatives of the cross-section. But, as it turns out, the second-derivative ter drops out, by virtue of (6), and so we are left with an algebraic in fact, quadratic equation in the first derivative, ( φ) α, of the cross-section. That is, we obtain an integrability condition for Eqn. (2). If this integrability condition holds as an algebraic consequence of Eqn. (2), then we say our constraint is integrable. Unfortunately, all this becoes soewhat ore coplicated when written out explicitly. Fix any (torsion-free) derivative operator, α, on the anifold b, such that the derivative of every vertical vector field is vertical. (Such always exists, at least locally, by the local-product character of the fibre bundle.) Extend 8 this operator to ixed fields on b by deanding 6 Note that this ter can be and in exaples (such as Klein-Gordon) frequently is less than the diension of the vector space of constraints. 7 We reark that there exist exaples (though apparently no physically interesting ones) of a tensor k A α aditting a hyperbolization, but whose constraints are not coplete. 8 This is done as follows. Any field ξ can be written in the for ( π) β ξ β, uniquely but for the freedo to add to ξ β any vertical vector field. Now define α ξ = ( π) β α ξ β, noting that the right side is invariant under this freedo. Note 15

16 β ( π) α = 0. Then the operator derivative along the cross-section is ( φ) n α α. Applying this operator to (2), and contracting with any constraint c An, we obtain c An ( β k A α )( φ) n β ( φ) n α + c An ( β j A )( φ) n β = 0. (10) In the derivation of Eqn. (10), there arises initially the ter [c An k A α ] [( φ) n β β ( φ) α ], involving the second derivative of the cross-section. To see that this ter vanishes, first note that the index α in the second factor is vertical (contracting with ( π) α s ), and so only the antisyetrization of this factor over n, contributes (by definition of a constraint), yielding zero (by the torsion-free character of α ). Eqn. (10) is our integrability condition. We say that constraint c An is integrable if Eqn. (10) is an algebraic consequence of Eqn. (2). What this eans, in ore detail, is that the left side of Eqn. (10) is soe ultiple of the left side of (2) plus soe ultiple of the difference between the two sides of the identity (1), where each of these two ultiplying factors is an expression linear in ( φ) a α. Writing this out and equating powers of ( φ) a α, we conclude: Constraint c An is integrable if and only if there exist tensors σ A α and σ a b α, with σ a a α = 0, such that α (c A ka n β ) + β (c An ka α ) = σ A α k A n β + σ An β k A α + σ s n β( π) α s + σ n s α( π) β s, (11) where we have set k A α = k A α j A( π) α. Applying a prie to both α and β in this equation, and using (6), we obtain 2 [α (c A n k A β ]) = σ A α k A n β + σan β k A α. (12) This part of (11) is anifestly gauge-invariant (involving only k A α, and not j A or the rest of k), and independent of the derivative operator α (involving only the vertical curl ). What reains of Eqn. (11) is essentially one scalar relation, expressing the divergence of c An j A in ters of other fields. Is there soe siple way to write this reaining relation, e.g., a way that separates its physical content fro the gauge freedo inherent in (k, j), α, and the σ s? In electroagnetis, to take one exaple, Eqn. (12) is satisfied with σ A α = 0. What reains of Eqn. (11) in this exaple is just the vanishing of the divergence of the electric charge-current. that this extension of α to fields with Latin indices is unique. 16

17 Failure of integrability would ean that we have soehow failed to include in (2) all the relevant conditions on the first derivative of the crosssection. The standard procedure, in such circustances, is, first, to enlarge Eqn. (2) to encopass the additional conditions on ( φ) α. Then look for any additional constraints arising fro this enlargeent, and if any of these fail to be integrable, enlarge Eqn. (2) further, etc. Unfortunately, it is not clear, in the present general context, how to ipleent this procedure. How do you enlarge a syste, (2), linear in ( φ) α, to encopass a quadratic relation (10)? Nowhere in this section so far have we introduced a hyperbolization. It is perhaps striking that so uch of the subject of constraints can be carried out at this level, for it is largely in their interaction with hyperbolizations that constraints coe to the fore. We turn now to this interaction. Fix, therefore, a hyperbolization, h A α, for Eqn. (2). Let c An be a constraint. Then Eqn. (8) holds for the restriction, φ 0 = φ S, of any solution, φ 0, of Eqn. (2) to any three-diensional subanifold, S, of M. So, in particular, this equation holds when (S, φ 0 ) are initial data, i.e., when the noral n a to S at each point κ lies in the cone s φ0 (x). Thus, given initial data, (S, φ 0 ), we have no hope of finding a corresponding solution of Eqn. (2) unless those data satisfy Eqn. (8) for every constraint c An. Eqns. (8) becoe constraint equations on initial data. Next, fix κ b and n a s κ. Then, we clai, for any constraint c An and any (vertical) vector v α, we can have n a c Aa = v α h A α only if each side is zero. Indeed, this equality iplies (n a c Aa )k A β v β n = (v α h A α )k A β vβ n. But the left side vanishes (by definition of a constraint), while vanishing of the right side iplies v α = 0 (by n a s κ ). What we have shown, in other words, is that the subspace of vectors of the for n a c Aa with c Aa a constraint, and that of vectors of the for v α h A α with vertical, have only the zero vector vα in coon. But this iplies that the left side of Eqn. (9) is less than or equal to the right side. That is, in the presence of a hyperbolization, half of Eqn. (9) is autoatic. Now suppose that the constraints are coplete, i.e., that the full equality (9) holds. It then follows that our two independent subspaces {n a c Aa } and {v α h A α } in fact span the space of all vectors of the for σa. What this eans, in geoetrical ters, is that the constraint coponents of Eqn. (2) the results of contracting it with vectors of the for n a c Aa and the dynaical coponents of Eqn. (2) the result of contracting it with h A α together coprise the whole of Eqn. (2). Copleteness, in the 17

