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Register a new userHamilton geometry: Phase space geometry from modified dispersion relations
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We describe the Hamilton geometry of the phase space of particles whose motion is characterised by general dispersion relations. In this framework spacetime and momentum space are naturally curved and intertwined, allowing for a simultaneous description of both spacetime curvature and non-trivial momentum space geometry. We consider as explicit examples two models for Planck-scale modified dispersion relations, inspired from the $q$-de Sitter and $kappa$-Poincare quantum groups. In the first case we find the expressions for the momentum and position dependent curvature of spacetime and momentum space, while for the second case the manifold is flat and only the momentum space possesses a nonzero, momentum dependent curvature. In contrast, for a dispersion relation that is induced by a spacetime metric, as in General Relativity, the Hamilton geometry yields a flat momentum space and the usual curved spacetime geometry with only position dependent geometric objects.
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Quantum gravity phenomenology suggests an effective modification of the general relativistic dispersion relation of freely falling point particles caused by an underlying theory of quantum gravity. Here we analyse the consequences of modifications of the general relativistic dispersion on the geometry of spacetime in the language of Hamilton geometry. The dispersion relation is interpreted as the Hamiltonian which determines the motion of point particles. It is a function on the cotangent bundle of spacetime, i.e. on phase space, and determines the geometry of phase space completely, in a similar way as the metric determines the geometry of spacetime in general relativity. After a review of the general Hamilton geometry of phase space we discuss two examples. The phase space geometry of the metric Hamiltonian $H_g(x,p)=g^{ab}(x)p_ap_b$ and the phase space geometry of the first order q-de Sitter dispersion relation of the form $H_{qDS}(x,p)=g^{ab}(x)p_ap_b + ell G^{abc}(x)p_ap_bp_c$ which is suggested from quantum gravity phenomenology. We will see that for the metric Hamiltonian $H_g$ the geometry of phase space is equivalent to the standard metric spacetime geometry from general relativity. For the q-de Sitter Hamiltonian $H_{qDS}$ the Hamilton equations of motion for point particles do not become autoparallels but contain a force term, the momentum space part of phase space is curved and the curvature of spacetime becomes momentum dependent.
The covariant understanding of dispersion relations as level sets of Hamilton functions on phase space enables us to derive the most general dispersion relation compatible with homogeneous and isotropic spacetimes. We use this concept to present a Planck-scale deformation of the Hamiltonian of a particle in Friedman-Lema^itre-Robertson-Walker (FLRW) geometry that is locally identical to the $kappa$-Poincare dispersion relation, in the same way as the dispersion relation of point particles in general relativity is locally identical to the one valid in special relativity. Studying the motion of particles subject to such Hamiltonian we derive the redshift and lateshift as observable consequences of the Planck-scale deformed FLRW universe.
We discuss a proposal on how gravitational collapse of a NEC (Null Energy Condition) violating spherically symmetric fluid distribution can avoid the formation of a zero proper volume singularity and eventually lead to a Lorentzian wormhole geometry. Our idea is illustrated using a time-evolving wormhole spacetime in which, we show how a collapsing sphere may never reach a zero proper volume end-state. The nature of geodesic congruences in such spacetimes is considered and analyzed. Our construction is inspired from a recently proposed static wormhole geometry, the multi-parameter Simpson-Visser line element, which is known to unite wormholes and black holes (regular and singular) in a single framework.
Relativistic quantum field theory in the presence of an external electric potential in a general curved space-time geometry is considered. The Fermi coordinates adapted to the time-like geodesic are utilized to describe the low-energy physics in the laboratory and to calculate the leading correction due to the curvature of the space-time geometry to the Schrodinger equation. The correction is employed to calculate the probability of excitation for a hydrogen atom that falls in or is scattered by a general Schwarzchild black hole. Since the excited states decay due to spontaneous photon emission, this study provides the theoretical base for detection of small isolated black holes by observing the decay of the excited states as neutral hydrogen atoms in the vacuum are devoured by the black hole.
Modified Newtonian Dynamics is an empirical modification to Poissons equation which has had success in accounting for the `gravitational field $Phi$ in a variety of astrophysical systems. The field $Phi$ may be interpreted in terms of the weak field limit of a variety of spacetime geometries. Here we consider three of these geometries in a more comprehensive manner and look at the effect on timelike and null geodesics. In particular we consider the Aquadratic Lagrangian (AQUAL) theory, Tensor-Vector-Scalar (TeVeS) theory and Generalized Einstein-{AE}ther (GEA) theory. We uncover a number of novel features, some of which are specific to the theory considered while others are generic. In the case of AQUAL and TeVeS theories, the spacetime exhibits an excess (AQUAL) or deficit (TeVeS) solid angle akin to the case of a Barriola-Vilenkin global monopole. In the case of GEA, a disformal symmetry of the action emerges in the limit of $gradPhirightarrow 0$. Finally, in all theories studied, massive particles can never reach spatial infinity while photons can do so only after experiencing infinite redshift.
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