The three-body problem shows us why we can’t accurately calculate the past

Our universe is driven by cause and effect. What happens now leads directly to what happens later. Because of this, many things in the universe are predictable. We can predict when a solar eclipse will occur, or how to launch a rocket that will take a spacecraft to Mars. This also works in reverse. By looking at events now, we can work backward to understand what happened before. We can, for example, look at the motion of galaxies today and know that the cosmos was once in the hot dense state we call the big bang.

This is possible thanks to a property of physics known as time symmetry. The laws of physics work the same way regardless of the direction of time. If you watch an animation of an orbiting planet, you have no way to know whether it is running forward or backward. The causality of physics allows for causes to be effects and effects to be causes. There is no preferred direction for time.

Is this animation running forward in time, or backward? Credit: Wikipedia user WillowW

But hold on, you might say, what about thermodynamics and entropy? My coffee always cools down while sitting on my desk, and if I drop my mug on the floor I can’t unshatter it. Doesn’t that give time a unique direction? Not quite.

Thermodynamics is statistical in nature. Entropy will indeed tend to increase over time, but that’s because there are many more possible disordered states than ordered ones. That’s a bit of an oversimplification, but it’s good enough for everyday life. If I toss a handful of sand in the air, the grains will almost certainly land on the ground in some random pattern. However, there is an infinitely small chance that they land in a perfect circle. The odds are so tiny we will never see it happen, but it isn’t impossible. Chaotic systems are nearly impossible to predict, but we could (in principle) predict them with enough information. Because of time symmetry, we could also work back to the initial state of a chaotic system.

This is known as retrodiction. It is the ability to “predict” past events from current ones, and it lies at the heart of fundamental physics. One thing we’ve learned about quantum physics, classical physics, and thermodynamics is that they all come down to information. The state of any system contains all the information you need to predict what will happen next. This means that information is a conserved quantity, and like energy can’t be created or destroyed.

Could a black hole keep your secrets safe? Credit: NASA

At least that’s what we think. One of the big unanswered questions is whether or not conservation of information applies to black holes. If I toss my personal diary into a black hole it can never escape. Once it crosses the event horizon, the diary can never escape. Does that mean my deepest secrets are forever safe? This information paradox has huge implications for quantum gravity, but that’s a story for another time.

But could retrodiction fail even without invoking event horizons or quantum physics? Since classical physics is deterministic, retrodiction should always be possible. But a new study argues against that idea.

In this work, the team ran computer models of three massive black holes in a gravitational dance. With each simulation, they shifted the initial positions of the black holes to see how similar or different their motions were over time. This kind of three-body problem is a classic example of a chaotic system. There is no exact solution for three-body problems, so it’s a great way to study how predictable a system might be.

As you might expect, when you vary the initial conditions you can get very different results. Small differences lead to large ones over time. We’ve known this about chaotic systems for decades. But the team found that the tiniest shifts can lead to big variations. When they made the shifts as small as a plank length, most of the simulations remained really consistent, but about 5% of them still varied widely.

Two systems (white and red) differ by a Planck length, but still diverge. Credit: Boekholt

This is interesting because a Plank length is about the limit of scale for quantum systems. Smaller than that and known laws of physics break down. Since the bodies in the mode are large black holes, this isn’t some effect of quantum uncertainty. It also isn’t some uncertainty in their simulation. The unpredictability of this three-body system seems to be intrinsic.

So we can’t always predict the future. What else is new? But since the laws of gravity are time-reversible, this also means for some systems we can’t know their origin. Not even in principle.

Before you think this throws all of science out the window, keep in mind that this is about the limit of what can be known. We can still understand the history of the universe by what we see today. But this could mean that information isn’t always conserved, even in a simple classical system. If that’s true, it could change the way we understand the most fundamental essence of physics.

Reference: Boekholt, T. C. N., S. F. Portegies Zwart, and Mauri Valtonen. “Gargantuan chaotic gravitational three-body systems and their irreversibility to the Planck length.”