Sjors Heefer’s doctoral research explores gravitational waves and spacetime using Finsler geometry, with the goal of reconciling general relativity with quantum mechanics. His findings support the Finslerian nature of spacetime, aligning with observations of gravitational waves. Credit: SciTechDaily.com
Investigations into
” data-gt-translate-attributes=”({“attribute”:”data-cmtooltip”, “format”:”html”})” tabindex=”0″ role=”link”>gravitational waves and their relationship to Finsler geometry provide new insights into spacetime, suggesting ways to harmonize relativity and quantum mechanics.
When talking about our universe, it is often said that “matter tells spacetime how to bend, and curved spacetime tells matter how to move.” This is the essence of Albert Einstein’s famous theory of general relativity, which describes how planets, stars, and galaxies move and influence the space around them. While general relativity captures much of what is big in our universe, it is at odds with what is small in physics as described by quantum mechanics. For his doctoral thesis, Sjors Heefer explored gravity in our universe, his research having implications for the exciting field of gravitational waves, and perhaps influencing how the big and the small in physics can be reconciled in the future.
Unveiling the Universe: Einstein’s Theories and Beyond
A little over a hundred years ago, Albert Einstein revolutionized our understanding of gravity with his theory of general relativity. “According to Einstein’s theory, gravity is not a force, but rather a result of the geometry of the four-dimensional space-time continuum, or spacetime for short,” Heefer explains. “And it’s at the heart of the emergence of fascinating phenomena in our universe, like gravitational waves.”
Massive objects, such as the Sun or galaxies, warp the spacetime around them, and other objects then move along the straightest possible paths – otherwise called geodesics – through this curved spacetime.
Due to their curvature, however, these geodesics are not at all straight in the usual sense of the term. In the case of the planets of the solar system, for example, they describe elliptical orbits around the Sun. General relativity thus elegantly explains the movement of the planets as well as many other gravitational phenomena, ranging from everyday situations to black holes and the Big Bang. It therefore remains a cornerstone of modern physics.
Solving the theories: quantum mechanics and general relativity
Although general relativity describes a multitude of astrophysical phenomena, it conflicts with another fundamental theory of physics: quantum mechanics.
“Quantum mechanics suggests that particles (like electrons or muons) exist in multiple states at the same time until they are measured or observed,” Heefer explains. “Once measured, they randomly select a state due to a mysterious effect called ‘wave function collapse.’”
In quantum mechanics, a wavefunction is a mathematical expression that describes the position and state of a particle, such as an electron. The square of the wave function gives a set of probabilities of locating the particle. The larger the square of the wave function at a given location, the higher the probability that a particle will be located at that location once observed.
“All matter in our universe appears to be subject to the strange probabilistic laws of quantum mechanics,” notes Heefer. “And the same goes for all forces of nature, except gravity. This divergence leads to profound philosophical and mathematical paradoxes, and resolving them is one of the main challenges of fundamental physics today.
Bridging the Gap with Finsler Geometry
One approach to resolving the conflict between general relativity and quantum mechanics is to expand the mathematical framework behind general relativity.
In mathematical terms, general relativity is based on pseudo-Riemannian geometry, which is a mathematical language capable of describing most of the typical shapes that space-time can take.
“Recent discoveries, however, indicate that the space-time of our universe may fall outside the scope of pseudo-Riemannian geometry and can only be described by Finsler geometry, a more advanced mathematical language,” explains Heefer.
It’s Finsler’s time to shine
In Finsler geometry – named after the German-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 is traveling from point A to point B or vice versa.
“Imagine that you are walking to a point on the top of a hill. Going up the steep hill to this point costs you a lot of energy to cover the distance, and it can take you a very long time. Going back, on the other hand, will be much easier and take much less time. In Finsler geometry, this can be taken into account by assigning a greater distance to the ascent than to the descent. »
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 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 would be temperature, for example; at any given time, each point in space is associated with 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 certain 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 location.
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 point in spacetime tells us the curvature of spacetime at that point, and it is this curvature that manifests itself in the form of gravity.
Discovery of new spatio-temporal geometries
Heefer then turned to Christian Pfeifer and Mattias N. R. Wohlfarth’s vacuum field equation, which is the equation that governs this gravitational field in empty space. In other words, this equation describes the possible forms that the geometry of spacetime could take in the absence of matter.
Heefer: “To a good approximation, this includes all interstellar space between stars and galaxies, as well as the empty space surrounding objects such as the Sun and Earth. By carefully analyzing the field equation, several new types of space-time geometries have been identified.
The era of gravitational waves
One 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 by colliding neutron stars or black holes, for example.
The first direct detection of gravitational waves on September 14th2015 marked the dawn of a new era in astronomy, allowing scientists to explore the universe in an entirely new way.
Since then, many observations of gravitational waves have been made. Heefer’s research indicates that all of this is consistent with the hypothesis that our spacetime has a Finslerian nature.
The Future of Finsler’s Gravity Research
Although Heefer’s results are promising, they only scratch the surface of the implications of Finsler’s field equation for gravity.
“The field is still young and further research in this direction is ongoing,” says Heefer. “I am optimistic that our results will contribute to a deeper understanding of gravity and I hope that they may eventually even shed light on the reconciliation between gravity and quantum mechanics.”
Title of doctoral thesis: Geometry, space-time and Finsler gravity: from the metrizability of Berwald spaces to exact vacuum solutions in Finsler gravity. Supervisors: Luc Florack and Andrea Fuster.