I am the type of person, like many around me, who is driven by curiosity and wonder. I marvel at questions that can’t be answered and seek the keys to understanding those which can. It is sensible, with that in mind, that I would have spent my entire young life absolutely intrigued by the intricacies of outer space. To me, the world outside that which I am familiar is a looming mystery, felt to its utmost degree in the pondering of what has always felt to me like the greatest secret of outer space: its expansion. The understanding of this, which I often found myself dismissing as nothing more than a mere concept, always seemed just a little bit too far out of reach. However, like much else that we experience in our lives, with effort, concentration, and a sizable amount of mathematics, comprehension is far from impossible.

In an effort to bridge the gap between myself and the grandeur of that which surrounds me, I’ve chosen to explore the mathematics that predicts and explains the growth of the universe. This will, of course, be only a small sample of what I understand to be endless possibilities for analysis and discovery; a tiny glimpse into how algebra, calculus, and geometry can begin to explain what is commonly disregarded as unexplainable. This discovery will begin with an analysis of Einstein’s Theory of General Relativity, which will precede an explanation of the equations modeling universal expansion by Alexander Friedmann, and lastly present an argument for the importance of the comprehension this mathematics allows us.

General Relativity is a geometric theory of gravity, published by Albert Einstein in 1915. With General Relativity, Einstein determined that what is perceived as the massive force of gravity arises from the curvature of spacetime. The word spacetime represents a mathematical model that interweaves the dimensions of space, and that of time, into a sole continuum. Its curve, which Einstein’s equations model, depend entirely on the matter and energy it contains. In other words, great masses do not carry smaller masses through any sort of empty space, as Isaac Newton’s previously published laws of gravitation did not contest. Rather, matter distorts spacetime, and that distortion is what bears the effect on other matter (Redd).

Illustrated above is matter bearing an effect on the physical dimensions of spacetime. While helpful in understanding how mass could have an affect on dimensional space, these images may be slightly misleading, as they do not captivate how the dimension of time is altered or affected by masses. Such a principle is difficult to imagine visually, it is however crucial to the development of Einstein’s theory and those which followed it.

General relativity can be summed up with a simple equation made up of three not-so-simple variables: G = RE. Where R consists of a matrix of 16 numbers and describes the distortion of spacetime; G is the gravitational constant, 6.67408 × 10-11; and E is a complicated tensor, or an arbitrarily complex mathematical object comparable to a vector, which represents the energy of an object.

Einstein’s theory had operated under the pretense that the universe was static, or experiencing no major changes. Not long after the publication of his field equations (those which modeled general relativity) this concept was refuted. It had been discovered that the universe was experiencing not only growth, but acceleration. Using the discovery of curved spacetime, Alexander Friedmann’s mathematics were able to model a growing universe.

Friedmann’s equations operate under the assumptions that the universe is both isotropic and homogeneous (Siegel). This means that it is identical in all directions and at all locations. In other words, regardless of what direction or angle it is studied from, the universe is exactly the same. This principle is known as the cosmological principle and implies something particular about the metric tensor of the universe. A metric tensor is a function used to define the distances and angles between tangent vectors, allowing one to essentially compute the lengths of curves along a set of points. The cosmological principle implies a metric tensor in the form of:

Where is a metric tensor of three dimensions, that may either represent a sphere with constant positive curvature, a space of hyperbolic shape with constant negative curvature, or a space that is flat.

Looking at the above image, one can realize how important geometry is to the understanding of the universe. These possible shapes with in which the universe may exist say just about everything regarding its expansion. They each, by their own geometric nature, alter that which the universes they model may contain. Think soley about the triangle inscribed within these shapes. On the flat surface, the interior angles of said triangle would add up to 180 degrees, as standard geometric principles state. However, on the surface of the saddle, or hyperbolic, shaped universe, the sum of the interior angles of the triangle would be less than 180 degrees, and those of the triangle inscribed on the surface of the spherical universe would have a sum greater than 180 degrees.

