Curved spacetimes with local κ-Poincaré dispersion relation
Leonardo Barcaroli, Lukas K. Brunkhorst, Giulia Gubitosi, Niccoló Loret, Christian Pfeifer
aa r X i v : . [ g r- q c ] O c t Curved spacetimes with local κ -Poincar´e dispersion relation Leonardo Barcaroli, ∗ Lukas K. Brunkhorst,
2, 3, † GiuliaGubitosi, ‡ Niccol´o Loret, § and Christian Pfeifer
5, 3, 2, ¶ Dipartimento di Fisica, Universit`a ”La Sapienza” andSez. Roma1 INFN, P.le A. Moro 2, 00185 Roma, Italy Center of Applied Space Technology and Microgravity (ZARM),University of Bremen, Am Fallturm, 28359 Bremen, Germany Institute for Theoretical Physics, Universit¨at Hannover,Appelstrasse 2, 30167 Hannover, Germany Theoretical Physics, Blackett Laboratory,Imperial College, London SW7 2AZ, United Kingdom. Laboratory of Theoretical Physics, Institute of Physics,University of Tartu, W. Ostwaldi 1, 50411 Tartu, Estonia
We use our previously developed identification of dispersion relations with Hamilton func-tions on phase space to locally implement the κ -Poincar´e dispersion relation in the momen-tum spaces at each point of a generic curved spacetime. We use this general constructionto build the most general Hamiltonian compatible with spherical symmetry and the Plank-scale-deformed one such that in the local frame it reproduces the κ -Poincar´e dispersionrelation. Specializing to Planck-scale-deformed Schwarzschild geometry, we find that thephoton sphere around a black hole becomes a thick shell since photons of different energywill orbit the black hole on circular orbits at different altitudes. We also compute the redshiftof a photon between different observers at rest, finding that there is a Planck-scale correctionto the usual redshift only if the observers detecting the photon have different masses. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] I. INTRODUCTION
The κ -Poincar´e algebra of symmetries, a quantum deformation of the Poincar´e algebra [1–3], isone of the most intensively studied phenomenological models relevant for quantum gravity research.This is mostly because it provides a mathematically consistent example of a relativistic theory withtwo invariants (the speed of light and the Planck length or energy) and it produces potentiallyobservable effects, such as an energy-dependent propagation velocity of massless particles whichmay be measured in the observation of γ -ray bursts at cosmological distances (see [4] and referencestherein). Geometrically, the motion of a particle admitting κ -Poincar´e symmetry can be interpretedas happening on a flat spacetime manifold with a curved momentum space enjoying de Sittersymmetry, the Planck scale being related to the curvature of the momentum space itself [5, 6].As already discussed in [7], in order to make the κ -Poincar´e model more suited to describequantum gravity effects in the cosmological framework, it is necessary to implement the κ -Poincar´edispersion relation on generally curved spacetimes. This entails building a model of intertwinedspacetime and momentum space such that in a local frame one recovers the flat spacetime κ -Poincar´e dispersion relation. In the local frame the κ -Lorentz symmetries hold, i.e. the κ -Poincar´esymmetries except translations. By now several steps towards this goal have been achieved. Theso called q -de Sitter dispersion relation implements the κ -Poincar´e dispersion relation on de Sitterspacetime geometry [8] and is associated to a quantum deformation of the de Sitter algebra ofspacetime symmetries. A first approach to a homogeneous and isotropic spacetime with κ -Poincar´edispersion relation was presented in [9] by gluing together slices of its de Sitter spacetime realization.Recently we could go even further. In [10] we interpreted dispersion relations as level sets ofHamilton functions on the cotangent bundle of a spacetime manifold and developed a precise notionof symmetries of dispersion relations. This enabled us to construct the most general homogeneousand isotropic dispersion relation and to identify what we called the qFLRW dispersion relation [7].It is constructed such that in a local frame the dispersion relation reduces to the κ -Poincar´e one,and, when the Planck-scale deformation vanishes it describes the motion of a relativistic particleon Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) spacetime.Here we show that in fact the κ -Poincar´e dispersion relation (and the associated κ -Lorentzsymmetries) can be realized locally on a general curved spacetime. Specifically, in section II weconstruct a Planck-scale-modified Hamiltonian on a general curved spacetime, such that at everypoint of spacetime there exists a basis of the cotangent space such that the covariantly-defineddispersion relation takes the standard κ -Poincar´e form. This characterization is similar to thefact that on every Lorentzian manifold there exist frames of the Lorentzian spacetime metric. Inthese frames the dispersion relation of point particles on curved spacetime takes the same form ason flat Minkowski spacetime. In II C we use Hamilton equations to work out the motion in phasespace of a particle with such Planck-scale-deformed dispersion relation on generic curved spacetimeand in II D we derive a general formula for the redshift between two observers. In section III wespecialize our model to the case of a spherically symmetric spacetime, presenting the most generalPlanck-scale-deformed dispersion relation compatible with these symmetries which reduces to the κ -Poincar´e one in the local frame. It contains four free functions depending on the time and radialcoordinate satisfying one algebraic constraint. Two of these functions are fixed by the undeformedmetric spacetime geometry, as we show in III B, where we deal with the Schwarzschild case. In III Cwe work out the Hamilton equations for the Schwarzschild case and finally in III D we computesome observable effects. In particular, we show that the radius of the circular photon orbits arounda black hole depend on the photon’s energy and that the observed redshift of a photon movingradially in the Schwarzschild geometry is modified with respect to the standard case only if thetwo observers detecting the photon have different masses.During this article we use the following notational conventions: Indices a, b, c, ... and µ, ν, .. runfrom 0 to 3. Latin indices denote tensor components in manifold induced coordinates, greek indicesdenote frame induced coordinates ( x, p ) ∼ P = p a dx a ∈ T ∗ x M of the cotangent bundle. Tensorialobjects on spacetime, like a spacetime metric g or a vector field Z are often interpreted as functionon the cotangent bundle g − ( p, p ) or Z ( p ) which are defined as these tensors action on the 1-form P : g − ( p, p ) = g ab p a p b , Z ( p ) = Z a p a . The signature convention for the spacetime metric we use is ( − , + , + , +). The manifold inducedcoordinates on the cotangent bundle satisfy the canonical Poisson relations { x a , p b } = δ ab , all othervanish. II. κ -POINCAR´E MOMENTUM SPACES ON CURVED SPACETIMES The general framework of Hamilton geometry applied to Planck-scale-modified dispersion rela-tions is discussed in detail in our previous publications [7, 10]. Here we only introduce the basicnotions, recalling the connection between dispersion relations and level sets of Hamilton functionson the cotangent bundle of a spacetime manifold. Subsequently, we write down the Hamiltonfunction which implements the κ -Poincar´e dispersion relation for free particles at every point of ageneric spacetime and introduce the notion of κ -Lorentzian symmetry. We then briefly discuss theequations of motion induced by such Hamiltonian and we compute the redshift between any twoobservers. A. Dispersion relations as level sets of Hamilton functions
