[Paper Review] General relativity with a topological phase: an action principle

Lee Smolin, Artem Starodubtsev|ArXiv.org|Nov 18, 2003

Black Holes and Theoretical Physics2 references41 citations

TL;DR

This paper proposes a unified action principle that dynamically unifies general relativity and topological field theory by introducing a scalar field that triggers phase transitions between distinct physical regimes: Palatini gravity, Ashtekar gravity, SO(5) BF theory, and F∧F theory. The key result is that phase boundaries resemble isolated horizons, suggesting a dynamical mechanism for gravitational phase transitions with implications for quantum gravity and black hole physics.

ABSTRACT

An action principle is described which unifies general relativity and topological field theory. An additional degree of freedom is introduced and depending on the value it takes the theory has solutions that reduce it to 1) general relativity in Palatini form, 2) general relativity in the Ashtekar form, 3) $F\wedge F$ theory for SO(5) and 4) $BF$ theory for SO(5). This theory then makes it possible to describe explicitly the dynamics of phase transition between a topological phase and a gravitational phase where the theory has local degrees of freedom. We also find that a boundary between adymnamical and topological phase resembles an horizon.

Motivation & Objective

• To develop a single action principle that dynamically unifies general relativity and topological field theories (TFTs), allowing for phase transitions between them.
• To address the long-standing question of whether high-energy topological phases could dynamically transition into low-energy gravitational phases.
• To explore the physical nature of phase boundaries between topological and dynamical gravitational phases, particularly their resemblance to black hole horizons.
• To generalize the framework to include non-constant scalar fields that could govern the shape and dynamics of phase boundaries.
• To provide a field-theoretic mechanism for the emergence of local degrees of freedom in gravity via spontaneous symmetry breaking from SO(5) to SO(4).

Proposed method

• Formulate a modified BF-theory action for SO(5) gauge group with a scalar field Γ introduced via a non-linear term in the second-order term of the action.
• Use γ-matrices and SO(5) Clifford algebra to express the action in terms of trace structures involving γA, γ5, and curvature FAB.
• Derive equations of motion from the action, showing that different solutions correspond to distinct physical theories depending on the value of the scalar field Γ.
• Analyze solutions by decomposing the SO(5) connection and fields into 4+1 dimensions, identifying GR in Palatini and Ashtekar forms, and BF/F∧F theories.
• Study phase boundaries by introducing a step-function profile for Γ across a hypersurface, leading to singular contributions in the action variation.
• Impose boundary conditions by requiring the variation to vanish, leading to FAB = 0 and torsionless conditions on the boundary, analogous to isolated horizon conditions.

Experimental results

Research questions

• RQ1 Can a single action principle dynamically unify general relativity and topological field theories through a scalar field parameter?
• RQ2 What are the physical conditions and equations of motion that arise when the scalar field takes different values, leading to distinct phases?
• RQ3 How do phase boundaries between topological and gravitational phases behave, and what constraints do they impose on the fields?
• RQ4 To what extent do the boundary conditions of such phase transitions resemble those of isolated horizons in general relativity?
• RQ5 Can the scalar field Γ be promoted to a dynamical field that governs the shape and dynamics of phase boundaries in a non-trivial spacetime background?

Key findings

• The action admits four distinct solutions: Palatini gravity, Ashtekar gravity (with Immirzi parameter 1), SO(5) BF theory, and SO(5) F∧F theory, depending on the value of the scalar field Γ.
• When Γ = 0, the theory reduces to BF-theory for SO(5), a topological field theory with no local degrees of freedom.
• When Γ = 1, the theory becomes Palatini gravity, a first-order formulation of general relativity with local degrees of freedom.
• When Γ = 1 and the self-dual projection is applied, the theory yields the Ashtekar formulation of Euclidean gravity with Immirzi parameter 1.
• Phase boundaries between topological and gravitational phases are governed by the condition FAB = 0 on the boundary, which corresponds to the isolated horizon boundary condition in general relativity.
• The singular variation at the phase boundary leads to a constraint that matches the zero-torsion and curvature conditions characteristic of isolated horizons, suggesting a deep physical analogy.