Gravitational Waves And The Geometry
When we talk about the universe, a common phrase is, “Matter tells spacetime how to curve, and curved spacetime tells matter how to move.” This idea lies at the heart of Albert Einstein’s general theory of relativity. It explains how massive objects like planets, stars, and galaxies move and interact with the space around them.
General relativity is excellent at describing the large-scale structure of the universe. However, it doesn’t mesh well with the principles of quantum mechanics, which governs the behavior of particles at the smallest scales. This discrepancy between the two theories remains one of the biggest challenges in modern physics.
For his Ph.D. research, Sjors Heefer has delved into the nature of gravity in our universe, focusing on its implications for the study of gravitational waves. His work could potentially bridge gaps between the large-scale and small-scale aspects of physics in the future.
Over a century ago, Albert Einstein transformed our understanding of gravity with his general theory of relativity.
“According to Einstein’s theory, gravity isn’t a force in the traditional sense. Instead, it arises from the curvature of the four-dimensional spacetime continuum, or simply spacetime,” Heefer explains. “This concept is crucial for understanding phenomena like gravitational waves.”
Massive objects, such as the sun or galaxies, distort the spacetime around them. As a result, other objects move along the straightest possible paths—known as geodesics—within this curved spacetime.
Due to the curvature, however, these geodesics are not straight in the usual sense at all. In the case of the planets in the solar system, for instance, they describe elliptical orbits around the sun. In this way, general relativity elegantly explains the movement of the planets as well as numerous 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 the theories
While general relativity describes a host 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,” says Heefer. “Once measured, they randomly select a state due to a mysterious effect referred to as the ‘collapse of the wave function.'”
In quantum mechanics, a wave function is a mathematical representation that describes the position and state of a particle, like an electron. The square of the wave function gives a probability distribution, indicating where the particle is likely to be found. The higher the square of the wave function at a particular point, the greater the probability of locating the particle there when it is observed.
Heefer explains, “All matter in our universe seems to follow the strange probabilistic laws of quantum mechanics. This is true for all fundamental forces of nature except for gravity. This discrepancy creates significant philosophical and mathematical challenges, and resolving these issues is one of the main goals of modern physics.”
Is expansion the solution?
One approach to resolving the clash of 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, which is a mathematical language capable of describing most of the typical shapes that spacetime can take.
“Recent discoveries indicate, however, that our universe’s spacetime might be outside the scope of pseudo-Riemannian geometry and can only be described by Finsler geometry, a more advanced mathematical language,” says Heefer.
Field equations
To delve into the possibilities of Finsler gravity, Heefer needed to tackle and solve a specific field equation.
Physicists often describe everything in nature in terms of fields. In physics, a field is something that has a value at each point in space and time.
A straightforward example is temperature; at any given moment, each point in space has a specific temperature.
A more complex example is the electromagnetic field. At any given moment, the value of the electromagnetic field at a certain point in space indicates the direction and strength of the electromagnetic force that a charged particle, such as an electron, would experience if it were there.
Similarly, the geometry of spacetime is also described by a field: the gravitational field. The value of this field at a point in spacetime tells us the curvature of spacetime at that point, and this curvature is what we perceive as gravity.
Heefer focused on the vacuum field equation developed by Christian Pfeifer and Mattias N. R. Wohlfarth, which governs the gravitational field in empty space. This equation describes the possible shapes that the geometry of spacetime can take in the absence of matter.
Heefer explains, “To a good approximation, this includes all interstellar space between stars and galaxies, as well as the empty space surrounding objects such as the sun and the Earth. By carefully analyzing the field equation, several new types of spacetime geometries have been identified.”
Gravitational waves confirmation
A particularly exciting discovery from Heefer’s work involves a class of spacetime geometries that represent gravitational waves. These waves are ripples in the fabric of spacetime that travel at the speed of light, often caused by events like the collision of neutron stars or black holes.
The first direct detection of gravitational waves on September 14, 2015, marked a new era in astronomy, allowing scientists to explore the universe in entirely new ways. Since then, numerous observations of gravitational waves have been made. Heefer’s research suggests that these observations are consistent with the hypothesis that our spacetime may have a Finslerian nature.
Scratching the surface
While Heefer’s findings are promising, they merely begin to explore the broader implications of the field equation of Finsler gravity.
“The field is still in its infancy, and there’s a lot more research to be done,” Heefer explains. “I’m hopeful that our results will play a key role in expanding our understanding of gravity. Ultimately, I aspire that this work might contribute to bridging the gap between gravity and quantum mechanics.”