In general relativity local Lorentz invariance is encoded in terms of symmetry transformationson the tangent, respectively cotangent spaces of spacetime. In particular this symmetry manifestsitself in the local invariance of the dispersion relation of fundamental point particles, which is givenby g ab ( x ) p a p b = − m , (1)where m is the invariant mass parameter. On each point x on spacetime, this dispersion relationis invariant under Lorentz transformations of the momenta p in the following sense: There existframes A of the metric g such that H g ( x, p ) ≡ g ab ( x ) p a p b = η µν A aµ ( x ) A bν ( x ) p a p b = η µν p µ p ν ≡ H η ( x, p ) . (2)Lorentz invariance manifests itself in the fact that the frame matrix A is not unique. In fact everytransformation ˆ A which is constructed from A via the application of a Lorentz transformation doesnot change the value of the function H g ( x, p ).In previous work [7, 10] we have demonstrated that one can interpret the dispersion relationof point particles as level sets of a Hamilton function H ( x, p ) on the cotangent bundle of space-time. Then the geometry of the particle’s phase space, i.e. the intertwined geometry of spacetimeand the point particle’s momentum space, can be derived from the Hamilton function. Here weschematically recall the most important features of the Hamiltonian construction of the phase spacegeometry (further details and explicit examples can be found in [7, 10]): • The Hamilton equations of motion are the autoparallel equations of the unique torsion-freeCartan non-linear connection. For non-homogeneous Hamiltonians they include a force-likesource term. • The Cartan non-linear connection, uniquely derived from the Hamiltonian, splits the tangentspaces of phase space covariantly in directions along spacetime and along momentum space. • Canonical linear connections, uniquely determined by the Hamiltonian and the non-linearconnection, define the curvature of spacetime and of momentum space, both of which ingeneral depend on both spacetime coordinates and momenta. • For the Hamiltonian H g ( x, p ) = g ab ( x ) p a p b one obtains the usual Lorentzian metric geometryof spacetime with a flat momentum space.A dispersion relation thus gives us access to observable predictions by determining the motion ofpoint particles obeying the dispersion relation through the Hamilton equations of motion, encodesthe local phase space symmetry in terms of its local invariances and determines the geometry ofphase space whose autoparallels coincide with the Hamilton equations of motion. As mentionedin the introduction, in this article we aim for observable predictions from a modification of thelocal Lorentz invariant point particle Hamilton function in general relativity. Specifically, we willconstruct a covariant Hamilton function on a generic curved spacetime whose local symmetrytransformations are generated by the κ -Poincar´e algebra [11]. B. The locally κ -Poincar´e Hamiltonian The κ -Poincar´e dispersion relation [11] can be represented as the level sets of the Hamiltonfunction H κ ( x, p ) = − ℓ sinh (cid:18) ℓ p t (cid:19) + e ℓp t ~p , (3)where p t and ~p are, respectively, the particle’s energy and spatial momentum and ℓ = κ − is thedeformation parameter, such that for ℓ = 0 the Hamiltonian reduces to the familiar expression ofspecial relativity H ( x, p ) = − p t + ~p = η ab p a p b . (4)The κ -Poincar´e Hamiltonian (3) is the κ -deformation of the flat Minkowski spacetime Hamilto-nian (4). The idea is that the Hamiltonian (3) determines the effective motion of point particlesin a semiclassical regime of quantum gravity, as discussed in [10]. Hamiltons equations of motionof both imply that all particles move force-free on straight lines in one and the same coordinatesystem, so both yield particle motion on flat spacetime. The difference is that if one derives themomentum space curvature of (3) and (4) according to the framework of Hamilton geometry out-line in the previous section, one finds a non-trivial momentum space curvature in the κ -Minkowskicase, but a vanishing momentum space curvature for Minkowski spacetime.As mentioned in the introduction, the phenomenological implications of this κ -deformed Hamil-tonian have been widely studied in the literature, however most of the observable effects wouldbe mostly apparent in a cosmological setting or in general in regimes where spacetime curvaturecan not be neglected. This provides motivation to look for ways to implement the κ -deformedHamiltonian on an arbitrarily curved spacetime ( M, g ), constructing a Hamilton function whichlocally takes the form (3), so that it is locally κ -Poincar´e invariant in the same sense as generalrelativity is locally Lorentz invariant. In [7] we focussed on homogeneous and isotropic FLRWspacetimes. Here we study the general case.Let ( M, g ) be a globally hyperbolic Lorentzian spacetime and let Z be a normalized globally-defined timelike vector field on ( M, g ), which can be interpreted as function Z ( p ) on the cotangentbundle of spacetime T ∗ M , g ( Z, Z ) ≡ g ab ( x ) Z a ( x ) Z b ( x ) = − , Z ( p ) = Z a ( x ) p a . (5)The κ -Poincar´e deformation of ( M, g ) is defined by changing the Hamiltonian H g to H Zg definedby H Zg ( x, p ) ≡ − ℓ sinh (cid:18) ℓ Z ( p ) (cid:19) + e ℓZ ( p ) ( g − ( p, p ) + Z ( p ) ) . (6)We label the deformed Hamiltonian by the vector field Z since in general different choices of Z leadto different κ -deformed Hamiltonians. In section III, where we discuss the spherically symmetric κ -Poincar´e phase space, we will see this freedom explicitly. This Hamiltonian can be consideredas κ -deformation of a local Lorentz invariant spacetime to a local κ -Lorentz invariant one and, inaddition, also be derived from a modified theory of electrodynamics as we discuss in Appendix A.Performing a power-series expansion in ℓ we find H Zg ( x, p ) = g − ( p, p ) + ℓZ ( p )( g − ( p, p ) + Z ( p ) ) + O ( ℓ ) . (7)Thus the zeroth order of H Zg is identical to the Hamilton function which determines the particlemotion and the geometry of spacetime in general relativity.It can be shown that the Hamiltonian (6) is locally κ -Poincar´e invariant via the followingargument. Since Z is a unit-timelike vector, there exists a frame A of the metric g such that A a ∂ a = Z , thus Z ( p ) = A a p a = p . Since A is a frame, we can express the metric square of themomenta in this frame as g ab p a p b = η µν p µ p ν = − p + ~ p . (8)Thus with respect to this frame the κ -Poincar´e (bicrossproduct basis) Hamiltonian we constructedbecomes H Zg ( x, p ) = − ℓ sinh (cid:18) ℓ p (cid:19) + e ℓp t ~ p = H Zη ( x, p ( x )) , (9)which is invariant under the transformations generated by the κ -Poincar´e algebra.The frame A induces a local and linear transformation on the momenta such that locally, atevery x ∈ M the Hamiltonian (6) takes the form (3). Again, as in the metric general-relativisticcase, this transformation is not unique. It can be combined with a κ -Poincar´e transformation andthe value of H Zg will not change. Thus we conclude that H Zg is locally κ -Poincar´e invariant in thesame sense as the metric Hamiltonian H g is local Lorentz invariant. To be precise observe that thislocal invariance on curved spacetimes excludes the translations from the full κ -Poincar´e algebraas transformations. To have a nomenclature for the allowed transformations available we call theremaining elements of the algebra, i.e. the κ -Poincar´e boosts and rotations, the κ -Lorentz algebra.The intertwined geometry of the smooth point particle phase space, i.e. its linear connectionsand curvatures, can now be derived according to the framework developed in [10]. This derivationis beyond the scope of this article which aims for phenomenological implications of the modifieddispersion relation induced by the Hamiltonian (6). We also do not study in detail the consequenceson this geometric picture of general non-linear momentum transformations. We only focus onthe non-linear momentum transformations that are symmetries of the model, i.e. that leave theHamiltonian invariant, and these are the transformations generated by the κ -Lorentz algebra.Before we continue with our analysis we would like to make a remark concerning different basesof the κ -Poincar´e algebra. The same line of argument applied above can be generalized to differentbases of the κ -Poincar´e Hopf algebra, which can be obtained by a nonlinear redefinition of thetranslation generators. Different bases have in general different Casimir operators and thus theyare associated to different Hamiltonians. In particular, there exists a basis with a classical Poincar´ealgebra sector and undeformed Hamiltonian. However such a basis is characterized by nontrivialcoproducts of the translation generators, that imply nontrivial composition law of momenta inparticles’ interactions [6]. Thus such a basis formalizes a physical model with undeformed singleparticle dynamics and nontrivial interaction vertices. Since in this paper we focus our phenomeno-logical analysis on the motion of a free single-particle in a Schwarzschild geometry with deformedlocal symmetries, it makes sense to specialize our investigation on the κ -Poincar´e bicrossproductbasis. C. Particle motion
