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Register a new userNew canonical analysis for higher order topologically massive gravity
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A detailed Gitman-Lyakhovich-Tyutin analysis for higher-order topologically massive gravity is performed. The full structure of the constraints, the counting of physical degrees of freedom, and the Dirac algebra among the constraints are reported. Moreover, our analysis presents a new structure of the constraints and we compare our results with those reported in the literature where a standard Ostrogradski framework was developed.
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To unify general relativity and quantum theory is hard in part because they are formulated in two very different mathematical languages, differential geometry and functional analysis. A natural candidate for bridging this language gap, at least in the case of the euclidean signature, is the discipline of spectral geometry. It aims at describing curved manifolds in terms of the spectra of their canonical differential operators. As an immediate benefit, this would offer a clean gauge-independent identification of the metrics degrees of freedom in terms of invariants that should be ready to quantize. However, spectral geometry is itself hard and has been plagued by ambiguities. Here, we regularize and break up spectral geometry into small finite-dimensional and therefore manageable steps. We constructively demonstrate that this strategy works at least in two dimensions. We can now calculate the shapes of 2-dimensional objects from their vibrational spectra.
Following the method of Buchbinder and Lyahovich, we carry out a canonical formalism for a higher-curvature gravity in which the Lagrangian density ${cal L}$ is given in terms of a function of the salar curvature $R$ as ${cal L}=sqrt{-det g_{mu u}}f(R)$. The local Hamiltonian is obtained by a canonical transformation which interchanges a pair of the generalized coordinate and its canonical momentum coming from the higher derivative of the metric.
Recent research has highlighted the non-uniqueness problem of energy-momentum tensors in linearized gravity; many different tensors are published in the literature, yet for particular calculations a unique expression is required. It has been shown that (A) none of these spin-2 energy-momentum tensors are gauge invariant and (B) the Noether and Hilbert energy-momentum tensors are not, in general, equivalent; therefore uniqueness criteria is difficult to specify. Conventional wisdom states that the various published energy-momentum tensors for linearized gravity can be derived from the canonical Noether energy-momentum tensor of spin-2 Fierz-Pauli theory by adding ad-hoc improvement terms (the divergence of a superpotential and terms proportional to the equations of motion), that these superpotentials are in some way unique or physically significant, and that this implies some meaningful connection to the Noether procedure. To explore this question of uniqueness, we consider the most general possible energy-momentum tensor for linearized gravity with free coefficients using the Fock method. We express this most general energy-momentum tensor as the canonical Noether tensor, supplemented by the divergence of a general superpotential plus all possible terms proportional to the equations of motion. We then derive systems of equations which we solve in order to prove several key results for spin-2 Fierz-Pauli theory, most notably that there are infinitely many conserved energy-momentum tensors derivable from the improvement method, and there are infinitely many conserved symmetric energy-momentum tensors that follow from specifying the Belinfante superpotential alone. $dots$ since there are infinitely many energy-momentum tensors of this form, no meaningful or unique connection to Noethers first theorem can be claimed by application of the canonical Noether improvement method.
Wolfgang Kummer was a pioneer of two-dimensional gravity and a strong advocate of the first order formulation in terms of Cartan variables. In the present work we apply Wolfgang Kummers philosophy, the `Vienna School approach, to a specific three-dimensional model of gravity, cosmological topologically massive gravity at the chiral point. Exploiting a new Chern-Simons representation we perform a canonical analysis. The dimension of the physical phase space is two per point, and thus the theory exhibits a local physical degree of freedom, the topologically massive graviton.
Extreme mass ratio in-spirals (EMRIs) are candidate events for gravitational wave detection in the millihertz range (by detectors like LISA and eLISA). These events involve a stellar-mass black hole, or a similar compact object, descending in the gravitational field of a supermassive black hole, eventually merging with it. Properties of the in-spiralling trajectory away from resonance are well known and have been studied extensively, however little is known about the behaviour of these binary systems at resonance, when the radial and lateral frequencies of the orbit become commensurate. We describe the two existing models, the instantaneous frequency approach used by Gair, Bender, and Yunes, and the standard two timescales approach implemented by Flanagan and Hinderer. In both cases, the exact treatment depends on the modelling of the gravitational self-force, which is currently not available. We extend the results in Gair, Bender and Yunes to higher order in the on-resonance flux modification, and argue that the instantaneous frequency approach is also a valid treatment of the resonance problem. The non-linear differential equations which arise in treating resonances are interesting from a mathematical view point. We present our algorithm for perturbative solutions and the results to third order in the infinitesimal parameter, and discuss the scope of this approach.
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