A relativistic gravity train

Abstract

A nonrelativistic particle released from rest at the edge of a ball of uniform charge density or mass density oscillates with simple harmonic motion. We consider the relativistic generalizations of these situations where the particle can attain speeds arbitrarily close to the speed of light; generalizing the electrostatic and gravitational cases requires special and general relativity, respectively. We find exact closed-form relations between the position, proper time, and coordinate time in both cases, and find that they are no longer harmonic, with oscillation periods that depend on the amplitude. In the highly relativistic limit of both cases, the particle spends almost all of its proper time near the turning points, but almost all of the coordinate time moving through the bulk of the ball. Buchdahl's theorem imposes nontrivial constraints on the general-relativistic case, as a ball of given density can only attain a finite maximum radius before collapsing into a black hole. This article is intended to be pedagogical, and should be accessible to those who have taken an undergraduate course in general relativity.

Experimental realization of a relativistic harmonic oscillator

“…The advent of special relativity suggested a simple question: what happens to this archetypal physical system when the maximum oscillator velocity approaches the speed of light? While theories of the relativistic harmonic oscillator have been discussed for decades [1][2][3][4][5], the combination of special relativity and harmonic motion has proven resistant to physical realization. The infeasibility of realizing harmonic traps with depths on the order of a particle's rest mass energy (Boltzmann's constant times nearly six billion degrees Kelvin for an electron) has motivated work studying relativistic phenomena in disparate physical contexts, including a measurement of the Dirac oscillator spectrum in an array of microwave resonators [6] and proposals and realizations of effective relativistic effects in trapped atoms [7][8][9][10][11][12][13], trapped ions [14][15][16], photonic waveguides [17], and graphene [18].…”

“…Before describing the experiments it is useful to briefly outline existing predictions for the behavior of a relativistic harmonic oscillator [1][2][3][4][5]. As the maximum velocity v max approaches c, relativistic effects are expected to modify the character of harmonic motion in several ways.…”

On the Relativistic Harmonic Oscillator

“…[52] [53] [55] [57]. This has led to the derivation of a closed-from solution for the equation of the RHO in terms of Jacobi elliptic functions [51] [56].…”

On the Relativistic Harmonic Oscillator

“…Oscillations in the weak-relativistic limit have been of interest in Plasma Physics [12][13][14][15][16][17][18][19][20][21][22][23][24]. Results have been attained by either generating numerical solutions, or by expansions in powers of β 2 (β = (Max| |)/c) in the weak-relativistic limit [25][26][27][28][29][30][31][32][33][34][35]. Owing to the limitation of the validity in time of approximate solutions obtained by asymptotic expansion methods, typically, the lowest-order approximation ((O(β 2 )) in such analyses has a 1-2% error relative to the numerical solution only up to β ≈ 0.2.…”