We use the geometric mean to parametrize metrics in the Hassan-Rosen ghost-free bimetric theory and pose the initial-value problem. The geometric mean of two positive definite symmetric matrices is a well-established mathematical notion which can be, under certain conditions, extended to quadratic forms having the Lorentzian signature, say metrics $g$ and $f$. In such a case, the null cone of the geometric mean metric $h$ is in the middle of the null cones of $g$ and $f$ appearing as a geometric average of a bimetric spacetime. The parametrization based on $h$ ensures the reality of the square root in the ghost-free bimetric interaction potential. Subsequently, we derive the standard $n+1$ decomposition in a frame adapted to the geometric mean and state the initial-value problem, that is, the evolution equations, the constraints, and the preservation of the constraints equation.
We explore spacetime symmetries and topologies of the two metric sectors in Hassan-Rosen bimetric theory. We show that, in vacuum, the two sectors can either share or have separate spacetime symmetries. If stress-energy tensors are present, a third case can arise, with different spacetime symmetries within the same sector. This raises the question of the best definition of spacetime symmetry in Hassan-Rosen bimetric theory. We emphasize the possibility of imposing ansätze and looking for solutions having different Killing vector fields or different isometries in the two sectors, which has gain…
We put forward a two-dimensional nonlinear sigma model that couples (bosonic) matter fields to topological Horava gravity on a nonrelativistic worldsheet. In the target space, this sigma model describes classical strings propagating in a curved spacetime background, whose geometry is described by two distinct metric fields. We evaluate the renormalization group flows of this sigma model on a flat worldsheet and derive a set of beta-functionals for the bimetric fields. Imposing worldsheet Weyl invariance at the quantum level, we uncover a set of gravitational field equations that dictate the dy…
We discuss the structure of horizons in spacetimes with two metrics, with applications to the Vainshtein mechanism and other examples. We show, without using the field equations, that if the two metrics are static, spherically symmetric, nonsingular, and diagonal in a common coordinate system, then a Killing horizon for one must also be a Killing horizon for the other. We then generalize this result to the axisymmetric case. We also show that the surface gravities must agree if the bifurcation surface in one spacetime lies smoothly in the interior of the spacetime of the other metric. These re…
The ghost-free bimetric theory describes interactions of gravity with another spin-2 field in terms of two Lorentzian metrics. However, if the two metrics do not admit compatible notions of space and time, the formulation of the initial value problem becomes problematic. Furthermore, the interaction potential is given in terms of the square root of a matrix which is in general nonunique and possibly nonreal. In this paper we prove that the reality of the square root matrix leads to a classification of the allowed metrics in terms of the intersections of their null cones. Then, the requirement …
We perform a canonical analysis of the bimetric theory in the metric formulation, computing the constraints and their algebra explicitly. In particular, we compute a secondary constraint, that has been argued to exist earlier, and show that it has the correct form to eliminate the ghost. We also identify a set of four first class constraints that generate the algebra of general covariance. The covariance algebra naturally determines a spacetime metric for the theory. However, in bimetric theory, this metric is not unique but depends on how the first class constraints are identified.
