[Paper Review] The Riemann-Lanczos Problem as an Exterior Differential System with Examples in 4 and 5 Dimensions

Patrick Dolan, A. Gerber|ArXiv.org|Dec 12, 2002

Quantum chaos and dynamical systems21 references3 citations

TL;DR

This paper investigates the Riemann-Lanczos problem in 4D and 5D spacetimes using exterior differential systems (EDS), showing it fails to be in involution and thus admits only singular solutions. It provides explicit examples of such solutions for Gödel, Kasner, and Debever-Hubaut spacetimes, demonstrating that symmetries can be inherited by singular potentials, and analyzes the non-involutive structure in 5D using Cartan characters and prolongation techniques.

ABSTRACT

The key problem of the theory of exterior differential systems (EDS) is to decide whether or not a system is in involution. The special case of EDSs generated by one-forms (Pfaffian systems) can be adequately illustrated by a 2-dimensional example. In 4 dimensions two such problems arise in a natural way, namely, the Riemann-Lanczos and the Weyl-Lanczos problems. It is known from the work of Bampi and Caviglia that the Weyl-Lanczos problem is always in involution in both 4 and 5 dimensions but that the Riemann-Lanczos problem fails to be in involution even for 4 dimensions. However, singular solutions of it can be found. We give examples of singular solutions for the Goedel, Kasner and Debever-Hubaut spacetimes. It is even possible that the singular solution can inherit the spacetime symmetries as in the Debever-Hubaut case. We comment on the Riemann-Lanczos problem in 5 dimensions which is neither in involution nor does it admit a 5-dimensional involution of Vessiot vector fields in the generic case.

Motivation & Objective

• To analyze the Riemann-Lanczos problem in 4 and 5 dimensions using the framework of exterior differential systems (EDS).
• To determine whether the Riemann-Lanczos system is in involution, given that the related Weyl-Lanczos problem is known to be in involution.
• To construct explicit singular solutions for the Riemann-Lanczos equations in specific spacetimes, including Gödel, Kasner, and Debever-Hubaut.
• To investigate whether singular solutions can preserve the isometries of the underlying spacetime, as seen in the Debever-Hubaut case.
• To examine the structure of the Riemann-Lanczos problem in 5 dimensions, particularly its failure to admit a 5-dimensional involution of Vessiot vector fields in the generic case.

Proposed method

• Formulates the Riemann-Lanczos problem as an exterior differential system (EDS) defined by the equation $ f^{(R)}_{abcd} := R_{abcd} - L_{abc;d} + L_{abd;c} - L_{cda;b} + L_{cdb;a} = 0 $, with $ L_{abc} $ satisfying index symmetries $ L_{[ab]c} = L_{abc} $ and $ L_{[abc]} = 0 $.
• Applies Cartan's theory of EDS, particularly the concept of involution and Cartan characters, to assess the integrability of the system.
• Uses the second Cartan-Kähler existence theorem to analyze the existence of integral manifolds, relying on normal form coordinates and prescribed initial data.
• Employs computational algebra (REDUCE code) to solve for constants in singular solutions, particularly for the Kasner spacetime, using constraints derived from the Riemann-Lanczos equations.
• Analyzes the prolongation of the Riemann-Lanczos system to second order to assess whether involution can be achieved, following Bampi and Caviglia’s approach.
• Compares the structure of the Riemann-Lanczos system with the Weyl-Lanczos problem, noting that while the latter is always in involution, the former is not, even in 4D.

Experimental results

Research questions

• RQ1 Is the Riemann-Lanczos problem in involution in 4 and 5 dimensions, and what are the implications for the existence of regular solutions?
• RQ2 Can singular solutions be constructed for the Riemann-Lanczos problem in specific spacetimes such as Gödel, Kasner, and Debever-Hubaut?
• RQ3 Do singular solutions of the Riemann-Lanczos problem inherit the isometries of the underlying spacetime, as observed in the Debever-Hubaut case?
• RQ4 What is the structure of the Riemann-Lanczos system in 5 dimensions, particularly regarding the existence of a 5-dimensional involution of Vessiot vector fields?
• RQ5 How do Cartan characters and the second Cartan-Kähler theorem apply to the Riemann-Lanczos problem, and what do they reveal about the number of arbitrary functions in the solution?

Key findings

• The Riemann-Lanczos problem is not in involution in 4 dimensions, precluding the existence of regular solutions and necessitating the use of singular solutions.
• Singular solutions exist for the Riemann-Lanczos problem in the Gödel, Kasner, and Debever-Hubaut spacetimes, with the latter showing that symmetries can be inherited by the potential.
• In 5 dimensions, the Riemann-Lanczos problem is neither in involution nor does it admit a 5-dimensional involution of Vessiot vector fields in the generic case.
• The Weyl-Lanczos problem is always in involution in both 4 and 5 dimensions, in contrast to the Riemann-Lanczos problem.
• The differential gauge condition $ L_{ab}^{ ext{ }s};s = 0 $ does not affect the existence or non-existence of solutions for either the Weyl- or Riemann-Lanczos problems.
• For the Kasner spacetime, the REDUCE code computes $ C_1, C_2, C_3 $ in terms of the Kasner parameters $ p_1, p_2, p_3 $, with $ p_1 + p_2 + p_3 = 1 $ and $ p_1^2 + p_2^2 + p_3^2 = 1 $, yielding explicit expressions for the singular solution constants.