Noninertial Relativistic Symmetry

TL;DR

This paper explores a new framework for relativistic symmetry incorporating noninertial states, using invariant metrics on phase space and revealing a noncompact unitary symmetry group that generalizes Einstein's proper time.

Contribution

It introduces a nondegenerate Born metric for invariant time, extending relativistic symmetry to noninertial states with a novel symmetry group structure.

Findings

The symmetry group for noninertial states is a noncompact unitary group.

Invariant phase space metrics unify relativistic and noninertial symmetries.

Causal cones in phase space bound the rate of change of physical quantities.

Abstract

The definition of invariant time is fundamental to relativistic symmetry. Invariant time may be formulated as a degenerate orthogonal metric on a flat phase space with time, position, energy and momentum degrees of freedom that is also endowed with a symplectic metric $\omega =-d t\wedge d \varepsilon +\delta _{i,j}d q^i\wedge d p^j$. For Einstein proper time, the degenerate orthogonal metric is $d \tau^{o 2}=d t^2-\frac{1}{c^2}d q^2$ and, in the limit $c\to \infty$, becomes Newtonian absolute time, $d t^2$. We show that the the resulting symmetry group leaving $\omega$ and $d t^2$ invariant is the Jacobi group that gives the expected transformations between noninertial states defined by Hamilton's equations. The symmetry group for $\omega$ and $d \tau^{o 2}$ is the semidirect product of the Lorentz and an abelian group parameterized by the time derivative of the energy-momentum tensor that characterizes noninertial states in special relativity. This leads to the consideration of invariant time based on a nondegenerate Born metric, $d \tau^2=d t^2-\frac{1}{c^2}d q^2-\frac{1}{b^2}d p^2+\frac{1}{b^2c^2}d \varepsilon^2$. $b$ is a universal constant with dimensions of force that, with $c,\hbar$ define the dimensional scales of phase space. We determine that the symmetry group for transformations between noninertial states is essentially a noncompact unitary group. It reduces to the noninertial symmetry group for Einstein proper time in the $b\to \infty$ limit and to the noninertial symmetry group for Hamiltonian mechanics in the $b,c\to \infty$ limit. The causal cones in phase space defined by the null surfaces $d\tau^2=0$ bound the rate of change of momentum as well as position. Furthermore, spacetime is no longer an invariant subspace of phase space but depends on the noninertial state; there is neither an absolute rest state nor an absolute inertial state that all observers agree on.