We propose a model of bimetric gravity in which the mixing of metrics naturally gives a mass to a graviton by the compactification with flux of two gauge fields in extra dimensions. We assume that each metric in the solution for the background geometry describes the four-dimensional Minkowski spacetime with an $S^2$ extra space, though the two radii of $S^2$ for two metrics take different values in general. The solution is derived by the effective potential method in the presence of the magnetic fluxes on the extra spheres. We find that the a massive graviton is governed by the Fierz-Pauli Lag…
The bimetric variational principle is a subtle reinterpretation of general relativity that assumes the spacetime connection to be generated by an independent metric. Unlike the so called Palatini formalism that promotes the connection into a fundamental field, the new variational principle results in a physically distinct theory since the potential for the connection carries new degrees of freedom. The connection-generating metric naturally allows also an antisymmetric component. This sets torsion propagating! It is also shown here that while in the most straightforward generalization of the E…
The existence of two metrics in massive gravity theories in principle allows solutions where there are singularities in new scalar invariants jointly constructed from them. These configurations occur when the two metrics differ substantially from each other, as in black hole and cosmological solutions. The simplest class of such singularities are determinant singularities. We investigate whether the dynamics of bimetric massive gravity -- where the second metric is allowed to evolve jointly with the spacetime metric -- can avoid these singularities. We investigate whether the dynamics of bimet…
We develop the "generalized Gordon ansatz" for the ghost-free versions of both massive and bimetric gravity, an ansatz which is general enough to include almost all spacetimes commonly considered to be physically interesting, and restricted enough to greatly simplify calculations. The ansatz allows explicit calculation of the matrix square root gamma = sqrt{g^{-1} f} appearing as a central feature of the ghost-free analysis. In particular, this ansatz automatically allows us to write the effective stress-energy tensor as that corresponding to a perfect fluid. A qualitatively similar "generaliz…
In this dissertation, we explore models based on the idea that there are two metrics in spacetime: One describes the standard gravity, and the other provides a geometry in which matter fields propagate. In order to do that, we provide the essentials of Finsler geometry and the rules to induce a metric for the propagation of matter. Such a description will cover some of the most critical features related to the field necessary to do the induction, these will arise in an attempt to build an action for this field. And finally, we provide an example to study the homogeneous limit of background FLR…
We report that a class of three-dimensional bimetric theories contain asymptotically flat solutions. These spacetimes can be cast in a set of asymptotic conditions at null infinity which are preserved under the infinite dimensional BMS group. Moreover, the algebra of the canonical generators exhibits a central extension. The possibility that these solutions describe regular black holes is also discussed.
The choice of gauge in numerical relativity is crucial in avoiding coordinate and curvature singularities. In addition, the gauge can affect the well-posedness of the system. In this work, we consider the mean gauges, established with respect to the geometric mean metric $h = g \, (g^{-1}f)^{1/2}$ in bimetric relativity. We consider three gauge conditions widely used in numerical relativity and compute them with respect to the geometric mean: The 1+log gauge condition and the maximal slicing for the lapse function of $h$, and the $Γ$-driver gauge condition for the shift vector of $h$. In addit…
A formulation of linearized gravity which is manifestly invariant under electric-magnetic duality rotations in the internal space of the metric and its dual, and which contains both metrics as basic variables (rather than the corresponding prepotentials), is derived. In this bimetric formulation, the variables have a more immediate geometrical significance, but the action is non-local in space, contrary to what occurs in the prepotential formulation. More specifically, one finds that: (i) the kinetic term is non-local in space (but local in time); (ii) the Hamiltonian is local in space and in …
We obtain a set of exact gravitational wave solutions for the ghost free bimetric theory of gravity. With a flat reference metric, the theory admits the vacuum Brinkmann plane wave solution for suitable choices of the coefficients of different terms in the interaction potential. An exact gravitational wave solution corresponding to a massive scalar mode is also admitted for arbitrary choice of the coefficients with the reference metric being proportional to the spacetime metric. The proportionality factor and the speed of the wave are calculated in terms of the parameters of the theory. We als…
We investigate the recently introduced geometric quench protocol for fractional quantum Hall (FQH) states within the framework of exactly solvable quantum Hall matrix models. In the geometric quench protocol a FQH state is subjected to a sudden change in the ambient geometry, which introduces anisotropy into the system. We formulate this quench in the matrix models and then we solve exactly for the post-quench dynamics of the system and the quantum fidelity (Loschmidt echo) of the post-quench state. Next, we explain how to define a spin-2 collective variable $\hat{g}_{ab}(t)$ in the matrix mod…
