Black Holes

The simplest picture of a black hole is that of a body whose gravity is so strong that nothing, not even light, can escape from it. Bodies of this type are already possible in the familiar Newtonian theory of gravity. The “escape velocity” of a body is the velocity at which an object would have to travel to escape the gravitational pull of the body and continue flying out to infinity. Because the escape velocity is measured from the surface of an object, it becomes higher if a body contracts down and becomes more dense. (Under such contraction, the mass of the body remains the same, but its surface gets closer to its center of mass; thus the gravitational force at the surface increases.) If the object were to become sufficiently dense, the escape velocity could therefore exceed the speed of light, and light itself would be unable to escape.

This much of the argument makes no appeal to relativistic physics, and the possibility of such classical black holes was noted in the late 18th Century by Michel (1784) and Laplace (1796). These Newtonian black holes do not precipitate quite the same sense of crisis as do relativistic black holes. While light hurled ballistically from the surface of the collapsed body cannot escape, a rocket with powerful motors firing could still gently pull itself free.

Taking relativistic considerations into account, however, we find that black holes are far more exotic entities. Given the usual understanding that relativity theory rules out any physical process going faster than light, we conclude that not only is light unable to escape from such a body: nothing would be able to escape this gravitational force. That includes the powerful rocket that could escape a Newtonian black hole. Further, once the body has collapsed down to the point where its escape velocity is the speed of light, no physical force whatsoever could prevent the body from continuing to collapse down further – for this would be equivalent to accelerating something to speeds beyond that of light. Thus once this critical amount of collapse is reached, the body will get smaller and smaller, more and more dense, without limit. It has formed a relativistic black hole; at its center lies a spacetime singularity.

For any given body, this critical stage of unavoidable collapse occurs when the object has collapsed to within its so-called Schwarzschild radius, which is proportional to the mass of the body. Our sun has a Schwarzschild radius of approximately three kilometers; the Earth's Schwarzschild radius is a little less than a centimeter. This means that if you could collapse all the Earth's matter down to a sphere the size of a pea, it would form a black hole. It is worth noting, however, that one does not need an extremely high density of matter to form a black hole if one has enough mass. Thus for example, if one has a couple hundred million solar masses of water at its standard density, it will be contained within its Schwarzschild radius and will form a black hole. Some supermassive black holes at the centers of galaxies are thought to be even more massive than this, at several billion solar masses.

The “event horizon” of a black hole is the point of no return. That is, it comprises the last events in the spacetime around the singularity at which a light signal can still escape to the external universe. For a standard (uncharged, non-rotating) black hole, the event horizon lies at the Schwarzschild radius. A flash of light that originates at an event inside the black hole will not be able to escape, but will instead end up in the central singularity of the black hole. A light flash originating at an event outside of the event horizon will escape, but it will be red-shifted strongly to the extent that it is near the horizon. An outgoing beam of light that originates at an event on the event horizon itself, by definition, remains on the event horizon until the temporal end of the universe.

General relativity tells us that clocks running at different locations in a gravitational field will generally not agree with one another. In the case of a black hole, this manifests itself in the following way. Imagine someone falls into a black hole, and, while falling, she flashes a light signal to us every time her watch hand ticks. Observing from a safe distance outside the black hole, we would find the times between the arrival of successive light signals to grow larger without limit. That is, it would appear to us that time were slowing down for the falling person as she approached the event horizon. The ticking of her watch (and every other process as well) would seem to go slower and slower as she got closer and closer to the event horizon. We would never actually see the light signals she emits when she crosses the event horizon; instead, she would seem to be eternally “frozen” just above the horizon. (This talk of “seeing” the person is somewhat misleading, because the light coming from the person would rapidly become severely red-shifted, and soon would not be practically detectable.)
From the perspective of the infalling person, however, nothing unusual happens at the event horizon. She would experience no slowing of clocks, nor see any evidence that she is passing through the event horizon of a black hole. Her passing the event horizon is simply the last moment in her history at which a light signal she emits would be able to escape from the black hole. The concept of an event horizon is a global concept that depends on how the events on the event horizon relate to the overall structure of the spacetime. Locally there is nothing noteworthy about the events at the event horizon. If the black hole is fairly small, then the tidal gravitational forces there would be quite strong. This just means that gravitational pull on one's feet, closer to the singularity, would be much stronger than the gravitational pull on one's head. That difference of force would be great enough to pull one apart. For a sufficiently large black hole the difference in gravitation at one's feet and head would be small enough for these tidal forces to be negligible.

