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When talking about our universe, it is often said that “matter tells spacetime how to curve, and curved 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. While general relativity accounts for much of the big 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 how the big and the small in physics can be reconciled in the future.

A little 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 occurs because of the geometry of the four-dimensional space-time continuum, or space-time for short,” says Heifer. “And it’s central to 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 follow the straightest possible paths—otherwise known as geodesics—through this curved spacetime.

Because of the curvature, however, these geodesic lines are not straight in the usual sense at all. In the case of the planets in the solar system, for example, they describe elliptical orbits around the sun. In this way, general relativity elegantly explains the motion of the planets as well as a host of other gravitational phenomena ranging from everyday situations to black holes and the Big Bang. As such, it remains a cornerstone of modern physics.

Clash of theories

While general relativity describes many astrophysical phenomena, it clashes with another fundamental theory of physics – quantum mechanics.

“Quantum mechanics suggests 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 state due to a mysterious effect called ‘wavefunction 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 for where the particle might be. The larger the square of the wave function at a particular location, the greater the probability that a particle will be located at that location after it is observed.

“All matter in our universe appears to obey the strange probabilistic laws of quantum mechanics,” notes Heifer. “And the same is true of all natural forces—with the exception of gravity. This inconsistency leads to profound philosophical and mathematical paradoxes, and resolving them is one of the major challenges facing fundamental physics today.”

Is expansion the solution?

One approach to resolving the clash between general relativity and quantum mechanics is to extend the mathematical framework behind general relativity.

In terms of mathematics, general relativity is based on pseudo-Riemannian geometry, which is 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 scope of pseudo-Riemannian geometry and can only be described by Finsler geometry, a more sophisticated mathematical language,” says Heifer.

Field equations

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 flight. In physics, a field is simply something that has a value at any point in space and time.

A simple example would be temperature, for example; at any given moment in time, every point in space has a certain temperature associated with it.

A slightly more complicated example is that of the electromagnetic field. At any given moment in 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 at that point.

As for the geometry of spacetime itself, this too 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.

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

Heifer explains: “To a good approximation, this includes all of the 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 space-time geometries have been identified.”

Confirmation of gravitational waves

One particularly exciting discovery from Heifer’s work involves 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 14, 2015 marked the dawn of a new era in astronomy, allowing scientists to explore the universe in an entirely new way.

Many observations of gravitational waves have been made since then. Heifer’s research shows that they are all consistent with the hypothesis that our spacetime has a Finslerian character.

Scratching the surface

While Heefer’s results are promising, they only scratch the surface of the implications of Finsler’s gravity field equation.

“The field is still young and further research in this direction is actively ongoing,” says Heifer. “I am optimistic that our results will prove instrumental in furthering our understanding of gravity, and I hope that eventually they may even shed light on the reconciliation of gravity with quantum mechanics.”