The geometric role of symmetry breaking in gravity
aa r X i v : . [ g r- q c ] D ec The geometric role of symmetry breaking in gravity
Derek K. Wise
Institute for Theoretical Physics III, University of Erlangen–N¨urnberg, Staudtstr. 7/B2, 91054Erlangen, GermanyE-mail: [email protected]
Abstract.
In gravity, breaking symmetry from a group G to a group H plays the role ofdescribing geometry in relation to the geometry the homogeneous space G/H . The deep reasonfor this is Cartan’s ‘method of equivalence,’ giving, in particular, an exact correspondencebetween metrics and Cartan connections. I argue that broken symmetry is thus implicit inany gravity theory, for purely geometric reasons. As an application, I explain how this kind ofthinking gives a new approach to Hamiltonian gravity in which an observer field spontaneouslybreaks Lorentz symmetry and gives a Cartan connection on space.
The success of spontaneous symmetry breaking in condensed matter and particle physics isfamous. It explains second order phase transitions, superconductivity, the origin of mass viathe Higgs mechanism, why there must be at least three generations of quarks, and so on. Theseapplications are by now standard material for modern textbooks.Much less famous is this: broken symmetry links the geometry of gauge fields to the geometryof spacetime . This, in my view, is the main role of symmetry breaking in gravity.An early clue came in 1977, when MacDowell and Mansouri wrote down an action for generalrelativity using a connection for the (anti-) de Sitter group, but invariant only under the Lorentzgroup [1]. Though their work was surely inspired by spontaneous symmetry breaking, it wasStelle and West [2] who first made their action fully gauge invariant, breaking the symmetrydynamically using a field y locally valued in (anti-) de Sitter space.Whether one breaks the symmetry dynamically or ‘by hand,’ the broken symmetry of theMacDowell–Mansouri connection plays the geometric role of relating spacetime geometry to thegeometry of de Sitter space. This is best understood using Cartan geometry , a generalization ofRiemannian geometry originating in the work of ´Elie Cartan, in which the geometry of tangentspaces is generalized—in this case, they become copies of de Sitter space [3]. But to explain howthis works, and how symmetry breaking is involved, it helps to back up further.In geometry, inklings of spontaneous symmetry breaking date from at least 1872, in thework of Felix Klein. Ironically, to study a homogeneous space Y , with symmetry group G , onefirst breaks its perfect symmetry, artificially giving special significance to some point y ∈ Y .This gives an isomorphism Y ∼ = G/G y as G -spaces, where G y is the stabilizer of y , allowingalgebraic study of the geometry. While Y itself has G symmetry, this description of it is onlyinvariant under the subgroup G y . Different algebraic descriptions of Y are however related in a G -equivariant way, since G gy = gG y g − .This is strikingly similar to spontaneous symmetry breaking in physics. There, one reallyhas a family of minimum-energy states, related in a G -equivariant way under the original gaugegroup G . Singling out any particular state | i as ‘the’ vacuum breaks symmetry to G | i .artan took Klein’s ideas a dramatic step further, getting an algebraic description ofthe geometry of a non homogeneous manifold M , by relating it ‘infinitesimally’ to one ofKlein’s geometries Y . Just as Klein geometry uses broken symmetry to get an isomorphism Y → G/G y , in Cartan geometry, the broken symmetry in a G connection induces an isomorphism e : T x M → g / g y for each tangent space. This is just the coframe field, also called the solderingform since identifying T x M with g / g y ∼ = T y Y effectively solders a copy of Y to M , at each point x . These copies of Y are then related via holonomy of the Cartan connection, which can beviewed as describing ‘rolling Y along M without slipping’ [3] (see also [5, Appendix B]).Physics history unfortunately skips over Cartan geometry. The Levi-Civita connection isadequate for the standard metric formulation of general relativity, and more general kinds ofconnections played no vital role in physics until some time later. When these eventually wereintroduced in Yang–Mill theory, they served a purpose far removed from spacetime geometry.Yang–Mills gauge fields are really just the principal connections of Ehresmann, who, buildingon Cartan’s ideas, liberated connections from their bondage to classical geometry. Ehresmann’sdefinition, which lacks the crucial ‘broken symmetry’ in Cartan’s original version, has just theflexibility needed for gauge fields in particle physics, which are concerned only with the geometryof an abstract ‘internal space’—a bundle over spacetime, rather than spacetime itself. On theother hand, Cartan’s original version is better when it comes to studying gravity.Concretely, a Cartan geometry may be thought of as a connection on a principal bundle (withEhresmann’s now standard definition) together with a section y of the associated Y bundle. Asan example, let us write a version of the MacDowell–Mansouri action, using de Sitter space Y ∼ = G/H = SO(4 , / SO(3 ,
1) as the corresponding Klein geometry: I [ A, y ] = Z tr( F y ∧ ⋆ y F y )The Cartan connection ( A, y ) consists of an SO(4 ,
1) connection A and a locally de Sitter-valuedfield y . F is the curvature of A , calculated by the usual formula, and F y is its g y -valued part,where g y ∼ = so (3 , ∼ = Λ R , has Hodge star operator ⋆ y .I have described additional examples of Cartan-geometric formulations of various gravitytheories elsewhere [4], and there are many more. But besides the diversity of specific examples,there are deep reasons that gravity, or any related ‘gauge theory of geometry,’ should be framedin the language of Cartan geometry. This is the subject of geometric ‘equivalence theorems.’In fact, if one believes semi-Riemannian metrics are fundamental in classical gravity, one isforced to accept Cartan connections as equally fundamental. The reason for this is Cartan’s method of equivalence , a process for proving that specified kinds of ‘raw geometric data’ areequivalent to corresponding types of Cartan geometry [6]. In the case of Riemannian geometry,solving the ‘equivalence problem’ leads to the following theorem: Theorem.