As in the case of singularties, alternative definitions of black holes have been explored. These definitions typically focus on the one-way nature of the event horizon: things can go in, but nothing can get out. Such accounts have not won widespread support, however, and we have not space here to elaborate on them further.

1 The Geometrical Nature of Black Holes

One of the most remarkable features of relativistic black holes is that they are purely gravitational entities. A pure black hole spacetime contains no matter whatsoever. It is a “vacuum” solution to the Einstein field equations, which just means that it is a solution of Einstein's gravitational field equations in which the matter density is everywhere zero. (Of course, one can also consider a black hole with matter present.) In pre-relativistic physics we think of gravity as a force produced by the mass contained in some matter. In the context of general relativity, however, we do away with gravitational force, and instead postulate a curved spacetime geometry that produces all the effects we standardly attribute to gravity. Thus a black hole is not a “thing” in spacetime; it is instead a feature of spacetime itself.

to if our definition is to make sense. The most common method of making this idea precise and rigorous employs the notion of “escaping to infinity.” If a particle or light ray cannot “travel arbitrarily far” from a definite, bounded region in the interior of spacetime but must remain always in the region, the idea is, then that region is one of no escape, and is thus a black hole. The boundary of the region is called the event horizon. Once a physical entity crosses the event horizon into the hole, it never crosses it again.

A careful definition of a relativistic black hole will therefore rely only on the geometrical features of spacetime. We'll need to be a little more precise about what it means to be “a region from which nothing, not even light, can escape.” First, there will have to be someplace to escape

Second, we will need a clear notion of the geometry that allows for “escape,” or makes such escape impossible. For this, we need the notion of the “causal structure” of spacetime. At any event in the spacetime, the possible trajectories of all light signals form a cone (or, more precisely, the four-dimensional analog of a cone). Since light travels at the fastest speed allowed in the spacetime, these cones map out the possible causal processes in the spacetime. If an occurence at an event A is able to causally affect another occurence at event B, there must be a continuous trajectory in spacetime from event A to event B such that the trajectory lies in or on the lightcones of every event along it. (For more discussion, see the Supplementary Document: Lightcones and Causal Structure.)

Figure 1 is a spacetime diagram of a sphere of matter collapsing down to form a black hole. The curvature of the spacetime is represented by the tilting of the light cones away from 45 degrees. Notice that the light cones tilt inwards more and more as one approaches the center of the black hole. The jagged line running vertically up the center of the diagram depicts the black hole central singularity. As we emphasized in Section 1, this is not actually part of the spacetime, but might be thought of as an edge of space and time itself. Thus, one should not imagine the possibility of traveling through the singularity; this would be as nonsensical as something's leaving the diagram (i.e., the spacetime) altogether.

Figure 1: A spacetime diagram of black hole formation
What makes this a black hole spacetime is the fact that it contains a region from which it is impossible to exit while traveling at or below the speed of light. This region is marked off by the events at which the outside edge of the forward light cone points straight upward. As one moves inward from these events, the light cone tilts so much that one is always forced to move inward toward the central singularity. This point of no return is, of course, the event horizon; and the spacetime region inside it is the black hole. In this region, one inevitably moves towards the singularity; the impossibility of avoiding the singularity is exactly like the impossibility of preventing ourselves from moving forward in time.

Notice that the matter of the collapsing star disappears into the black hole singularity. All the details of the matter are completely lost; all that is left is the geometrical properties of the black hole which can be identified with mass, charge, and angular momentum. Indeed, there are so-called “no-hair” theorems which make rigorous the claim that a black hole in equilibrium is entirely characterized by its mass, its angular momentum, and its electric charge. This has the remarkable consequence that no matter what the particulars may be of any body that collapses to form a black hole—it may be as intricate, complicated and Byzantine as one likes, composed of the most exotic materials—the final result after the system has settled down to equilibrium will be identical in every respect to a black hole that formed from the collapse of any other body having the same total mass, angular momentum and electric charge. For this reason Chandrasekhar (1983) called black holes “the most perfect objects in the universe.”

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