We study, as a model of Lorentz symmetry breaking, the quantisation and renormalisation of an extension of QED in a flat spacetime where the photons and electrons propagate differently and do not share the same lightcone. We will refer to this model as Bimetric QED (BIMQED). As a preliminary we discuss the formulation of electrodynamics in a pre-metric formalism showing nevertheless that there is, on the basis of a simple criteron, a preferred metric. Arising from this choice of metric is a Weyl-like tensor (WLT). The Petrov classification of the WLT gives rise to a corresponding classificatio…
The possibility of matter coupling to two metrics at once is considered. This appears natural in the most general ghost-free, bimetric theory of gravity, where it unlocks an additional symmetry with respect to the exchange of the metrics. This double coupling, however, raises the problem of identifying the observables of the theory. It is shown that if the two metrics couple minimally to matter, then there is no physical metric to which all matter would universally couple, and that moreover such an effective metric generically does not exist even for an individual matter species. By studying p…
Deformed relativistic kinematics have been considered as a way to capture residual effects of quantum gravity. It has been shown that they can be understood geometrically in terms of a curved momentum space on a flat spacetime. In this article we present a systematic analysis under which conditions and how deformed relativistic kinematics, encoded in a momentum space metric on flat spacetime, can be lifted to curved spacetimes in terms of a self-consistent cotangent bundle geometry, which leads to purely geometric, geodesic motion of freely falling point particles. We comment how this construc…
In theories of gravity where the cosmological constant defines a thermodynamic variable, the pressure, it has been shown that solutions of Einstein's equations have a corresponding thermodynamic volume. In general, the expression for the volume is not the same as the one arising from naive geometrical considerations. Both rotation and nut charge are properties known in the literature to give non-geometric thermodynamic volumes, and so we combine the two in a single example and compute the volume for the Kerr-bolt-AdS spacetime, presenting a new expression that generalizes the previously known …
The coupling between the electromagnetic and gravitational fields results in "faster than light" photons and invalids the Lorentz invariance and some laws of physics. A typical example is that the first and third laws of geometric optics are invalid in the usual spacetime. By introducing an effective spacetime, we find that the wave vector can be casted into null and then it obeys the geodesic equation, the polarization vector is perpendicular to the rays, and the number of photons is conserved. That is to say, the laws of geometric optics are still valid for the modified theory in the effecti…
Classical geometry can be described either in terms of a metric tensor $g_{ab}(x)$ or in terms of the geodesic distance $σ^2(x,x')$. Recent work, however, has shown that the geodesic distance is better suited to describe the quantum structure of spacetime. This is because one can incorporate some of the key quantum effects by replacing $σ^2$ by another function $S[σ^2]$ such that $S[0]=L_0^2$ is non-zero. This allows one to introduce a zero-point-length in the spacetime. I show that the geodesic distance can be an emergent construct, arising in the form of a correlator $S[σ^2(x,y)]=\langle J(x…
Scalar fields on SL(2,R) and H^2 x R geometric spacetimes and linear perturbations
利用适当的调和函数，研究了一类宇宙真空时空中无质量标量场的未来渐近行为。假设空间流形是高亏格曲面上具有局部齐次度量的圆丛。这样的流形对应于SL（2，R）-几何（Bianchi VIII型）或H^2 x R-几何（Bianchi III型）。经过技术准备，包括引入合适的谐波圆纤维比安奇八分离变量，我们推导出系统的常微分方程的标量场。我们提出了未来的渐近解这些方程在一个特殊的情况下，并发现有一个密切的相似之处，圆纤维比安奇III时空。我们讨论这种相似性的影响，特别是（引力）线性扰动。我们还指出，这种相似性可以解释为“纤维项主导的行为”的两个模型。
Theories of gravity are fundamentally a relation between matter and the geometric structure of the underlying spacetime. So once we put some additional restrictions on the spacetime geometry, the theory of gravity is bound to get the impact, irrespective of whether it is general relativity or the modified theories of gravity. In the present article, we consider two curvature-based constraints, namely the almost pseudo-Ricci symmetric and weakly Ricci symmetric condition. As a novel result, such spacetimes with non-null associated vectors are entirely classified, and then applying the obtained …
In the framework of open quantum systems, we study the geometric phase acquired by freely falling and static two-level atoms interacting with quantized conformally coupled massless scalar fields in de Sitter-invariant vacuum. We find that, for the freely falling atom, the geometric phase gets a correction resulting from a thermal bath with the Gibbons-Hawking temperature, thus it clearly reveals the intrinsic thermal nature of de Sitter spacetime from a different physical context. For the static atom, there is a correction to the geometric phase coming from both the intrinsic thermal nature of…
