Questions: Black Holes (Schwarzschild)

Question 1

Inside the event horizon of a Schwarzschild black hole, the radial coordinate r becomes timelike. What is the physical consequence?

• A. An observer inside can still orbit at constant r if they accelerate hard enough
• B. All future-directed worldlines lead to decreasing r, making the singularity at r = 0 inevitable regardless of the observer's actions
• C. The observer experiences time running backward
• D. The distinction between space and time disappears entirely

Answer: (B)

Inside the horizon, the signs of g_{tt} and g_{rr} swap: r becomes the timelike coordinate and t becomes spacelike. Since all observers must move forward in time (toward decreasing r inside the horizon), reaching r = 0 is as inevitable as reaching tomorrow. No amount of rocket thrust can reverse the inward 'fall' because it is not motion through space but progress through time. The observer's proper time remains well-defined and forward-moving — it is the coordinate roles that have swapped, not the experience of time.

Question 2

A distant observer watching an astronaut fall into a Schwarzschild black hole will see the astronaut cross the event horizon in finite time.

• T. True
• F. False

Answer: False

In the distant observer's Schwarzschild coordinate time t, the astronaut asymptotically approaches the horizon but never crosses it. Light signals from the astronaut become increasingly redshifted and delayed — the last photon emitted just before crossing takes (in principle) infinite coordinate time to reach the distant observer. However, the astronaut's own proper time is finite: they cross the horizon and reach the singularity in a finite, often quite short, proper time. The 'infinite time' is a coordinate effect in the distant frame, not a physical barrier.

Question 3

Explain why the singularity at r = 0 inside a Schwarzschild black hole is better described as a moment in time than a point in space.

Model answer: Inside the horizon, the Schwarzschild coordinate r is timelike (its metric coefficient changes sign from positive to negative). The singularity r = 0 is therefore a surface of constant 'time' — it lies in the future of every observer who has crossed the horizon, regardless of their spatial position or direction of motion. You cannot point toward the singularity or try to avoid it by moving sideways; it is not located at a particular place in space but rather occurs at a particular moment along every possible future worldline. This is fundamentally different from a Newtonian point mass, which occupies a definite spatial location.

This reinterpretation of the singularity is one of the most conceptually striking features of black hole physics. It means that inside the horizon, the question 'where is the singularity?' is ill-posed — the correct question is 'when will I reach it?'

Question 4

What is the maximal analytic extension of the Schwarzschild spacetime, and why is most of it not physically realized?

Model answer: The Kruskal-Szekeres coordinates reveal that the full analytic extension contains four regions: (I) the exterior of the black hole, (II) the black hole interior (future singularity), (III) a 'parallel' asymptotically flat exterior, and (IV) a white hole interior (past singularity). The white hole is the time-reverse of a black hole — matter can emerge from it but nothing can enter. Regions III and IV are connected to I and II through the Einstein-Rosen bridge (wormhole). However, in a realistic gravitational collapse, a star forms the black hole at a finite time, so the spacetime before collapse is not vacuum Schwarzschild. The white hole, the parallel universe, and the wormhole are artifacts of the eternal vacuum solution and are not present in the spacetime of a collapsing star.

The maximal extension is a mathematical curiosity of the exact vacuum solution. Penrose diagrams make the causal structure transparent and show that the wormhole is non-traversable even in the extended solution — no timelike or null path can travel from region I to region III.