Geometry of Area Without Length

TL;DR

This paper explores a generalized geometric framework based on area metrics instead of traditional length-based metrics, aiming to describe space-time in scenarios where length is ill-defined, and derives solutions analogous to classical Einstein solutions.

Contribution

It introduces a novel formulation of gravity using area metrics, extending Einstein's theory to contexts lacking a well-defined notion of length.

Findings

Derived analogues of connections, curvatures, and Einstein tensor for area metric geometry.

Found static spherical solutions including a generalized Schwarzschild solution.

Demonstrated the applicability of the area metric approach to vacuum space-times.

Abstract

To define a free string by the Nambu-Goto action, all we need is the notion of area, and mathematically the area can be defined directly in the absence of a metric. Motivated by the possibility that string theory admits backgrounds where the notion of length is not well defined but a definition of area is given, we study space-time geometries based on the generalization of metric to area metric. In analogy with Riemannian geometry, we define the analogues of connections, curvatures and Einstein tensor. We propose a formulation generalizing Einstein's theory that will be useful if at a certain stage or a certain scale the metric is ill-defined and the space-time is better characterized by the notion of area. Static spherical solutions are found for the generalized Einstein equation in vacuum, including the Schwarzschild solution as a special case.