18 presence of a hyperbolization, eans the absence of any stray equations in (2). Finally, we return to the issue, raised in the previous section, of when there is an initial-value forulation for the full equation (2). Fix a hyperbolization h A α for this equation, and suppose that its constraints are both coplete and integrable. Let (S, φ 0 ) be initial data, and suppose that these data satisfy all the constraint equations of the type (8) (for if not, then there is certainly no evolution of these data). By the general existence-uniqueness theore (Sect. 3 and Appendix B), there exists a solution, U φ b, of the evolution Eqn. (5), with φ S = φ 0, where U is an appropriate neighborhood of the three-diensional subanifold S of M. We now clai that, under certain conditions, this cross-section φ satisfies our full syste, (2), of partial differential equations. It is convenient, for purposes of this paragraph, to introduce upper-case Greek indices to lie in the vector space of constraints; so, in this notation, we have a single constraint tensor, c Aa Γ. Denote the left side of Eqn. (2), evaluated on the cross-section φ, by I A. Thus, we have h A α I A = 0 everywhere in U (by (5)), and I A = 0 on S (by copleteness); and we wish to show I A = 0 everywhere in U. To this end, consider the expression (c A Γ + σ α Γh A α ) I A, (13) where σ α Γ is any field on b. We clai that this expression is, everywhere in U, a ultiple of I A. Indeed, the first ter in parentheses leads to such a ultiple since, by integrability, c A Γ I A is a ultiple of I A ; and the second ter also leads to such a ultiple, using h A α I A = 0 and differentiating by parts. To suarize, we have shown so far that I A vanishes on S, and satisfies a certain first-order, quasilinear (in fact, linear) partial differential equation arising fro the expression (13). Since this differential equation clearly has as one solution I A = 0, we can conclude that I A = 0 in a neighborhood of S if we can show local uniqueness of its solutions. The ost direct way to prove local uniquenss of solutions of a partial differential equation is to show that it adits a hyperbolization. In the present instance, tensor h AΓ is a hyperbolization of the differential equation resulting fro (13) for soe choice of σ α A provided h AΓ has the following property: The expression h AΓ c B Γv A w B, for all v, w with v A h A α = w Ah A α = 0, is syetric under interchange of v and w, and, contracted with soe n, is positive-definite. When in ters of the original k A α and its hyperboliza- 18

19 tion h A α does such a hyperbolization haγ exist? In physical exaples e.g., in electroagnetis such an h AΓ does indeed exist, and so we have in these exaples uniqueness of solutions of the equation resulting fro (13), and so an initial-value forulation for the full syste (2). Is there any siple, reasonably general, condition on k A α, h A α that will guarantee existence of a hyperbolization h AΓ? Are there interesting cases in which uniqueness of solutions of the differential equation arising fro (13) ust be shown by soe other ethod? 5 The Cobined Syste Diffeoorphiss In the preceeding sections, we have been analyzing the structure of the partial differential equation describing a single physical syste. This analysis was to be applied separately to the electroagnetic field, a perfect fluid, or whatever. However, in the real world, all these systes coexist on M, norally in interaction with each other. We now consider the syste that results fro cobining all these subsystes. Once again, we have a sooth fibre bundle, B Π M, over the fourdiensional space-tie anifold M. Now, however, the fibre in B over x M represents the possible values at x of all possible physical fields in the universe. Thus, this fibre would include an antisyetric tensor (for electroagnetis), a Lorentz etric and derivative operator (for gravity), two scalar and one vector field (for a perfect fluid), etc. Note that we are iplicitly assuing that the space that results fro cobining these fields is finite-diensional, and that we are soehow capable of finding it. One or both of these assuptions ay be incorrect. In any case, we iagine that we have constructed such a bundle. Again, a cross-section, M Φ B, of B represents an assignent of a coplete physical state (of everything) to each point of space-tie, i.e., a stateent of the entire dynaics of the universe. And, again, we ipose on such cross-sections the general first-order, quasilinear partial differential equation, K A α ( Φ) α + J A = 0. (14) In the analogous equation for a single syste, (2), we allowed the coefficients, k A α and j A, to be arbitrary (sooth) fields on the bundle anifold b. That is, we allowed these coefficients to depend on both the point of the 19

