“If black holes truly had no entropy, then any time an object fell into a black hole, its entropy would effectively be deleted, reducing the entropy of the universe and violating the second law of thermodynamics.”

Black holes are mystifying objects that have captivated our imaginations since their existence was first proposed. The most striking feature of a black hole is its event horizon—a boundary from within which nothing can escape. Objects can cross the event horizon from outside to inside, but once they do, they can never cross back, nor can any information about them; anything that crosses the event horizon of a black hole is cut off entirely from the outside universe.

For many years, the existence of black holes seemed to threaten a fundamental tenet of modern physics called the second law of thermodynamics. This law helps us to distinguish the past from the future, thus defining an “arrow of time.” To understand why black holes posed this threat, we need to discuss time reversal and entropy.

Entropy and the Arrow of Time

Based on our observations, the laws of physics are (mostly) invariant under time reversal. What does this mean? Imagine a friend shows you the following video: A pendulum swings across the screen from left to right. Is this video being played normally or in reverse? Well, you have certainly seen a pendulum swing the other way before. If the laws of physics do not change under time reversal, then there is actually no way to tell: Physics looks the same with time playing forward or backward.

However, this doesn’t seem to jibe with our daily experience. Consider another video in which a bunch of ceramic shards fly up off the floor and assemble themselves into a coffee mug before coming to rest on a table. Is this video being played forward or backward? Most people would reasonably guess that the video was being played in reverse. If the laws of physics are truly invariant under time reversal, why does this intuition seem so obvious to us? The reason is that, although the laws of physics technically allow for this bizarre process to occur as shown in the video, the fact that the broken mug is made up of many, many particles means that it is essentially impossible for it to spontaneously reassemble itself.

This concept is formalized by the second law of thermodynamics, which tells us that a certain quantity, the entropy S, of any isolated system cannot decrease over time (but can increase). In other words, the change in entropy cannot be negative:

≥ 0 .

The entropy S is a statistically defined notion measuring our lack of knowledge about the underlying state of a system when we only know “macroscopic” (large-scale) information about it. By “state” here, we mean the exact configuration of each particle making up the entire system. For example, consider a box filled with gas. While we can easily measure the temperature and pressure of the gas, it is practically impossible for us to know the position and velocity of each gas particle within the box. In fact, there are many configurations of particle positions and velocities, i.e. states, that would give rise to the same temperature and pressure. The entropy encodes our ignorance of which specific state the system is actually in.

The greater the number of states that are consistent with the same temperature and pressure, the greater the entropy.

The fact that the entropy cannot decrease over time—but can increase—follows from invariance under time reversal combined with an additional property called causality. Together, these tell us that any single state of a system corresponds to exactly one state at any point in the past or in the future—no more, and no fewer. For example, one state cannot become two states at some point in the future, and two states cannot become one state.

Now consider what happens when we open our box of gas into a large room. If the gas starts in the box and then flows out to fill the room, as in Figure 1a, then we can easily satisfy the rule that each initial state in the box evolves to a unique final state in the room. If we were paying close attention to every particle in the room during this process, the entropy couldn’t increase because each initial state evolves to a single final state, but we can’t keep track of so many variables; all we are able to do is measure the temperature and pressure after opening the box, and we will find that there are many, many more possible states of the gas in the whole room that are consistent with the new temperature and pressure. During this process, we lose information about the exact configuration of the particles, and thus the entropy increases. If instead the gas starts in the room and then flows into the box, as in Figure 1b, then the vast majority of the initial states in the room have nowhere to go—there simply aren’t enough states in the box. Thus, the entropy cannot decrease!

The second law of thermodynamics now gives us some sense of an “arrow of time.” Despite the fact that the laws of physics are time-reversible, the statistical notion of entropy allows us to define a forward direction for time: Time flows forward in the direction that entropy increases! This is why we feel that a video of a spontaneously reassembling coffee mug must be playing in reverse.

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