[Paper Review] Gravitational fields of lightons and helixons in General Relativity

А. М. Baranov|arXiv (Cornell University)|May 31, 2011

Relativity and Gravitational Theory4 citations

TL;DR

This paper investigates the lightlike limit of massive particles (Schwarzschild, Kerr, NUT) in General Relativity, showing that as their velocity approaches light speed and rest mass vanishes while energy and angular momentum remain constant, their gravitational fields converge to massless wave-like solutions of Petrov types N and III. The limiting objects are identified as scalar lightons and spinning helixons, with a phase transition in gravitational field symmetry described via catastrophe theory.

ABSTRACT

A lightlike limit procedure for massive particles of Schwarzschild, Kerr and NUT which are sources of the exterior gravitational fields of an algebraic type D is introduced. It is shown when a velocity of fast moving massive particles along z axis tends towards the light velocity axis and the total energy of each particle is constant (i.e. a rest mass of the particle tends towards zero) together with Kerr's angular momentum along z axis and with NUT's parameter which also tend to constants then the gravitational fields of these fast moving particles tend towards the wave's fields of the N and III algebraic types as their limits. The lightlike limit of massive particle may be described as cusp catastrophe on the level of Weyl's matrix with a change of gravitational field's symmetry of such source. In considered cases such limit is a phase transition of the gravitational field from D type into N type or III type (transition from one "phase" to another). Petrov's algebraic types are different "phases" of gravitational field. As result of the lightlike limit procedure the lightlike scalar massless particle ({\it lighton}) and a vector lightlike massless particle with a helicity ({\it helixon}) are obtained. It is shown the lightlike sources in General Relativity "have no hairs"

Motivation & Objective

• To investigate the behavior of massive particle gravitational fields under extreme relativistic conditions approaching the speed of light.
• To analyze the limiting gravitational fields of Schwarzschild, Kerr, and NUT solutions when rest mass tends to zero but total energy and angular momentum remain constant.
• To identify the resulting massless, lightlike particles (lightons and helixons) emerging from this limit.
• To interpret the transition in gravitational field algebraic type (from D to N or III) as a phase transition using catastrophe theory.
• To demonstrate that these lightlike sources possess no conserved parameters beyond energy and helicity, supporting the 'no-hair' conjecture in gravity.

Proposed method

• Applies a lightlike limit procedure where velocity v → c (v → 1), rest mass m₀ → 0, and total energy E → const, while keeping Kerr angular momentum L_z and NUT parameter b constant.
• Uses Weyl’s 3×3 traceless complex matrix representation of the gravitational field to analyze the limit on the level of eigenvalues and symmetry.
• Transforms the Weyl matrix of massive particles (Schwarzschild, Kerr, NUT) under Lorentz boosts into a moving frame using orthogonal transformation matrices T.
• Derives the limiting field as a distributional solution involving Dirac δ-functions, specifically δ(z + t), representing a plane-fronted gravitational wave.
• Applies catastrophe theory to model the transition: the characteristic equation λ³ + pλ + q = 0 with control parameters p(ε), q(ε) exhibits a cusp catastrophe at p = q = 0.
• Identifies the phase transition point (p = q = 0) as the critical point where the gravitational field symmetry changes from type D to N or III, corresponding to the emergence of lightlike sources.

Experimental results

Research questions

• RQ1 What happens to the gravitational field of a massive particle (Schwarzschild, Kerr, NUT) as its velocity approaches the speed of light and its rest mass vanishes while total energy is conserved?
• RQ2 How does the algebraic type of the gravitational field (Petrov type) change under such a lightlike limit, and what does this imply for the field’s symmetry?
• RQ3 What are the resulting massless, lightlike particles that emerge from this limit, and how are they characterized in terms of spin and energy?
• RQ4 Can the transition from type D to type N or III gravitational fields be described using catastrophe theory, and what does this imply about the nature of the phase transition?
• RQ5 Do these lightlike sources in General Relativity truly possess no conserved parameters beyond energy and helicity, supporting the 'no-hair' principle?

Key findings

• The lightlike limit of the Schwarzschild solution yields a gravitational field described by a metric gμν = ημν − 8H lμ lν with H = −2E δ(z + t) ln(ρ²), representing a scalar massless particle (lighton).
• The limiting field of the Kerr solution with angular momentum along z-axis results in a superposition of type N and III Weyl matrices, yielding a spinning massless particle with helicity JE, identified as a helixon.
• The NUT solution under the same limit reduces to the same lighton field after a rotation in the complex plane that absorbs the NUT parameter B, confirming the field is independent of B in the limit.
• The transition from type D to type N or III is interpreted as a cusp catastrophe in catastrophe theory, with the critical point at p = q = 0 marking a second-order phase transition in the gravitational field.
• All physical parameters except total energy E and helicity JE are lost in the limit, confirming that lightlike sources in GR 'have no hairs', consistent with the no-hair conjecture.
• The resulting gravitational fields are exact solutions of Einstein’s equations with a singular energy-momentum tensor Tμν = 2E δ(z + t) δ(x) δ(y) lμ lν, describing null radiation.