Gravitational waves and the geometry of space-time

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When talking about our universe, it is often said that “matter tells space-time how to curve, and curved space-time tells matter how to move.” This is the essence of Albert Einstein’s famous theory of general relativity, and describes how planets, stars, and galaxies move and affect the space around them. While general relativity captures a lot of the big things in our universe, it conflicts with the small things in physics as described by quantum mechanics.

for him Ph.D. researchSewers Heifer discovered gravity in our universe, and his research has implications for the exciting field of gravitational waves, and may influence how large and small physics are reconciled in the future.

Just 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 arises due to the geometry of the four-dimensional space-time continuum, or space-time for short,” Heffer says. “It is essential for the emergence of wonderful phenomena in our universe, such as gravitational waves.”

Massive objects, such as the Sun or galaxies, distort the spacetime around them, and then other objects move along the straightest possible paths – known as geodesics – through this curved spacetime.

However, due to curvature, these geodesics 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 planetary motion as well as many 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 general relativity describes a range of astrophysical phenomena, it conflicts with another fundamental theory in physics – quantum mechanics.

“Quantum mechanics suggests that particles (such as electrons or muons) exist in multiple states at the same time in order to be measured or observed,” says Heffer. “Once measured, they randomly choose a state due to a mysterious effect referred to as ‘wave function collapse.’

In quantum mechanics, a wave function is a mathematical expression that describes the position and state of a particle, such as an electron. The square of the wave function gives rise to a set of probabilities about where the particle is. The larger the square of the wave function at a given location, the more likely it is that the particle will be at that location once it is observed.

“All matter in our universe seems to obey the strange probabilistic laws of quantum mechanics,” Heffer says. “The same is true for all forces of nature, except gravity. This contradiction gives rise to profound philosophical and mathematical paradoxes, and resolving these paradoxes is one of the fundamental challenges in fundamental physics today.”

Is expansion the solution?

One approach to resolving the conflict between general relativity and quantum mechanics is to expand 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.

“Recent discoveries suggest that spacetime in our universe may be beyond the scope of pseudo-Riemannian geometry and can only be described by Fensler geometry, a more advanced mathematical language,” says Heifer.

Field equations

To explore the possibilities of Fensler gravity, Heffer needed to analyze and solve a specific field equation.

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

A simple example of this is temperature, for example; At any given point in time, every point in space has a specific temperature associated with it.

A slightly more complex example is that of an electromagnetic field. At any given point 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, it is also described by a field, which is 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 is manifested in gravity.

Heffer turned to the vacuum field equation developed by Christian Pfeiffer and Matthias N. R. Wohlfahrt, which 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.

“To a good approximation, this includes all of the interstellar space between stars and galaxies, as well as the empty space surrounding objects like the Sun and Earth,” Heffer explains. “By carefully analyzing the field equation, several new types of space-time geometry have been identified.”

Confirmation of gravitational waves

One particularly exciting discovery from Heffer’s work involves a class of space-time geometry that represents gravitational waves, which are ripples in the fabric of space-time that propagate at the speed of light and can be caused by collisions 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, many observations of gravitational waves have been made. Heffer’s research suggests that these are all consistent with the hypothesis that our space-time has a Venslerian nature.

Scratch the surface

While Heffer’s results are promising, they only scratch the surface of the implications of the Fensler gravitational field equation.

“This is still a young field, and there is more research being done in this direction,” says Heifer. “I am optimistic that our results will prove effective in deepening our understanding of gravity, and I hope that they will ultimately shed light on the reconciliation of gravity and quantum mechanics.”

more information:
sj heffer, Fensler geometry, space-time, and gravity (2024)

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