Having implemented the κ -Poincar´e dispersion relation locally on a general curved spacetime aslevel sets of the Hamilton function (6), we study the particle motion in phase space which is deter-mined by the Hamilton equations of motion derived from (6). These are eight first-order ordinarydifferential equations which are equivalent to four second order ordinary differential equations, theEuler-Lagrange equations of the Lagrangian corresponding to the Hamiltonian in consideration.The transformation of the Hamiltonian representation of the theory to its Lagrangian counterpartis the starting point for finding a Finsler geometric formulation of the κ -deformed geometry ofspacetime, which is investigated in several articles [12–14].The Hamilton equations of motion of the general κ -deformed Hamiltonian imply the followingrelation between velocities and momenta:˙ x a = ¯ ∂ a H Zg (10)= Z a (cid:20) − ℓ sinh (cid:0) ℓZ ( p ) (cid:1) + ℓe ℓZ ( p ) ( g − ( p, p ) + Z ( p ) ) + 2 e ℓZ ( p ) Z ( p ) (cid:21) + 2 e ℓZ ( p ) g ab p b , while the evolution of momenta is given by˙ p a = − ∂ a H Zg (11)= p q ∂ a Z q (cid:20) ℓ sinh (cid:0) ℓZ ( p ) (cid:1) − ℓe ℓZ ( p ) ( g − ( p, p ) + Z ( p ) ) − e ℓZ ( p ) Z ( p ) (cid:21) − e ℓZ ( p ) p b p c ∂ a g bc . The latter can be written in an explicitly covariant form with respect to manifold induced coordi-nate transformation by introducing the Levi-Civita connection of the Lorentzian metric g :˙ p a = p q ∇ a Z q (cid:20) ℓ sinh (cid:0) ℓZ ( p ) (cid:1) − ℓe ℓZ ( p ) ( g − ( p, p ) + Z ( p ) ) − e ℓZ ( p ) Z ( p ) (cid:21) + 2 e ℓZ ( p ) p c p b Γ cba + p q Γ qab Z b (cid:20) ℓ sinh (cid:0) ℓZ ( p ) (cid:1) − ℓe ℓZ ( p ) ( g − ( p, p ) + Z ( p ) ) − e ℓZ ( p ) Z ( p ) (cid:21) . Observe that this spacetime metric is used here as an available mathematical tool to check thecovariance of the equations of motion explicitly. It is not what fundamentally determines thegeometry of spacetime, momentum space nor the motion of particles (in fact we can not reallyseparate spacetime and momentum space within the phase space). The fundamental ingredient isthe Hamilton function itself and when the Planck-scale corrections are introduced spacetime andmomentum space are intertwined so that it is not possible to talk about a spacetime metric on itsown.Reshuffling the terms in the above equations we find p q ∇ a Z q (cid:20) ℓ sinh (cid:18) ℓZ ( p ) (cid:19) − ℓe ℓZ ( p ) ( g − ( p, p ) + Z ( p ) ) − e ℓZ ( p ) Z ( p ) (cid:21) = (12)˙ p a − e ℓZ ( p ) p c p b Γ cba − p q Γ qab Z b (cid:20) ℓ sinh (cid:18) ℓZ ( p ) (cid:19) − ℓe ℓZ ( p ) ( g − ( p, p ) + Z ( p ) ) − e ℓZ ( p ) Z ( p ) (cid:21) . Since the Hamilton equations of motion are covariant, i.e. behave tensorial under manifold inducedcoordinate changes, and since the left hand side of these equations are covariant as well, the righthand side must be covariant. For ℓ → x a = 2 g ab p b , ˙ p a − p c p b Γ cba = 0 ⇒ ¨ x a + Γ abc ˙ x b ˙ x c = ∇ ˙ x ˙ x = 0 . (13)We do not transform the Hamilton equations of motion of the κ -deformed Hamiltonian into theirEuler-Lagrange form explicitly since this is a lengthy calculation not needed for the scope of thisarticle. If needed they can be directly calculated from the Lagrangian corresponding to the κ -deformed Hamiltonian. Surprisingly it is not too difficult to derive the Legendre transformation L ( x, ˙ x ) = ˙ x a p a ( x, ˙ x ) − H Zg ( x, p ( x, ˙ x )) of H Zg explicitly. The calculations are discussed in ap-pendix B and yield ˙ x a p a = g ( ˙ x, ˙ x ) + g ( ˙ x, Z ) ℓg ( ˙ x, Z ) ± p ℓ g ( ˙ x, Z ) + ℓ g ( ˙ x, ˙ x ) + 4 − g ( ˙ x, Z ) ℓ ln (cid:18)
12 ( ℓ ( g ( ˙ x, Z ) ± p ℓ g ( ˙ x, Z ) + ℓ g ( ˙ x, ˙ x ) + 4) (cid:19) (14) H Zg ( x, p ( x, ˙ x )) = 2 ℓ − g ( ˙ x, Z ) ℓ − ℓ ℓg ( ˙ x, Z ) ± p ℓ g ( ˙ x, Z ) + ℓ g ( ˙ x, ˙ x ) + 4) . (15)Even though these expressions are quite involved one can calculate their ℓ → x a p a = 12 g ( ˙ x, ˙ x ) , H Zg ( x, p ( x, ˙ x )) = 14 g ( ˙ x, ˙ x ) ⇒ L ( x, ˙ x ) = 14 g ( ˙ x, ˙ x ) (16)as expected.The main qualitative difference between the Hamilton equations in general relativity and theones for the κ -deformed Hamiltonian is that in general relativity the ˙ p a equation is only sourcedby a term proportional to the Christoffel symbols, while in the κ -deformed case there are extrasource terms. This means that, unlike in general relativity, there exists no coordinate systemaround every point q of spacetime such that ˙ p a = 0 at q (i.e. it is not possible to define normalcoordinates around every point). This nicely demonstrates what we already discussed in Theorem2 of [10], namely that for non-homogeneous Hamiltonians a force-like term appears in the Hamiltonequations dragging particles away from auto-parallel motion.0 D. Observers and redshift
One prominent feature of physics on curved spacetimes is the gravitational redshift. Followingour previous analysis done for homogeneous and isotropic models [7], here we investigate how theamount of redshift between two observers in a generic curved spacetime is influenced by the κ -deformation. In order to do so we need a notion of the frequency ν σ ( γ ) of a light ray γ measuredby an observer σ .A light ray is a solution γ ( τ ) = ( x γ ( τ ) , p γ ( τ )) of the Hamilton equations of motion which satisfies H ( γ ) = 0. An observer is a curve σ ( λ ) = ( x σ ( λ ) , p σ ( λ )) to which a tangent vector is associatedvia ˙ x aσ = ¯ ∂ a H ( σ ) and which satisfies the following properties:1. The energy of an observer is real for all masses and spatial momenta, i.e. H ( σ ) < H ( σ ) = − m σ = constant ,These conditions are the same conditions observers satisfy in general relativity, which can be real-ized in the ℓ → p σ andthe observer’s tangent ˙ x σ is given via the first Hamilton equation of motion. So the observer is alsosubject to the κ -deformed dynamics, in contrast to other models considered [9, 15, 16], in which theobserver is formalized just as a low-energetic (classical) worldline. In our case, however, since weare describing deformations to the particles’ dynamics in a Schwarzschild-like framework later, themass of the observer plays a crucial role, being proportional to the influence of the κ -deformationdetected in the observer’s reference frame, as we will see explicitly in III D 2.The frequency an observer associates to the light ray is given by ν σ ( γ ) = p γa ˙ x aσ m σ = p γa ¯ ∂ a H ( σ ) m σ . (17)Surely this expression only makes sense when the light ray and the observer intersect at a certainpoint on spacetime.For the κ -Poincar´e Hamiltonian ˙ x is displayed in (10) so ν σ ( γ ) m σ = Z ( p γ ) (cid:20) − ℓ sinh (cid:18) ℓZ ( p σ ) (cid:19) + ℓe ℓZ ( p σ ) ( g − ( p σ , p σ ) + Z ( p σ ) )+ 2 e ℓZ ( p σ ) Z ( p σ ) (cid:21) + e ℓZ ( p σ ) g − ( p σ , p γ ) (18)1with correct classical limit ℓ → ν σ ( γ ) = 2 m σ g − ( p σ , p γ ) . (19)We demanded that H ( σ ) = − m σ is constant thus we can use − ℓ sinh (cid:18) ℓ Z ( p σ ) (cid:19) + e ℓZ ( p σ ) ( g − ( p σ , p σ ) + Z ( p σ ) ) = − m σ (20)to simplify the frequency to ν σ ( γ ) = 1 m σ Z ( p γ ) (cid:20) ℓ e − ℓZ ( p σ ) − ℓ − ℓm σ + 2 e ℓZ ( p σ ) Z ( p σ ) (cid:21) + e ℓZ ( p σ ) m σ g − ( p σ , p γ ) . (21)This last expression can easily be used to calculate the redshift between two different observers σ and σ who intersect the light ray at different spacetime positions z + 1 = ν σ ( γ ) ν σ ( γ ) . (22)In section III D 2 we will use this formula to derive the deformation of the gravitational redshift ina κ -deformation of Schwarzschild geometry. III. SPHERICALLY SYMMETRIC κ -DEFORMED PHASE SPACE In our previous article [10] we gave a detailed account of the notion of symmetry in Hamiltongeometry. Summarizing, a Hamiltonian H ( x, p ) is invariant under the action of certain diffeomor-phisms Φ on phase space if the vector field X Φ which induces this diffeomorphism annihilates theHamiltonian X Φ ( H ) = 0 . (23)Particularly interesting are those diffeomorphisms of phase space which are induced by a diffeo-morphism of the spacetime manifold. In this case the symmetry condition becomes X C ( H ) ≡ ( ξ a ∂ a − p q ∂ a ξ q ¯ ∂ a ) H = 0 , (24)where X = ξ a ( x ) ∂ a is the vector field which induces the diffeomorphism of spacetime. The detailsof the derivation of this symmetry condition can be found in [10], while an application in thecontext of homogeneous and isotropic geometries is discussed in [7]. In the following we use thisconstruction to define general spherically symmetric Hamiltonians.2 A. The general case
