Visualizing curved spacetime

TL;DR

This paper introduces a visualization method for curved spacetime by representing it as a curved surface with local Minkowski systems, aiding understanding of phenomena like black hole horizons and cosmological models.

Contribution

It proposes a novel visualization technique that maps curved spacetime to a surface with local coordinate systems, enhancing conceptual understanding of general relativity.

Findings

Visualizations of black hole horizons are possible.

The method illustrates gravitational time dilation effects.

The approach can represent cosmological models in 3D space.

Abstract

I present a way to visualize the concept of curved spacetime. The result is a curved surface with local coordinate systems (Minkowski Systems) living on it, giving the local directions of space and time. Relative to these systems, special relativity holds. The method can be used to visualize gravitational time dilation, the horizon of black holes, and cosmological models. The idea underlying the illustrations is first to specify a field of timelike four-velocities. Then, at every point, one performs a coordinate transformation to a local Minkowski system comoving with the given four-velocity. In the local system, the sign of the spatial part of the metric is flipped to create a new metric of Euclidean signature. The new positive definite metric, called the absolute metric, can be covariantly related to the original Lorentzian metric. For the special case of a 2-dimensional original metric, the absolute metric may be embedded in 3-dimensional Euclidean space as a curved surface.