Lifting General Relativity to Observer Space
aa r X i v : . [ g r- q c ] M a y Lifting General Relativityto Observer Space
Steffen Gielen
Perimeter Institute for Theoretical Physics31 Caroline St. N.Waterloo ON, N2L 2Y5, Canada [email protected]
Derek K. Wise
Institute for Quantum GravityUniversit¨at Erlangen–N¨urnbergStaudtstr. 7/B2, 91058 Erlangen, Germany [email protected]
May 4, 2013
Abstract
The ‘observer space’ of a Lorentzian spacetime is the space of future-timelike unit tangentvectors. Using Cartan geometry, we first study the structure a given spacetime induces on its ob-server space, then use this to define abstract observer space geometries for which no underlyingspacetime is assumed. We propose taking observer space as fundamental in general relativity,and prove integrability conditions under which spacetime can be reconstructed as a quotient ofobserver space. Additional field equations on observer space then descend to Einstein’s equa-tions on the reconstructed spacetime. We also consider the case where no such reconstructionis possible, and spacetime becomes an observer-dependent, relative concept. Finally, we dis-cuss applications of observer space, including a geometric link between covariant and canonicalapproaches to gravity.
General relativity is about understanding that physics does not take place against the backdrop ofa fixed geometry. Rather, geometry itself is a dynamical entity, bending and curving in responseto matter, just as matter is subject to geometric rules of the space it inhabits. There are, however,different possible interpretations of such statements. In particular, do we mean the geometry of spacetime , the geometry of space , or something else?This question is the root of tension between ‘covariant’ and ‘canonical’ approaches to gravity.The ‘covariant’ approach focuses on the geometry of spacetime , given ‘all at once’. This is elegant,but unfortunately rather far removed from our actual experience of the world, in which space andtime appear quite distinct. The so-called ‘canonical’ picture focuses instead on the geometry of space and how this geometry evolves in time, and is thus more clearly related to our spatiotemporalintuition. On the other hand, the notion of ‘time’ is fixed arbitrarily from the outset, goingagainst the spirit of relativity, even when the final result is independent of this choice. Worseyet, showing this independence in some formulations is decidedly nontrivial. The term ‘canonicalgravity’, stemming from the ‘canonically conjugate’ variables in Hamiltonian mechanics, thus standsin ironic contrast with standard mathematical use of the word ‘canonical’, where it means involvingno arbitrary choices. In brief, canonical gravity is not canonical.n this paper, we reformulate general relativity in a way that maintains the best of both ap-proaches. To do this, we pass from spacetime to observer space —a 7-dimensional manifold of allpossible observers. On one hand, this perspective offers a clear-cut distinction between spatial andtemporal directions. On the other, it acknowledges the local, observer-dependent nature of timeand space in general relativity. Because we consider the space of all possible observers, there areno arbitrary choices to be made.But how do we describe the ‘geometry of observer space’, and how is it related to the geometryof spacetime or of space? To make this precise, it is helpful to use the approach to geometry rootedin the works of Felix Klein and ´Elie Cartan.Klein’s Erlangen Program was about understanding geometry in terms of symmetry . In its orig-inal form, it applied to the ‘Platonic ideals’ of geometry—homogeneous spaces, such as Euclidean,hyperbolic, or projective geometry, in which any two points look essentially the same. Cartangeneralized Klein’s ideas to the setting of differential geometry, giving a precise characterizationof spaces with only ‘infinitesimal’ symmetry. His approach involves ‘infinitesimally modeling’ ageneral manifold on one of Klein’s homogeneous geometries.This outlook meshes nicely with relativistic physics: because geometry is locally , dynamicallydetermined, our Platonic ideals of classical geometry carry over to the real world only as in-finitesimal approximations. In passing from special relativity to general relativity, for example,Minkowski spacetime R , survives only as the tangent space to a more general Lorentzian mani-fold M . The global Lorentz symmetry of R , itself becomes a local gauge symmetry of orthonormalframes, which essentially implements Einstein’s equivalence principle. A connection describes howthe frames at different points are related, in a way that generally depends on the path betweenpoints. So, in some sense, general relativity describes spacetimes that are infinitesimally modeledon Minkowski space .To make Cartan’s idea of infinitesimal modeling more precise, recall that a homogeneousspace , or Klein geometry , is a manifold Z equipped with a smooth transitive action of a Liegroup G . If H ⊆ G is the subgroup fixing some point z ∈ Z , then Z may be identified with thecoset space G/H . On the other hand, a
Cartan geometry on M , modeled on Z , locally amountsto a g -valued 1-form which, when composed with the projection g → g / h , gives an identification ofthe tangent space T x M with g / h ∼ = T z Z . This 1-form transforms under gauge transformations asa connection, but the geometry is only invariant under the subgroup H . There is thus an intrinsic‘symmetry breaking’ aspect to any Cartan geometry.Hints of Cartan geometry are to be found in gauge theoretic formulations of general relativity.Kibble [19] may have been first to notice that the usual Lorentz connection and coframe field, orvierbein, can be viewed as pieces of a single connection for the Poincar´e group ISO(3 , ,
1) or anti-de Sitter group SO(3 , G of the theory is broken to the Lorentz group H = SO(3 , G/H has: G = SO(4 , de Sitter group ISO(3 , Poincar´e group
SO(3 , anti-de Sitter group H = SO(3 , Lorentz group.
MacDowell–Mansouri gravity and Poincar´e gauge theory thus provide Cartan-geometric descrip-tions of the ‘covariant’ picture of general relativity. [27, 28]2imilarly, on the ‘canonical’ side, Ashtekar variables—especially in their ‘real’ form originallygiven by Barbero [7]—also hint at Cartan geometry. The key variables are an SO(3) connectionon space together with a spatial coframe field. This makes it tempting to think of these fields ascoming from a Cartan connection with model geometry H ′ /K where H ′ ∼ = SO(4) spherical group
ISO(3)
Euclidean group
SO(3 , hyperbolic group K = SO(3) rotation group, depending on whether space is modeled on the 3-sphere, Euclidean space or hyperbolic space.Presumably, the choice of H ′ should be related to the choice of G , since the model spacetimes havedifferent associated spatial geometries.However, while tempting, the precise relationship between real Ashtekar variables and Cartangeometry modeled on H ′ /K is not immediately apparent. In an effort to sort out this relation-ship, we recently obtained a version of Ashtekar variables using ‘spontaneous breaking’ of Lorentzsymmetry [14]. The idea is to introduce a field of ‘local observers’—each with their own preferredlocal notions of space and time—to extract from the spacetime Lorentz connection and coframefield a ‘spatial’ SO(3) connection and triad. These pieces can be assembled into a ‘spatial Cartanconnection’, giving a system of evolving spatial Cartan geometries, or ‘Cartan geometrodynamics’.But the role of H ′ is still unclear: breaking Lorentz symmetry has more to do with the coset space H/K than with the spatial geometry H ′ /K .This becomes clear when we consider observer space . The key geometric idea is to combine thetwo levels of ‘symmetry breaking’ we have just described: picking not just a point but a particular observer in homogeneous spacetime breaks symmetry not just to H or H ′ but all the way to K ina single step. Hence G/K is the observer space of the model spacetime
G/H . H and H ′ still playgeometric roles in observer space geometry, as do each of the possible coset spaces: G H ′ G/H ′ space of‘spaces’ K H ′ /K space H H/K velocity space
G/H spacetime
G/K observer space
In particular, to get from G to K we can just as well go through H ′ , first choosing a homogeneous‘spatial slice’ and then a point in this submanifold breaking the symmetry group down to K . All ofthis generalizes from homogeneous to Cartan geometry, where we can view general observer spacesas a deformation of the homogeneous models.One point of this paper is that spacetime Cartan geometry and Cartan geometrodynamics arejust two aspects of the Cartan geometry of observer space . In more physical language, the geometryof observer space links the covariant and canonical pictures of general relativity. Our study of this3dea began with understanding how Ashtekar variables arise from breaking Lorentz symmetry usinglocal observers [14, 15]. The present paper is, in part, a continuation of this story, completing andclarifying the geometric picture underlying our previous work.However, we also have independent physical motives for studying observer space, especially forpotential applications beyond general relativity. First, since the group stabilizing a given observer is K , we can use observer space to investigate Lorentz-violating physical theories which take K as thefundamental symmetry group. Such a violation of Lorentz symmetry is a possibility that continuesto be investigated experimentally, and is also inherent in several theoretical models. For instance,several proposals for gravitational theories beyond general relativity, such as Hoˇrava-Lifshitz gravity[18], causal dynamical triangulations [3], and shape dynamics [8], involve a preferred foliationof spacetime. In our framework, a choice of foliation corresponds to a field of local observers,locally breaking Lorentz symmetry; on observer space we consider all such observers at once. Morepractically, spacetime geometry can only be probed by ‘observers’, and to build a phenomenologicalmodel to be confronted with observation, one might prefer to assume gauge invariance only under K . While many such proposals for Lorentz violation are motivated by attempts to quantize gravity,our framework is, at this stage, purely classical.More interestingly, passing from spacetime to observer space as the arena for physics could allowus to discard spacetime as a fundamental concept. This is the main message of the ‘relative locality’proposal [4]: the notion of spacetime itself may be observer-dependent . The proposal of [4] isessentially a modification of special relativity based on such an observer-dependent spacetime, withan absolute momentum space common to all observers. In a more general setting one would expectboth momentum space and spacetime to be observer-dependent concepts; this will be precisely theinterpretation given to a general observer space geometry in this paper. Our construction henceprovides a natural framework to move from ‘special’ to ‘general’ relative locality. Plan of the paper
In section 2, we explain how Cartan geometry is used in ordinary spacetime physics, including areview of the Cartan-geometric underpinnings of MacDowell–Mansouri gravity.In section 3, we study the geometry of observer space, first with a Lorentzian spacetime givenat the outset, and then from the perspective of Cartan geometry, where the Cartan connectioninduces the geometry of observer space. We explain our recent ‘Cartan geometrodynamics’ pictureof Ashtekar variables from the perspective of observer space.Section 4 contains some of our main results. We consider the possibility that observer space ismore fundamental than spacetime, and derive conditions (Thm. 19) for the existence of spacetime ,i.e. the ability to reconstruct an observer-independent spacetime from observer space. We give anaction on observer space that allows the reconstruction of spacetime, and whose solutions includeall solutions of vacuum general relativity.In section 5 we discuss the general scenario where no such reconstruction is possible and space-time is relative . We explain observer-dependent notions of coincidence of observers and of spacetime.As a special case we consider the possibility that velocity space, rather than spacetime, is absolute.We explain how the proposal of relative locality [4] is naturally described in these terms.We conclude with some remarks about the fundamental status of spacetime, and suggest somedirections for further investigation. For the convenience of the reader, we summarize our notationin appendix A. 4
Cartan geometry in spacetime physics
We begin with a brief overview of Cartan geometry and of its use in the MacDowell-Mansouridescription of gravity. This material is not new, and partly overlaps with our previous expositionsof Cartan geometry in physics [27, 28]. While Cartan geometry is a broad subject (see e.g. [25]),we include here a brief introduction sufficient for our main goal: the construction of observer spacegeometries. We find it best to do this before leaving the familiar world of spacetime physics.