In order to study spherically symmetric phase spaces it is most convenient to use sphericalcoordinates ( t, r, θ, φ, p t , p r , p θ , p φ ). The generators of rotations of spacetime are X = sin φ ∂ θ + cot θ cos φ ∂ φ (25) X = − cos φ ∂ θ + cot θ sin φ ∂ φ (26) X = ∂ φ . (27)Their complete lifts are displayed in the appendix C. Evaluating equation (24) we find, with thesame techniques already used in the homogeneous and isotropic case [7], that the most generalspherically symmetric Hamiltonian must take the form H ( x, p ) = H ( t, p t , r, p r , w ( θ, p θ , p φ )) with w = p θ + 1sin θ p φ . (28)As one could expect, the form of the Hamiltonian is less constrained compared to the homogeneousand isotropic case [7]. This freedom translates to the appearance of several free functions in themost general third-order polynomial expansion around the standard metric dispersion relation: H ( x, p ) = − A ( t, r ) p t + C ( t, r ) p t p r + B ( t, r ) p r + R ( t, r ) w (29)+ ℓ (cid:0) D ( t, r ) p t + E ( t, r ) p t p r + F ( t, r ) p t p r + G ( t, r ) p r + J ( t, r ) p t w + K ( t, r ) p r w (cid:1) + O ( ℓ ) . Since we are interested in building a Hamiltonian that reduces to the κ -Poincar´e one in thelocal frame we will have a reduced freedom compared to this general case. In particular, wewant to construct a Hamiltonian that, besides having spherical symmetry, can be written in theform (6). The general κ -deformed Hamiltonian (6) is built out of two elements: a spacetime metricterm g − ( p, p ) and a vector field term Z ( p ). The mostly considered spherically-symmetric metricterm, which contains all spherically symmetric vacuum solutions of the Einstein equations, can bewritten, after an appropriate choice of coordinates, as g − ( p, p ) = − a ( t, r ) p t + b ( t, r ) p r + 1 r w . (30)On the other hand, in order to respect spherical symmetry, the vector field term must take theform Z ( p ) = c ( t, r ) p t + d ( t, r ) p r , (31) The most general version of the term would be g − ( p, p ) = − a ( t, r ) p t + c ( t, r ) p t p r + b ( t, r ) p r + d ( t, r ) w . In casethe gradient of d ( r, t ) is spacelike or timelike the form we displayed can be achieved, however in case the gradientof d ( r, t ) is vanishing or a null vector, this may not be possible [17]. g ( Z, Z ) = −
1, which yields − c ( t, r ) a ( t, r ) + d ( t, r ) b ( t, r ) = − . (32)Plugging these objects into the κ -deformed Hamiltonian (6) results in the most general sphericallysymmetric κ -deformed Hamiltonian: H Zg = − ℓ sinh (cid:18) ℓ cp t + dp r ) (cid:19) + e ℓ ( cp t + dp r ) (( − a + c ) p t + 2 cdp r p t + ( b + d ) p r + 1 r w ) , (33)where we suppressed the arguments of the functions a, b, c, d for the sake of readability.The functions c and d , intertwined by (32), identify a family of κ -deformations of the phasespace of a spherically symmetric spacetime. One could hope that some fundamental mechanismderived from a complete theory of quantum gravity would single out one specific correct form ofthe deformation.One the other hand, if one restricts to specific spherically-symmetric spacetimes, it is not alwaysthe case that there exists such freedom in the definition of the κ -deformation. For example,including further symmetries like in the homogeneous and isotropic case discussed in [7], the onlynormalized homogeneous and isotropic vector field evaluated on a 1-form P = p a dx a is Z ( p ) = p t . (34)Then the unique homogeneous and isotropic κ -deformed Hamiltonian was found to be H qF LRW = − ℓ sinh (cid:18) ℓ p t (cid:19) + e ℓp t a ( t ) − (cid:18) (1 − kr ) p r + 1 r w (cid:19) . (35)Here no additional degrees of freedom in addition to the scale factor of the F LRW metric, whichis determined by the Einstein equations, appear.In the following we specialize to the κ -deformation of the most famous spherically symmetricsolution of Einstein’s equations, the Schwarzschild geometry. B. The κ -deformation of Schwarzschild geometry In the Schwarzschild solution of general relativity the functions which determine the spacetimemetric are a ( t, r ) = 11 − r s r , b ( t, r ) = a ( t, r ) − = 1 − r s r , (36)4where r s is the Schwarzschild radius. Thus the functions c and d appearing in the timelike vectorfield Z which defines the deformation of the classical phase space, eq. (31), must satisfy − (cid:18) − r s r (cid:19) c ( t, r ) + d ( t, r ) (cid:18) − r s r (cid:19) = − , (37)according to equation (32). Following the discussion of the previous section we can write down thegeneral spherically symmetric κ -deformation of the phase space of Schwarzschild spacetime H ZSchw ( x, p ) = − ℓ sinh (cid:18) ℓ cp t + dp r ) (cid:19) + e ℓ ( cp t + dp r ) (cid:20)(cid:18) − − r s r + c (cid:19) p t + 2 cdp r p t + (cid:18) − r s r + d (cid:19) p r + 1 r w (cid:21) . (38)In the rest of this section we omit the subscript ZSchw for the sake of readability. As alreadymentioned we find a family of deformations defined by the function c and d subject to the con-dition (37). This result demonstrates the importance of our general construction in section II B,since without the insight that a vector field parametrizes the possible κ -Poincar´e deformations wemay not have found this general class of κ -deformations of Schwarzschild geometry. C. Motion in phase space
To study observable consequence of the κ -deformation of Schwarzschild geometry we now discussthe equations of motion for point particles.In general relativity the Einstein vacuum equations guarantee that every spherically symmetricsolution of the equations is static, also known as Birkhoff’s theorem. Since so far we have notdeveloped further the dynamics which the κ -deformation of a classical spacetime geometry has tosatisfy, in the following we assume for simplicity that c and d do not depend on t , i.e. that ∂ t induces yet another symmetry of H .Due to the symmetry of the geometry which we are studying there exist several constants ofmotion, one for each generator of symmetry X I , displayed in equations (25) to (27), to which weadd the generator of time translations ∂ t . The constants of motion are found as X I ( P ) = X aI ( x ) p a .In fact, it is easy to see that this object is constant along the solutions to the Hamilton equationsof motion. One then finds the constants of motion: E = p t , L = p φ , K = sin φp θ + cot θ cos φp φ , K = − cos φp θ + cot θ sin φp φ . (39)We can use these constants to restrict the motion of particles to the equatorial plane, fixing θ = π and p θ = 0. For this case L = p φ = w . Moreover H itself is another constant of motion representing5the dispersion relation − m = − ℓ sinh (cid:18) ℓ cp t + dp r ) (cid:19) + e ℓ ( cp t + dp r ) (cid:20)(cid:18) − − r s r + c (cid:19) p t + 2 cdp r p t + (cid:18) − r s r + d (cid:19) p r + 1 r w (cid:21) . (40)Under these conditions the non-trivial Hamilton equations of motion are˙ t = ¯ ∂ t H, ˙ p r = − ∂ r H, ˙ r = ¯ ∂ r H, ˙ φ = ¯ ∂ φ H . (41)Solving analytically the equations of motion is not possible, so, in order to get a first impressionof the sort of effects caused by κ -deformations of Schwarzschild spacetime geometry we choose c = √ | − rsr | in the region r > r s , i.e. outside the classical horizon, for which equation (37) implies d = 0. A thorough analysis of the implications of general κ -deformations of Schwarzschild geometry,parametrized by the functions c and d , will be discussed in an upcoming separate article. D. Observable effects in d = 0 κ -deformed Schwarzschild geometry Choosing c = √ | − rsr | ≡ √ A , r > r s and thus d = 0, the κ -deformed Schwarzschild Hamiltoniantakes the form: H ( x, p ) = − ℓ sinh (cid:18) ℓ p t √ A (cid:19) + e ℓpt √ A (cid:18) Ap r + 1 r w (cid:19) . (42)Using this specific choice of the free functions allows to study some relevant features of the modelexplicitly. In the following we focus on the effects of the deformation on the circular orbits aroundthe origin with radius larger than r s , and on the redshift between stationary observers.
