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arXiv:1712.05960v1 [gr-qc] 16 Dec 2017 sdtdywsdvlpdb ia,adapidb i ogaiy[] Dir g replacing [1]. by form to convenient him more constra by a of applied to analysis and results The Dirac, Rosenfeld’s appa by bring geometry.” in the developed to Riemannian also the was why but of today reason (1950) form used particular work the no important probably in of is year appear timing “there 20 the [5] almost [5, to in of program according only gap a not into a constraints [9] After of Dirac analysis inc constraints. o the relativity, case of turned general the collaborators role Hamiltonian in important of is theor the still analysis Maxwell’s it of detailed as to a formulation, presented their Starting canonical [4] of covariance applied field. the had was electromagnetic 3] issue the [2, of Pauli analysis and the about questions ueo ia’ nihs ttesm ie eipoeorfundament our th improve confirm we can , we same — the cont recent At together -time. some insights. put and Dirac’s we of older If ture some — recen picture. but mathematicians updated conclusion, and this an on Ham provides elaborate the not fundamental did upon and no He look be theory.” to then the would permissible in there be try and would one, “It prot fundamental by outstanding the the [1] as of in one analysis — detailed instance p his for to Dirac, led has heretical. affairs the are of histo on state interesting under-appreciated an this the gravity, Throughout However, tonian pictures. Hamiltonian geometry. independen and and grangian coordinate guarantees physics which between symmetries links of set large a ∗ h itr fHmloingaiyhdbgnwl eoeDrcsentr Dirac’s before well begun had gravity Hamiltonian of history The eea eaiiyi itnuse yiscvrac ne space-tim under covariance its by distinguished is relativity General -aladdress: e-mail ln ihtehsoyo hi eiainada introducti here. an discussed and symmetrie derivation are their its support, of and history space-time the becau with of rights along nature own the their about in Lagr statements fundamental the are in formulations found both versions covariant manifestly and elegant h qain fHmloingaiyaeotncniee u considered often are gravity Hamiltonian of equations The [email protected] 0 ae a,Uiest ak A182 USA 16802, PA Park, University Lab, Davey 104 ymtiso Space-time of Symmetries nttt o rvtto n h Cosmos, the and Gravitation for Institute h enyvnaSaeUniversity, State Pennsylvania The atnBojowald Martin Abstract 1 ∗ no eetmathematical recent of on ninter.However, theory. angian ete aedifferent make they se .Teeimplications, These s. ooneet htverge that ronouncements ordmninlsymme- four-dimensional gnss—accompanied — agonists ,7 ] nprle with parallel in 8], 7, 6, eadsple fruitful supplies and ce iuin yrelativists by ributions l osn fthe of cousins gly oki mathematics in work t r frltvt must relativity of ory ,Brmn n his and Bergmann s, yo oko Hamil- on work of ry n12,Heisenberg 1929, in ieeti h La- the in different rvt.Rosenfeld gravity. f ludrtnigof understanding al nrlttaswith tetrads eneral uigadiscussion a luding diffeomorphisms, e .A important An y. nsms widely most ints rsin na- prescient e ,sandby spawned y, etheresies: rent loinform iltonian cwsable was ac metric variables adapted to a spatial foliation. The final step was made by Arnowitt, Deser and Misner in the 1960s [10], introducing a powerful parameterization of the space-time a metric by lapse N, shift M and the metric qab on a spatial . The resulting ADM formulation is widely used in numerical relativity, cosmology, and quantum gravity. An important question for Rosenfeld, following Heisenberg and Pauli, was the role of symmetries. He was able to show that covariance implies constraints on the fields, which are equivalent to some components of Einstein’s equation. However, he did not encounter the characteristic of Hamiltonian gravity because his variables were not adapted to a space-time foliation. Dirac was the first to introduce this crucial condition and to derive the symmetries. In modern ADM notation, there are infinitely many generators GN,M a , subject to commutator relations

a a a ab [GN1,M1 ,GN2,M2 ]= GLM1 N2−LM2 N1,[M1,M2] +q (N1∂bN2−N2∂bN1) . (1)

While the first few terms show the typical form of Lie derivatives as infinitesimal spatial diffeomorphisms, the last term is fundamentally different. In particular, it contains the inverse spatial metric qab, which is not a structure constant and not one of the generators. A satisfactory mathematical formulation requires some care. It was provided only recently [11], concluding that the brackets (1) belong to a Lie algebroid. In physics terminology, the relations (1) have “structure functions” depending on qab. As realized by Hojman, Kuchaˇrand Teitelboim [12], they present a new symmetry de- forming spatial , tangentially (along M a) and normally (along Nnµ, with the unit normal nµ). The symmetry agrees with space-time diffeomorphisms “on shell” when hold. However, it is not identical with space-time diffeomorphisms. Off-shell properties are relevant when we talk about the Riemannian structure underlying , or the 4-dimensional symmetries of space-time. Is the symmetry gener- ated by (1) more fundamental, vindicating Dirac’s heresy? Or does it lead to departures from Riemannian structures, justifying Bergmann’s iconoclasticism? Unfortunately, the importance of the new symmetry is often obscured by the messy derivation of its rela- tions (1). Dirac first found them by brute- computations of Poisson brackets. Kuchaˇr [13, 14, 15] rederived them in terms of commutators of derivatives by the functions that embed a spatial hypersurface in space-time. Such derivations are long and do not easily suggest intuitive pictures. More recently, in 2010, a new derivation has been given by Blohmann, Barbosa Fernan- des, and Weinstein [11]. Even though it derives a central statement of Hamiltonian gravity, their method does not require an explicit implementation of the 3 + 1 split which often hides the elegance of covariant theories. As presented in [11], spread over several proofs of other results, the new derivation is not easy to access. The following two paragraphs present a remodeled version in compact form, painted in notation cherished by relativists. Choose a Riemannian space-time with signature ǫ = ±1, pick a spatial foliation, and introduce Gaussian coordinates adapted to one of the spatial hypersurfaces. The resulting 2 2 a b line element ds = ǫdt + qabdx dx depends only on the spatial metric qab. Its general ρ µ form is preserved by any vector field v which satisfies n Lvgµν = 0, using the unit normal