Klein geometry studies homogeneous spaces via their symmetry groups. If a Lie group G actstransitively on a manifold Z , we can identify Z with the coset space G/H , where H is the stabilizerof an arbitrarily chosen point z ∈ Z : H = { g ∈ G : gz = z } , (2.1)a topologically closed (and hence Lie) subgroup. The isomorphism Z ∼ = G/H is G -equivariant.Conversely, the coset space G/H is a manifold with smooth G action, provided H is closed in G .It is thus convenient to define a Klein geometry to be such a pair of groups, even though we thinkof these as algebraic tools for studying the geometry of the corresponding homogeneous space. Definition 1. A Klein geometry ( G, H ) is a Lie group G with closed subgroup H . For spacetime geometry, the obvious Klein geometries are the standard family of maximallysymmetric solutions of the vacuum Einstein equations: de Sitter, Minkowski, or anti-de Sitter,depending on the cosmological constant Λ. In 3 + 1 dimensions, the corresponding groups ofisometries preserving both orientation and time orientation are G = SO o (4 , de Sitter (Λ > o (3 , Minkowski (Λ = 0)SO o (3 , anti-de Sitter (Λ < H = SO o (3 , Lorentz group . (2.2)where the subscript o denotes the connected component. Throughout the rest of this paper, unlessotherwise noted, the letters G and H will refer to the particular groups in (2.2), and Z will denotethe corresponding homogeneous spacetime, where H is the stabilizer of an arbitrary z ∈ Z , fixedonce and for all. Since little of what follows depends on the sign of Λ, we treat all cases in parallel,noting exceptions as necessary.In any of these spacetime Klein geometries, G acts irreducibly on its Lie algebra g via theadjoint representation, but restricting this to the group H , we have an invariant direct sum g = h ⊕ z (2.3)where h is the Lie algebra of H , and the complement z ∼ = R , may be identified in a canonical waywith the tangent space at z : z = T z Z. (2.4)Geometrically, this direct sum breaks the ‘infinitesimal symmetries’ of Z up into ‘infinitesimalLorentz transformations’, which preserve z , and ‘infinitesimal translations’ of z . In fact, this givesa Z / g with even part h and odd part z :[ h , h ] ⊆ h [ h , z ] ⊆ z [ z , z ] ⊆ h (2.5)5 reductive geometry in which h ⊕ z is a Z / symmetric space [17].In fact, even the metric on these homogeneous spacetimes can essentially be recovered from Lietheory. Any semi-Riemannian metric on Z invariant under G is induced by some nondegenerate H -invariant symmetric bilinear form on z . In each of the spacetime geometries Z , there is only onesuch invariant bilinear form up to scale, so the geometry of Z reduces to knowing the groups G and H , plus a unit of length. To each Klein geometry, there is an associated type of Cartan geometry. Here we give the generaldefinition, before specializing to the spacetime Klein geometries just discussed.
Definition 2. A Cartan geometry ( π : P → M , A ) modeled on the Klein geometry ( G, H ) is aprincipal right H bundle π : P → M equipped with a g -valued 1-form A on PA : T P → g (2.6) called the Cartan connection , satisfying three properties: C1 . For each p ∈ P , A p : T p P → g is a linear isomorphism; C2 . ( R h ) ∗ A = Ad( h − ) ◦ A ∀ h ∈ H ; C3 . A restricts to the Maurer–Cartan form on vertical vectors. To be more precise in property C3, note that we can pull back forms on H along any localtrivialization f : P | U → U × H . On each fiber P x , this gives an isomorphism of right H -spaces, butonly the pullback of the left -invariant Maurer–Cartan form on H —defined by A H ( v ) = ( L h − ) ∗ v for v ∈ T h H —is independent of which trivialization we use. This is the Maurer–Cartan form on P that property C3 refers to.The curvature of a Cartan connection is the ( P × H g )-valued 2-form F = dA + 12 [ A, A ] (2.7)and a Cartan geometry is called flat if F = 0. Example 3.
Any Klein geometry ( G, H ) becomes a Cartan geometry in a canonical way. Themap G → G/H is principal right H -bundle, and the Maurer–Cartan form A : T G → g is a Cartanconnection. This Cartan geometry is flat by the Maurer–Cartan equation. Conversely, any flat Cartan geometry is locally isomorphic to a Klein geometry. (See e.g. [25] .) From Lorentzian geometry to Cartan geometry
We now focus on spacetime Cartan geometry and, in particular, give the precise correspondence toLorentzian geometry in the familiar sense.Fix any one of the three spacetime Klein geometries (
G, H ) defined in section 2.1. Startingwith any manifold M equipped with a Lorentzian metric, orientation and time orientation, we willcanonically construct a Cartan geometry associated to it.6s the principal bundle, we take the bundle of all ways to glue Z to M by identifying z = T z Z with some tangent space of M in a way that respects all of the relevant structure on these tangentspaces. More precisely, we define: F M = { proper linear isometries f : z → T x M, x ∈ M } , (2.8)where we call a map proper if it preserves orientation and time orientation. F M is isomorphicto the usual oriented and time oriented orthonormal frame bundle, so we refer to it as the framebundle and to its elements as frames . Since H acts on z , it acts on frames via composition, f f ◦ h , making F M into a principal right H bundle.The Cartan connection is a certain g -valued 1-form on the bundle F M , but since
F M is aprincipal H bundle, H -invariance of the splitting (2.3) means a g -valued 1-form is simply an h -valued 1-form together with a z -valued 1-form. In fact, we have a canonical z -valued 1-form on F M : e : T F M → z (2.9)called the soldering form . Given the canonical maps T F M ̟ π ∗ F M T M (2.10)where π ∗ : T F M → T M is the differential of the projection π : F M → M and ̟ maps v ∈ T f F M to f , the soldering form is given by e ( v ) = ̟ ( v ) − ( π ∗ ( v )) . (2.11)Moreover, the metric has a canonical Levi-Civita connection: the unique torsion-free connectionon T M . This corresponds to a torsion-free connection ω on F M . Together, we have 1-forms ω : T F M → h and e : T F M → z , which assemble to give: A : T F M → g (2.12)unique up to gauge transformations of the principal H bundle F M → M . One may check that( π : F M → M, A ) is a Cartan geometry modeled on (
G, H ). From Cartan geometry back to Lorentzian geometry
We have just seen how to get a Cartan geometry starting from a Lorentzian spacetime, for any ofthe models (
G, H ) from section 2.1. To describe general relativity as a gauge theory for a Cartanconnection, however, we need just the opposite: we want to see how an H -bundle with Cartanconnection gives us a Lorentzian geometry.In a general Cartan geometry ( π : P → M , A ) modeled on the spacetime geometry (
G, H ), wecan still think of the bundle P as a stand-in for the bundle (2.8) of oriented orthonormal frames.We often call it a fake frame bundle . From it, we can construct the associated vector bundle T = P × H z fake tangent bundle .An important observation is that the fake tangent bundle inherits a metric from the Kleingeometry ( G/H ): since η is an H -invariant inner product on z , it induces a metric on T . It shouldcause little confusion if we also call this metric η . Since T is only the fake tangent bundle, wecannot use the metric η to measure lengths and angles for tangent vectors to M . However, we cantransfer this metric to T M if we have a coframe field —a vector bundle isomorphism:
T M e T M This is the global analog of the local coframe fields e : T M → R , often used in gravitationaltheory, and induces a metric on spacetime via pullback: g ( v, w ) = η ( e ( v ) , e ( w )) . (2.13)Most importantly for our purposes, a coframe field can be obtained as part of a g -valued 1-form ona principal H bundle over M : Lemma 4.
Let p : P → M be a principal H -bundle, T the corresponding fake tangent bundle.Then there is a canonical one-to-one correspondence between: • vector bundle morphisms e : T M → T , and • z -valued 1-forms ε on P that are: – horizontal: ε vanishes on ker( dp ) – H -equivariant: R ∗ h ε = h − ◦ ε for all h ∈ H .Moreover, the first of these is an isomorphism precisely when the second is nondegenerate, meaningthat each restriction ε : T f P → z has maximal rank. Proof:
The proof is straightforward. See [6] for details.From this lemma, it is immediate that the z part of the Cartan connection is equivalent to acoframe field. In particular, P is isomorphic to the frame bundle (2.8) for the metric (2.13). The h part of the Cartan connection is then an Ehresmann connection on P , which corresponds to ametric-compatible connection.Note that ‘fake frame bundles’ and ‘fake tangent bundles’ are equivalent. While we started withthe principal bundle P and built the vector bundle T = P × H z , we could just as well start withan arbitrary ‘fake tangent bundle’ T —a vector bundle T ∼ = T M equipped with a metric η and anorientation, and mimic the construction of F M to get the principal H bundle P ∼ = { proper linear isometries f : z → T x , x ∈ M } . (2.14)8 .3 MacDowell–Mansouri gravity The formulation of general relativity in which spacetime Cartan geometry plays the most conspic-uous role is the action introduced by MacDowell and Mansouri [21]. MacDowell–Mansouri gravityworks only with a nonzero cosmological constant, so here we take (
G, H ) to be either the de Sitteror anti-de Sitter model. Fixing a fake frame bundle over spacetime M , the only field in the theoryis a Cartan connection A . The action is S MM [ A ] = Z M κ h ( F h ∧ F h ) , (2.15)where F = F [ A ] is the curvature, F h is its h -valued part, and κ h is a non-degenerate H -invariantinner product on h . There is a two-parameter family of such products, of the form κ h ( X, Y ) = tr h ( X ( c + c ⋆ ) Y ) , X, Y ∈ h , (2.16)where ⋆ is a Hodge star operator on h ∼ = Λ R , , and tr h is the Killing form on h .To see how this gives general relativity, first note that the splitting g = h ⊕ z reduces A to an H connection ω together with a coframe field e . The h part of the curvature is then F h = R + 12 [ e, e ]where R = R [ ω ] is the curvature of ω . Substituting this into the action, we get: S MM [ A ] = c Z M tr h (cid:18) [ e, e ] ∧ R + 14 [ e, e ] ∧ [ e, e ] + R ∧ R (cid:19) + c Z M tr h (cid:18) [ e, e ] ∧ ⋆R + 14 [ e, e ] ∧ ⋆ [ e, e ] + R ∧ ⋆R (cid:19) . (2.17)In the second line, with appropriate normalization, the first two terms are just the Palatiniaction for general relativity; the final term is a topological ‘Gauss–Bonnet’ term, which does notaffect the classical theory. In fact, the resulting field equations for general relativity can also beneatly summarized as [ e, ⋆F ] = 0 (2.18)where ⋆ acts only on the h part of the full curvature F . In the first line, tr h ( R ∧ R ) is also topological(the ‘Hirzebruch signature’), while the second term vanishes under the trace (see [28]). The firstterm is then the ‘Holst term’ added to the Palatini action to introduce the ‘Immirzi parameter’ γ = c /c , and also does not modify the vacuum equations of motion (2.18).It is worth noting that while G symmetry is broken ‘by hand’ to H in this action, the symmetrycan also be broken ‘spontaneously’, as first observed by Stelle and West [26]. Discussion of thisidea and its geometric significance can also be found in [13, 29]. However, we will not need thishere. Much more detail on the Cartan geometric underpinnings of MacDowell-Mansouri gravityand related theories can be found in our previous work [13, 27, 28].9 .4 Geodesics and development One of the central statements of general relativity is that test particles move along geodesics for theLevi-Civita connection on spacetime. In order to define the geodesics for a given Cartan geometry( π : P → M, A ), the central notion is development , a mapping of paths in M into the model space Z ∼ = G/H . We first define the development using paths in P , but then see immediately that itdepends only on the projection of the path to M . Definition 5.
