Gravitational waves and space-time geometry

When we talk about our universe, we often say that “matter tells space-time how to curve, and curved space-time tells matter how to move.” It is the essence of Albert Einstein’s famous theory of general relativity and describes how planets, stars and galaxies move and influence the space around them. Although general relativity captures much of the large of our universe, it is at odds with the small of 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 perhaps influencing how the big and small of physics can be reconciled in the future.

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 emerges due to the geometry of the four-dimensional space-time continuum, or space-time for short,” explains Heefer. “And it is at the heart of the emergence 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 called geodesics – through that spacetime curve.

However, due to the curvature, these geodesics are not at all straight in the usual sense. 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. As such, it remains a cornerstone of modern physics.

Clash of theories

If general relativity describes a multitude of astrophysical phenomena, it comes into conflict 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,” explains Heefer. “Once measured, they select a state at random 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. And the square of the wave function leads to a set of probabilities as to the possible location of the particle. The larger the square of the wave function at a particular location, the higher the probability that a particle will be at that location when observed.

“All matter in our universe appears to be subject to the strange probabilistic laws of quantum mechanics,” notes Heefer. “And the same is true for all forces of nature except gravity. This divergence leads to deep philosophical and mathematical paradoxes, and resolving them is one of the main challenges of fundamental physics today today.”

Is expansion the solution?

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.

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 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 a given moment, 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 point.

When it comes to the geometry of spacetime 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 itself as gravity.

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

Heefer explains: “As a good approximation, this includes all interstellar space between stars and galaxies, as well as 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. “.

Confirmation of gravitational waves

A particularly interesting discovery from Heefer’s work concerns a class of space-time geometries that represent gravitational waves – ripples in the fabric of space-time that propagate at the speed of light and can be caused by the collision of 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, numerous 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.

Scratch the surface

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 underway,” says Heefer. “I am optimistic that our results will help deepen our understanding of gravity and I hope that ultimately they may even shed light on reconciling gravity with quantum mechanics.”