The Cauchy horizon and the event horizon are two distinct concepts related to black holes in general relativity. Let's explore each of them:
Event Horizon: The event horizon of a black hole is a boundary in spacetime beyond which no information or signals can escape to the outside universe. It is defined as the region from which the escape velocity exceeds the speed of light. Once an object crosses the event horizon, it is inevitably drawn into the black hole's singularity at its center. From an external observer's perspective, any information about events inside the event horizon is forever hidden.
Cauchy Horizon: The Cauchy horizon, also known as the Cauchy surface, is a boundary that lies inside a black hole and separates the black hole's region of spacetime from the region outside the black hole. It is a surface in which both future and past-directed light cones converge, meaning that events on the Cauchy horizon have a unique causal connection to events both inside and outside the black hole.
The Cauchy horizon is derived from the mathematical concept of a Cauchy surface, which is a surface that intersects every time slice in a spacetime manifold. It plays a role in the mathematical formulation of the initial value problem in general relativity, which involves specifying the initial conditions for the gravitational field equations.
However, the Cauchy horizon is a concept that is closely tied to idealized, simplified models of black holes. In reality, the formation of a Cauchy horizon is highly dependent on the specific assumptions made about the black hole's properties, such as its mass, charge, and rotation. In many realistic scenarios, it is believed that the Cauchy horizon is unstable and prone to the formation of singularities or other physical phenomena that invalidate its existence.
It is important to note that the Cauchy horizon and the event horizon are distinct entities with different properties. The event horizon represents the boundary beyond which no information can escape, while the Cauchy horizon is a mathematical construct that plays a role in certain theoretical formulations but may not be realized in physically realistic black holes.