Exploring Spacetime with Finsler Geometry

Investigations into Gravitational waves And their relationship to Finsler geometry provides new insights into spacetime, suggesting ways to reconcile relativity and quantum mechanics.

When talking about our universe, it is often said that 'matter tells us how to bend space-time, and curved space-time tells us 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. While general relativity captures the large in our universe, it contrasts with the small in physics described by quantum mechanics. For his PhD research, Sjors Heifer investigated gravity in our universe, his research has implications for the exciting field of gravitational waves, and may influence how large and small physics can be reconciled in the future.

Uncovering the Universe: Einstein’s Theories and Beyond

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 emerges due to the geometry of the four-dimensional space-time continuum, or short space of time,” says Heifer. „And it’s central to the origin of fascinating phenomena like gravitational waves in our universe.”

Massive objects, such as the Sun or galaxies, warp the space-time around them, and other objects move through this curved space-time in straight possible paths – otherwise known as geodesics.

However, due to curvature, these geodesics are not straight in the conventional 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 many gravitational phenomena, from planetary motion and everyday situations to black holes and the Big Bang. Hence it is the cornerstone of modern physics.

Solving Theories: Quantum Mechanics vs. General Relativity

General relativity describes many astrophysical phenomena that collide with another fundamental theory of physics – quantum mechanics.

„Quantum mechanics states that particles (such as electrons or muons) exist in multiple states at the same time until they are measured or observed,” says Heifer. „Once measured, they randomly choose a position due to a mysterious effect referred to as 'decay of the wave function’.”

In quantum mechanics, the 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 the location of the particle. The larger the square of the wave function at a particular location, the greater the probability that a particle is located at that location.

„All matter in our universe appears to be subject to the strange probabilistic laws of quantum mechanics,” notes Heifer. „The same is true for all forces of nature – except gravity. This paradox leads to profound philosophical and mathematical conundrums, and resolving these is one of the foremost challenges in fundamental physics today.

Gap reduction 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 terms of mathematics, general relativity is based on pseudo-Riemannian geometry, a mathematical language capable of describing most of the general shapes that spacetime can take.

„However, recent discoveries indicate that the spacetime of our universe may be beyond the bounds of pseudo-Riemannian geometry and can only be described by the more advanced mathematical language of Finsler geometry,” says Heifer.

Finsler’s time to shine

In Finsler geometry – named after the German and Swiss mathematician Paul Finsler, the distance between two points – A and B – depends not only on the location of the two points. It depends on whether one is traveling from A to B or the other way around.

“Imagine walking towards a point on the top of a mountain. Walking on a steep incline towards the point will cost you more energy to cover the distance and may take you longer. On the other hand, the way back down is much easier and takes much less time. In Finsler geometry, this can be accounted for by assigning a greater distance to the upward path than the downward path.”

Rewriting general relativity using the mathematics of Finsler geometry leads to Finsler gravitation, a very powerful theory of gravity that captures everything in the universe explained by general relativity, and potentially much more.

Finsler explores the possibilities of gravity

To investigate the possibility of Finsler gravity, Heifer needed to analyze and solve a specific field equation.

Physicists like to describe everything in nature in terms of fields. In physics, a field has a value at every point in space and time.

A simple example is temperature, for example; At any point in time, every point in space has a specific temperature associated with it.

A slightly more complicated example is the electromagnetic field. At any given time, the value of the electromagnetic field at a particular 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 located at that point.

When it comes to the geometry of spacetime, it is 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 as gravity.

Discovery of a new space-time geometry

Heifer turned to the vacuum field equation of Christian Pfeiffer and Matthias NR Wohlbarth, the equation governing 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.

Heifer: “As a good approximation, this includes all the interstellar space between stars and galaxies, and the empty space around objects like the Sun and Earth. By carefully analyzing the field equation, many new types of spacetime geometry have been discovered.

The era of gravitational waves

A groundbreaking discovery from Heifer’s work involves a class of spacetime geometries that represent gravitational waves—for example, ripples in spacetime caused by colliding neutron stars or black holes that travel at the speed of light.

First direct detection of gravitational waves on September 14^Th2015 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. Heifer’s research indicates that all this is consistent with the hypothesis that our spacetime has a Finslerian character.

The Future of Finsler Gravity Research

Although Heifer’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 Heifer. „I am confident that our results will be instrumental in deepening our understanding of gravity and, ultimately, they may shed light on the reconciliation of gravity with quantum mechanics.”

PhD Thesis Title: Finsler Geometry, Spacetime & Gravity: From Metrability of Perwald Spaces to Exact Vacuum Solutions in Finsler Gravity. Supervisors: Luke Florak and Andrea Fuster.