Given a Cartan connection A on P , the development of a path C : [0 , → P is apath z C : [0 , → G/H defined by z C ( t ) = g C ( t ) H , where g C ( t ) solves the differential equation g − C ˙ g C = A ( ˙ C ) (2.19) where ˙ denotes differentiation with respect to t . Clearly g C is unique up to choosing g C (0), hence z C is unique after fixing an origin z C (0) ∈ G/H ,usually the identity coset. Note that g − ˙ g is shorthand for L g − ∗ : T g G → g applied to ˙ g . Lemma 6.
Let C and C ′ be two paths in P with the same projection into the base manifold: π ◦ C = π ◦ C ′ . Then the developments z C and z C ′ are equal. Proof: If π ◦ C = π ◦ C ′ , we can write C ( t ) = C ′ ( t ) h ( t ) for some path h in H . If g C is a solutionto the equation g − C ˙ g C = A ( ˙ C ), then g ′ := g C h solves g ′− ˙ g ′ = h − A ( ˙ C ) ˙ h + h − ˙ h (2.20)Now ˙ C ′ = ˙ C h + C ˙ h and A satisfies properties C2 and C3 in definition 2: ( R h ) ∗ A = Ad( h − ) ◦ A and A restricts to the Maurer-Cartan form on vertical vectors. Hence the right-hand side of (2.20) is A ( ˙ C ′ ), and the development of C ′ is z C ′ = g ′ H = g C H = z C .This lemma lets us define development of a path in M using an arbitrary lifting to P . Geomet-rically, the development z C of C into Z represents the path traced out on the ‘model’ space Z as itis rolled along π ◦ C in the manifold M . Proposition 7.
Let ( π : P → M , A ) be a Cartan geometry for one of the models Z ∼ = G/H in (2.2) , and let A = ω + e be the reductive splitting of the Cartan connection. A path γ in M is ageodesic for the connection on T M induced by ω if and only if its development z γ is a geodesic in Z . Proof:
We consider the case where the fake frame bundle P is the actual frame bundle F M , andthe Cartan geometry is the canonical one constructed in section 2.2. The general case is no harder,but requires translating between ‘fake’ and ‘real’ bundles using the coframe.Since ω is an Ehresmann connection on F M , γ has a unique horizontal lift ˜ γ to F M . At eachvalue of t we have a frame ˜ γ ( t ) : z → T γ ( t ) M , and the condition for parallel transport along γ isthat ˜ γ ( t ) − ( ˙ γ ( t )) =: z ∈ z is independent of t . On the other hand, since ˜ γ is horizontal, i.e. ω ( ˙˜ γ ( t )) = 0, the differentialequation for calculating the development reduces to g ( t ) − ˙ g ( t ) = e ( ˙˜ γ ( t ))10ut e is the soldering form (2.11), so the right hand side becomes e ( ˙˜ γ ( t )) = ˜ γ − ( ˙ γ ( t )) = z . Thuswe find that ˙ g ( t ) = g ( t ) z , or more properly˙ g ( t ) = L g ( t ) ∗ z. That is, g ( t ) = exp( tz ) for a constant z ∈ z . This is just the condition t g ( t ) H be a geodesicstarting at the identity coset in the symmetric space G/H (cf. Theorem IV.3.3 of Helgason [17]).This shows a geodesic in M has geodesic development; for the converse, simply reverse this argu-ment. We now move on to the main goal of this paper: to understand general relativity in terms of observer space , rather than spacetime, setting the geometric stage for modifications that coulddescribe physics beyond general relativity. We do this using the basic machinery of Cartan geometrydescribed in section 2, which will allow us to define observer space geometries without relying onspacetime. First, however, we describe in general terms what sort of geometric features observerspace inherits from the geometry of a Lorentzian spacetime and its tangent bundle.
Given a Lorentzian spacetime (
M, g ) equipped with a time orientation, we define the observerspace O of M to be the space of all unit future timelike vectors, also known as the future unit tan-gent bundle of M . We review here what sort of geometric features such an observer space naturallyhas: a contact structure, a Sasaki metric, and other canonical distributions. Mathematically, thisis standard material (see e.g. [5, 23]); what is new here is the interpretation in terms of observerspace. Contact geometry
The observer space O has a canonical contact structure . Recall that on any (2 n + 1)-dimensionalmanifold, a contact form is a 1-form α that is maximally nonintegrable in the sense that the (2 n +1)-form α ∧ dα ∧ · · · ∧ dα is nowhere-vanishing, and hence is a volume form. In contrast, note thatthe Frobenius integrability condition is that α ∧ dα vanishes identically, so a contact form is indeedhighly nonintegrable. A contact structure on a manifold is the hyperplane distribution given bythe kernel of a contact form. The kernel does not change if we multiply the contact form by anynowhere-vanishing function, so we can also define a contact structure as an equivalence class ofcontact forms under multiplication by nonvanishing functions.In fact, given a Lorentzian manifold M , its observer space O has not only a canonical contactstructure, but a canonical contact form inducing this structure. This contact form, α : T O → R is given by: α ( v ) = g ( p ( v ) , π ∗ v ) , (3.1)11here p : T O → O ⊆ T M is the tangent bundle of O , and π ∗ : T O → T M is the differential of theobserver bundle π : O → M .The contact form α induces a Reeb vector field r on observer space, the unique vector fieldnormalized by α and whose flow preserves α , that is α ( r ) = 1 and £ r α = 0 , (3.2)where £ denotes the Lie derivative. The Reeb vector field is the restriction to observer space ofthe ‘geodesic spray’, the vector field on T M whose integral curves give geodesics on M for theLevi-Civita connection of g (see e.g. [23, Ch. V]). Thus “inertial observers”, who follow timelikegeodesics in spacetime, simply follow the flow of the Reeb vector field in observer space. The Sasaki metric
Besides being a contact manifold, the observer space of a Lorentzian spacetime is also naturallya semi-Riemannian manifold. In fact, it has natural metrics of both Lorentzian and Riemanniansignature.The key to this is that the double tangent bundle of any semi-Riemannian manifold (
M, g ) hasa pair of projections down to the tangent bundle:
T T M π ∗ κ T M T M (3.3)The first of these, π ∗ , is just the differential of the tangent bundle π : T M → M , while the second, κ is the ‘connection mapping’ defined using the Levi-Civita connection ∇ on M . More precisely,since a vector field on M is, in particular, a map ξ : M → T M , its differential is ξ ∗ : T M → T T M ,and κ is uniquely determined by requiring κ ( ξ ∗ X ) = ∇ X ξ (3.4)for every vector field ξ and every tangent vector X to M (see e.g. [11]). One can show that T T M is a direct sum of the distributions given by the kernels of these projections:
T T M = ker π ∗ ⊕ ker κ called respectively the vertical and horizontal distributions on T M . Moreover,˜ g ( v, w ) = g ( π ∗ v, π ∗ w ) + g ( κv, κw )is thus a nondegenerate metric on T M . In our case, where g has signature (3 , g has signature(6 , O ⊆ T M is a submanifold with induced metric of Lorentzian signature(6 , Sasaki metric on observer space.On the other hand, we can also get a
Riemannian metric on observer space. The Reeb vectorfield is a nonvanishing timelike vector field orthogonal to the symplectic structure under the Sasakimetric. We can thus flip the sign of the Sasaki metric in the direction of the Reeb vector field,while fixing the metric on the contact distribution.12 patial, temporal and boost distributions
The contact structure on observer space is a hyperplane distribution. But there are other canonicaldistributions on observer space as well. First, there is the distribution complementary to thecontact structure, the 1-dimensional distribution spanned by the Reeb vector field, which we callthe temporal distribution . Timelike geodesics on M correspond to curves on O tangent to thetemporal distribution.But also, the 6-dimensional contact distribution splits into a pair of 3-dimensional subdistri-butions. The boost distribution is the kernel of π ∗ , where π : O → M is the restriction ofthe tangent bundle to observer space. A path tangent to this distribution corresponds to simplychanging observers at a fixed point in spacetime. By definition of the contact form (3.1), the boostdistribution is clearly a 3-dimensional sub-distribution of the contact distribution. The spatialdistribution can be defined either as the orthogonal complement of the boost distribution in thecontact distribution, or as the kernel of κ restricted to O . In this way, each tangent space splitscanonically into a direct sum: T o O = (boost vectors) ⊕ (spatial vectors) ⊕ (temporal vectors) (3.5)where the first two summands constitute the contact structure. Momentum space
In spacetime physics, whereas the velocity of a subluminal particle is a unit future-timelike vector,the momentum is an arbitrary future-timelike cotangent vector. The space of momenta is thus asubspace of T ∗ M . A particle’s momentum can be dualized and then normalized via the metric toget the corresponding velocity, and conversely, the dual of the velocity times the particle’s massgives the momentum.In observer space, on the other hand, the velocity of a particle is not a vector but just a point.However, we can still reconstruct the space of all momenta of particles directly from observer space,without appealing to spacetime in any direct way: it is the ‘symplectification’ of observer space.Any (2 n +1)-dimensional contact manifold extends canonically to a (2 n +2)-dimensional symplecticmanifold (see e.g. [5]).To recall how this works, first define a contact element at a point o in a contact manifold O tobe a covector β o ∈ T ∗ o O whose kernel is the contact hyperplane at o . Then the symplectification S of O is the space of all contact elements on O . In other words, S is the subbundle of T ∗ O whosesections are contact forms for the contact structure. The symplectic structure on S : ω : T S × T S → R is the differential of the canonical 1-form α : T S → R given by: α ( v ) = β o ( π ∗ v ) v ∈ T β o S where π : S → O is the obvious bundle and β o is a contact element at o ∈ O .While the contact distribution alone determines the symplectification, in the cases of interesthere we always have a specified contact form, and this gives a preferred section of the symplecti-fication. Physically speaking, this section selects, at each point of observer space, the momentumcorresponding to a particle with unit mass. So far, we have defined O as the unit future tangentbundle of a Lorentzian spacetime; in this case the contact form (3.1) provides the normalizingsection. 