Gravitational Waves and Conformal Time Transformations

Abstract

Recent interest in the "memory effect" prompted us to revisit the relation of gravitational aves and oscillators. 50 years ago Niederer [1] found that an isotropic harmonic oscillator with a constant frequency can be mapped onto a free particle. Later Takagi [2] has shown that "time-dependent scaling" extends the oscillator versus free particle correspondence to a time-dependent frequency when the scale factor satisfies a Sturm-Liouville equation. More recently Gibbons [3] pointed out that time redefinition is conveniently studied in terms of the Schwarzian derivative. The oscillator versus free particle correspondence "Eisenhart-Duval lifts" to a conformal transformation between Bargmann spaces [4-7]. These methods are extended to spacetimes which are not conformally flat and have a time-dependent profile, and can then be applied to the geodesic motion in a plane gravitational wave.

affiliation corrected

References in corpus (25)

• Celestial Mechanics, Conformal Structures, and Gravitational Waves
• Solving the Schwarzian via the Conformal Bootstrap
• The Memory Effect for Plane Gravitational Waves
• (2+1)D Exotic Newton-Hooke Symmetry, Duality and Projective Phase
• Anisotropic harmonic oscillator, non-commutative Landau problem and exotic Newton-Hooke symmetry
• Velocity Memory Effect for Polarized Gravitational Waves
• Eisenhart lifts and symmetries of time-dependent systems
• Conformal Newton-Hooke algebras, Niederer's transformation and Pais-Uhlenbeck oscillator
• Conformal and projective symmetries in Newtonian cosmology
• Scaling and conformal symmetries for plane gravitational waves
• Newton-Hooke type symmetry of anisotropic oscillators
• Conformal bridge between asymptotic freedom and confinement
• Memory effect, conformal symmetry and gravitational plane waves
• Trapping and guiding bodies by gravitational waves endowed with angular momentum
• Dark Energy and the Schwarzian Derivative
• Ion Traps and the Memory Effect for Periodic Gravitational Waves
• Niederer's transformation, time-dependent oscillators and polarized gravitational waves
• Conformal symmetries and integrals of the motion in pp waves with external electromagnetic fields
• Dynamics, symmetries, anomaly and vortices in a rotating cosmic string background
• Geometry of the isotropic oscillator driven by the conformal mode
• Conformal generation of an exotic rotationally invariant harmonic oscillator
• Schwarzian derivative treatment of the quantum second-order supersymmetry anomaly, and coupling-constant metamorphosis
• Particle motion in circularly polarized vacuum pp waves
• Lukash plane waves, revisited
• Time-dependent conformal transformations and the propagator for quadratic systems

Cited by in corpus (5)

• Schrödinger symmetry: a historical review
• Eisenhart lift of Koopman-von Neumann mechanics
• Memory Effect and Carroll Symmetry: 50 years later
• Conformally related vacuum gravitational waves, and their symmetries
• Evading Quantum Mechanics à la Sudarshan: quantum-mechanics-free subsystem as a realization of Koopman-von Neumann mechanics