1. Circular particle motion
The relevant Hamilton equations in the study of circular motion are the ones associated to theradial coordinate and momentum. Moreover, the on-shell condition H = − m relates the particle’senergy p t to the radial and angular momenta: p t √ A = − ℓ ln (cid:18) ℓ m ± ℓ s m (cid:18) ℓ m (cid:19) + w r + p r A (cid:19) → − ℓ ln (cid:18) ℓ m ℓ s m (cid:18) ℓ m (cid:19) + w r + p r A (cid:19) , (43)where the sign was chosen so to have ( p t √ A ) = − m for observers with p r = w = 0 in the ℓ = 0limit.6A circular orbit is characterized by constant radial coordinate, ˙ r = 0. Then from the Hamiltonequation for ˙ r it follows that the radial momentum must be constantly vanishing:0 = ˙ r = ¯ ∂ r H = 2 A p r e ℓpt √ A ⇒ p r = 0 . (44)This of course also implies that ˙ p r = 0. Using the Hamilton equation for the radial momentum:0 = ˙ p r = − ∂ r H = − ℓ sinh (cid:18) ℓp t √ A (cid:19) r s r p t A − e ℓpt √ A (cid:18) r s r p r − w r (cid:19) + ℓ p t A r s r e ℓpt √ A (cid:18) Ap r + w r (cid:19) (45)= − ℓ sinh (cid:18) ℓp t √ A (cid:19) r s r p t A + e ℓpt √ A w r + ℓ p t A r s r w e ℓpt √ A , (46)where in the second line we used p r = 0. Before solving for r , we can simplify this expressionfurther by using the mass-shell constraint (43) to remove the p t dependence: ℓ ln (cid:18) ℓ m + ℓP m (cid:19) A r s rP m (cid:18) ℓ m ℓ P m (cid:19) − w = 0 , (47)where we multiplied everything by 2 r e − ℓpt √ A and we defined P m = s m (cid:18) ℓ m + 1 (cid:19) + w r . In themassless limit this becomes: ℓ ln (cid:18) ℓ wr (cid:19) A r s w (cid:16) ℓ wr (cid:17) − w = 0 , (48)In general, the equations (47) and (48) are not solvable analytically, so we continue our studyperturbatively. The above equations read, up to first order in ℓ : r s r − r s m r + 2 w (cid:18) − rr s (cid:19) + ℓ (cid:0) w + 2 m r (cid:1) r m + w r ! = 0 , (49)for the massive case, and w (cid:18) − r s r − r s + ℓw r s r ( r − r s ) (cid:19) = 0 (50)for the massless case. Solving for r one finds the radius of circular orbits for massive particles: r m = w m r s (cid:18) − r − (cid:16) r s mw (cid:17) (cid:19) + ℓ w m s (cid:18) wm r m (cid:19) (4 r m − r s )( w − r m r s m ) , (51)where r m = lim ℓ → r m . In the massless limit this becomes: r m =0 = 32 r s + ℓ w . (52)This last results indicates that the photon sphere, which is universal in Schwarzschild geometry,is in fact dependent on the angular momentum of the photons once the Planck-scale deformation7is introduced, so that photons with different energy are allowed to orbit a black hole at differentaltitudes. Such a modification of the geometry of the photon sphere of spherically symmetricblack holes would immediately have an influence on further observables like lensing [18] and theobservation of the shadows of black holes [19]. These subjects go beyond the scope of this articleand will be investigated in the future.
2. Redshift
Our goal here is to compute the change in the energy of a photon as measured by two differentobservers, σ and σ , at rest. The observers are characterized by their spacetime coordinates andmomenta: σ i = ( x σ i , p σ i ), i = 1 ,
2. Since the observers are at rest only the time component of theirfour-momentum is nonzero: p σ i = ( p σ i t , , , H ( x σ i , p σ i ) = − ℓ sinh (cid:18) ℓ p σ i t p A ( x σ i ) (cid:19) = − m σ i . (53)This constraint implies that the four-momentum of the observers is related to their position andmass via p σ i t = p A ( x σ i ) Q σ i , with Q σ i ≡ − ℓ ln (cid:18) ℓ m σi + ℓm σ i q ℓ m σi (cid:19) being a constant.Having defined the observers, we can use equation (21) to obtain the frequencies that each ofthem associates to the photon: ν σ i ( γ ) = 1 m σ i p γt | σ i p A ( x σ i ) (cid:20) ℓ e − ℓQ σi − ℓ − ℓm σ i (cid:21) = p γt | σ i p A ( x σ i ) " r ℓ m σ i (54)The time component of the momentum of the photon at the position of the observer σ i is givenby p γt | σ i . Since the light trajectory γ is a solution of the Hamilton equations of motion, p γt isconstant along γ . In particular, p γt has the same value at the intersection point with σ and atthe intersection point with σ , so p γt | σ = p γt | σ = p γt . The redshift of the photon between thetwo observers is thus given by z + 1 = ν σ ( γ ) ν σ ( γ ) = √ A √ A q ℓ m σ q ℓ m σ = s − r s r − r s r q ℓ m σ q ℓ m σ (55) ≃ s − r s r − r s r (cid:18) ℓ m σ − m σ )( m σ + m σ ) (cid:19) , (56)where in the last step we only kept the lowest order ℓ -correction. Thus for two static observersthe redshift of a photon is identical to the one in Schwarzschild geometry to all orders in ℓ , if8the observers have the same mass. Otherwise, if the observers have different masses, then theymeasure a redshift which departs from the standard result to second order in ℓ . This influence ofthe mass of the observers on the redshift is due to the fact that we assumed that the observers arealso subject to the κ -deformed dynamics. If one were to assume that observers follow the dynamicsof the general relativistic Hamiltonian (2), or that the observers have negligible masses, then therewould be again no additional effect compared to the usual redshift in Schwarzschild geometry.Surely the results of this section highly depend on the specific choice of observers and of the vec-tor field Z (remember that the possible deformations of Schwarzschild geometries encoded by thevector field Z depend on two free functions of spacetime coordinates, which we fixed at the begin-ning of this subsection III D in order to have a workable example). In general we would expect thatthe Planck-scale deformation would alter the gravitational redshift of photons in spherical symme-try also for equal-mass observers, as it is the case in the homogeneous and isotropic cosmologicalsituation discussed in [7]. IV. DISCUSSION
We used the insights we gained in the local implementation of the κ -Poincar´e dispersion relationon homogeneous and isotropic spacetimes [7] to extend our findings to general curved spacetimes.The key result of our work is the construction of a phase space in which locally one can identifya spacetime with κ -Lorentz symmetry, i.e. κ -Poincar´e symmetries excluding translations. Theimplementation of this local symmetry via the level sets of a Hamilton function on the pointparticle phase space causes the geometry of spacetime and the geometry of momentum space to beintertwined into a geometry of phase space.In equation (6) we presented the locally κ -Poincar´e Hamilton function which deserves its nameby the fact that at every point on spacetime there exists a local basis of the cotangent spaces ofthe spacetime manifold such that the level sets of the Hamilton function assume the form of the κ -Poincar´e dispersion relation. This is the direct generalization of local Lorentz invariance of thegeometry of spacetime to local κ -Lorentz invariance. The explicit construction of the κ -Poincar´eHamilton function will allow us to study the mathematical differential geometric structure of thephase space geometry in the future. In particular, the local frame bundle properties of spacetimeare of interest since equivalent frames are no longer identified with linear transformations likeLorentz transformations but with the partly non-linear κ -Lorentz transformations, the κ -Poincar´eboosts and rotations.9Having established the notion of a general κ -deformed phase space we studied the motion oftest particles on such a background. The modification of the geodesic equation was presentedin equation (11). As already stated when we introduced Hamiltonian geometry in [10], thereappears a force-like term in the equations of motion which can not be absorbed into