13 ightlike particles At this point, we may wonder how massless particles fit into the observer space picture. Since apoint in observer space corresponds to the instantaneous velocity of a subluminal particle, thereseems to be no room in observer space for particles traveling at the speed of light. The answer,however, is clear: lightlike particles live in the ‘boundary’ of observer space.More precisely, each fiber in the observer space bundle O → M is a copy of hyperbolic space,namely the space of all timelike velocity vectors at the same spacetime event. If we adjoin to eachfiber the space of lightrays at that event, we obtain an extension of observer space in which eachfiber is a compactification of hyperbolic space, diffeomorphic to a 3-dimensional ball.This extended observer space, which includes ‘lightlike observers’ on its boundary, deservesfurther study. However, for the remainder of this paper, we focus on ordinary subluminal observersonly. To describe observer space Cartan geometry, we first need Klein geometries to serve as homogeneousmodels of observer space. Fortunately, there are three obvious choices: The model spacetimes Z ∼ = G/H described in section 2.1 are not only homogeneous, but also isotropic , meaning that G acts transitively on observers as well. Thus, to each of these models there is a corresponding modelobserver space.Recall that our chosen spacetime event, z ∈ Z , has stabilizer H and z = T z Z . The observersat z , the unit timelike vectors, are thus elements of hyperbolic 3-space , which we define to be asubmanifold of z ∼ = R , in the usual way:H := { y ∈ z : η ( y, y ) = − , y > } ∼ = H/K , (3.6)where we are using a G -invariant metric η on Z of signature ( − +++), and the stabilizer of y ∈ H is K ∼ = SO(3) . (3.7)The natural projection from observer space down to spacetime: G/K → G/H (3.8)is a G -equivariant fiber bundle with standard fiber H . The fiber over an event is of course just thehyperbolic space of all velocities that observers at that event can have.As representations of K , both h and z in the reductive splitting (2.3) are further reducible, eachsplitting into a direct sum of two irreducible K representations: h = k ⊕ y and z = ~ z ⊕ z o . (3.9)Here k is the Lie algebra of K , and the complement y ∼ = R is canonically isomorphic to thetangent space to H/K at the basepoint y . The representation z , corresponding to spacetimetranslations, naturally splits into spatial translations ~ z and temporal translations z o , from theobserver’s perspective.Thus the adjoint representation of K ⊆ G on g splits into a direct sum of four irreduciblerepresentations, based on our choice of basepoint z ∈ Z and observer y at z : g = k ⊕ ( y ⊕ ~ z ⊕ z o ) (3.10)14he parenthesized part is naturally identified with the tangent space to observer space at the chosenobserver; the three summands correspond respectively to those in the canonical splitting (3.5) of atangent space to observer space.We can interpret the sum (3.10) of K representations in terms of infinitesimal symmetries ofobserver space: k ∼ rotations around the observer y ∼ boosts, changing the observer but not the base event ~ z ∼ spatial translations of the event/observer z o ∼ time translations of the event/observer.It is a straightforward exercise to work out the Lie brackets. For any of the models, we have[ k , k ] ⊆ k [ y , y ] ⊆ k [ ~ z ,~ z ] ⊆ k [ z o , z o ]= 0[ k , y ] ⊆ y [ y ,~ z ] ⊆ z o [ ~ z , z o ] ⊆ y [ k ,~ z ] ⊆ ~ z [ y , z o ] ⊆ ~ z [ k , z o ] = 0 (3.11)where the third column is actually zero in the case Λ = 0. The geometric interpretation of these isclear: for example, [ y ,~ z ] ⊆ z o says tiny boosts and tiny spatial translations commute up to a timetranslation.From the above chart we see that [ k , y ⊕ z ] ⊆ y ⊕ z , but [ y ⊕ z , y ⊕ z ] generally has parts in both k and its complement. Thus our model observer geometries are reductive but not symmetric. Notethat g = k ⊕ y ⊕ ~ z ⊕ z o makes g into a ( Z / × Z / H ′ ⊆ G , which acts transitively onthe hypersurface. The stabilizer of a point on this spatial slice under the action of H ′ is the same asthe stabilizer of an observer, namely K . This lets us describe spatial geometry using these groups: H ′ ∼ = SO(4) spherical group
ISO(3)
Euclidean group
SO(3 , hyperbolic group K = SO(3) rotation group. From this perspective, we get a rather different interpretation of the decomposition of g in(3.10): k ∼ rotations around the basepoint on the spatial slice y ∼ changes of spatial slice, without changing the basepoint ~ z ∼ translations of the basepoint, within the same spatial slice z o ∼ time translations of the basepoint, changing the spatial slice.It is worth mentioning that the models discussed in this section are not the only possiblemodels of observer space. For example, starting from one of the models ( G, K ), one could forma new model ( G ′ , K ) where G ′ = K ⋉ ( y ⊕ z ). This example—which is nothing but the observerspace of Galilean spacetime—is a ‘mutation’ of the models derived from homogeneous Lorentzian15pacetimes, meaning that the Lie algebra g ′ ∼ = g , not as Lie algebras but as representations of K .Mutations carry the same essential geometric information [25]. In this paper, we use only the threemodels ( G, K ) described above.
Contact structure
The homogeneous models of observer space are homogeneous as contact manifolds, meaning thatthe symmetries preserve the contact structure. We now describe the contact structure directly inthe language of Kleinian geometry.
Proposition 8.
Consider the observer space Cartan geometry ( π : G → G/K, A ) where A is theMaurer–Cartan form. The projection of A into z o : T G A g z o induces a 1-form on G/K . This 1-form coincides with the negative of the contact form (3.1) inducedby the structure of
G/K as the unit future tangent bundle of
G/H . Proof:
First, by a slight variation of Lemma 4, the projection of the Maurer–Cartan form into y ⊕ z is the same as an isomorphism T ( G/K ) → G × K ( y ⊕ z ) . Similarly, projecting further to z o , the z o part of A is the same as a 1-form on G/K with values inthe associated vector bundle G × K z o over G/K . But this latter vector bundle is trivial, since K acts trivially on z o , so we have just a z o -valued 1-form. Taking advantage of the above isomorphismfor T ( G/K ), this 1-form is given by: G × K ( y ⊕ z ) → z o [ g, ( a, w )] w o where w o is the z o part of w ∈ z . Namely, using the metric η on G/K , w o = − η ( y, w ) y since z o is by definition the span of y ∈ z . Using the isomorphism z o ∼ = R given by y
1, we thusget a real-valued 1-form ˜ α on G/H : ˜ α ([ g, ( a, w )]) = − η ( y, w ) . On the other hand, note that the differential of the bundle map (3.8) at the observer y (cor-responding to the identity coset) is just the obvious projection y ⊕ z → z . Thus, the contact form(3.1) is given by: α y : G × K ( y ⊕ z ) → R [ g, ( a, w )] η ( y, w )This differs from ˜ α by a minus sign.For the momentum space corresponding to the model observer space, the symplectificationof the contact manifold G/K is canonically isomorphic to the space of future-timelike cotangentvectors to
G/H . 16 .3 Cartan geometry of observer space
Now that we understand the homogeneous models, the general definition of Cartan geometry (def. 2)lets us define abstractly what it means for a manifold to have the geometry of an ‘observer space’.
Definition 9. An observer space geometry ( π : P → O, A ) is a Cartan geometry modeled on ( G, K ) , where G is SO(4 , , ISO(3 , or SO(3 , and K is SO(3) , as described in section 3.2.
This definition is intrinsic in the sense that it describes the geometry of an observer spacedirectly, without using any underlying notion of ‘spacetime’. We now turn to describing observerspace Cartan geometries in more detail, and give some examples.
Geometry of observer space connections
According to the splitting (3.10) of the Lie algebra g , the Cartan connection on observer spacebreaks up into four irreducible pieces: g = k ⊕ y ⊕ ~ z ⊕ z o A = Ω + b + ~e + e o (3.12)This can be interpreted geometrically using ‘rolling without slipping’: Locally, as we begin to movealong some path γ in O , rolling the model observer space along as we go, the transformation of themodel space breaks up into: • Ω( γ ′ (0)) —a tiny rotation around the model observer • b ( γ ′ (0)) —a tiny boost of the model observer • ~e ( γ ′ (0)) —a tiny spatial translation of the model observer • e o ( γ ′ (0)) —a tiny time translation of the model observerUsing the commutation relations of the algebra, the curvature is: F = dA + [ A, A ]= ( d Ω + [Ω , Ω] + [ b, b ] + [ ~e, ~e ]) + ( d Ω b + [ ~e, e o ]) + ( d Ω ~e + [ b, e o ]) + ( de o + [ b, ~e ]) , (3.13)where the four parenthesized terms live respectively in k , y , ~ z , and z o . Distributions
In Cartan geometry based on any model (
G, K ), a K -invariant structure on the tangent space g / k of the model Klein geometry yields the same type of structure on tangent spaces of the model. Foran observer space Cartan geometry, according to (3.10) we have a K -invariant splitting g / k ∼ = y ⊕ ~ z ⊕ z o and the Cartan connection thus gives a splitting of each tangent space. To see how this works,first note that the Cartan connection gives four distributions on the total space P , just by taking17reimages of the components along A : T P → g . It will be convenient to denote the distributioncorresponding to a particular subalgebra using an underline, for example: k = A − ( k ) y = A − ( y ) h = k ⊕ y = A − ( h )...Note that by definition A ( q ) = q for any K -invariant subspace q ⊆ g .Because of K -invariance, these distributions descend to observer space in a gauge invariant way.The distribution k is just the vertical distribution of the bundle, and so is trivial on the base; thedistributions corresponding to y , ~ z and z o descend to give the local directions corresponding toboosts, spatial translations, and time translations, according to each observer. Observer space geometry from spacetime geometry
Of course, the most obvious way to construct an observer space geometry is to start with a spacetimegeometry. Given a spacetime Cartan geometry, the principal H bundle P over spacetime determinesan isomorphism between the (fake) observer space O and the associated bundle P × H H/K . Definingthe fake tangent bundle T = P × H z and viewing P as the fake frame bundle P = { linear isometries f : z → T x , x ∈ M } , (3.14)we can define fake observer space O as the bundle of unit future-directed timelike vectors in thefake tangent bundle. We have a canonical bundle isomorphism P × H H/K → O [ f x , y ] f x ( y ) . (3.15)Starting from P , we can fix an observer y ∈ H/K which gives us a projection map P → O f x f x ( y ) (3.16)allowing us to identify P as a principal K bundle over O . The original Cartan connection onspacetime becomes a Cartan connection on observer space: Lemma 10 (Observer space geometry from spacetime geometry). If ( π : P → M, A ) isa Cartan geometry with model ( G, H ) , then ( π : P → O , A ) , where O = P × H H/K is a Cartangeometry with model ( G, K ) . Proof:
First note that a Cartan connection on P → M and a Cartan connection on P → O areboth g -valued 1-forms on P . If A is a Cartan connection on P → M , it has properties C1-C3 indefinition 2: It is a linear isomorphism T p P → g at each p ∈ P , it transforms under the adjointof H , and is the Maurer-Cartan form on invariant vector fields associated to the Lie algebra h .But the second and third properties imply the same properties for the subgroup K ⊆ H and thesubalgebra k . Hence A is also a Cartan connection on P → O .18he converse is of course not true: the K action on a principal K bundle need not extend to an H action, and even if it does, properties C2 and C3 for the group K do not imply those for H .As explained in section 2.2, a Cartan connection on the bundle P → M allows us to reconstructthe ‘real’ bundles from the fake ones, since its z part defines a coframe and hence an isomorphismbetween the bundles T M and T . From the observer space perspective, the geometry of spacetime is always viewed locally in relationto some particular observer. To study spacetime geometry over extended regions, it is thus helpfulto single out one observer at each point by specifying a ‘field of observers’. As in our previousdiscussions we will start with a Lorentzian spacetime with its associated Cartan connection on theframe bundle
F M . Definition 11. If M is a Lorentzian spacetime with observer space O , a field of observers is asection of the bundle O → M . In other words an observer field is a unit future-directed timelike vector field. These exist on anytime-oriented Lorentzian manifold.An observer field u gives a vector field on M , and hence a 1-form ˆ u , defined by ˆ u ( v ) = g ( u, v ).However, we can also obtain this dual 1-form by pulling back the contact form α along u : Proposition 12.
