Gravitational waves and the geometry of spacetime

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 core 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 much of the big in our universe, it is at odds with the small in physics as described by quantum mechanics.

For his Ph.D. research, Sjors Heefer has explored gravity in our universe, with his research having implications for the exciting field of gravitational waves and possibly influencing how large and small physics may 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 appears due to the geometry of the four-dimensional continuum of space-time, or space-time for short,” says Heefer. “And it is essential for the occurrence of fascinating phenomena in our universe, such as gravitational waves.”

Massive objects, such as the sun or galaxies, warp spacetime around them, and other objects then move along the straightest possible paths—otherwise known as geodesics—through this curved spacetime.

However, due to curvature, these geodesics are by no means straight in the usual sense. 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 many 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 a host 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 at the same time until they are measured or observed,” says Heefer. “Once measured, they randomly choose a state due to a mysterious effect referred to as ‘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 collection of probabilities of where the particle can be found. The larger the square of the wave function at a given location, the higher the probability that a particle will settle at that location after being observed.

“All matter in our universe appears to obey the strange probabilistic laws of quantum mechanics,” notes Heefer. “And the same is true of all the forces of nature—except gravity. This inconsistency leads to profound philosophical and mathematical paradoxes, and their resolution is one of the major challenges in fundamental physics today.”

Is expansion the solution?

One approach to resolving the collision of 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.

“Recent discoveries show, however, that the spacetime of our universe may be outside the realm of pseudo-Riemannian geometry and can only be described by Finsler geometry, a more advanced mathematical language,” says Heefer.

Field equations

To explore the possibilities of Finsler’s gravity, Heefer needed 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 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 complex example is that of the electromagnetic field. At any given moment in 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 located at that point .

When it comes to the geometry of space-time itself, this is also described by a field, namely the gravitational field. The value of this field at a point in space-time tells us the curvature of space-time at that point, and it is this curvature that manifests as gravity.

Heefer turned to the vacuum field equation of Christian Pfeifer and Mattias NR Wohlfarth, which is the equation governing this gravitational field in empty space. In other words, this equation describes the possible forms that the geometry of spacetime can take in the absence of matter.

Heefer explains, “To a good approximation, this includes all of the 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 geometries have been identified space.”

Confirmation of gravitational waves

One particularly exciting discovery from Heefer’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.

Since then, many observations of gravitational waves have been made. Heefer’s research shows that all of this is consistent with the hypothesis that our space has a Finslerian nature.

Scratching the surface

While 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 ongoing,” says Heefer. “I am optimistic that our results will be useful in deepening our understanding of gravity and hope that, eventually, they may also illuminate the reconciliation of gravity with quantum mechanics.”