The three-body problem is called the oldest unsolved problem in Mathematics while the problem of Arrow of Time is referred to as the oldest unsolved problem in Physics. The former one goes back to 1687 and the famous English mathematician Isaac Newton, while the latter was highlighted by the Austrian physicist Ludvig Boltzmann two hundred years later. A string of the brightest mathematicians and physicists have tackled these problems over the years, including the French mathematicians Lagrange, Laplace and Poincare, and the German mathematicians and physicists Euler, Jacobi, Dirichlet and Einstein.

The three-body problem has also been quoted as the most important problem of mankind. Briefly, the problem is what happens to a planet like the Earth when it is pulled by two astronomical bodies to different directions at the same time. The pull of the Sun dominates in the case of the Earth and therefore the Earth circles around the Sun. But at the same time, planet Venus pulls the Earth further in, and planets Mars and Jupiter pull the Earth further out. Are these planets able to displace the Earth from its orbit and to make it dive into the Sun, or to leave the vicinity of the Sun and escape to the cold outer space? In either case, it would be the greatest imaginable climatic change on Earth!

The Academy of Science of Paris set special prizes for the solution of the problem in the 18th century, while the king of Sweden Oscar II tried again with a prize in the late 19th century. No solution emerged in spite of the financial incentives. Rather, at the closing of the 19th century it was entirely unclear what the solution of the problem might be, and whether we could trust the Earth taking us around the Sun at a constant distance and in relative comfort, climate wise.

Early in the 20th century there were two opinions: first, that there is no solution at all, and that three body systems are totally unpredictable. This view championed by Henri Poincare led to considerable uncertainty about the ultimate fate of the Earth. The other view led by the Finnish mathematician Karl Sundman claimed that there was a solution which could be represented by a mathematical series, even though with an infinite number of terms. In this view the Earth could be trusted to “behave well”, at least in principle. The work of both scientists was rewarded by special prizes even though the final answer was still elusive.

The computer technology and calculation methods advanced to the stage where the orbits of the three bodies could be finally calculated in late 1960s. Szebehely and Peters in University of Texas calculated the first solution of a special problem in 1967. It showed that the three-body system is unstable. It was unclear if this result was a special case or whether it had general validity. In 1974 I published in my thesis work at Cambridge University the solutions of 25,000 three-body problems, and proved that the special solution of Szebehely and Peters was generally applicable. Thus Poincare’s view prevailed in the sense that any individual solution is unpredictable. However, Poincare did not anticipate that the solutions are predictable in statistical sense, that is, there are well defined probabilities of what might happen, even though no certainty. The thesis work was supervised by Sverre Aarseth and William Saslaw at Cambridge.

Later in the same year Joseph Monaghan arrived from Australia to spend a sabbatical year at Cambridge. He became immediately interested in trying to explain the statistical nature of the three-body problem starting from the first principles of physics. His work showed promise, but also disagreed with my statistical results in important respects. In 2000 I came back to this problem at the University of the West Indies, in Barbados, where I started to enquire why Monaghan’s approach had failed. Was there something wrong with the first principles, or an error in calculation? Since the calculation was extremely complex, it was not immediately clear what the problem was. However, after long pondering on the problem, I realized that Monaghan had neglected an important term in the first equation of his long calculation. When it was properly added, Monaghan’s idea worked magnificently. Thus this represented a solution of the three-body problem at last!

What does this mean to the stability of Earth’s orbit? Unfortunately, even at this stage we cannot say surely what the answer is. This is because the Earth is influenced by more than two bodies. The problem is more than a three-body problem. However, computer calculations seem to indicate that even though the Earth’s orbit is less stable than we think, the Earth will not dive into the Sun nor will it leave the Solar system. Instead the orbit varies a lot, and that means great natural variations in the Earth’s climate. For example, in the median latitudes of the Earth the amount of solar radiation in a given season varies naturally by as much as 20%. For example in Finland, at latitudes between 60 and 70 degrees north, the climate is sometimes like in central Greenland today, i.e. the country is covered by 2 km of ice, while at other times the climate is mild like in today’s France. These cycles influence also the Caribbean, primarily by variations in the level of the ocean, as one can easily witness in Barbados where ancient sea levels are preserved in the levels of the “shelves” around the shoreline.

Let us now come to the second problem, the question of Arrow of Time. Simply, the question is why time flows forward, never backward. Why cannot we turn the clocks (i.e. the time) back for example to correct an error that we regret? After all, time is just a coordinate, fundamentally similar to space coordinates, as Albert Einstein explained already in 1905. I can step sideways, I can step forward, I can jump up and down, but I cannot take a step back in time even if I wish to do so. Why?

Ludvig Boltzmann had an answer to this question: there is a quantity called entropy which can only stay constant or increase. The probability that entropy decreases is so small as to be negligible. Entropy could be described as measure of disorder. For example, if I have a carbonated water bottle under some amount of extra pressure, the gas (carbon dioxide molecules) will escape from the bottle as soon as I open the bottle cover. It is entirely possible, but completely unlikely that the same gas molecules will fly back into the bottle and settle so that I could close the cover and have a “fresh” carbonated water bottle. The process is one way: there is clearly an Arrow of Time there. The gas in the bottle under pressure represents a state of low entropy; the gas released to the surroundings represents a state of high entropy. Thus in this case (as always) the direction of increase of entropy gives the direction of Arrow of Time.

However, as Einstein pointed out, all basic laws of physics are symmetric with respect to time. What happens in forward direction of time could just as well happen if time is reversed. If we make a movie of particles moving around in a container, we could run the movie backward or forward without knowing which is the correct direction. At least, this is what has been thought so far.

Now we come to the solution of the three-body problem which we found to be statistical, not deterministic. Even at microscopic level, three bodies is the simplest system that we could study. In this system there is already a sense of time. This seems odd, since we could run a movie of three bodies moving backward or forwards, and due to the time symmetry of Newton’s law of gravity, we should not be able to discover what is the correct sense of running the movie. But we do. This is because there is a form of entropy called Kolmogorov – Sinai entropy associated with every three-body system, as I have demonstrated with my colleagues Harry Lehto and Pekka Heinamaki of University of Turku, and Arthur Chernin of Moscow University. The eminent Russian mathematicians Kolmogorov and Sinai introduced this modified definition of entropy already in 1950s. When applied to the three-body problem, this concept of entropy gives us the Arrow of Time even at the most basic level, in spite of the time symmetry of the governing law of physics. And fortunately, this time runs in the same direction as the thermodynamic time first introduced by Boltzmann.

So why cannot we step backward in time? Actually, if the connection between Arrow of Time and entropy is correct, then it is not impossible to go back in time, only it is totally unlikely that it would ever happen. Entropy is all about probabilities, and thus it appears that Arrow of Time is also a concept of probability.

We see that the Boltzmann’s Arrow of Time, which applies at macroscopic level, can be extended to the microscopic level, and that the problem of the origin of Arrow of Time is solved. The solution of the oldest problem in mathematics shows the way to the solution of the oldest problem in physics.