A Riemannian metric determines a unique torsion-free Cartan geometry modeledon Euclidean space; conversely, a torsion-free Cartan geometry modeled on Euclidean spacedetermines a Riemannian metric up to overall scale (on each connected component).
Proof.
See Sharpe’s book [5].Physically, the ‘overall scale’ in the converse just represents a choice of length unit. One canalso show that deformed versions (or ‘mutations’) of Euclidean geometry, namely hyperbolicand spherical geometry, lead to Cartan geometries that carry the same information [5]. TheLorentzian analogs of these results are the real reason de Sitter and anti de Sitter geometrieswork in MacDowell–Mansouri gravity.Riemannian geometry is but one application of the equivalence method. There are analogoustheorems, for example, in conformal geometry or Weyl geometry, relating various types ofonformal structures to Cartan geometries that take the model Y to be an appropriate kindof homogeneous conformal model. Sharpe’s book [5] contains some such theorems, and somesignificant work has been done on applications of conformal Cartan geometry—which often goesby the name ‘tractor calculus’—in physics. (See, e.g. [7] and references therein.)For now, I just want to describe one more application of Cartan geometric thinking ingravitational theory. Besides spacetime geometry, one can also use Cartan’s ideas to describethe geometry of space .Wheeler’s term ‘geometrodynamics’ originally referred to the of evolution of spatialgeometries in the metric sense. This has sometimes been contrasted with ‘connection dynamics’[9]. In light of the above equivalence theorem, however, there seems little point in establishingany technical distinction between geometrodynamics and connection dynamics, at least if wemean connections in the Cartan-geometric sense. The metric and connection pictures have theirown advantages, but the equivalence theorem suggests we should be able to translate exactlybetween the two.In recent work with Steffen Gielen [8], we take an explicitly Cartan-geometric approach toevolving spatial geometries. In this case, the symmetry breaking field y lives in 3d hyperbolicspace SO(3 , / SO(3), and can be interpretated as a field of observers , since the spacetimecoframe field converts it into a unit timelike vector field. This can be dualized via the metric toa unit covector field, which we might call a field of co-observers . Just as observers determinea local time direction, co-observers determine local space directions, by taking their kernel. Ourstrategy in the Hamiltonian formulation is to fix a field of co-observers—the infinitesimal analogof picking a spacetime folitation—but let the field of observers be determined dynamically, aspart of determining the metric.The result is a model in which the observer field plays a two part symmetry breakingrole: first splitting spacetime fields into spatial and temporal parts, but then also acting asthe symmetry breaking field in Cartan geometry of space . This gives a Cartan-geometricHamiltonian framework in which the spatial fields fit neatly and transparently into theirspacetime counterparts and transform in an equivariant way under local Lorentz symmetry.Thanks to the equivalence theorem, this may be viewed as a concrete link between connectiondynamics and geometrodynamics in the original sense.It is conceivable that gravity descends from a more fundamental theory with larger gaugegroup, and so fits into the tradition of symmetry breaking in gauge theories. Such ideas areclearly worth pursuing (see, e.g. [10, 11]). At the same time, we should not ignore the lesson ofCartan geometry: broken symmetry is the means to establishing exact correspondence betweengeometric structures living on tangent spaces on one hand and connections on the other.I refer the interested reader to the bibliography of [4] for many additional Cartan geometryreferences. I would like to thank John Baez, Julian Barbour, James Dolan and Andy Randonofor helpful discussions, and especially Steffen Gielen for collaboration on [8].
References [1] S. W. MacDowell and F. Mansouri, Unified geometric theory of gravity and supergravity,
Phys. Rev. Lett. ,739 (1977). Erratum, ibid. , 1376 (1977).[2] K. S. Stelle and P. C. West, De Sitter gauge invariance and the geometry of the Einstein–Cartan theory, J. Phys. A: Math. Gen. L205 (1979); Spontaneously broken de Sitter symmetry and the gravitationalholonomy group,
Phys. Rev. D , 1466 (1980).[3] D. K. Wise, MacDowell–Mansouri gravity and Cartan geometry, Class. Quant. Grav. , 155010 (2010).arXiv:gr-qc/0611154.[4] D. K. Wise, Symmetric space Cartan connections and gravity in three and four dimensions, SIGMA , 080(2009). arXiv:0904.1738.[5] R. W. Sharpe, Differential geometry: Cartan’s Generalization of Klein’s Erlangen Program (Springer-Verlag,New York, 1997).6] R. B. Gardner,
The Method of Equivalence and its Applications (Capital City Press, Montpelier, Vermont,1989).[7] A. Shaukat and A. Waldron, Weyl’s gauge Invariance: conformal geometry, spinors, supersymmetry, andinteractions, arXiv:0911.2477.[8] S. Gielen and D. K. Wise, Spontaneous symmetry breaking for Hamiltonian gravity, arXiv:1111.7195.[9] J. D. Romano, Geometrodynamics vs. connection dynamics,
Gen. Rel. Grav. (1993) 759-854.arXiv:gr-qc/9303032; K. Kuchaˇr, Canonical quantum gravity, arXiv:gr-qc/9304012[10] R. Percacci, Gravity from a particle physicist’s perspective, PoS ISFTG011