The 1-form dual to u is ˆ u = u ∗ α . Proof:
Directly calculating the pullback of α along u , we get: u ∗ α ( v ) = α ( u ∗ v )= g ( p ( u ∗ v ) , π ∗ u ∗ v )= g ( u, v )= ˆ u ( v ) , since by definition ˆ u , the dual of u , is the 1-form on M given by ˆ u ( v ) = g ( u, v ).A field of observers u and its corresponding field of co-observers ˆ u give us a canonical way tosplit differential forms into spatial and temporal parts. First interior multiplication by u is thegrade -1 map: ι u : Ω p ( M ) → Ω p − ( M )defined on 1-forms X by ι u X = X ( u ), and on higher forms by requiring it to be a graded derivation: ι u ( X ∧ Y ) = ( ι u X ) ∧ Y + ( − ) p X ∧ ι u Y , (3.17)where X is a p -form. Definition 13.
We say a differential form X on M is temporal if ˆ u ∧ X = 0 , and spatial if ι u X = 0 .
19t is then easy to check that any form X splits into spatial and temporal parts as: X = ( X − ˆ u ∧ ι u X ) | {z } spatial + (ˆ u ∧ ι u X ) | {z } temporal =: X ⊥ + X k (3.18)and that the spatial and temporal projections are grade 0 derivations. Similarly, we can definespatial and temporal differentials that act on differential forms as d ⊥ X = dX − ˆ u ∧ £ u X , d k X = ˆ u ∧ £ u X , (3.19)where £ u = ι u d + d ι u is the usual Lie derivative.So far, all of this assumes a metric given from the outset. As we have seen, we can avoid thisby starting with a ‘fake frame bundle’ P and its associated fake tangent bundle T with metric η . Definition 14.
Given a fake frame bundle P with associated ‘fake observer space’ O ∼ = P × H H/K ,a field of internal observers is a section of the bundle O → M . Such an internal observer field y reduces P to a principal K bundle: P is a principal K bundleover O , and this can be pulled back along y to a principal K bundle Q y → M : Q y i y PM y O (3.20)Thinking of P as a fake frame bundle, Q y corresponds to the bundle of frames that map a given(fixed) observer y o ∈ z into y ( x ) ∈ T x , with an obvious inclusion map i y into P .The internal observer field also lets us split fields living in any associated vector bundle of P into various components, according to how the relevant H representation splits when pulled backto K . For instance, the fake tangent bundle T splits into internal ‘temporal’ and ‘spatial’ parts: T ∼ = P × H z ∼ = ( Q y × K z o ) ⊕ M ( Q y × K ~ z ) , (3.21)where ⊕ M denotes the fiberwise direct sum of vector bundles. Similarly, the internal observer splitsfields valued in the bundle Ad( P ) = P × H h into a k part and a y part, according to (3.9).Next, suppose we have not only a fake frame bundle, but a Cartan connection on it. The z partgives us a coframe field, hence a specific isomorphism T M ∼ = T . Using this isomorphism, a fieldof internal observers obviously corresponds to a field of observers for the metric and orientationinduced by the coframe field. This same linking of ‘internal’ and ‘spacetime’ observers is the keyto our construction of covariant Ashtekar variables [14], which we also review in section 3.5 withthe benefit of observer space. But first, we consider what happens to the Cartan connection whenwe reduce the frame bundle to a K bundle via an observer field.For a given observer field y : M → O , we can use the inclusion i y in (3.20) to pull back a Cartanconnection A on P → M to Q y : ˜ A = i ∗ y A : T Q y → g . (3.22)It is clear that ˜ A cannot be a Cartan connection for the bundle Q y → M , since T q Q y and g do nothave the same dimension. But, it has all other essential features of a Cartan connection:20 roposition 15. Let ( π : P → M, A ) be a spacetime Cartan geometry with model ( G, H ) , andlet i : Q → P be a reduction of P to a principal K -bundle. The pullback ˜ A := i ∗ A of the Cartanconnection A satisfies:1. For each q ∈ Q , the projection of ˜ A to k ⊕ z , ( ˜ A q ) k ⊕ z : T q Q → k ⊕ z is a linear isomorphism;2. R ∗ k ˜ A = Ad( k − ) ˜ A ∀ k ∈ K ;3. ˜ A restricts to the Maurer–Cartan form on vertical vectors. Proof:
Note that the inclusion i is K -equivariant: i ◦ R k = R k ◦ i . It then follows from ( R k ) ∗ A =Ad( k − ) A that ( R k ) ∗ ˜ A = ( R k ) ∗ i ∗ A = i ∗ ( R k ) ∗ A = i ∗ (Ad( k − ) A ) = Ad( k − ) ˜ A , which shows 2).Similarly i ∗ maps vertical vectors in Q → M to vertical vectors in P → M , which shows 3) for ˜ A since it holds for A . Lastly, to show the first property it suffices to show that ( ˜ A q ) k ⊕ z is injective.If ( ˜ A q ) k ⊕ z ( V q ) = 0, then A i ( q ) ( i ∗ ( V q )) ∈ y . But the preimage of y under A is the space of vectorscanonically associated with X ∈ y . For i ∗ ( V q ) to be in this space we must have V q = 0.This proposition has an immediate corollary that works only for the Minkowski observer spacemodel, with G ∼ = ISO(3 , k ⊕ z is a Lie subalgebra—the Lie algebra of the subgroup J = SO(3) ⋉ R consisting of rotations and spacetime translations. The proposition thereforeimplies ( Q → M, ˜ A k ⊕ z ) is a Cartan geometry with model ( J, K ). For the other models, there is nosubgroup analogous to J : according to (3.11), the bracket of a spatial translation in ~ z and a timetranslation in z o is a boost, so that any subgroup including all spacetime ‘translations’ must alsoinclude boosts. Thus, for the models with Λ = 0, we do not get a Cartan geometry of this sort.On the other hand, using only spatial translations, and discarding the time translations, we canget a Cartan connection for any of the three models—not on spacetime but on space . This leadsto the subject of ‘Cartan geometrodynamics’. Assume we have a Cartan connection on P → M and an internal observer field y . This gives us theobserver field u , as well as the principal K bundle Q y → M . Also assume that there is a totallyspatial hypersurface S (“space”) of M , meaning a codimension 1 submanifold such that each T x S consists entirely of spatial vectors: ˆ u ( v ) = 0 for any v ∈ T x S . Such S exists whenever the Frobeniuscondition ˆ u ∧ d ˆ u = 0 holds.Pulling back the fake frame bundle P along the embedding of S into M defines a spatial fakeframe bundle P S over S , a principal K bundle. Because of the way we defined S , this is the bundleof fake frames P S ∼ = { proper linear isometries E : R → ( T x ) y , x ∈ S} . (3.23)where ( T x ) y is the subspace of T x orthogonal to y ( x ) in the metric η on T .We have an inclusion map i y, S : P S → Q y which maps E to its unique extension to an isometry z → T x respecting time orientation, resulting in the following diagram: P S i y, S Q y i y P S M y O (3.24)21ulling back the 1-form ˜ A = i ∗ y A on Q y along i y, S gives a g -valued 1-form on P S which now alwaysdefines a Cartan geometry. Proposition 16.
The projection A of the 1-form i ∗ y, S ˜ A = i ∗S i ∗ y A , the pullback of the 1-form ˜ A on Q y → M to the bundle P S → S , to k ⊕ ~ z is a Cartan connection; ( P S → S , A ) is a Cartan geometrymodeled on the Klein geometry ( H ′ , K ) where H ′ is the Lie group with Lie algebra k ⊕ ~ z . Proof:
Again we need to show that A satisfies properties C1-C3 in definition 2. As in proposition15, the second property follows from compatibility of the inclusion with the action of K , and thethird from the fact that the inclusion maps vertical vectors to vertical vectors. For the first property,by proposition 15, the projection of the image of ( i y, S ) ∗ T r P S (for r ∈ P S ) under ˜ A to k ⊕ z is a six-dimensional subspace of k ⊕ z . It contains all of k by property 3, and a three-dimensional spacelikesubspace of z in the metric η . The projection of such a subspace to ~ z must also be three-dimensional.The Cartan connection A on P S → S is the basic ingredient for the picture of Ashtekar variablesas Cartan geometrodynamics. If we have not just one spatial hypersurface S but a foliation of M by totally spatial hypersurfaces of identical topology, we can, just as in usual geometrodynamics,identify the hypersurfaces and view them as spatial geometries evolving in time, where ‘time’ is thefunction t associated to the field of co-observers ˆ u which is of the form ˆ u = N dt if the Frobeniuscondition ˆ u ∧ d ˆ u = 0 holds. On each of these ‘constant time slices’ S t we have a Cartan geometrygiven by ( P S t → S t , A t ) modeled on H ′ /K ; we call this Cartan geometrodynamics .In our previous discussion of Cartan geometrodynamics [14] the Cartan connection on eachspatial slice was modeled on hyperbolic space H since, without the backdrop of observer space,this was the only 3-dimensional observer space in sight. Here, we clearly see the role of the group H ′ , which depends on the cosmological constant chosen for the model spacetime Z ∼ = G/H . TheCartan geometry on each spatial slice is most naturally modeled on hyperbolic space, Euclideanspace or the sphere, according to whether the spacetime model is anti-de Sitter, Minkowski, or deSitter, respectively.We now detail the construction of Lorentz-covariant Ashtekar variables [14] in order to clarifythe relation between Ashtekar variables and Cartan geometrodynamics. The g -valued connection A on the K -bundle P → O breaks up into the irreducible pieces as in (3.12): g = k ⊕ y ⊕ ~ z ⊕ z o A = Ω + b + ~e + e o Pulling back A along the internal observer field y , we obtain the connection ˜ A on Q y → M whichsplits similarly: ˜ A = y ∗ Ω + y ∗ b + E + ˆ u . (3.25)The parts of ˜ A valued in ~ z and z o play a special role: they determine the notions of spatial andtemporal vectors and forms. In particular, spatial vectors live in the kernel of ˆ u , while temporalvectors live in the kernel of the triad E ; notions of spatial and temporal forms follow from this.We can define the field of observers u by requiring it to be spatial and normalized by ˆ u : E ( u ) = 0 , ˆ u ( u ) = 1 . (3.26)All this agrees with definition 13. 