aa r X i v : . [ g r- q c ] D ec Symmetries of Space-time
Martin Bojowald ∗ Institute for Gravitation and the Cosmos,The Pennsylvania State University,104 Davey Lab, University Park, PA 16802, USA
Abstract
The equations of Hamiltonian gravity are often considered ugly cousins of theelegant and manifestly covariant versions found in the Lagrangian theory. However,both formulations are fundamental in their own rights because they make differentstatements about the nature of space-time and its symmetries. These implications,along with the history of their derivation and an introduction of recent mathematicalsupport, are discussed here.
General relativity is distinguished by its covariance under space-time diffeomorphisms,a large set of symmetries which guarantees coordinate independence and supplies fruitfullinks between physics and geometry. However, the symmetries are different in the La-grangian and Hamiltonian pictures. Throughout an interesting history of work on Hamil-tonian gravity, this under-appreciated state of affairs has led to pronouncements that vergeon the heretical. Dirac, for instance — one of the outstanding protagonists — accompaniedhis detailed analysis in [1] by “It would be permissible to look upon the Hamiltonian formas the fundamental one, and there would then be no fundamental four-dimensional symme-try in the theory.” He did not elaborate on this conclusion, but recent work in mathematicsand physics provides an updated picture. If we put together contributions by relativistsand mathematicians — some older and some recent — we can confirm the prescient na-ture of Dirac’s insights. At the same time, we improve our fundamental understanding ofspace-time.The history of Hamiltonian gravity had begun well before Dirac’s entry, spawned byquestions about the analysis of the electromagnetic field. Starting in 1929, Heisenbergand Pauli [2, 3] had applied canonical quantization to Maxwell’s theory. An importantissue was the covariance of their formulation, as it still is in the case of gravity. Rosenfeld[4] presented a detailed analysis of Hamiltonian general relativity, including a discussionof the important role of constraints. After a gap of almost 20 years, Bergmann and hiscollaborators turned the analysis of constraints into a program [5, 6, 7, 8], in parallel withDirac [9] not only in the timing of important work (1950) but also in apparent heresies:according to [5] “there is probably no particular reason why the theory of relativity mustappear in the form of Riemannian geometry.” The analysis of constraints most widelyused today was developed by Dirac, and applied by him to gravity [1]. Dirac was ableto bring Rosenfeld’s results to a more convenient form by replacing general tetrads with ∗ e-mail address: [email protected] N , shift M a and the metric q ab on a spatial hypersurface. The resultingADM formulation is widely used in numerical relativity, cosmology, and quantum gravity.An important question for Rosenfeld, following Heisenberg and Pauli, was the role ofsymmetries. He was able to show that covariance implies constraints on the fields, whichare equivalent to some components of Einstein’s equation. However, he did not encounterthe characteristic symmetry of Hamiltonian gravity because his variables were not adaptedto a space-time foliation. Dirac was the first to introduce this crucial condition and toderive the symmetries. In modern ADM notation, there are infinitely many generators G N,M a , subject to commutator relations[ G N ,M a , G N ,M a ] = G L M N −L M N , [ M ,M ] a + q ab ( N ∂ b N − N ∂ b N ) . (1)While the first few terms show the typical form of Lie derivatives as infinitesimal spatialdiffeomorphisms, the last term is fundamentally different. In particular, it contains theinverse spatial metric q ab , 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 q ab .As realized by Hojman, Kuchaˇr and Teitelboim [12], they present a new symmetry de-forming spatial hypersurfaces, tangentially (along M a ) and normally (along N n µ , with theunit normal n µ ). The symmetry agrees with space-time diffeomorphisms “on shell” whenequations of motion hold. However, it is not identical with space-time diffeomorphisms.Off-shell properties are relevant when we talk about the Riemannian structure underlyinggeneral relativity, 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 departuresfrom Riemannian structures, justifying Bergmann’s iconoclasticism? Unfortunately, theimportance of the new symmetry is often obscured by the messy derivation of its rela-tions (1). Dirac first found them by brute-force computations of Poisson brackets. Kuchaˇr[13, 14, 15] rederived them in terms of commutators of derivatives