20 space-tie anifold M and the value of the field assigned to that point. But, in the context of this cobined syste, an explicit dependence of the coefficients, K A α and J A, on the point of M is, we suggest, inappropriate. After all, we identify the points of the anifold M, not by soehow perceiving the directly, but rather ore indirectly, by observing the various physical fields on M. So, for instance, the physical distinction between two points, x and y, of M rests on the difference between the values of soe physical field at x and at y. (This issue did not arise in the context of a single syste, for there x-dependence of k and j could arise through other physical fields, not included in the dynaics of (2).) In any case, we expect that the coefficients of K and J in (14) will depend explicitly only on the fibres of B, with any dependence on the point of M arising only iplicitly through the cross-section. Unfortunately, this expectation at least, as it is stated above does not ake atheatical sense! The proble is that our fibre bundles are not naturally products, and so there is no such thing as a function only of the fibre-variables, independent of the base-space variables. We ust therefore proceed in a different way. We now deand that, as part of the physical content of the bundle B, there be given on it the following additional structure. To each diffeoorphis D on the anifold M, there is to be assigned 9 a lifting of it to a diffeoorphis, ˆD, on the anifold B. By lifting, we ean that ˆD ust satisfy Π ˆD = D Π, i.e., that ˆD ust take entire fibres to fibres, such that the induced diffeoorphis on M is precisely the original D. We further require of these liftings that they respect the group structure of the M-diffeoorphiss, i.e., that ( id M ) = (id B ) and ( D D ) = ˆD ˆD. In short, we ust specify how the physical fields transfor under diffeoorphiss on M. In the exaples of Appendix A, the fibres consist of tensors, spinors, derivative operators, etc., and on such geoetrical objects there is a natural action of M-diffeoorphiss. Indeed, we clai that it is this transforation behavior that endows such fields with a geoetrical content in ters of M. For instance, given a point of the tangent bundle of M, the direction in M in which the vector points can be read out fro the M-diffeoorphiss 9 This deand rules out ost gauge theories other than electroagnetis (Appendix A). What happens in these theories is that to each diffeoorphis on M is assigned a nuber of liftings to B. Indeed, the collection of all liftings assigned to the identity diffeoorphis on M is called the gauge group. Presuably, uch of what follows could be generalized to include such gauge theories. 20

21 whose lifts leave this point invariant. Consider now the bundle B that results fro cobining all the exaples of Appendix A. Lift diffeoorphiss fro M to this B by cobining these liftings for all the individual exaples. We now have the achinery to express the idea that the coefficients in Eqn. (14) be functions only of the physical fields. We deand that, for every diffeoorphis D on M, its lifting ˆD leave K A α and J A invariant (noting that this akes sense for fields having indices in both B and M), up to gauge. It then follows that, for Φ any cross-section satisfying Eqn. (14) and D any diffeoorphis on M, the transfored cross-section, ˆD Φ D, is also a solution. The infinitesial version (with diffeoorphiss replaced by vector fields) of all this is the following. To each (sooth) vector field ξ a on M, there is to be assigned a lifting of it to a vector field, ˆξ α, on B. By lifting we ean that ˆξ α ( Π) a α = ξ a. We require of these liftings that they be linear (i.e., that ( cξ + η) = cˆξ + ˆη, for c constant), and Lie-bracket preserving (i.e., that [ξ, η] = [ˆξ, ˆη]). Invariance of the coefficients in (14) under these infinitesial diffeoorphiss now becoes 10 LˆξK A α = Λ A b ( Π) α b, LˆξJ A = Λ A, (15) for soe field Λ A b on B, and for every vector field ξ a on M. We shall further assue that ˆξ α results fro ξ a through the action of soe differential operator 11 : ˆξ α = δ α 1 sr 1 s ξ r +. (16) In (16), we have written out only the highest-order ter. Its coefficient, δ α 1 sr = δ α ( 1 s) r, is soe sooth field on B, independent of the derivative operator eployed in (16). Note that the index α of δ is vertical, as follows fro the definition of a lifting. As an exaple, consider the syste resulting fro cobining all the exaples of Appendix A. In this case the highest order appearing in the expression, (16), for ˆξ α is s = 2, and this order occurs only for the derivative operator of general relativity. A typical vertical vector in the bundle of derivative operators is given by δφ α = δγ a bc 10 The infinitesial version of the transforation of solutions of (14) under diffeoorphiss becoes that, for every ξ a, its lifting be a linearized solution of (14). 11 It is possible that this assuption follows already fro the general properties above of the liftings of diffeoorphiss. 21