22e can then split y ∗ Ω and y ∗ b into their spatial and temporal parts, Ξ := y ∗ Ω( u ) , Ω := y ∗ Ω − ˆ u Ξ , ξ := y ∗ b ( u ) , K := y ∗ b − ˆ u ξ , (3.27)to obtain the variables needed to describe generalized canonical gravity in [14] (where we onlydefined the h -valued scalar Ξ = Ξ + ξ ). Note that Ω ( u ) = K ( u ) = 0, as required.Starting from the Palatini action for general relativity as a functional of the h and z parts ω and e of a Cartan connection (identified with ˜ A ) and decomposing the variables and their derivativesfurther according to (3.25), (3.27), and (3.19), we then recover the usual formulation of canonicalgravity in connection variables. The dynamical variable conjugate to the triad (or rather the inversetriad [ E, E ]) is the k -valued Ashtekar-Barbero connection A A . B . := Ω + γ ⋆ K (3.28)where ⋆ is the Hodge dual on h and γ is the Immirzi parameter, giving the relative weight betweenthe two terms in the inner product used to define the Palatini action; cf. (2.16) and the discussionbelow. The variables E, Ω and K must satisfy the Gauss constraints[ K, [ E, E ]] = 0 , d ⊥ Ω [ E, E ] = 0 , (3.29)where the spatial covariant differential d ⊥ Ω acts as d ⊥ Ω X = d ⊥ X + [ Ω , X ]. They are also subject tothe diffeomorphism and Hamiltonian constraints which can be given in the form (cid:20) E, R [ Ω ] + 12 [ K, K ] + d ⊥ ˆ u Ξ (cid:21) = 0 , h E, d ⊥ Ω K + d ⊥ ˆ u ξ i = 0 (3.30)where R [ Ω ] := d ⊥ Ω + Ω ∧ Ω is the spatial curvature of Ω . If the temporal 1-form ˆ u defines afoliation of spacetime, d ⊥ ˆ u = 0 and the terms involving d ⊥ ˆ u disappear; then one recovers theconstraint formulation of Ashtekar-Barbero variables. See [14] for more details.By proposition 16, if ˆ u defines a foliation, the fields Ω and E can be assembled into an h ′ -valuedCartan connection A that defines spatial geometry as a Cartan geometry modeled on ( H ′ , K ).Using the Ashtekar-Barbero connection instead of Ω defines a different spatial Cartan geometry ofthe same type with Cartan connection A A . B . = ( A A . B . , E ).In [14] we argued that Cartan geometrodynamics gives an understanding of the Ashtekar-Barbero formulation of canonical gravity as a theory of spontaneously broken H -symmetry. Fromthe perspective of observer space advocated in the present paper, it would be more appropriate tospeak of a spontaneously broken G -symmetry, where group G is spontaneously broken to K by achoice of internal observer field. We have seen in lemma 10 that spacetime Cartan geometry automatically gives us Cartan geometryon observer space. We now deal with the more interesting converse question: Given just an observerspace O , under what conditions can we sensibly construct a spacetime M for which O is the unitfuture tangent bundle? 23o address this question, first note that in the principal H bundle P → M , the spacetimemanifold M itself actually contains only redundant information: it is just the space of H -orbits P/H . In the observer space picture, however, we do not, a priori, have any action of H on P .Rather, we have only a K action, since P is a principal K bundle over observer space. However,while there is no action of the group H , we will see that under certain conditions, we get an actionof the Lie algebra h . Just as an action of a Lie group H on a manifold M is a group homomorphism H → Diff( M ) , an action of a Lie algebra h on a manifold M is a Lie algebra homomorphism h → Vect( M )where Vect( M ) is the Lie algebra of Diff( M ), the Lie algebra of smooth vector fields on M . Any Liealgebra action α : h → Vect( M ) is integrable in the sense that the distribution α ( h ) is an integrabledistribution. The integral submanifolds of this distribution are the orbits of the h -action, and thespace of orbits is denoted M/ h . [1]In our situation, we have an action of K on the manifold P , and this induces an action of the Liealgebra k of K , by differentiation. Using the Cartan connection A , it is clear that the distribution A ( k ) on P is just the distribution tangent to the fibers of P → P/K . So, the spaces of orbitscoincide: P/ k = P/K .
To reconstruct spacetime, we can try to extend this k -action to an h -action. Whenever this works,we can immediately define: spacetime M := P/ h . In the best case, the h -action will integrate to an H -action, the map P → M will become a principal H bundle, and the observer space Cartan connection will therefore give the geometry of spacetime.To see how Cartan connections are related to Lie group actions, it is helpful to first view Cartanconnections in a different way, using the nondegeneracy of a Cartan connection A : T P → g to turnthis map around. More precisely, suppose that A is any g -valued 1-form on a manifold P such that A p : T p P → g is a linear isomorphism (i.e., property C1 in the definition of a Cartan connection).Then we get a map A : g → Vect( P )where for X ∈ g , the value of A ( X ) at p ∈ P is A ( X ) p := ( A p ) − ( X ) . (4.1)It is easy to check that the properties C1, C2, and C3 are respectively equivalent to:C1 ′ . For each p ∈ P , the composite g A −→ Vect( P ) → T p P is a linear isomorphism;C2 ′ . R h ∗ ◦ A = A ◦ Ad( h − ) ∀ h ∈ H ;C3 ′ . For X in h , A ( X ) is the canonical vertical vector field associated to h .24e recall that any element of h , thought of as a left-invariant vector field on H , pulls back canon-ically to a vector field on P along any local trivialization.If A is a Cartan connection, A is generally not a Lie algebra homomorphism. In fact, the failureof A to be a homomorphism corresponds to the curvature F of A : Proposition 17.
Suppose ( π : P → M , A ) is a Cartan geometry modeled on the Klein geometry ( G, H ) and define A by (4.1) . Then A ([ X, Y ]) − [ A ( X ) , A ( Y )] = A (cid:0) F ( A ( X ) , A ( Y )) (cid:1) . for all X, Y ∈ g . Proof:
For any 1-form ω , dω ( X, Y ) = X ( ω ( Y )) − Y ( ω ( X )) − ω ([ X, Y ]), where X ( ω ( Y )) denotesthe directional derivative of the function ω ( Y ) along X . Applying this formula to dA ( A ( X ) , A ( Y )),the first two terms are directional derivatives of constant functions, and thus vanish, leaving simply: dA ( A ( X ) , A ( Y )) = − A ([ A ( X ) , A ( Y )]) . Hence, with F = dA + [ A, A ], F ( A ( X ) , A ( Y )) = dA ( A ( X ) , A ( Y )) + [ X, Y ]= − A ([ A ( X ) , A ( Y )]) + [ X, Y ]After applying A to both sides, we have the result.This proposition implies that A restricts to a homomorphism on the subalgebra h ⊆ g . In fact,we get a bit more: [ A ( X ) , A ( Y )] = A ([ X, Y ]) (4.2)whenever at least one of X, Y is in h ⊆ g . To see this note that curvature F is horizontal: it vanisheson any vertical vector. Thus, if either X or Y is in h , then the left hand side of the equation inthe lemma vanishes. In particular, A | h : h → Vect( P ) is an action of the Lie algebra h on P . Theseobservations lead to a more general notion of Cartan connection studied by Alekseevsky and Michor[2] where the principal bundle structure is discarded, defining a g / h -Cartan connection to be a1-form on a manifold P for which (4.2) holds whenever at least one argument is in h . A significantpart of Cartan’s theory carries over to this case.In our case, we start with a Cartan geometry for the model G/K and want a Cartan geometryfor the model
G/H . Definition 18.
Suppose ( π : P → O, A ) is a Cartan geometry modeled on the Klein geometry ( G, K ) , and h is a Lie algebra with k ⊆ h ⊆ g . We say A is h -flat if [ A ( X ) , A ( Y )] = A ([ X, Y ]) (4.3) whenever at least one of X, Y is in h ⊆ g . We say A is h -complete if A ( X ) is a complete vectorfield (i.e. generates a global flow) for all X ∈ h . G, K ) is trivially k -flat, and g -flat if and only if it is flat.We can now give conditions under which spacetime geometry can be reconstructed from observerspace. To state these conditions it is most convenient to modify the model geometry slightly, byreplacing the groups G , H and K with their universal covering groups. In each of the cases we areinterested in, the universal cover is just the double cover: e G = Spin o (4 , o (3 , o (3 , e H = Spin o (3 , e K = Spin(3)where ISpin(3 ,
1) := Spin(3 , ⋉ R , , and the subscript o denotes the connected component, asbefore. Note that passing to these covering groups does not change the geometry at all on theinfinitesimal level, since the Lie algebras are unchanged. Theorem 19 (Reconstruction of spacetime).
Suppose ( π : P → O, A ) is an observer spacegeometry, with model ( e G, e K ) .1. If A is h -flat, then A ( h ) spans an integrable distribution of constant rank.2. If in addition A is h -complete, it induces a locally free action of e H on P .3. If this locally free action is free and proper, then the quotient M = P/ e H is a smooth manifold,and ( π : P → M , A ) is a Cartan geometry with model ( e G, e H ) . In the third part, recall that a proper G action on X is one for which the graph G × X → X × X defined by ( g, x ) ( x, gx ) is a proper map: the preimages of compact sets are compact. Proof:
Since A is a Cartan connection for a geometry on observer space, property C1 implies A ( h ) has constant rank equal to the dimension of h . If A is h -flat, then in particular A | h is a Liealgebra action, and the span of any Lie algebra action is integrable.If A is also h -complete, then a result of Palais [24] implies the Lie algebra action A | h is thederivative of a locally free action of some Lie group H with Lie algebra h . But the action of H induces an action of its universal cover e H .If e H has a free and proper action on P , then there exists a unique smooth structure on P/ e H such that P → P/ e H is a submersion; with this smooth structure, P → P/ e H is a principal e H -bundle. (see e.g. [22, Thm. 1.21]). It remains to check the properties of a Cartan connection. Thenondegeneracy property C1 is already satisfied. Since e H is connected, equivariance C2 follows from h -flatness. For property C3, note that restricted to any fiber of P → P/ e H , A is a complete h -valued1-form that is an isomorphism on each tangent space and satisfies the Maurer–Cartan equation.Hence, the fiber is an e H torsor and A restricts to the Maurer–Cartan form. (See e.g. [25, Thm 8.7]for uniqueness of the Maurer–Cartan form with respect to these properties.)It will be convenient to rephrase h -flatness for an observer space Cartan connection A in termsof A itself, rather than A : Proposition 20.
Suppose ( π : P → O, A ) is a Cartan geometry on observer space, modeled on theKlein geometry ( G, K ) . Then A is h -flat if and only if F vanishes on any v ∈ T P for which A ( v ) ∈ y . roof: Since A is nondegenerate at each point, it is clear from Prop. 17 that A is h -flat if andonly if F ( A ( X ) , A ( Y )) = 0 whenever X ∈ h . Since A is a Cartan connection modeled on ( G, K )this equation holds already for X ∈ k , and hence we require only that F ( A ( X ) , A ( Y )) = 0 ∀ X ∈ y . But A ( g ) spans the tangent space at each point of P , and A ( y ) spans the subspace of ‘boost’ vectors v . So, the condition on F is equivalently written F ( v, w ) = 0 ∀ v ∈ T P such that A ( v ) ∈ y , as we wished to show.To rephrase this proposition, A is h -flat if and only if F ( y , v ) = 0 for all vectors v , where y = A ( y ) is the distribution on P corresponding to Lorentz boosts, discussed in section 3.3. While F need not vanish on y , it is interesting to note that the y part of F always does: Proposition 21.