Together with collaborators, we introduced a noncommutative Riemannian geometry over Moyal algebras and systematically developed it for noncommutative spaces embedded in higher dimensions in the last few years. The theory was applied to construct a noncommutative version of general relativity, which is expected to capture some essential structural features of spacetime at the Planck scale. Examples of noncommutative spacetimes were investigated in detail. These include quantisations of plane-fronted gravitational waves, quantum Schwarzschild spacetime and Schwarzschild-de Sitter spacetime, and…
The homogeneous and isotropic radiation dominated universe, following the inflationary stage, is expressed as a spherically symmetric and inhomogeneous spacetime upon a power law type conformal transformation of the null (cosmological) coordinates. This new spacetime metric has many interesting properties. While the static observers, at a fixed position in this new spacetime, do not see any horizon, some non-static observers encounter a horizon due to their motion which is analogous to the situation of Rindler observers in Minkowski spacetime. The symmetry of the new metric offers a unitarily …
By starting from the modified Maxwell theory coupled to gravity, the arising of geometric quantum phases in the relativistic and nonrelativistic quantum dynamics of a Dirac neutral particle from the effects of the violation of the Lorentz symmetry in the cosmic string spacetime is investigated. It is shown that the Dirac equation can be written in terms of an effective metric and a relativistic geometric phase stems from the topology of the cosmic string spacetime and the Lorentz symmetry breaking effects. Further, the nonrelativistic limit of the Dirac equation is discussed and it is shown th…
The ten-dimensional supergravity theory is a geometric low-energy effective theory and the equations of motion for its fields can be obtained from string theory by computing $β$ functions. With $d$ compact dimensions, we can add to it an $O(d, d;\mathbb{Z})$ geometric structure and construct the supergravity theory inspired by double field theory through the use of a suitable commutative star product. The latter implements the weak constraint of the double field theory on its fields and gauge parameters in order to have a closed gauge symmetry algebra. The consistency of the action here propos…
We give a set of local geometric conditions on a spacetime metric which are necessary and sufficient for it to be a null electrovacuum, that is, the metric is part of a solution to the Einstein-Maxwell equations with a null electromagnetic field. These conditions are restrictions on a null congruence canonically constructed from the spacetime metric, and can involve up to five derivatives of the metric. The null electrovacuum conditions are counterparts of the Rainich conditions, which geometrically characterize non-null electrovacua. Given a spacetime satisfying the conditions for a null elec…
We study the consistency of a model which includes torsion as well as the metric as dynamical fields and has massive spin-2 particle in its spectrum. The massive spin-2 mode resides in the torsion, rather than in the metric. It is known that this model is tachyon- and ghost-free in Minkowski background. We show that this property remains valid and no other pathologies emerge in de Sitter and anti-de Sitter backgrounds, with some of our results extending to arbirary Einstein space backgrounds. This suggests that the model is consistent, at least at the classical level, unlike, e.g., the Fierz--…
In this paper we study a unified formalism for Thermal Quantum Field Theories, i.e., for the Matsubara approach, Thermo Field Dynamics and the Path Ordered Method. To do so, we employ a mechanism akin to the Hawking effect which explores a relationship between the concept of temperature and spacetimes endowed with event-horizons. In particular, we consider an eight dimensional static spacetime, the so-called eta-xi spacetime, which we show to form an appropriate geometric background for generic Thermal Quantum Field Theories. Within this framework, the different formalisms of Thermal Field The…
An information-geometrical interpretation of AdS3/CFT2 correspondence is given. In particular, we consider an inverse problem in which the classical spacetime metric is given in advance and then we find what is the proper quantum information that is well stored into the spacetime. We see that the Fisher metric plays a central role on this problem. Actually, if we start with the two-dimensional hyperbolic space, a constant-time surface in AdS3, the resulting singular value spectrum of the quantum state shows power law for the correlation length with conformal dimension proportional to the curva…
We present a recent definition of symmetry generating vector fields on manifolds equipped with a first-order reductive Cartan geometry. We apply this definition to a number of spacetime geometries used in gravity theories and show that it agrees with the usual notions of symmetry of affine, Riemann-Cartan, Riemannian, Weizenböck and Finsler spacetimes.
Einstein Equivalence Principle (EEP) requires all matter components to universally couple to gravity via a single common geometry: that of spacetime. This relates quantum theory with geometry as soon as interactions with gravity are considered. In this work, I study the geometric theory of coupling a spin-1/2 particle to gravity in a twofold expansion scheme: First with respect to the distance based on Fermi normal coordinates around a preferred worldline (e.g., that of a clock in the laboratory), second with respect to 1/c (post-Newtonian expansion). I consider the one-particle sector of a ma…
