Particles with Negative Energies in Nonrelativistic and Relativistic Cases

Andrey A Grib,Yuri V Pavlov

doi:10.3390/sym12040528

Abstract

States of particles with negative energies are considered for the nonrelativistic and relativistic cases. In the nonrelativistic case it is shown that the decay close to the attracting center can lead to the situation similar to the Penrose effect for a rotating black hole when the energy of one of the fragments is larger than the energy of the initial body. This is known as the Oberth effect in the theory of the rocket movement. The realizations of the Penrose effect in the non-relativistic case in collisions near the attracting body and in the evaporation of stars from star clusters are indicated. In the relativistic case similar to the well known Penrose process in the ergosphere of the rotating black hole it is shown that the same situation as in ergosphere of the black hole occurs in rotating coordinate system in Minkowski space-time out of the static limit due to existence of negative energies. In relativistic cases differently from the nonrelativistic ones, the mass of the fragment can be larger than the mass of the decaying body. Negative energies for particles are possible in the relativistic case in cosmology of the expanding space when the coordinate system is used with a nondiagonal term in metrical tensor of the space-time. Friedmann metrics for three cases: open, close and quasieuclidian, are analyzed. The De Sitter space-time is shortly discussed.

Highlights

The energy of the second part becomes larger than the energy of the initial body
This is some realization of the Penrose effect [2,3] on getting the energy from the rotating black hole due to the decay of some body in the ergosphere
Note that negative values of the energy E(ζ ) can be obtained for the positive energy density Tik if the Killing vector becomes spacelike as it is the case for the ergosphere of the rotating black hole and in our case in rotating coordinates in region out of the static limit

Summary

Negative Energies and the Penrose Effect in Nonrelativistic Case

It is well known that in the non-relativistic case, energy is determined with accuracy to the additive constant. Note that in the nonrelativistic case one has in the equation of energy conservation for the considered process (4) the term This is the energy necessary for the flight of the fragment with the relative velocity u. As one can see from (9) to get the energy profit comparable with the relativistic rest mass of the fuel m1 c2 one needs relativistic values of the fuel expiration velocity and use of gravitational field close to the gravitational radius In this case one must do calculations using relativistic theory (changing the Newton potential on the Schwarzschild metric and taking into account that relative velocities are close to the velocity of light). For example such decays occur for three interacting stars or evaporation of stars clusters [5]

Negative Energies in Rotating Coordinates

Negative Energies and Static Limit in Expanding Universe