Noninertial Relativistic Symmetry

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.