the geometryof spacetime. Thus there exists no local coordinate system such that the equations of motionlocally reduce to ¨ x + O ( x ) = 0 as they do in normal coordinates in the undeformed spacetimegeometry. Also generalizations of normal coordinates, as they were discussed in the context ofFinsler geometry in [20] and [21], do not exist. To complete the discussion on particle motionon the κ -deformed phase space geometry we derived the Lagrangian formulation of point particlemotion. This can be used as starting point for the derivation of a Finslerian version of the locally κ -deformed spacetime geometry in the future, as it was done for particular κ -deformed geometriesin [12–14].In the second half of this article we derived the most general form of the locally κ -Poincar´eHamilton function compatible with spherical symmetry. We obtained a Hamilton function definedin terms of four free functions of the time and radial coordinate, two of which are fixed by thespecific spacetime geometry on which the deformation is based. The presence of the other twofree functions is due to the fact that the timelike vector field which is necessary to define theHamilton function is not fixed by the available symmetry constraints. This is to be contrastedwith the homogeneous and isotropic case [7], where the symmetry constraints were sufficient tofully determine the form of the deformation.We studied observable predictions of the model in the special case of deformations of theSchwarzschild geometry, where the vector field defining the deformation was chosen as the tan-gent of the standard observer at rest in Schwarzschild geometry. In an upcoming article we willinvestigate the influence of the choice of this vector field on observables in more detail. The freedomin the choice of the vector field defining the deformed Hamiltonian may be related to the deformedboosts which underly the κ -deformed spacetime geometry, in the sense that the deformed boostmay map one choice of Z to another. This will be matter of investigation in future work. Forour choice of κ -deformed Schwarzschild geometry we studied two possibly observable features: theradius of photon orbits around the spherical symmetric black hole (known as photon sphere inthe standard case) and the gravitational redshift between two observers at rest with respect toeach other and with respect to the black hole horizon. For the first observable we found that thephoton sphere, which is universal for all photons in Schwarzschild geometry, becomes momentumdependent. In particular, photons with a different angular momentum have circular orbits at dif-0ferent altitudes (52). For the redshift we found that corrections to the standard Schwarzschild caseemerge only at the second order in the deformation parameter, (55). Moreover, these correctionsare proportional to the difference of the masses of the observers measuring the frequency of thephoton and they only exist if one assumes that the observers enjoy the same deformed symmetriesas the photon itself.In an upcoming work we will study the spherically symmetric κ -Poincar´e deformed spacetimegeometry in further detail to derive observable implications in solar system and black hole obser-vations, like perihelion shifts, light deflections, the horizon and the singularity. Further interestingstudies which are now in reach are locally κ -deformed spacetime geometries with any desired sym-metry, like axial symmetry, as generalization of the spherically symmetric case.Besides these phenomenological studies, one can further develop our method to locally imple-ment more general dispersion relations on curved spacetime, generalizing the κ -Poincar´e case thatwas studied here. The procedure to be applied would be to identify four basis vector fields { Z i } i =0 on spacetime which represent, when applied to a four momentum Z i ( p ), the different Cartesian mo-mentum components p i = Z i ( p ). This sort of generalization would be particularly interesting sinceit would allow to compare predictions concerning black hole physics obtained in the framework ofHamilton geometry to the ones obtained using rainbow gravity as a formalization of Planck-scaleeffects [22–25]. ACKNOWLEDGMENTS
CP gratefully thanks the Center of Applied Space Technology and Microgravity (ZARM) at theUniversity of Bremen for their kind hospitality and acknowledges partial support of the EuropeanRegional Development Fund through the Center of Excellence TK133 “The Dark Side of theUniverse”. GG acknowledges support from the John Templeton Foundation. LKB acknowledgesthe support by a Ph.D. grant of the German Research Foundation within its Research TrainingGroup 1620
Models of Gravity . NL aknowledges partial support from the 000008 15 RS
Avvio allaricerca
Appendix A: Theories of electrodynamics leading to κ -Poincar´e light propagation To demonstrate that the κ -deformed Hamiltonian we constructed in equation (6) can be ob-tained as the geometric optics limit of a theory of electrodynamics we summarize here the argumentsleading to such a theory.Theories of electrodynamics are rooted in field equations which yield charge and magnetic fluxconservation. Following the axiomatic approach to electrodynamics presented in [26] a most generalway to formulate such a theory requires two 2-form fields, the electric field strength F and themagnetic excitation H , and a closed current 3-form J subject to the equations dF = 0 , dH = J. (A1)In four dimensions these are eight equations which shall determine the components of the fields F and H , which are twelve in total. Thus this set of equations is not sufficient to yield a predictivetheory of electrodynamics. In addition a so called constitutive relation H = F ) . (A2)Combining these equations on a contractable spacetime one obtains F = dA , which makes thetheory a theory with a 1-form potential A as fundamental field and gauge invariance. The re-maining field equation, the second in (A1), becomes d dA = J as dynamical equation for thepotential. All theories of electrodynamics constructed according to this scheme are gauge invariantby construction.The most famous examples of theories of electrodynamics are local and linear, i.e. H is alinear function of F . In Maxwell vacuum electrodynamics on curved spacetime the constitutiverelation is given by the Hodge star operator of the metric H = ⋆F , or in components H ab = ǫ abcd g ce g df F ef , while for example electrodynamics in media is described by a general local andlinear constitutive relation H ab = ǫ abcd χ cdef F ef . Here ǫ abcd is the Levi-Civita symbol and χ abcd theso called constitutive density, where we omit explicit displaying density factors like determinantsof the metric, for the sake of a compact presentation of the arguments. Details on this approachto electrodynamics can be found for example in [26–28] and further references therein.To obtain a theory of electrodynamics which implies propagation of light (resp. propagation ofsingularities in the language of partial differential equations [29–32]) governed by the κ -deformed2Hamiltonian we consider the following class of linear higher derivative constitutive relations H ab = 12 ǫ abcd G ec ( Q,S ) ( x, ∂ ) G df ( Q,S ) ( x, ∂ ) F df (A3)with G ab ( Q,S ) ( x, ∂ ) = 4 ℓ sinh (cid:18) i ℓ Z ( ∂ ) (cid:19) Q ab Q ( ∂, ∂ ) − e − ℓiZ ( ∂ ) ( g − ( ∂, ∂ ) + Z ( ∂ ) ) S ab S ( ∂, ∂ ) , (A4)where Z ( ∂ ) = Z a ( x ) ∂ a , g − ( ∂, ∂ ) = g − ab ( x ) ∂ a ∂ b , Q ( ∂, ∂ ) = Q ab ( x, ∂ ) ∂ a ∂ b and S ( ∂, ∂ ) = S ab ( x, ∂ ) ∂ a ∂ b . The operators Q and S parametrize different constitutive relations and thus differenttheories of electrodynamics. Simple choices, not involving