Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1503.07802

The Eisenhart lift: a didactical introduction of modern geometrical concepts from Hamiltonian dynamics

Marco Cariglia·Filipe Kelmer Alves
View original source ↗Read source document ↗Export metadata ↓

Abstract

This work originates from part of a final year undergraduate research project on the Eisenhart lift for Hamiltonian systems. The Eisenhart lift is a procedure to describe trajectories of a classical natural Hamiltonian system as geodesics in an enlarged space. We point out that it can be easily obtained from basic principles of Hamiltonian dynamics, and as such it represents a useful didactical way to introduce graduate students to several modern concepts of geometry applied to physics: curved spaces, both Riemannian and Lorentzian, conformal transformations, geometrisation of interactions and extra dimensions, geometrisation of dynamical symmetries. For all these concepts the Eisenhart lift can be used as a theoretical tool that provides easily achievable examples, with the added benefit of also being a topic of current research with several applications, among which the study of dynamical systems and non-relativistic holography.

Explore related subjects

gr-qc

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Marco Cariglia, Filipe Kelmer Alves. 2015-03-02. The Eisenhart lift: a didactical introduction of modern geometrical concepts from Hamiltonian dynamics. https://doi.org/10.1088/0143-0807%2F36%2F2%2F025018

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quantum Correlations of Neutrinos in the Kerr-Newman Space-time

Quantum phases establish a connection between gravitation and quantum information, offering a novel avenue for exploring the properties of space-time. In this paper, we investigate the quantum correlations (QCs) of neutrinos in the Kerr--Newman space-time for both zero- and nonzero-angular-momentum propagation. The results show that, for zero-angular-momentum propagation, the oscillation periods of the survival probability and QCs progressively decrease with propagation distance in the inward direction. In the outward direction, increasing $M$ lengthens the oscillation periods of $P_{ν_e\rightarrowν_e}$, entanglement, and the monogamy of nonlocality, whereas increasing the angular momentum $a$ or charge $Q$ shortens them. For nonzero-angular-momentum propagation, the metric parameters also generate local profile modulations through additional two-path interference terms, rather than merely rescaling the oscillation period. Furthermore, we find that, despite differences in their ranges of variation, entanglement and coherence exhibit highly consistent oscillatory behavior in both propagation cases. These findings provide a comprehensive understanding of neutrino-based relativistic quantum information.

gr-qc↗
arXiv2026

Dynamical tidal response of neutron stars: From effective field theory to gravitational waveforms

We investigate the fully relativistic dynamical tidal response of neutron stars up to second order in the frequency. Combining the worldline effective field theory for extended gravitating bodies with perturbation theory of relativistic stellar models, we derive the tidal deformation induced by an external time-dependent field, including a universal logarithmic running term. In the effective theory, we work in dimensional regularization and, through a consistent matching procedure, obtain for the first time the complete leading-order dynamical tidal corrections to both the conservative dynamics and the gravitational-wave signal of compact binaries, including the scheme-dependent finite terms in addition to the running. We show that, in the relativistic regime, dynamical effects cannot be fully captured by mode excitations alone. The magnitude of the additional contribution depends on the stellar compactness, the equation of state, and the running term. Dynamical Love numbers are significantly enhanced with respect to their static counterparts for relatively small compactness. As a result, although they formally enter the gravitational-wave phase at eighth post-Newtonian order, dynamical tidal effects yield a nonnegligible contribution during the late inspiral. Using a Fisher-matrix analysis, we show that third-generation detectors such as the Einstein Telescope could measure dynamical Love numbers for a range of neutron-star masses and equations of state. Conversely, neglecting these effects can lead to significant biases in the inference of static Love numbers, and hence on the nuclear equation of state. Our results highlight the importance of dynamical tidal effects for high-precision gravitational-wave modeling with future detectors.

gr-qc↗
arXiv2026

Probing quantum chaos near a wormhole throat with a circular string

We investigate whether quantum fluctuations of a circular probe string develop a quantum-chaotic response while traversing a wormhole throat. The classical circular-string embedding is periodic and radially stable, but its two physical transverse polarizations experience time-dependent tidal potentials. Expanding the world-sheet action to quadratic order, we canonically quantize these modes and construct out-of-time-ordered correlator(OTOC) amplitudes from their unequal-time commutators. For the Ellis--Bronnikov wormhole, both polarizations exhibit finite intervals of approximately exponential OTOC growth associated with the first throat passage. The corresponding dimensionless rate measured with respect to physical time is positive over the parameter range studied and generally decreases as the probe energy is increased relative to the throat scale. In the global-monopole extension, increasing the solid-angle deficit narrows the band of locally amplifiable modes and suppresses the extracted rates; a sufficiently strong defect can nearly quench the radial signal, while the angular channel retains a polarization-dependent non-monotonic structure when the energy-to-throat-scale ratio is small. These quantities characterize finite-time dynamical sensitivity in the Gaussian fluctuation sector and should not be identified with asymptotic many-body chaos or a thermodynamic phase transition. Although the numerical analysis uses two representative wormhole geometries, the construction depends only on covariant world-sheet fluctuations and real-time commutators. It therefore provides a transferable, non-holographic framework for applying quantum-chaos diagnostics directly to quantum probes in curved spacetimes.

gr-qc↗