Suppose ( π : P → O, A ) is an observer space geometry. Then the y part of thecurvature, F y , is a spacetime 2-form, i.e. it vanishes on any vector in the boost distribution y . Proof:
From (3.13) we have F y = d Ω b + [ ~e, e o ]. We must show that F y ( v, w ) = 0 whenever A ( v ) ∈ y . It suffices to pick v ∈ y and w ∈ g and define v = A ( v ), w = A ( w ). In fact, it suffices tocheck the case where w ∈ y ⊕ z , since we know that F ( v, w ) vanishes whenever w ∈ k , since A is aCartan connection with model ( G, K ). We have: F y ( v, w ) = ( db + [Ω , b ] + [ ~e, e o ])( v, w )= v ( b ( w )) − w ( b ( v )) + [Ω( v ) , b ( w )] − [Ω( w ) , b ( v )] + [ ~e ( v ) , e o ( w )] − [ ~e ( w ) , e o ( v )] . The first two terms are directional derivatives of constant functions, and so vanish. Moreover, since v ∈ A ( y ) and w ∈ A ( y ⊕ z ), we have Ω( v ) = Ω( w ) = ~e ( v ) = e o ( v ) = 0. We have seen in section 2.3 how to define general relativity in terms of the MacDowell-Mansouriaction (2.15) for a Cartan connection A on spacetime M , corresponding to a Cartan geometrymodeled on ( G, H ). Here we give a new action on observer space O , starting with a Cartanconnection A corresponding to a Cartan geometry modeled on ( G, K ). In order to use theorem19, our action enforces h -flatness of A ; the rest of the action is a straightforward extension of theMacDowell-Mansouri action (2.15). Using a new field to enforce h -flatness, we find that this fieldin general appears as a source to Einstein’s equations, so that only a certain class of solutions toour action will correspond to vacuum general relativity.We define the following action on observer space: S [ A, λ, χ ] = Z O κ h ( F h ∧ F h ) ∧ τ y ( b ∧ b ∧ b )+ [tr g ( F ( λ )) + tr h ⊗ g ( χ [ e, e ]( λ ))] tr h ([ e, e ] ∧ ⋆ [ e, e ]) ∧ τ y ( b ∧ b ∧ b ) . (4.4)27ere, κ h is a general H -invariant inner product on h (as defined in (2.16)), by ‘tr’ we again meanthe Killing form, and A is the Cartan connection on observer space, which splits according to thesplitting of representations (3.12) which we repeat here: g = k ⊕ y ⊕ ~ z ⊕ z o A = Ω + b + ~e + e o In particular, b denotes the y part and e is the part valued in z = ~ z ⊕ z o . The other two fieldsinclude a bivector valued in g λ ∈ Λ T O ⊗ g (4.5)and a scalar valued in h ⊗ g : χ : O → h ⊗ g . (4.6)Since F is a g -valued 2-form, we thus have F ( λ ) ∈ g ⊗ g , and the bilinear form tr g : g ⊗ g → R gives us a scalar. Similarly, since e ∈ T ∗ O ⊗ z , the commutation relation [ z , z ] ⊆ h gives us[ e, e ] ∈ Ω ( O ) ⊗ h . Feeding in the bivector λ , we thus get [ e, e ]( λ ) ∈ h ⊗ g , and hence a scalar afterapplying tr h ⊗ g : h ⊗ g ⊗ h ⊗ g → R to χ [ e, e ]( λ ). We also need to fix a K -invariant trilinear form τ y on y , which as a representation of K is isomorphic to the adjoint representation of K ; in componentnotation, one may take this to be τ y ( b ∧ b ∧ b ) := ǫ ijk b i ∧ b j ∧ b k . As one may have expected, theaction (4.4) is only invariant under K , not H .First consider the equations of motion resulting from variation with respect to χ and λ :[ e, e ]( λ ) = 0 ,F = − tr h ( χ [ e, e ]) . (4.7)To interpret these equations, note that since A is a Cartan connection corresponding to a Cartangeometry modeled on ( G, K ), its projection on y ⊕ z defines a ‘siebenbein’, an isomorphism betweeneach tangent space to observer space and y ⊕ z . In particular, e defines a basis of ‘spacetime’ 1-forms at each point in observer space. Then tr h ([ e, e ] ∧ ⋆ [ e, e ]) is just the usual ‘spacetime’ volumeform induced by the vierbein e , and the wedge product with τ y ( b ∧ b ∧ b ) defines a volume form onobserver space.The second equation in (4.7) then says that the curvature F vanishes on any ‘boost’ vector,i.e. on any vector in the 3-dimensional subspace annihilated by the span of e . This is precisely thesubspace of vectors that are mapped into y by the Cartan connection A , and gives us the conditionof h -flatness (4.3). The remaining components of F are left arbitrary; they are given in terms of theLagrange multiplier χ . The first equation in (4.7) is a restriction on λ ; it requires its ‘spacetime’components to vanish.On the solutions of (4.7), we therefore have a spacetime M whose observer space is O , and A is a Cartan connection defining a Cartan geometry modeled on ( G, H ).To obtain the equations of motion satisfied by the Cartan connection A , one can split thevariations of (4.4) with respect to the connection A into variations with respect to its ‘spacetime’form part and the rest. Variation with respect to the ‘spacetime’ form part will give, schematically, δS = Z O δ (cid:16) κ h ( F h ∧ F h ) (cid:17) ∧ τ y ( b ∧ b ∧ b )+ δ (cid:16) [tr g ( F ( λ )) + tr h ⊗ g ( χ [ e, e ]( λ ))] (cid:17) tr h ([ e, e ] ∧ ⋆ [ e, e ]) ∧ τ y ( b ∧ b ∧ b )+ [tr g ( F ( λ )) + tr h ⊗ g ( χ [ e, e ]( λ ))] δ (cid:16) tr h ([ e, e ] ∧ ⋆ [ e, e ]) ∧ τ y ( b ∧ b ∧ b ) (cid:17) . (4.8)28he first line reproduces the equations of motion of MacDowell-Mansouri gravity, wedged with theeverywhere non-zero 3-form τ y ( b ∧ b ∧ b ). The contribution from the third line vanishes once (4.7) isimposed, since then the terms in square brackets are zero. But the variation of these terms need notbe zero on-shell; the second line in general gives a contribution to the equations of motion involvingthe covariant divergence in ‘velocity space’ directions of the mixed ‘spacetime’/‘velocity space’components of λ , which thus appears as a source in Einstein’s equations. Here we are facing well-known issues with trying to enforce nonholonomic constraints through Lagrange multipliers; seee.g. [12] for a general discussion. In order to restrict to vacuum general relativity (with cosmologicalconstant), we must assume this divergence to vanish everywhere. Physically, we could impose therequirement that the field λ on observer space is really a ‘spacetime’ field, parallelly transportedalong ‘velocity’ directions by the connection A . We did not however find an elegant way to imposethis condition directly through the action.The remaining variation is with respect to the ‘velocity’ form parts of A which gives δS = Z O κ h ( F h ∧ F h ) ∧ δ (cid:16) τ y ( b ∧ b ∧ b ) (cid:17) + δ (cid:16) tr g ( F ( λ )) (cid:17) tr h ([ e, e ] ∧ ⋆ [ e, e ]) ∧ τ y ( b ∧ b ∧ b ) ; (4.9)if we restrict to solutions where the source term involving λ vanishes, the 4-form tr h ( F h ∧ ⋆F h )vanishes, so that there is no contribution from the first term and we only get further restrictionson the field λ . Up to now, while the definition of an observer space geometry does not presuppose the existenceof spacetime, we have been largely concerned with recovering spacetime from observer space. Insection 4.1 we showed an h -flat Cartan connection on observer space gives a concept of spacetimethat all observers agree on: spacetime is the quotient of observer space obtained by integratingout the ‘boost’ distribution on observer space. But this is a rather special situation: the boostdistribution of an arbitrary observer space geometry is generically nonintegrable. We now considerthe more general situation. When the boost distribution is not integrable, spacetime itself is atbest an observer-dependent approximation. Briefly, spacetime is relative.In considering the idea of relative spacetime, we are leaving the conceptual foundations ofgeneral relativity, and we make no attempt at specific prescriptions for how the general relativisticobserver space should be deformed. Rather, we emphasize generic features of observer space, whichreduce to familiar notions in the case of an integrable boost distribution, but are just as sensiblein general.In a general observer space geometry, each observer has a set of observers perceived as being atthe same ‘spacetime point’. Definition 22.
Let ( π : P → O, A ) be an observer space Cartan geometry. The set of observers coincident to a given observer o ∈ O is the union of points along geodesics starting at o withinitial velocity v such that A ( v ) ∈ y . The notion of ‘coincidence’ thus becomes relative, analogously with the relativity of simultaneity inspecial relativity: For a nonintegrable distribution of ‘boost’ vector fields, two different observers29ight view the same third observer as coincident without viewing each other as coincident. Forobserver spaces in which the deviation from h -flatness is appropriately small, such effects would benoticeable only at very high relative velocities.Likewise, even when no absolute spacetime exists, each observer can reconstruct a local notionof ‘spacetime’, not as a quotient, but as a subspace of observer space, much as ‘space’ in specialrelativity is a particular observer-dependent slice through spacetime. Definition 23.
