Causal relation between regions I and IV of the Kruskal extension

TL;DR

This paper explores a novel extension of the Schwarzschild spacetime using Kruskal coordinates, revealing that regions I and IV are causally connected parts of the same spacetime rather than separate universes.

Contribution

It introduces a 4-dimensional Kruskal extension showing regions I and IV are causally connected within the same spacetime, challenging traditional interpretations.

Findings

Regions I and IV are causally connected.

The extended spacetime includes the conventional Kruskal extension.

Region IV is part of the same exterior Schwarzschild spacetime.

Abstract

By extending the exterior Schwarzschild spacetime in two opposite directions with the Kruskal method, we get an extension which has the same T-X spacetime diagram as has the conventional Kruskal extension, while allowing its regions I and IV to correspond to different directions of the original spacetime. We further extend the exterior Schwarzschild spacetime in all directions and get a 4-dimensional form of the Kruskal extension. The new form of extension includes the conventional one as a part of itself. From the point of view of the 4-dimensional form, region IV of the conventional extension does not belong to another universe but is a portion of the same exterior Schwarzschild spacetime that contains region I. The two regions are causally related: particles can move from one to the other.