further derivatives, may be Q ab = g ab = S ab , or Q ab = g ab , S ab = g ab + Z a Z b . (A5)As a remark recall that using constitutive laws which involve derivative operators is somethingknown in the literature. The most famous example of such a higher derivative theory of electro-dynamics may be Bopp-Podolski electrodynamics [33, 34], which is studied as a candidate theoryof electrodynamics which yields a finite self force of charged particles [35].We will now demonstrate that all theories of electrodynamics which are constructed from aconstitutive law of the form (A3) yield wave propagation governed by the Hamiltonian (6).It is well known that for local and linear constitutive laws the wave propagation is governed bythe Fresnel polynomial, first derived in [27], G ( x, p ) = 14! ǫ c a a a ǫ d b b b χ a c b d ( x ) χ a c b d ( x ) χ a c b d ( x ) p d p c p d p c . (A6)It serves as Hamiltonian which determines the motion of light along those solutions of Hamiltonsequations of motion which satisfy G ( x, p ) = 0. Technically speaking it is the principal polynomialof the dynamical equation d dA ) = J , obtained from its Fourier space representation. Or, inother words, the highest derivative term in the equation where the partial derivatives are exchangedwith − ip . Since only this highest order derivative term is relevant for the geometric optics limit ofthe theory we do not need to worry about using covariant or partial derivatives when defining (A3)in terms of (A4). Terms involving connection coefficient, which covariantize the field equations,are of lower order derivatives acting on the dynamical field and thus do not contribute.In Maxwell electrodynamics with χ abcd ∼ g a [ c g d ] b one obtains G ( x, p ) = ( g ab p a p b ) while forexample in an uniaxial crystal with χ abcd ∼ g a [ c g d ] b + U [ a X b ] U [ c X d ] , where X denotes the crystalaxis and U the rest frame of the crystal, one obtains birefringent light propagation from the bi-metric Fresnel polynomial G ( x, p ) = g ab p a p b ( g cd − ( g ij X i X j ) U c U d + X c X d ) p c p d .3Algebraically we are dealing with the same situation as in the general local and linear case,except that our constitutive relation is a partial differential operator. Following the usual derivationof the Fresnel polynomial, again see [26–28] for details, we find the same algebraic expression for theprincipal symbol except that our constitutive density now depends on momenta, by the interchangeof ∂ → − ip , when going from configuration to Fourier space G ( p ) = 14 ǫ c a a a ǫ d b b b χ a c b d ( − ip ) χ a c b d ( − ip ) χ a c b d ( − ip ) p d p c p d p c . (A7)Since the specific constitutive relation is constructed from an operator which has the same indexstructure as the Hodge star of a metric it is simple to calculate the Fresnel polynomial (A7) forthe constitutive relation (A3) and we find G ( p ) = ( G ab ( x, − ip ) p a p b ) = ( H Zg ( x, p )) , (A8)for any choice of Q and S . An explicit calculation can be found in [28] for the standard Maxwellcase.Thus there exists a huge class of higher derivative gauge invariant theories of electrodynamics,parametrized by Q and S , whose geometric optics limit is governed by the locally κ -deformedHamiltonian we constructed in this article and is thus invariant under local κ -Poincar´e transfor-mations.As final remark we like to point put that the symmetries of the geometric optic limit and the fullfield theory may very well differ. In the case of the uniaxial crystal, the field equations are defined interms of a metric and two vector fields. In the geometric optics limit these building blocks combineto a bi-metric Fresnel polynomial, thus the geometric optics posses all the symmetries these metricsshare. The full field theory however can not be formulated in terms of the two metrics alone andhence may posses different symmetries. Thus whether the theory of electrodynamics leading to κ -deformed geometric optics must be locally κ -Poincar´e invariant itself is an open issue. It mayvery well be that local κ -Poincar´e invariance is only a geometric optics feature and not one ofthe full field theory. This would in particular depend on the full quantum gravity theory whosesemiclassical limit can be described in terms of the local κ -Poincar´e symmetries. Appendix B: The κ -Poincar´e Lagrangian In section II C we discussed the Hamilton equations of motion of the general κ -Poincar´e Hamil-tonian. Here we demonstrate how the corresponding Lagrangian can be obtained from which4one can derive the second oder Euler-Lagrange equations. The Legendre transformation form theHamiltonian to the Lagrangian involves the terms L ( x, ˙ x ) = ˙ x ( p ( x, ˙ x )) − H ( x, p ( x, ˙ x )) (B1)which we will derive now.In (10) we already found˙ x a = ¯ ∂ a H (B2)= Z a (cid:20) − ℓ sinh (cid:18) ℓZ ( p ) (cid:19) + ℓe ℓZ ( p ) ( g − ( p, p ) + Z ( p ) ) + 2 e ℓZ ( p ) Z ( p ) (cid:21) + e ℓZ ( p ) g ab p b . Contracting this equation with Z yields g ( ˙ x, Z ) = 2 ℓ sinh (cid:18) ℓZ ( p ) (cid:19) − ℓe ℓZ ( p ) ( g − ( p, p ) + Z ( p ) ) (B3)which allows us to write˙ x a = Z a (cid:20) − g ( ˙ x, Z ) + 2 e ℓZ ( p ) Z ( p ) (cid:21) + e ℓZ ( p ) g ab p b , (B4)and ˙ x a p a = − Z ( p ) g ( ˙ x, Z ) + 2 e ℓZ ( p ) ( Z ( p ) + g − ( p, p )) . (B5)as well as g ( ˙ x, ˙ x ) = − g ( ˙ x, Z ) + 2 e ℓZ ( p ) ( Z ( p ) g ( ˙ x, Z ) + ˙ x ( p )) (B6)= − g ( ˙ x, Z ) + 4 e ℓZ ( p ) ( Z ( p ) + g − ( p, p )) (B7)= − g ( ˙ x, Z ) + 4 e ℓZ ( p ) (cid:18) ℓ sinh( ℓZ ( p )) − g ( ˙ x, Z ) ℓ (cid:19) (B8)= − g ( ˙ x, Z ) − ℓ e ℓZ ( p ) g ( ˙ x, Z ) + 4 ℓ ( e ℓZ ( p ) −
1) (B9)The last equation can be reformulated as quadratic equation for e ℓZ ( p ) e ℓZ ( p ) − ℓe ℓZ ( p ) g ( ˙ x, Z ) − ℓ g ( ˙ x, ˙ x ) + g ( ˙ x, Z ) ) − . (B10)with solution e ℓZ ( p ) = ℓ g ( ˙ x, Z ) ± r ℓ g ( ˙ x, Z ) + ℓ g ( ˙ x, ˙ x ) + 1 (B11)= 12 (cid:0) ℓg ( ˙ x, Z ) ± p ℓ g ( ˙ x, Z ) + ℓ g ( ˙ x, ˙ x ) + 4 (cid:1) (B12) Z ( p ) = 1 ℓ ln (cid:18) (cid:0) ℓg ( ˙ x, Z ) ± p ℓ g ( ˙ x, Z ) + ℓ g ( ˙ x, ˙ x ) + 4 (cid:1)(cid:19) . (B13)5Finally we can use the terms we found to solve (B4) for p ( x, ˙ x ) p a ( x, ˙ x ) = 12 g ab ˙ x b e − ℓZ ( p ) − g ab Z b (cid:20) − e − ℓZ ( p ) g ( ˙ x, Z ) + 2 Z ( p ) (cid:21) (B14)= 12 e − ℓZ ( p ) (cid:18) g ab ˙ x b + g ab Z b g ( ˙ x, Z ) (cid:19) − g ab Z b Z ( p ) (B15)= g ab ˙ x b + g ab Z b g ( ˙ x, Z ) ℓg ( ˙ x, Z ) ± p ℓg ( ˙ x, Z ) + ℓg ( ˙ x, ˙ x ) + 4 (B16) − g ab Z b ℓ ln (cid:18) (cid:0) ℓg ( ˙ x, Z ) ± p ℓg ( ˙ x, Z ) + ℓg ( ˙ x, ˙ x ) + 4 (cid:1)(cid:19) . (B17)Contracting this expression with ˙ x a yields the desired equation (14). Equation (15) is obtained bysolving (B3) for e ℓZ ( p ) ( g − ( p, p ) + Z ( p ) ) = 2 ℓ sinh (cid:18) ℓZ ( p ) (cid:19) − g ( ˙ x, Z ) ℓ , (B18)plugging this result into the Hamiltonian (6) and inserting (B12) afterwards. Appendix C: The lifts of the symmetry generating vector fields to phase space
In section III A we used the lifts of the vector fields which generate spherical symmetry onspacetime to derive the most general spherically symmetric Hamilton function on phase space.These lifts X CI = ξ a ∂ a − p q ∂ a ξ q ¯ ∂ a (C1)of the vector fields X I = ξ aI ( x ) ∂ a , I = 1 , , X C = sin φ∂ θ + cot θ cos φ∂ φ + cos φ sin θ p φ ¯ ∂ θ − (cid:18) cos φp θ − cot θ sin φp φ (cid:19) ¯ ∂ φ , (C2) X C = − cos φ∂ θ + cot θ sin φ∂ φ + sin φ sin θ p φ ¯ ∂ θ − (cid:18) sin φp θ + cot θ cos φp φ (cid:19) ¯ ∂ φ , (C3) X C = ∂ φ . (C4)6It can be easily checked by direct calculation that the Hamiltonian (28) satisfies X CI ( H ) = 0 forall I = 1 , , [1] J. Lukierski, H. Ruegg, A. Nowicki, and V. N. Tolstoi, “Q deformation of Poincare algebra,” Phys.Lett.
