Are Simple Real Pole Solutions Physical?

TL;DR

This paper investigates simple real pole solutions generated by inverse scattering, revealing that they inherently contain negative energy and cannot be obtained as limits of complex conjugate pole solutions, challenging previous assumptions.

Contribution

It demonstrates that simple real pole solutions cannot be derived from complex conjugate poles and that negative energy density is intrinsic to these solutions.

Findings

Real pole solutions cannot be obtained as limits of complex conjugate poles.

Negative energy density is an inherent feature of these solutions.

Coordinate transformations do not remove the negative energy component.

Abstract

We consider exact solutions generated by the inverse scattering technique, also known as the soliton transformation. In particular, we study the class of simple real pole solutions. For quite some time, those solutions have been considered interesting as models of cosmological shock waves. A coordinate singularity on the wave fronts was removed by a transformation which induces a null fluid with negative energy density on the wave front. This null fluid is usually seen as another coordinate artifact, since there seems to be a general belief that that this kind of solution can be seen as the real pole limit of the smooth solution generated with a pair of complex conjugate poles in the transformation. We perform this limit explicitly, and find that the belief is unfounded: two coalescing complex conjugate poles cannot yield a solution with one real pole. Instead, the two complex conjugate poles go to a different limit, what we call a ``pole on a pole''. The limiting procedure is not unique; it is sensitive to how quickly some parameters approach zero. We also show that there exists no improved coordinate transformation which would remove the negative energy density. We conclude that negative energy is an intrinsic part of this class of solutions.