This principle is evidenced by the images above, as well as by a useful analogy regarding a globe. Picture an object, even a human being, standing at the very north point of the earth: the North Pole. This object moves southward in a straight line until it arrives at the equator, at which point it turns and moves in a westward direction. Eventually the object turns once more and travels north until it has arrived back at its initial starting point, the furthest north point of the globe. They’ve created a shape with three sides and three angles, namely a triangle. However, each interior angle of the triangle was 90 degrees, making their sum 270 degrees. This is precisely why the assumptions under the cosmological principle are so important.

The equations that these assumption led to are known as the Friedmann equations, which tell us just exactly how the universe expands quantitatively to contain the matter, energy, and radiation within it (Siegel).

The first can be modeled by:

This equation refers directly to the speed ( ) of the universe and how it relates to its contents. In addition to the aforementioned gravitation constant G, the first Friedmann equation also contains the universal constants and c. Known as the cosmological constant,, refers to the energy density (the amount of energy stored in a particular unit of volume) of space. While the value of c is that of the speed of light in a vacuum.

Scale factor a is used to parametrize the universe’s rate of change. So the left side of the equation is essentially modeling the universe’s speed. Similar to variables p and ⍴, a is a function of time.

p = pressure

⍴ = density

The Curvature Parameter k describes whether the expansion rate is increasing or decreasing.

If k = 0, then the density is equivalent to a critical value at which expansion is infinite but at a decreasing rate

If k > 0, then density must be great enough that gravitational attraction, pulling matter inwards, will eventually cease and the universe will collapse in onto itself.

If k < 0, then there is sufficient density to never halt gravitational attraction, and thus expansion will not be stopped (Nave).

The second Friedmann equation can be modeled by:

This equation can be studied in a slightly more intuitive manner.

It refers to the acceleration of the universe ( ) as a function of time. As the term which constrains the gravitational constant and the variables for pressure and density are negative, it tells us that a greater amount of matter will yield more universal contraction. This makes sense, given our understanding of gravity and general relativity. The term which includes the cosmological constant, however, is positive. This means that greater energy density will result in greater acceleration.

These equations can and have been derived and combined to exist in many different forms. Nonetheless, their purpose and magnitude is sustained. They are not only the results of great strides in the fields of mathematics and physics, but also representations of what proficiency in these areas of study are able to achieve. The application of Friedman’s tensor calculus to Einstein’s geometry has yielded something innately rewarding, the understanding of one’s physical surroundings, and the key to solving the mysteries with which their existence is associated.

## Works Cited

- Esteban, J Gabàs. “Space Time Curvature.” Flickr, 19 Apr. 2015, www.flickr.com/photos/[email protected]/17015996680.
- “Geometry of the Universe.” University of Oregon, abyss.uoregon.edu/~js/cosmo/lectures/lec15.html.
- Nave, R. “The Friedmann Equation.” Total Internal Reflection, hyperphysics.phy-astr.gsu.edu/hbase/Astro/fried.html.
- Redd, Nola Taylor. “Einstein’s Theory of General Relativity.” Space.com, Space.com, 8 Nov. 2017, www.space.com/17661-theory-general-relativity.html.
- Siegel, Ethan. “Spacetime Curvature Around Massive Object.” Ask Ethan: If Mass Curves Spacetime, How Does It Un-Curve Again?, 16 June 2018, medium.com/starts-with-a-bang/ask-ethan-if-mass-curves-spacetime-how-does-it-un-curve-again-ce51a391cdc4.
- Siegel, Ethan. “The Most Important Equation In The Universe.” Forbes, Forbes Magazine, 17 Apr. 2018, www.forbes.com/sites/startswithabang/2018/04/17/the-most-important-equation-in-the-universe/#3a7234b960da.
- Weebly, Steve. “Spherical Geometry.” Math With Steve, mathwithsteve.weebly.com/non-euclidean-geometry.html.
- Weebly, Steve. “Hyperbolic Geometry.” Math With Steve, mathwithsteve.weebly.com/non-euclidean-geometry.html.