B264 (1991) 331–338.[2] J. Lukierski, A. Nowicki, and H. Ruegg, “New quantum Poincare algebra and k deformed fieldtheory,”
Phys.Lett.
B293 (1992) 344–352.[3] J. Lukierski and H. Ruegg, “Quantum kappa Poincare in any dimension,”
Phys.Lett.
B329 (1994) 189–194, arXiv:hep-th/9310117 [hep-th] .[4] G. Amelino-Camelia, “Quantum-Spacetime Phenomenology,”
Living Rev.Rel. (2013) 5, arXiv:0806.0339 [gr-qc] .[5] J. Kowalski-Glikman, “De sitter space as an arena for doubly special relativity,” Phys.Lett.
B547 (2002) 291–296, arXiv:hep-th/0207279 [hep-th] .[6] G. Gubitosi and F. Mercati, “Relative Locality in κ -Poincar´e,” Class. Quant. Grav. (2013) 145002, arXiv:1106.5710 [gr-qc] .[7] L. Barcaroli, L. K. Brunkhorst, G. Gubitosi, N. Loret, and C. Pfeifer, “Planck-scale-modifieddispersion relations in homogeneous and isotropic spacetimes,” Phys. Rev.
D95 (2017) no. 2, 024036, arXiv:1612.01390 [gr-qc] .[8] L. Barcaroli and G. Gubitosi, “Kinematics of particles with quantum-de Sitter-inspired symmetries,”
Phys. Rev.
D93 (2016) no. 12, 124063, arXiv:1512.03462 [gr-qc] .[9] G. Rosati, G. Amelino-Camelia, A. Marciano, and M. Matassa, “Planck-scale-modified dispersionrelations in FRW spacetime,”
Phys. Rev.
D92 (2015) no. 12, 124042, arXiv:1507.02056 [hep-th] .[10] L. Barcaroli, L. K. Brunkhorst, G. Gubitosi, N. Loret, and C. Pfeifer, “Hamilton geometry: Phasespace geometry from modified dispersion relations,”
Phys. Rev.
D92 (2015) no. 8, 084053, arXiv:1507.00922 [gr-qc] .[11] S. Majid and H. Ruegg, “Bicrossproduct structure of kappa Poincare group and noncommutativegeometry,”
Phys.Lett.
B334 (1994) 348–354, arXiv:hep-th/9405107 [hep-th] .[12] G. Amelino-Camelia, L. Barcaroli, G. Gubitosi, S. Liberati, and N. Loret, “Realization of doublyspecial relativistic symmetries in Finsler geometries,”
Phys. Rev.
D90 (2014) no. 12, 125030, arXiv:1407.8143 [gr-qc] .[13] I. P. Lobo, N. Loret, and F. Nettel, “Investigation of Finsler geometry as a generalization to curvedspacetime of Planck-scale-deformed relativity in the de Sitter case,”
Phys. Rev.
D95 (2017) no. 4, 046015, arXiv:1611.04995 [gr-qc] .[14] M. Letizia and S. Liberati, “Deformed relativity symmetries and the local structure of spacetime,”
Phys. Rev.
D95 (2017) no. 4, 046007, arXiv:1612.03065 [gr-qc] . [15] U. Jacob and T. Piran, “Lorentz-violation-induced arrival delays of cosmological particles,” JCAP (2008) 031, arXiv:0712.2170 [astro-ph] .[16] G. Amelino-Camelia, L. Barcaroli, G. D’Amico, N. Loret, and G. Rosati, “IceCube and GRBneutrinos propagating in quantum spacetime,”
Phys. Lett.
B761 (2016) 318–325, arXiv:1605.00496 [gr-qc] .[17] J. Plebanski and J. Krasinski,
An Introduction to General Relativity and Cosmology . CambridgeUniversity Press, 2006.[18] K. S. Virbhadra and G. F. R. Ellis, “Schwarzschild black hole lensing,”
Phys. Rev.
D62 (2000) 084003, arXiv:astro-ph/9904193 [astro-ph] .[19] A. Grenzebach, V. Perlick, and C. Lmmerzahl, “Photon Regions and Shadows of Accelerated BlackHoles,”
Int. J. Mod. Phys.
D24 (2015) no. 09, 1542024, arXiv:1503.03036 [gr-qc] .[20] C. Pfeifer, “The tangent bundle exponential map and locally autoparallel coordinates for generalconnections on the tangent bundle with application to Finsler geometry,”
Int. J. Geom. Meth. Mod. Phys. (2016) no. 03, 1650023, arXiv:1406.5413 [math-ph] .[21] E. Minguzzi, “Special coordinate systems in pseudo-Finsler geometry and the equivalence principle,” J. Geom. Phys. (2017) 336–347, arXiv:1601.07952 [gr-qc] .[22] Y. Ling, X. Li, and H.-b. Zhang, “Thermodynamics of modified black holes from gravity’s rainbow,”
Mod. Phys. Lett.
A22 (2007) 2749–2756, arXiv:gr-qc/0512084 [gr-qc] .[23] A. F. Ali, “Black hole remnant from gravity’s rainbow,”
Phys. Rev.
D89 (2014) no. 10, 104040, arXiv:1402.5320 [hep-th] .[24] C. Leiva, J. Saavedra, and J. Villanueva, “The Geodesic Structure of the Schwarzschild Black Holes inGravity’s Rainbow,”
Mod. Phys. Lett.
A24 (2009) 1443–1451, arXiv:0808.2601 [gr-qc] .[25] Y. Gim and W. Kim, “Thermodynamic phase transition in the rainbow Schwarzschild black hole,”
JCAP (2014) 003, arXiv:1406.6475 [gr-qc] .[26] F. W. Hehl and Y. N. Obukhov,
Foundations of Classical Electrodynamics . Birkh¨auser, 2003.[27] F. W. Hehl, Y. N. Obukhov, and G. F. Rubilar, “Light propagation in generally covariantelectrodynamics and the Fresnel equation,”
International Journal of Modern Physics A (2002) no. 20, 2695–2700, arXiv:gr-qc/0203105 .[28] C. Pfeifer and D. Siemssen, “Electromagnetic potential in pre-metric electrodynamics: Causalstructure, propagators and quantization,” Physical Review D (2016) no. 10, 105046, arXiv:1602.00946 [math-ph] .[29] L. H¨ormander, The Analysis of Linear Partial Differential Operators I: Distribution Theory andFourier Analysis . No. 256 in Grundlehren der mathematischen Wissenschaften. Springer, 1983.[30] L. H¨ormander,
The Analysis of Linear Partial Differential Operators II: Differential Operators withConstant Coefficients . No. 257 in Grundlehren der mathematischen Wissenschaften. Springer, 1983.[31] L. H¨ormander,
The Analysis of Linear Partial Differential Operators III: Pseudo-DifferentialOperators . No. 274 in Grundlehren der mathematischen Wissenschaften. Springer, 1985. [32] N. Dencker, “On the propagation of polarization sets for systems of real principal type,” Journal of Functional Analysis (1982) no. 3, 351–372.[33] F. Bopp, “Eine lineare theorie des elektrons,” Annalen der Physik (1940) no. 5, 345–384. http://dx.doi.org/10.1002/andp.19404300504 .[34] B. Podolsky, “A generalized electrodynamics part i¯non-quantum,”
Phys. Rev. (Jul, 1942) 68–71. https://link.aps.org/doi/10.1103/PhysRev.62.68 .[35] J. Gratus, V. Perlick, and R. W. Tucker, “On the self-force in BoppPodolsky electrodynamics,” J. Phys.
A48 (2015) no. 43, 435401, arXiv:1502.01945 [gr-qc]arXiv:1502.01945 [gr-qc]