The local spacetime according to a given observer o ∈ O is the union of pointsin observer space along geodesics starting at o with initial velocity v such that A ( v ) ∈ z . We emphasize that this is a local definition: far away from the initial observer both topological andgeometric problems may arise. Notice that for an observer space constructed from a spacetime,such geodesics project down to geodesics in spacetime; these geodesics may not fill all of spacetime,but they fill at least some neighborhood of the spacetime point corresponding to the observer.We now turn to examples of observer spaces with no underlying notion of spacetime. As wehave emphasized, this is the generic situation, and only special observer spaces have a notion ofspacetime. However it is worth considering examples that are special in other ways.Let us consider observer spaces with no absolute spacetime, but with an absolute notion of velocity space . Such examples are motivated by the recently proposed ‘principle of relative locality’[4]. Much like in our description of observer space, in this proposal phase space is deformed in sucha way that spacetime is no longer a natural quotient but rather an observer-dependent subspace.This ‘phase space’—presumably the symplectification of some observer space not associated to anyspacetime—has so far mostly been studied mostly under the simplifying assumption that there isan absolute momentum space . Underlying this momentum space is an absolute velocity space , thespace of momenta with unit mass.To construct such examples, we choose a three-dimensional Riemannian manifold V to serve asvelocity space. To fit experimental constraints, its geometry should deviate only slightly from thatof hyperbolic space. We then proceed similarly to the construction in section 3 of observer spacesfrom spacetime. We use the Minkowski observer space model, ISO o (3 , / SO(3), since this is theonly one with an absolute velocity space. In particular, the velocity space of the model isISO o (3 , /J ∼ = SO o (3 , / SO(3) , (5.1)where J ∼ = SO(3) ⋉ R is the group of rotations for a fixed observer and spacetime translations.Using the model spacetime as a guide, we arrive at a canonical procedure for constructing anobserver space from our velocity space V . Recall that an observer space with absolute spacetimeis a sub-bundle of the tangent bundle of spacetime, with three-dimensional fiber at each spacetimepoint. Likewise, an observer space with absolute velocity space will be an extension of the tangentbundle of velocity space, with four-dimensional fiber at each point. In the Minkowski model, thisjust means we can view the observer space R , × H as a bundle over either spacetime R , orvelocity space H .More precisely, our Riemannian 3-manifold V has a canonical hyperbolic Cartan geometry,with model H/K ∼ = SO o (3 , / SO(3), built from the Levi–Civita connection and coframe, just asin section 2.2. The Cartan connection is an h -valued 1-form on the principal K bundle F V oforthonormal frames. Using the isomorphism (5.1), we extend this canonically to a 1-form A on theprincipal J bundle P = F V × K J over V , with values in the Lie algebra g ∼ = iso (3 , V : it uses the model of H asan ISO o (3 , o (3 , O := P × J R ∼ = P × J J/K is an affine extension of the tangent bundle of V , the projection P → O is a principal K bundle,and the Cartan connection A gives an observer space geometry on O (cf. lemma 10). This observerspace is j -flat by construction and has absolute velocity space P/ j ∼ = V . However, ‘spacetime’ for aparticular observer is the fiber in O over that observer’s velocity; there is in general no canonicalway to identify these notions of spacetime.This procedure suggests a way to construct further examples: whenever a particular modelobserver space has some particular absolute feature, we can consider Cartan-geometric deformationswhich preserve that feature. For example, in the de Sitter models, there is an absolute notion of‘conformal infinity’. This is given by an SO o (4 , o (4 , As indicated in the introduction, one of our motivations has been to relate covariant and canonicalapproaches to gravity. The discord between these two pictures has led some physicists, beginningwith Dirac (see [20], p. 290) to doubt the ultimate significance of the spacetime picture:This result has led me to doubt how fundamental the four-dimensional requirement inphysics is. . . . [I]t seems that four-dimensional symmetry is not of overriding importance,since the description of nature sometimes gets simpler when one departs of it.This doubt has perhaps been carried furthest by Barbour, whose work has culminated in an al-ternative to general relativity in which only space is fundamental and spacetime emerges from thetheory itself [8]. Other theories under current investigation, including the anisotropic gravity ofHoˇrava [18], and causal dynamical triangulations [3] start from a spacetime picture, but introducea preferred spatial foliation, restoring an absolute notion of simultaneity.In this paper, we have argued for a complementary approach: rather than assuming any funda-mental concept of ‘space’, we take the notion of an observer seriously, as ontologically prior to eitherspace or spacetime. As observers, we do not experience spacetime directly. From our collectiveexperience, we notice that: • Each of us can organize the objects near us by describing their positions using three coordi-nates. In brief, each observer sees ‘space’ as three-dimensional. • We each experience things changing in time. Each observer sees ‘time’ as one-dimensional.31
We can relate other observers to us according to their (relative) velocity. We see ‘velocityspace’—the space of all observers coincident with us—as three-dimensional.Admittedly, these do not appear to be independent, since we use our notions of space and timeto measure velocities. However, measuring a velocity requires nonlocal measurements, and evenspecial relativity shows that the ‘obvious’ relationship among space, time, and velocity is onlyapproximate. Most important is relativity’s lesson that we disagree on these notions of space, time,and velocity space. If we suppose that an observer is uniquely determined by its perceived notionsof these three spaces, then we need, a priori, some seven-dimensional space of observers. It is notobvious that this can be reduced precisely to some 4-dimensional ‘spacetime’.Observer space provides a new way of understanding the geometry of general relativity. Wehave also argued that the flexibility of this new perspective provides a natural setting for studyingproposed modifications of general relativity, and their relationships. This leaves much to be done,both on the subject of observer space itself, and on applications.First, it will be interesting to study more particular examples of observer space, both thosearising from solutions of general relativity and observer spaces without an underlying spacetime.For instance, one immediate question is how the existence of an event horizon in the spacetime ofa black hole, or of a cosmological horizon in an expanding universe, is encoded in the geometry ofobserver space.Second, while lightlike particles play an obviously important role in general relativity, we haveso far mostly ignored the extension of observer space that includes them, as described at the endof section 3.1. The action of the group H on hyperbolic space H ∼ = H/K can be extended to anaction on the compactification H , with two orbits: H itself and the boundary. The respectivestabilizers are K and K ′ ∼ = SIM(2), the stabilizer of a light ray through the origin of Minkowskispacetime. A general extended observer space geometry should include not only Cartan geometrymodeled on G/K , as we have defined in this paper, but also Cartan geometry modeled on
G/K ′ ,describing the boundary of observer space. In the same way that the G/K geometry is related tostandard Hamiltonian methods, as described in section 3.5, the
G/K ′ geometry should presumablybe related to light-front methods [16], in which the splitting of fields is done from the perspective of‘lightlike observers’. A deeper study of this geometry may reveal connections to other theories, suchas the ‘very special relativity’ proposal, which uses SIM(2) as the fundamental spacetime symmetrygroup [10].To discuss possibly observable Lorentz violation, it will be useful to couple matter fields togravity on observer space. For gravity itself, as we have mentioned in the introduction, thereexist several competing ideas that are not covariant under changes of observer. The perspective ofobserver space could allow studying those ideas from a new geometric angle.On the phenomenological side, it would be interesting to tighten the relationship between ob-server space and the relative locality proposal. While the two seem clearly related, and both lead tothe idea that locality, or coincidence, is a relative notion, the two frameworks have different start-ing points. In particular, the idea behind relative locality involves building up spacetime geometryfrom the interactions of particles in the universal velocity (or rather momentum) space.The idea that spacetime geometry is ‘velocity-dependent’ or ‘momentum-dependent’ appears inseveral approaches going beyond usual Lorentzian geometry as the basic framework for gravitationalphysics, some of which might be related to observer space Cartan geometries. The most obviousexample is Finsler geometry, sometimes referred to as ‘Riemannian geometry without the quadraticrestriction’ [9]—one replaces the metric by general length functional , whose second derivative with32espect to velocities can be viewed as a ‘velocity-dependent metric’. Observer spaces naturallydescribe a ‘velocity-dependent’ geometry, although it is not completely obvious how to relate ourconnection-based approach to an essentially metric-based approach.Relating our framework to possible predictions for physical measurements will involve clarifyingsome interpretational and foundational issues. Measurements should be made by inertial observersmoving on certain geodesics on observer space (as discussed in section 3.1) and compared to thosemade by other observers. We then have to understand why the assumption that spacetime exists iscompatible with our experience to such excellent precision. We saw that it is not obvious to recoveran underlying spacetime from observer space when trying to give an action for general relativity onobserver space. Similarly, we must explain why matter fields are not arbitrary functions on observerspace, but to a good approximation just fields on ‘spacetime’. We leave all of this to future work. Acknowledgments
SG thanks the Institute for Quantum Gravity of the University of Erlangen-N¨urnberg for supportinga visit during which some of this work was carried out. Likewise, DW thanks the Perimeter Institutefor Theoretical Physics for supporting a visit, and especially Jim Dolan and Josh Willis for valuablediscussions that helped lead up to our study of observer space. Research at Perimeter Institute issupported by the Government of Canada through Industry Canada and by the Province of Ontariothrough the Ministry of Research & Innovation.
A Notation
Here are the letters used in this paper for various things. M spacetime O observer space x point in Mo point in O T fake tangent bundle P fake orthonormal frame bundle O fake observer space Z model spacetime z point in Z z T z Z References [1] D. V. Alekseevsky and P. W. Michor, Differential geometry of g -manifolds, Differential Geom.Appl. (1995) 371–403, arXiv:math/9309214.[2] D. V. Alekseevsky and P. W. Michor, Differential geometry of Cartan connections, Publ. Math.Debrecen (1995) 349–375, arXiv:math/9412232.[3] J. Ambjorn, J. Jurkiewicz, and R. Loll, Reconstructing the universe, Phys. Rev. D (2005)064014, arXiv:hep-th/0505154. 334] G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman, and L. Smolin, The principle of relativelocality, Phys. Rev. D (2011) 084010, arXiv:1101.0931.[5] V. I. Arnold, Mathematical Methods of Classical Mechanics , 2ed. (Springer, New York, 1989).[6] J. C. Baez and D. K. Wise, Teleparallel gravity as a higher gauge theory, arXiv:1204.4339.[7] J. F. Barbero G., Real Ashtekar variables for Lorentzian signature space times,
Phys. Rev. D , 5507–5510 (1995). arXiv:gr-qc/9410014.[8] J. Barbour, Shape dynamics, an introduction, in Quantum Field Theory and Gravity (Springer,Basel, 2012), arXiv:1105.0183; H. Gomes, S. Gryb, and T. Koslowski, Einstein gravity as a 3Dconformally invariant theory,
Class. Quant. Grav. , 045005 (2011), arXiv:1010.2481.[9] S. S. Chern, Finsler geometry is just Riemannian geometry without the quadratic restriction, Notices AMS (1996) 959–63.[10] A. G. Cohen and S. L. Glashow, Very special relativity, Phys. Rev. Lett. (2006) 021601.[11] H. M. Dida and A. Ikemakhen, A class of metrics on tangent bundles of pseudo-Riemannianmanifolds, Arch. Math. (Brno) (2011) 293–308.[12] M. R. Flannery, The enigma of nonholonomic constraints, Am. J. Phys. (2005) 265-272.[13] G. W. Gibbons and S. Gielen, Deformed general relativity and torsion, Class. Quant. Grav. (2009) 135005, arXiv:0902.2001.[14] S. Gielen and D. K. Wise, Spontaneously broken Lorentz symmetry for Hamiltonian gravity, Phys. Rev. D (2012) 104013, arXiv:1111.7195.[15] S. Gielen and D. K. Wise, Linking covariant and canonical general relativity via local observers, Gen. Relativ. Grav. (2012) 3103-3109, arXiv:1206.0658.[16] A. Harindranath, An introduction to light-front dynamics for pedestrians, in Light-Front Quan-tization and Non-Perturbative QCD , edited by J. P. Vary and F. Woelz (International Instituteof Theoretical and Applied Physics, Ames, 1997), arXiv:hep-ph/9612244.[17] S. Helgason,
Differential geometry, Lie groups, and symmetric spaces (Academic Press, NewYork, 1978).[18] P. Hoˇrava, Quantum gravity at a Lifshitz point,
Phys. Rev. D (2009) 084008,arXiv:0901.3775.[19] T. W. B. Kibble, Lorentz invariance and the gravitational field, J. Math. Phys. (1961)212-221.[20] H. Kragh, Dirac: A Scientific Biography (Cambridge University Press, Cambridge, 2005).[21] S. W. MacDowell and F. Mansouri, Unified geometric theory of gravity and supergravity,
Phys.Rev. Lett. Topics in Differential Geometry (Graduate Studies in Mathematics Vol. 93) (American Mathematical Society, 2008).[24] R. S. Palais,
A global formulation of the Lie theory of transformation groups , Memoirs of theAmerican Mathematical Society, Vol. 22 (American Mathematical Society, 1957).[25] R. W. Sharpe,
Differential Geometry: Cartan’s Generalization of Klein’s Erlangen Program (Springer, 1997).[26] K. S. Stelle and P. C. West, Spontaneously broken de Sitter symmetry and the gravitationalholonomy group,
Phys. Rev. D (1980) 1466-1488.[27] D. K. Wise, MacDowell-Mansouri gravity and Cartan geometry, Class. Quant. Grav. (2010)155010, arXiv:gr-qc/0611154.[28] D. K. Wise, Symmetric space Cartan connections and gravity in three and four dimensions, SIGMA (2009) 080, arXiv:0904.1738.[29] D. K. Wise, The geometric role of symmetry breaking in gravity, J. Phys.: Conf. Ser.360