# Instanton representation of Plebanski gravity. Consistency of the initital value constraints under time evolution

## TL;DR

This paper proves that the initial value constraints in the instanton representation of Plebanski gravity remain consistent under time evolution, using Lagrangian equations rather than the Dirac procedure, and compares with Ashtekar formulation.

## Contribution

It demonstrates the preservation of initial value constraints in the instanton representation of Plebanski gravity through a novel Lagrangian approach and contextualizes it with Ashtekar variables.

## Key findings

- Constraint surface is preserved under time evolution.
- Lagrangian equations effectively analyze constraint consistency.
- Comparison with Ashtekar formulation clarifies the results.

## Abstract

The instanton representation of Plebanski gravity provides as equations of motion a Hodge self-duality condition and a set of `generalized' Maxwell's equations, subject to gravitational degrees of freedom encoded in the initial value constraints of general relativity. The main result of the present paper will be to prove that this constraint surface is preserved under time evolution. We carry this out not using the usual Dirac procedure, but rather the Lagrangian equations of motion themsleves. Finally, we provide a comparison with the Ashtekar formulation to place these results into overall context.