Gravitational waves and the geometry of space-time

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 it 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. Sjors Heefer has been investigating gravity in our universe, with his research having implications for exciting gravitational wave fields and possibly influencing how large and small physics can be reconciled in the future.

More than a century 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 arises due to the geometry of the four-dimensional space-time continuum, or space-time for short,” says Heefer. “And it is fundamental to the origin of fascinating phenomena in our universe, such as gravitational waves.”

Massive objects like the sun or galaxies warp space-time around them, and other objects then move along this curved space-time along the straightest possible paths—otherwise known as geodesics.

However, due to curvature, these geodesics are not at all 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, the general theory of relativity elegantly explains the motion of the planets as well as a number of other gravitational phenomena, from everyday situations to black holes and the Big Bang. As such, it remains a cornerstone of modern physics.

Clash of theories

While the general theory of relativity describes a number of astrophysical phenomena, it clashes 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 says. “After the measurement, they randomly choose a state because of 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 collection of probabilities of where the particle could be. The larger the square of the wave function at a particular location, the higher the probability that the particle will be at that location once it is 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 of all natural forces—except gravity. This contradiction leads to profound philosophical and mathematical paradoxes, and their resolution is one of the major challenges of fundamental physics today.”

Is expansion the solution?

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

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

“However, recent discoveries suggest that the spacetime of our universe may be outside the scope of pseudo-Riemann geometry and can only be described by Finsler geometry, a more advanced mathematical language,” says Heefer.

Equation of the field

To explore the possibilities of Finsler gravity, Heefer needed to analyze and solve a certain field equation.

Physicists like to describe everything in nature using fields. In physics, a field is simply something that has a value at every point in space and time.

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

A slightly more complicated example is the electromagnetic field. At any given moment, the value of the electromagnetic field at a 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, it is also described by a field, specifically a 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 as gravity.

Heefer turned to the vacuum field equation of Christian Pfeifer and Mattias NR Wohlfarth, the equation that governs 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.

Heefer explains: “To a good approximation, this includes all the interstellar space between stars and galaxies, as well as the empty space surrounding objects such as the Sun and Earth. Through careful analysis of the field equation, several new types of space-time geometries have been identified.” .”

Confirmation of gravitational waves

One particularly exciting discovery from Heefer’s work involves a class of space-time geometries that represent gravitational waves—ripples in the fabric of space-time that travel at the speed of light and can be caused, for example, by colliding neutron stars or black holes.

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 a whole new way.

Many observations of gravitational waves have been made since then. Heefer’s research suggests that all of this is 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 the Finsler gravity field equation.

“The field is still young and further research in this direction is actively underway,” says Heefer. “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 alignment of gravity with quantum mechanics.”