[Paper Review] Wonderful Consequences of the Kerr Theorem

Alexander Burinskii|ArXiv.org|Jun 1, 2005

Relativity and Gravitational Theory3 citations

TL;DR

This paper proposes a multi-particle extension of the Kerr-Newman solution using the Kerr theorem with a generating function $ F $ as a product of individual particle functions $ F_i $. It reveals that gravitational and electromagnetic interactions occur via shared singular twistor lines—lightlike strings connecting the outgoing (+) sheet of one particle to the incoming (−) sheet of another—resulting in a 'dressed' multi-particle solution with twistorial, string-like vacuum structure, suggesting a geometric pathway to quantum gravity.

ABSTRACT

Kerr's multi-particle solution is obtained on the base of the Kerr theorem. Choosing generating function of the Kerr theorem $F$ as a product of partial functions $F_i$ for spinning particles i=1,...k, we obtain a multi-sheeted, multi-twistorial space-time over $M^4$ possessing unusual properties. Twistorial structures of the i-th and j-th particles do not feel each other, forming a type of its internal space. Gravitation and electromagnetic interaction of the particles occurs via a singular twistor line which is common for twistorial structures of interacting particles. The obtained multi-particle Kerr-Newman solution turns out to be `dressed' by singular twistor lines linked to surrounding particles. We conjecture that this structure of space-time has the relation to a stringy structure of vacuum and opens a geometrical way to quantum gravity.

Motivation & Objective

• To generalize the single-particle Kerr-Newman solution to a multi-particle system using the Kerr theorem.
• To explore how interacting spinning particles manifest through shared twistorial structures in a multi-sheeted space-time.
• To investigate the role of singular twistor lines in mediating gravitational and electromagnetic interactions between particles.
• To examine the implications of this structure for a geometric formulation of quantum gravity.
• To analyze the renormalization-like effects in mass and charge due to the presence of surrounding particles in the generating function.

Proposed method

• Construct the generating function $ F = \prod_{i=1}^k F_i(Y|q_i) $, where each $ F_i $ corresponds to a spinning particle with parameters $ q_i $.
• Apply the Kerr theorem to derive the null congruence $ k^{(i)}_\mu $ from the roots $ Y^\pm_i $ of $ F_i = 0 $, ensuring geodesic and shear-free properties.
• Use the Kerr-Schild formalism with $ g_{\mu\nu} = \eta_{\mu\nu} + 2h k_\mu k_\nu $, where $ h $ and $ P $ are derived from $ F $ via $ \tilde{r} = -dF/dY $ and $ P = \partial_{\lambda_1}F - \bar{Y}\partial_{\lambda_2}F $.
• Introduce a normalization factor $ \mu_i = \prod_{j \neq i} F_j(Y) $, which modifies the metric and gauge field, leading to effective mass and charge renormalization.
• Identify singular interaction lines as common twistor rays where $ Y^+_i = Y^-_j $, signaling a lightlike string connecting the (+) sheet of particle $ i $ to the (−) sheet of particle $ j $.
• Demonstrate that the resulting solution satisfies the Einstein-Maxwell equations and exhibits a multi-twistorial, multi-sheeted space-time structure.

Experimental results

Research questions

• RQ1 How can the Kerr theorem be generalized to describe a system of multiple interacting spinning particles?
• RQ2 What is the geometric origin of gravitational and electromagnetic interactions in this multi-particle framework?
• RQ3 How do the twistorial structures of individual particles interact, and what role does the singular twistor line play?
• RQ4 What are the implications of the normalization factor $ \mu_i $, which depends on external particles, for the physical parameters like mass and charge?
• RQ5 Can this multi-particle solution provide a geometric foundation for quantum gravity through its string-like vacuum structure?

Key findings

• The multi-particle solution is obtained by setting the generating function $ F = \prod_{i=1}^k F_i(Y|q_i) $, leading to a multi-sheeted, multi-twistorial space-time over $ M^4 $.
• The metric and gauge field are modified by a normalization factor $ \mu_i $, resulting in effective mass $ m_i(Y) = m_0 \mu_i^3 $ and charge $ e/\mu_i $, indicating a form of renormalization.
• Singular twistor lines emerge where $ Y^+_i = Y^-_j $, forming lightlike strings that mediate interactions between particles, with fields singular along these lines.
• The interaction is geometrically realized as a common twistor line connecting the (+) sheet of one particle to the (−) sheet of another, resembling a pp-wave structure.
• The solution preserves the geodesic and shear-free nature of the Kerr congruence on each sheet, ensuring consistency with the original Kerr-Schild formalism.
• The structure suggests a twistorial, string-like texture of vacuum, offering a potential geometric route to quantum gravity through the multi-particle Fock space-like configuration.