Causal relation between regions I and IV of the Kruskal extension

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.