### Paid Science / Physics

# General Relativity and the Geometry of the Universe

Exploring the fascinating realm of general relativity and the geometry of the universe.

## Transcript

Welcome to today's episode of Listen Learn Pods, where we explore various intriguing topics that spark curiosity and broaden our understanding of the world around us. On today's show, we'll be delving into the fascinating realm of general relativity and the geometry of the universe. Without further ado, let's get started!

General relativity is a cornerstone of modern physics, and it's a theory that has reshaped our understanding of not just the universe, but reality itself. Formulated by Albert Einstein in 1915, general relativity is a theory of gravitation that describes how objects with mass interact with the fabric of spacetime. It replaces the notion of gravity as a force between masses with the idea that massive objects warp spacetime around them, causing other objects to move on curved paths.

One key insight of general relativity is the equivalence principle, which states that the gravitational mass of an object (which determines how strongly it's attracted to other objects) is equal to its inertial mass (which affects how much it resists being accelerated). This principle underlies the famous thought experiment of an observer in freefall, who wouldn't be able to tell whether they were in a uniform gravitational field or whether they were being accelerated by a non-gravitational force, such as the force exerted by a rocket engine. This idea of the equivalence between gravity and acceleration provided the foundation for Einstein's development of general relativity.

Now, let's talk about the geometry of the universe, which is intimately connected to general relativity. The universe, as described by general relativity, consists of four dimensions: three dimensions of space (height, width, and length) and one dimension of time. Together, they form spacetime, a four-dimensional "fabric" that can be deformed by massive objects.

The geometry of spacetime is a critical aspect of general relativity, as it determines the paths taken by objects as they move through the universe. To describe the geometry of spacetime mathematically, Einstein and his contemporaries turned to the field of differential geometry and the concept of curved manifolds. Manifolds are mathematical spaces that, when examined on a smaller scale, resemble the more familiar three-dimensional Euclidean space or flat space. When mass is distributed through spacetime, it causes the manifold to curve, changing the geometry of the universe.

A simple analogy for understanding curvature in spacetime is to imagine a trampoline, with the fabric representing two dimensions of space. When you place a heavy object like a bowling ball onto the trampoline, it causes the fabric to curve. If you then roll a marble on the trampoline, it will follow the curved path created by the bowling ball's influence on the fabric. This is similar to how massive objects like stars and planets curve spacetime and cause the trajectory of other objects—like smaller planets or comets—to follow curved paths as well.

In terms of the broader structure of the universe, general relativity has played a pivotal role in our understanding of cosmology. It has helped us to appreciate the extent to which the universe's geometry can be influenced by the distribution and properties of the matter and energy it contains. According to the current leading model of the universe's evolution, the Big Bang Theory, the universe started as an extremely hot, dense state roughly 13.8 billion years ago. Since then, it has been expanding and cooling, with its geometry evolving over time.

One of the major ramifications of general relativity for our understanding of the universe is its link to the phenomenon of black holes. Black holes are regions of spacetime where the gravitational field is so intense that not even light can escape from it. They are formed when massive stars exhaust their nuclear fuel and collapse under their own gravity, giving rise to these singular entities that warp spacetime to an extreme degree. Black holes are described by the famous Schwarzschild solution of Einstein's equations of general relativity, which provides the metric—a mathematical description of spacetime geometry—around a spherically symmetric mass.

Aside from black holes, general relativity also predicts the existence of gravitational waves, which are ripples in spacetime created by accelerating masses. These waves travel at the speed of light and were first detected by the LIGO and Virgo observatories in September 2015. This groundbreaking discovery confirmed a key prediction of Einstein's theory and opened up an entirely new way of observing the universe, using gravitational waves as a means of "listening" to cosmic events.

Now, what does all this mean for our understanding of the shape of the universe? Researchers use various models to describe the geometry of the cosmos, such as the Friedmann–Lemaître–Robertson–Walker (FLRW) metric. This model represents the large-scale structure of the universe as a homogeneous and isotropic space, which means it appears the same in all directions and has a consistent matter distribution throughout.

Within the context of the FLRW metric, there are three possible geometric models for the universe: open, flat, and closed. In an open universe, the geometry resembles a hyperbolic space, in which the sum of the angles of a triangle is less than 180 degrees. In a closed universe, the geometry is spherical, and the sum of the angles of a triangle is more than 180 degrees. In a flat universe, the geometry is Euclidean, and triangles follow the familiar rules of plane geometry, such as having angles adding up to 180 degrees.

Through the use of various cosmic observational tools, such as the cosmic microwave background radiation, scientists have determined that the universe appears to be remarkably flat. This suggests that the large-scale cosmological geometry adheres closely to Euclidean principles.

In conclusion, general relativity and its treatment of the geometry of the universe have had far-reaching impacts on our understanding of the cosmos. From the equivalence principle to the curvature of spacetime, this overarching theory continues to refine and challenge our comprehension of reality. As we further explore the universe and develop new technologies, the insights provided by general relativity will continue to inform our investigations and guide us in our quest for knowledge.