2 nµ = (dt)µ in the Gaussian system. We expand this condition by writing out the Lie derivative: µ µ ρ µ ρ µ ρ 0= n Lvgµν = n v ∂ρgµν + n gνρ∂µv + n gµρ∂ν v . (2) µ ρ ρ ρ µ In the first term, we use n v ∂ρgµν = v ∂ρnν − gµν v ∂ρn and manipulate the last term to µ ρ µ ρ ρ 2 n gµρ∂ν v = ∂ν (n v gµρ)−v ∂ν nρ. Combining these equations and using dnµ = (d t)µ = 0, we arrive at µ µ µ ρ 0= n Lvgµν = [n, v] gµν + ∂ν(n v gµρ) . (3) We now decompose vµ = Nnµ + M µ into components normal and tangential to the µ a µ µ foliation. (We have M = M sa if sa , a = 1, 2, 3, is a spatial basis.) Equation (3) then µ µ µν implies n ∂µN = 0 and [n, M] = −ǫq ∂ν N. These new equations, together with linearity µ µ µ and the Leibniz rule, allow us to write the Lie bracket of two vector fields, v1 = N1n +M1 µ µ µ and v2 = N2n + M2 , as

µ µ µ µν [v1, v2] =(LM1 N2 −LM2 N1) n + [M1, M2] − ǫq (N1∂ν N2 − N2∂νN1) . (4)

The result agrees with (1) for ǫ = −1, while ǫ = 1 corresponds to the version of (1) in Euclidean general relativity. We are left with the problem of structure functions in the brackets. They are not constant because the spatial metric changes under our symmetries. If we cannot fix qab, we have to deal with the abundance of infinitely many copies of the brackets (4), one for each qab. Our new-found riches can be invested in a fancy mathematical structure: The brackets are defined on sections of an infinite-dimensional vector bundle with fiber (N, M a) and as base the space of spatial metrics. A heuristic argument shows that this viewpoint is fruitful: Assume finitely many con- straints CI, I = 1,...n, defined on a phase space B, with Poisson brackets {CI,CJ } = K cIJ (x)CK for x ∈ B. Extend the generators by introducing, iteratively, CHIJ··· := {CH ,CIJ···}. The new system has infinitely many generators with structure constants because {CI ,CJ } = CIJ and so on. All these generators can be written as the original constraints multi- plied with functions on B. They are examples of a new kind of vector field, or sections I I α = α CI of a vector bundle over B with fiber coordinates α (x). There is a Lie bracket I J [α1,α2] = {α CI ,α CJ }, and the linear map ρα = LX I from α to the Lie derivative 1 2 α CI I along the Hamiltonian vector field of α CI is a Lie-algebra homomorphism. It cooper-

ates with the bracket in a Leibniz rule: [α1,gα2]= g[α1,α2]+(ρα1 g)α2. These properties characterize the vector bundle as a Lie algebroid [16]. The brackets of Hamiltonian gravity form a Lie algebroid. It is the infinitesimal ver- sion of the Lie groupoid of finite evolutions, pasting together whole chunks of space-time between spatial hypersurfaces [11]. At this point, two important research directions are merging, the physical analysis of Hamiltonian gravity and the mathematical study of Lie algebroids. The link remains rather unexplored, but it shows great promise. And it could help us to illuminate Dirac’s statement. As for the promise, a good understanding of the right form of Lie algebroid representa- tions could show the way to a consistent theory of canonical quantum gravity. It is already

3 clear that there is fascinating physics behind the math. Lie algebroids can be deformed more freely than Lie algebras. A gravitational example is given by the relations (1), where a free phase-space function β multiplying qab can be inserted. We do not always obtain new versions of space-time: generators can be redefined so as to absorb β [17], but only if this function does not change sign anywhere. If it does, for instance at large curvature in models of quantum gravity [18, 19, 20], a smooth transition from ǫ = −1 to ǫ = 1 in (4) implies a passage from Lorentzian space-time to Euclidean 4-space [21, 22, 23]. Such a model with non-singular signature change cannot be Riemannian. Bergmann’s expectation has been confirmed. What about Dirac’s heresy? Is the Hamiltonian form more fundamental than the Lagrangian one? It is hard to realize space-time structures with β-modified brackets in 4 Lagrangian form: An action principle needs a measure factor, such as d xq| det g|, but a non-Riemannian version corresponding to brackets with β =6 1 remains unknown. The Hamiltonian version has no such problems, and may well be considered more fundamental. But is it realized in nature? Only a consistent version of canonical quantum gravity can give a final answer.

Acknowledgements

This work was supported in part by NSF grant PHY-1307408.

References

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