Gravitation, electromagnetism and the cosmological constant in purely affine gravity

TL;DR

This paper investigates the purely affine formulation of gravity and electromagnetism, revealing limitations in combining these fields with the cosmological constant and highlighting differences from metric-based theories.

Contribution

It demonstrates that the sum of affine Lagrangians for gravity and electromagnetism is not equivalent to their metric counterparts and is only valid for weak fields.

Findings

Purely affine formulation cannot simply sum gravity and electromagnetism Lagrangians.

The combined affine Lagrangian is only valid for weak electromagnetic fields.

Electromagnetism in affine gravity appears unphysical unless coupled to the cosmological constant.

Abstract

The Eddington Lagrangian in the purely affine formulation of general relativity generates the Einstein equations with the cosmological constant. The Ferraris-Kijowski purely affine Lagrangian for the electromagnetic field, which has the form of the Maxwell Lagrangian with the metric tensor replaced by the symmetrized Ricci tensor, is dynamically equivalent to the Einstein-Maxwell Lagrangian in the metric formulation. We show that the sum of the two affine Lagrangians is dynamically inequivalent to the sum of the analogous Lagrangians in the metric-affine/metric formulation. We also show that such a construction is valid only for weak electromagnetic fields. Therefore the purely affine formulation that combines gravitation, electromagnetism and the cosmological constant cannot be a simple sum of terms corresponding to separate fields. Consequently, this formulation of electromagnetism seems to be unphysical, unlike the purely metric and metric-affine pictures, unless the electromagnetic field couples to the cosmological constant.