by the functions thatembed a spatial hypersurface in space-time. Such derivations are long and do not easilysuggest 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 oftenhides the elegance of covariant theories. As presented in [11], spread over several proofsof other results, the new derivation is not easy to access. The following two paragraphspresent 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, andintroduce Gaussian coordinates adapted to one of the spatial hypersurfaces. The resultingline element d s = ǫ d t + q ab d x a d x b depends only on the spatial metric q ab . Its generalform is preserved by any vector field v ρ which satisfies n µ L v g µν = 0, using the unit normal2 µ = (d t ) µ in the Gaussian system. We expand this condition by writing out the Liederivative: 0 = n µ L v g µν = n µ v ρ ∂ ρ g µν + n µ g νρ ∂ µ v ρ + n µ g µρ ∂ ν v ρ . (2)In the first term, we use n µ v ρ ∂ ρ g µν = v ρ ∂ ρ n ν − g µν v ρ ∂ ρ n µ and manipulate the last term to n µ g µρ ∂ ν v ρ = ∂ ν ( n µ v ρ g µρ ) − v ρ ∂ ν n ρ . Combining these equations and using d n µ = (d t ) µ = 0,we arrive at 0 = n µ L v g µν = [ n, v ] µ g µν + ∂ ν ( n µ v ρ g µρ ) . (3)We now decompose v µ = N n µ + M µ into components normal and tangential to thefoliation. (We have M µ = M a s µa if s µa , a = 1 , ,
3, is a spatial basis.) Equation (3) thenimplies n µ ∂ µ N = 0 and [ n, M ] µ = − ǫq µν ∂ ν N . These new equations, together with linearityand the Leibniz rule, allow us to write the Lie bracket of two vector fields, v µ = N n µ + M µ and v µ = N n µ + M µ , as[ v , v ] µ = ( L M N − L M N ) n µ + [ M , M ] µ − ǫq µν ( N ∂ ν N − N ∂ ν N ) . (4)The result agrees with (1) for ǫ = −
1, while ǫ = 1 corresponds to the version of (1) inEuclidean general relativity.We are left with the problem of structure functions in the brackets. They are notconstant because the spatial metric changes under our symmetries. If we cannot fix q ab , wehave to deal with the abundance of infinitely many copies of the brackets (4), one for each q ab . Our new-found riches can be invested in a fancy mathematical structure: The bracketsare defined on sections of an infinite-dimensional vector bundle with fiber ( N, M a ) and asbase manifold the space of spatial metrics.A heuristic argument shows that this viewpoint is fruitful: Assume finitely many con-straints C I , I = 1 , . . . n , defined on a phase space B , with Poisson brackets { C I , C J } = c KIJ ( x ) C K for x ∈ B . Extend the generators by introducing, iteratively, C HIJ ··· := { C H , C IJ ··· } .The new system has infinitely many generators with structure constants because { C I , C J } = C IJ 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 α = α I C I of a vector bundle over B with fiber coordinates α I ( x ). There is a Lie bracket[ α , α ] = { α I C I , α J C J } , and the linear map ρ α = L X αICI from α to the Lie derivativealong the Hamiltonian vector field of α I C I is a Lie-algebra homomorphism. It cooper-ates with the bracket in a Leibniz rule: [ α , gα ] = g [ α , α ] + ( ρ α g ) α . These propertiescharacterize 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-timebetween spatial hypersurfaces [11]. At this point, two important research directions aremerging, the physical analysis of Hamiltonian gravity and the mathematical study of Liealgebroids. The link remains rather unexplored, but it shows great promise. And it couldhelp 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 already3lear that there is fascinating physics behind the math. Lie algebroids can be deformedmore freely than Lie algebras. A gravitational example is given by the relations (1), wherea free phase-space function β multiplying q ab can be inserted. We do not always obtainnew versions of space-time: generators can be redefined so as to absorb β [17], but onlyif this function does not change sign anywhere. If it does, for instance at large curvaturein models of quantum gravity [18, 19, 20], a smooth transition from ǫ = − ǫ = 1 in(4) implies a passage from Lorentzian space-time to Euclidean 4-space [21, 22, 23]. Such amodel with non-singular signature change cannot be Riemannian. Bergmann’s expectationhas been confirmed.What about Dirac’s heresy? Is the Hamiltonian form more fundamental than theLagrangian one? It is hard to realize space-time structures with β -modified brackets inLagrangian form: An action principle needs a measure factor, such as d x q | det g | , buta non-Riemannian version corresponding to brackets with β = 1 remains unknown. TheHamiltonian 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 gravitycan give a final answer. Acknowledgements
This work was supported in part by NSF grant PHY-1307408.
