On Distance and Area

TL;DR

This paper proposes replacing the metric tensor in four-dimensional Riemannian manifolds with a geometrical simplex whose face areas encode metric information, potentially simplifying the concept of distance to area.

Contribution

It introduces a novel approach to geometry by substituting the metric tensor with a simplex's face areas, offering an alternative foundation for understanding distances in manifolds.

Findings

Metric tensor components correspond to simplex face areas.

Distance concept can be reduced to area in four-dimensional manifolds.

Potential applications in quantum gravity thermodynamics.

Abstract

We seek for an alternative to the metric tensor $g_{\mu\nu}$ as a fundamental geometrical object in four-dimensional Riemannian manifolds. We suggest that the metric tensor $g_{\mu\nu}(P)$ at a given point $P$ of a manifold may be replaced by a four-dimensional geometrical simplex $\sigma^^4(P)$ embedded to the tangent space $T_P$ of the point $P$. The number of two-faces, or triangles, of $\sigma^4(P)$ is the same as is the number of independent components of $g_{\mu\nu}(P)$, and hence we may replace the components of $g_{\mu\nu}$ by the two-face areas of $\sigma^4(P)$. In this sense the concept of distance may, in four-dimensional Riemannian manifolds, be reduced to the concept of area. This result may find some applications in the thermodynamical approaches to quantum gravity.