[Paper Review] A mathematicians' view of geometrical unification of classical physics in high-dimensional space-time

Michel Vaugon, Benoit Vaugon+3 more|arXiv (Cornell University)|Oct 5, 2010

Cosmology and Gravitation Theories8 references3 citations

TL;DR

This paper presents a differential geometric unification of classical gravity and electromagnetism in higher-dimensional spacetime, using a five-dimensional Lorentzian manifold where charged fluid dynamics emerge purely from curvature and Bianchi identities. By reinterpreting the Einstein tensor and introducing geometric concepts of 'type' and 'rigidity', the authors derive the Einstein-Maxwell-Lorentz equations in 4D as projections of geodesic motion in 5D, achieving a purely geometric formulation of classical physics without additional physical postulates.

ABSTRACT

We propose in this paper a mathematicians' view of the Kaluza-Klein idea of a five dimensional space-time unifying gravitation and electromagnetism, and extension to higher-dimensional space-time. By considering the classification of positive Einstein curvature tensors and the classical Cauchy-Choquet-Bruhat theorems in general relativity, we introduce concepts of types and rigidity. Then, abandoning the usual requirement of a Ricci-flat five dimensional space-time, we show that a unified geometrical frame can be set for gravitation and electromagnetism, giving, by projection on the classical 4-dimensional space-time, the known Einstein-Maxwell-Lorentz equations for charged fluids. Thus, although not introducing, at least at this stage, new physics, we get a very aesthetic presentation of classical physics in the spirit of general relativity. The usual physical concepts, such as mass, energy, charge, trajectory, Maxwell-Lorentz law, are shown to be only various aspects of the geometry, for example curvature, of space-time considered as a Lorentzian manifold; that is no physical objects are introduced in space-time, no laws are given, everything is only geometry. We will then extend this setting to more than 5 dimensions, giving a precise mathematical frame for possible additional physical effects, preserving gravitation and electromagnetism.

Motivation & Objective

• To reformulate classical unification of gravity and electromagnetism using differential geometry, avoiding ad hoc physical postulates.
• To extend Kaluza-Klein theory by abandoning the Ricci-flatness requirement in 5D, enabling a natural geometric derivation of charged fluid dynamics.
• To introduce the mathematical concepts of 'type' and 'rigidity' for Lorentzian manifolds to classify energy-momentum structures and constrain physical realizability.
• To generalize the 5D framework to 5+m dimensions, preserving electromagnetism and gravity while allowing for hidden pressures and potentials.
• To show that all classical laws—equations of motion, conservation laws, Maxwell-Lorentz dynamics—arise as geometric consequences of the Bianchi identities in higher-dimensional spacetime.

Proposed method

• Classify Lorentzian manifolds by their Einstein tensor types using positivity conditions on symmetric two-tensors, establishing a geometric classification of energy-momentum structures.
• Apply Cauchy-Choquet-Bruhat theorems to analyze initial value problems and derive rigidity theorems for Lorentzian manifolds, linking causality to geometric constraints.
• Define a 'v-perfect fluid' in 5D as a geometric object with energy-momentum tensor proportional to the square of a timelike vector field, generalizing dust fluids.
• Construct a 5D spacetime metric where the 4D projection of geodesic motion yields the Einstein-Maxwell-Lorentz equations for charged fluids.
• Use the Bianchi identity ∇·G = 0 to derive the equations of motion and conservation laws as geometric identities, without external physical axioms.
• Extend the framework to 5+m dimensions using fiber bundle structures with m hidden dimensions, introducing visible and hidden pressure tensors to model additional physical effects.

Experimental results

Research questions

• RQ1 Can classical electromagnetism be fully geometrically encoded in a 5-dimensional spacetime without introducing additional physical fields?
• RQ2 What geometric constraints (types and rigidity) limit the possible physical realizations of energy-momentum tensors in 4D spacetime?
• RQ3 How can the motion of a charged fluid in 4D spacetime be derived purely from geodesic motion in a 5D Lorentzian manifold?
• RQ4 What is the role of the Ricci curvature in higher-dimensional unification, and can it be non-zero while preserving consistency with 4D physics?
• RQ5 How can higher-dimensional spacetimes (5+m dimensions) be structured to preserve gravity and electromagnetism while allowing for new physical effects like hidden pressures or potentials?

Key findings

• The Einstein-Maxwell-Lorentz equations for charged fluids in 4D spacetime emerge as projections of geodesic motion in a 5D Lorentzian manifold, with no additional physical laws required.
• The equations of motion and conservation of mass and charge are derived purely from the Bianchi identity ∇·G = 0, which holds identically in the 5D geometric framework.
• The electromagnetic field strength F is geometrically defined via the Killing vector Y associated with the fifth dimension, and its divergence ∇·F is shown to relate to curvature and scalar curvature via ∇·F = eX₀ + ½|F|_g Y − P(Y).
• In the 5D setting, the energy-momentum tensor T = μX⊗X + e(X⊗Y + Y⊗X) + γY⊗Y + P describes a charged fluid, where μ is mass density, e is charge density, and P is a traceless tensor representing pressure.
• The model generalizes to 5+m dimensions, where hidden pressure tensors and additional potentials (Newtonian and electromagnetic) emerge naturally from the fiber bundle structure of the total space.
• The paper establishes that the 4D spacetime cannot geometrically describe electromagnetism without additional dimensions, as shown by the rigidity theorem in 4D, which fails to support the required type of energy-momentum tensor for electromagnetism.