You might have heard of the three-body problem, how there is no exact solution to predicting the behavior of three interacting gravitational celestial bodies under Newton’s laws. The best you can do is to have solutions for specific cases or work with numerical solutions and approximations. But if you are working in general relativity, not even the two-body problem has a general solution, just approximations. Now, researchers have found a way to get better solutions to the interactions between black holes as seen in the collisions that have produced detectable gravitational waves.
According to Einstein’s theory of general relativity, when two heavy mass objects like black holes or neutron stars collide, they emit gravitational waves, ripples in space-time that can be detected with observatories we have here on Earth, like LIGO or Virgo in Italy. However, interpretations of these detections rely on accurately modeling what the signal would look like in the first place. This new, highly precise prediction of the gravitational waves produced by colliding black holes could lead to better models, especially as new detectors come online.
These results represent the most accurate solution to Einstein’s field equations generated to date.
Professor Jan Plefka
The team approached the problem using perturbation theory, which starts by solving a simple approximation to the problem and then solving the details in increasing complexity.
They provided mathematical expressions for how the momentum of two interacting black holes changes, the angle at which they might deflect, and the total energy radiated. It is in this latter quantity that curious mathematical structures surprisingly popped up.
“Unlike its Newtonian counterpart, which is solved by Kepler’s Laws, the relativistic two-body problem cannot be solved exactly. These results represent the most accurate solution to Einstein’s field equations generated to date,” Professor Jan Plefka, from Humboldt-Universität zu Berlin, told IFLScience.
“The radiated energy, in particular, we found contains a new class of mathematical functions known as Calabi-Yau periods. While these functions are well-known in algebraic geometry and string theory, this marks the first time they have been shown to describe a genuine physical process.”
Calabi–Yau manifolds are the six-dimensional analogous of a torus, a donut-shaped shape. While theoretical physicists are familiar with them, this is the first time that they have been linked to a genuine measurable quantity. However, they only appear in the complex version of the solution, not in the simpler approximations.
“These analytic results give us new insights into the structure of the two-body problem in General Relativity. Most immediately, they will be used to build high-precision models for both current-generation (LIGO, Virgo, KAGRA) and future-generation gravitational wave detectors (LISA, Einstein Telescope, Cosmic Explorer), which will measure gravitational waves with unprecedented resolution.” Professor Plefka told IFLScience.
Yet, it’s not just about the observations. This work might lead to a deeper understanding of the mathematical underpinnings of reality.
“As physicists, we are also intrigued by what kinds of mathematical structures and geometries are necessary to describe nature. In this regard, the appearance of Calabi-Yau periods represents a step into a whole new class of mathematical functions. We expect these periods to show up in other branches of physics, including collider physics, and the mathematical techniques we employed to calculate the relevant integrals will no doubt also apply there,” Professor Plefka told IFLScience.
A paper describing this work is published in the journal Nature.
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