[Paper Review] D=4 Einstein gravity from higher D CS and BI gravity and an alternative to dimensional reduction

Horaƫiu Năstase|ArXiv.org|Mar 5, 2007

Black Holes and Theoretical Physics18 references3 citations

TL;DR

This paper proposes a novel dimensional reduction mechanism for gravity, embedding 4D Einstein gravity with a topological term (Born-Infeld gravity) into higher-dimensional Chern-Simons (CS) gravity in odd dimensions, particularly 5D and 11D. By formulating gravity in the vielbein-spin connection formalism and generalizing descent-like reduction, it shows that 4D Einstein gravity coupled to a topological term arises naturally from 5D CS gravity, offering a potential path to defining quantum 4D gravity via topological higher-dimensional theories, analogous to Witten's 3D CS approach.

ABSTRACT

An alternative to usual dimensional reduction for gravity is analyzed, in the vielbein-spin connection formulation. Usual 4d Einstein gravity plus a topological term (the "Born-Infeld" Lagrangian for gravity), is shown to be obtained by a generalized dimensional reduction from 5d Chern-Simons gravity. Chern-Simons gravity in d=2n+1 is dimensionally reduced to CS gravity in d=2n-1 via a mechanism similar to descent equations. The consistency of the dimensional reduction in both cases is analyzed. The dimensional reduction of d=2n+2 Born-Infeld gravity to d=2n BI gravity, as well as d=2n BI gravity to d=2n-1 CS gravity is hard to achieve. Thus 4d gravity (plus a topological term) can be embedded into d=2n+1 CS gravity, including 11d CS, whose supersymmetric version could possibly be related to usual 11d supergravity. This raises the hope that maybe 4d quantum Einstein gravity could be embedded in a well defined quantum theory, similar to Witten's treatment of 3d quantum Einstein gravity as a CS theory.

Motivation & Objective

• To develop an alternative to standard dimensional reduction for gravity, particularly in the vielbein-spin connection formulation.
• To show that 4D Einstein gravity with a topological term (Born-Infeld gravity) can be derived from 5D Chern-Simons gravity through a generalized dimensional reduction mechanism.
• To explore the consistency and feasibility of reducing higher-dimensional topological theories (CS and BI gravity) to lower-dimensional ones, especially from d=2n+2 BI to d=2n BI and d=2n BI to d=2n-1 CS.
• To investigate the potential embedding of 4D quantum gravity into a well-defined higher-dimensional topological quantum theory, such as 11D CS gravity.
• To connect this construction to 11D supergravity and the OSp(1|32)×OSp(1|32) gauge group, suggesting a possible link to M-theory.

Proposed method

• Formulates gravity using the vielbein and spin connection, treating the Einstein-Hilbert action as a gauge-like theory for the Poincaré group.
• Applies a generalized dimensional reduction procedure inspired by descent equations, reducing d=2n+1 Chern-Simons gravity to d=2n-1 CS gravity via intermediate d=2n topological theory.
• Introduces a specific reduction ansatz: setting the extra-dimensional component of the spin connection to a constant, ω_{2n+1}^{2n+1,2n+2} = c, to reduce 5D CS to 4D Born-Infeld gravity.
• Analyzes the consistency of the reduction by checking whether the derivative of the reduced action reproduces the lower-dimensional action, particularly for BI gravity.
• Considers the signature (2n-2,2) for the intermediate d=2n theory, allowing interpretation as a pure spin connection theory with SO(2n-2,2) gauge group.
• Explores the possibility of embedding 4D Born-Infeld gravity into 11D Chern-Simons gravity with SO(10,2) gauge group, and relates this to 11D supergravity via group contraction in the λ→0 limit.

Experimental results

Research questions

• RQ1 Can 4D Einstein gravity with a topological term be consistently derived from 5D Chern-Simons gravity via a non-geometric dimensional reduction?
• RQ2 Is there a generalized dimensional reduction mechanism that allows reducing d=2n+1 CS gravity to d=2n-1 CS gravity without relying on geometric compactification?
• RQ3 Can d=2n+2 Born-Infeld gravity be reduced to d=2n BI gravity, and if so, under what conditions?
• RQ4 Does the intermediate d=2n theory resulting from the reduction of d=2n+1 CS gravity to d=2n-1 CS gravity have a consistent topological or gauge-theoretic interpretation?
• RQ5 Can the embedding of 4D Born-Infeld gravity into higher-dimensional topological theories like 11D CS gravity provide a framework for defining quantum 4D Einstein gravity?

Key findings

• 4D Einstein gravity with a topological term (Born-Infeld gravity) can be obtained from 5D Chern-Simons gravity via a generalized dimensional reduction that does not rely on geometric compactification.
• The reduction mechanism is analogous to descent equations, allowing d=2n+1 CS gravity to reduce to d=2n-1 CS gravity through an intermediate d=2n topological theory.
• The reduction from d=2n+2 BI gravity to d=2n BI gravity is not straightforward and requires specific conditions on the action and gauge group reduction, which are not fully satisfied in the current construction.
• The intermediate d=2n theory resulting from the reduction of d=2n+1 CS gravity is a topological theory with SO(2n-2,2) gauge group, which can be interpreted as a pure spin connection theory in (2n-2,2) signature.
• The 4D Born-Infeld theory can be embedded into 11D Chern-Simons gravity with SO(10,2) gauge group, suggesting a possible connection to 11D supergravity and M-theory.
• In the λ→0 limit, the OSp(1|32)×OSp(1|32) gauge group of 11D Chern-Simons supergravity contracts to the d’Auria-Fré group, recovering standard 11D supergravity.