References [1] P. A. M. Dirac, The theory of gravitation in Hamiltonian form,
Proc. Roy. Soc. A
246 (1958) 333–343[2] W. Heisenberg and W. Pauli, Zur Quantendynamik der Wellenfelder,
Z. Phys.
Z. Phys.
Annalen Phys.
Phys. Rev.
75 (1949) 680–685[6] P. G. Bergmann and J. H. M. Brunings, Non-Linear Field Theories II. CanonicalEquations and Quantization,
Rev. Mod. Phys.
21 (1949) 480–487[7] P. G. Bergmann, R. Penfield, R. Schiller, and H. Zatzkis, The Hamiltonian of theGeneral Theory of Relativity with Electromagnetic Field,
Phys. Rev.
80 (1950) 81–88[8] J. L. Anderson and P. G. Bergmann, Constraints in Covariant Field Theories,
Phys.Rev.
83 (1951) 1018–1025 49] P. A. M. Dirac, Generalized Hamiltonian dynamics,
Can. J. Math.
Gravitation: An Introduction to Current Research , Wiley, New York,1962, Reprinted in [24][11] C. Blohmann, M. C. Barbosa Fernandes, and A. Weinstein, Groupoid symmetry andconstraints in general relativity. 1: kinematics,
Commun. Contemp. Math.
15 (2013)1250061, [arXiv:1003.2857][12] S. A. Hojman, K. Kuchaˇr, and C. Teitelboim, Geometrodynamics Regained,
Ann.Phys. (New York)
96 (1976) 88–135[13] K. V. Kuchaˇr, Geometry of hypersurfaces. I,
J. Math. Phys.
17 (1976) 777–791[14] K. V. Kuchaˇr, Kinematics of tensor fields in hyperspace. II,
J. Math. Phys.
17 (1976)792–800[15] K. V. Kuchaˇr, Dynamics of tensor fields in hyperspace. III,
J. Math. Phys.
17 (1976)801–820[16] J. Pradines, Th´eorie de Lie pour les groupo¨ıdes diff´erentiables. Calcul diff´erenetieldans la cat´egorie des groupo¨ıdes infinit´esimaux,
Comptes Rendus Acad. Sci. ParisS´er. A–B
264 (1967) A245–A248[17] R. Tibrewala, Inhomogeneities, loop quantum gravity corrections, constraint algebraand general covariance,
Class. Quantum Grav.
31 (2014) 055010, [arXiv:1311.1297][18] J. D. Reyes,
Spherically Symmetric Loop Quantum Gravity: Connections to 2-Dimensional Models and Applications to Gravitational Collapse , PhD thesis, ThePennsylvania State University, 2009[19] T. Cailleteau, J. Mielczarek, A. Barrau, and J. Grain, Anomaly-free scalar perturba-tions with holonomy corrections in loop quantum cosmology,
Class. Quant. Grav.
Phys. Rev. D
90 (2014) 025025,[arXiv:1402.5130][21] M. Bojowald and G. M. Paily, Deformed General Relativity and Effective Actionsfrom Loop Quantum Gravity,
Phys. Rev. D
86 (2012) 104018, [arXiv:1112.1899][22] J. Mielczarek, Signature change in loop quantum cosmology,
Springer Proc. Phys.
157 (2014) 555, [arXiv:1207.4657][23] M. Bojowald and J. Mielczarek, Some implications of signature-change in cosmologicalmodels of loop quantum gravity,
JCAP
08 (2015) 052, [arXiv:1503.09154]524] R. Arnowitt, S. Deser, and C. W. Misner, The Dynamics of General Relativity,