Sjors Heefer’s PhD research investigates gravitational waves and spacetime using Finsler geometry, with the aim of reconciling general relativity with quantum mechanics. His findings support the Finslerian nature of spacetime, in line with observations of gravitational waves. Credit: SciTechDaily.com
Investigations gravitational waves and their relationship with Finsler geometry provide new insights into spacetime and suggest ways to harmonize relativity and quantum mechanics.
When talking about our universe, it is often said that “matter tells spacetime how to bend, and bent spacetime tells matter how to move.” This is the essence of Albert Einstein’s famous general theory of relativity and describes how planets, stars and galaxies move and affect the space around them. Although general relativity captures much of the large in our universe, it is at odds with the small in physics, as described by quantum mechanics. For his PhD research, Sjors Heefer investigated gravity in our universe, with his research having implications for the exciting field of gravitational waves, and perhaps influencing the way in which the large and the small of physics can be reconciled in the future .
Unveiling the Universe: Einstein’s Theories and Beyond
Just over a hundred years ago, Albert Einstein revolutionized our understanding of gravity with his general theory of relativity. “According to Einstein’s theory, gravity is not a force, but arises as a result of the geometry of the four-dimensional spacetime continuum, or spacetime for short,” says Heefer. “And it is crucial for the emergence of fascinating phenomena in our universe, such as gravitational waves.”
Large objects, such as the sun or galaxies, warp the spacetime around them, and other objects then move along the straightest possible paths – known as geodesics – through this curved spacetime.
However, due to the curvature, these geodesics are not straight at all in the usual sense. For example, in the case of the planets in the solar system, they describe elliptical orbits around the sun. In this way, general relativity elegantly explains the motion of the planets and many other gravitational phenomena, ranging from everyday situations to black holes and the Big Bang. As such, it remains a cornerstone of modern physics.
Resolving Theories: Quantum Mechanics vs. General Relativity
Although general relativity describes a large number of astrophysical phenomena, it conflicts with another fundamental theory of physics: quantum mechanics.
“Quantum mechanics suggests that particles (such as electrons or muons) exist in multiple states simultaneously until they are measured or observed,” says Heefer. “Once measured, they randomly select a state due to a mysterious effect called ‘wave function collapse’.”
In quantum mechanics, a wave function is a mathematical expression that describes the position and state of a particle, such as an electron. And the square of the wave function leads to a set of probabilities of where the particle might be. The greater the square of the wave function at a given location, the greater the probability that a particle will be at that location after it has been observed.
“All matter in our universe appears to be subject to the strange probabilistic laws of quantum mechanics,” Heefer notes. “And the same goes for all the forces of nature – except gravity. This discrepancy leads to deep philosophical and mathematical paradoxes, and resolving them is one of the most important challenges in fundamental physics today.”
Bridging the gap with Finsler geometry
One approach to resolving the clash between general relativity and quantum mechanics is to expand the mathematical framework behind general relativity.
In terms of mathematics, general relativity is based on pseudo-Riemannian geometry, a mathematical language capable of describing most of the typical shapes that spacetime can take.
“However, recent discoveries indicate that the spacetime of our universe may be beyond the reach of pseudo-Riemannian geometry and can only be described by Finsler geometry, a more advanced mathematical language,” says Heefer.
Time for Finsler to shine
In Finsler geometry – named after the German and Swiss mathematician Paul Finsler – the distance between two points – A and B – does not only depend on the location of the two points. It also depends on whether one travels from A to B or vice versa.
“Imagine you are walking to a point at the top of a hill. Climbing the steep slope to the point will take a lot of energy to cover the distance, and it can take a very long time. The way back down, on the other hand, will be much easier and take much less time. In the Finsler geometry this can be taken into account by assigning a greater distance to the way up than to the way down.”
Rewriting general relativity using the mathematics of Finsler geometry leads to Finsler gravity, a more powerful theory of gravity, which captures everything in the universe explained by general relativity, and potentially much more than that.
Exploring the possibilities of Finsler gravity
To explore the possibilities of Finsler’s gravity, Heefer had to analyze and solve a certain field equation.
Physicists like to describe everything in nature in terms of fields. In physics, a field is simply something that has a value at every point in space and time.
A simple example is temperature; at any given time, every point in space has a certain temperature.
A slightly more complex example is that of the electromagnetic field. At any given time, the value of the electromagnetic field at a given point in space tells us the direction and magnitude of the electromagnetic force that a charged particle, such as an electron, would experience if it were at that point.
And when it comes to the geometry of spacetime itself, it is also described by a field, namely the gravitational field. The value of this field at a given point in spacetime tells us the curvature of spacetime at that point, and it is this curvature that manifests itself as gravity.
Discovery of new spacetime geometries
Heefer turned to the vacuum field equation of Christian Pfeifer and Mattias NR Wohlfarth, the equation that governs this gravitational field in empty space. In other words, this equation describes the possible shapes that the geometry of spacetime could take in the absence of matter.
Heefer: “In a good approximation, this includes the entire interstellar space between stars and galaxies, as well as the empty space around objects such as the Sun and Earth. By carefully analyzing the field equation, several new types of spacetime geometries have been identified.”
The age of gravitational waves
A particularly exciting discovery from Heefer’s work concerns a class of spacetime geometries that represent gravitational waves: ripples in the fabric of spacetime that propagate at the speed of light and can be caused, for example, by the collision of neutron stars or black holes.
The first direct detection of gravitational waves on September 14e2015 marked the beginning of a new era in astronomy, allowing scientists to explore the universe in a whole new way.
Since then, many observations of gravitational waves have been made. Heefer’s research shows that these are all consistent with the hypothesis that our spacetime has a Finslerian character.
The future of Finsler gravity research
Although Heefer’s results are promising, they only scratch the surface of the implications of the Finsler gravity field equation.
“The field is still young and further research in this direction is actively underway,” says Heefer. “I am optimistic that our results will play an important role in deepening our understanding of gravity and I hope that they may eventually even shed light on reconciling gravity with quantum mechanics.”
Title of thesis: Finsler geometry, spacetime and gravity: from metrizability of Berwald spaces to exact vacuum solutions in Finsler gravity. Supervisors: Luc Florack and Andrea Fuster.