Albert Einstein (1879-1955) was the 20th century’s most notable scientist. Specializing in theoretical physics, Einstein revolutionized our understanding of the physical universe. Amongst his many accomplishments are his theory of relativity, his equation regarding mass and energy (famously known as E = mc²), and his work on the photoelectric effect. He also articulated the central goal of modern physics, which is a unified field theory that encompasses all physical phenomena in the universe. In Relativity, Einstein lays out in clear terms his theory of relativity, which inaugurated the present era of modern physics.

Before the 20th century, physics understood space and time as independent structures of the universe against which all particular objects are measured. Thanks to the work of Galileo and Newton, one could conceive of space according to an abstract coordinate system, an objective grid upon which every physical object has a specific position. Likewise, changes in position could be measured by time, which itself could be quantified in seconds according to ticking clocks.

The assumption of objective, absolute measurement according to space and time emphasizes universal order, simplicity, and unity. The special theory of relativity, however, has revealed a universe far stranger than we realized. When our tools of measurement such as rulers or clocks are examined, we realize that they do not contain absolute space or time. They are useful devices for measuring distance and duration, but distance and duration relative to their observers. The position and motion of the person recording any given object limits their powers of observation. Far from having an objective gaze according to abstract space and time, any observer can only evaluate a physical event or object relative to their own position. This erects a division between the absolute concepts of mathematics and the relative positions of physical observers.

Moreover, this problem became especially apparent when considering the speed of light. Theoretical physics has established that the speed of light in a vacuum is constant, and that speed has become a law in the field. Light is known to have the fastest velocity, such that nothing can catch up to or exceed it. Yet according to the absolute conceptions of space and time, light is irrelevant for observation. If space, for instance, exists apart from any physical object, including light, then it becomes possible to say that one can independently measure objects. Such mathematical assumptions, however, are contradicted by experimental data regarding light. Measurements can only occur under relative motion, and all motion is relative to the speed of light.

Rather than understanding space and time as an abstract mathematical graph or coordinate system, early theoretical and later experimental research revealed these are outdated notions which could not comport with the principle of relativity. No matter how mathematically pleasing, the relativity of motion and the maximum speed of light indicated that time and space were useful—but limited—concepts for measurement.

The short equation E = mc²  is one of the greatest discoveries of relativity. What often goes unnoticed, however, is that this equation represents the unification of two fundamental physical laws. Prior to the special theory of relativity, physics espoused both a conservation law of matter and a conservation law of energy. These laws separately state that mass and energy cannot be created or destroyed, only changed.

What relativity predicted, and nuclear physics later confirmed experimentally, is the relation between energy and mass. As a physical object takes in energy, its inertial mass likewise increases. As that object decomposes or changes, it releases energy precisely in the amount that its mass decreases. This equivalence between mass and energy enabled physicists to simplify the two conservation laws into one law governing this newfound relationship. The correspondence between mass and energy was reflected in Einstein’s work on velocity and light. As the fastest thing in the universe, light is massless energy.

Thus, E = mc². Energy is equal to its mass multiplied by the speed of light squared. In this elegant equation, mass can be understood as an object’s current energy before it is affected by an outside force, thus gaining new energy. Likewise, energy can be understood as some object’s released mass. With this new insight in mind, theoretical physics opened up a new frontier in its investigation of the physical universe.

As groundbreaking as special relativity was for theoretical physics, it was constrained in its range and could only account for uniform, rectilinear motion. It could easily account for a singular object moving in a vacuum at a consistent velocity, but such conditions are very unlikely or “special” (hence the name). What special relativity could not account for was the constant, asymmetric collisions between objects and forces that occur all throughout the universe. The main issue for special relativity was that it could not account for the force of gravity, which operates constantly between macro and microscopic physical objects throughout the known universe.

A general theory of relativity must be sought which could account for all states of motion, no matter what they may be. Moreover, accounting for gravity in this theory augmented prevailing knowledge about the velocity of light. Not only was the speed of light different in non-vacuous conditions, but it was theorized and then experimentally confirmed that gravity causes rays of light to curve according to their fields. Though the curvature of light is insignificant in most situations (black holes being the major exception) this data revealed that relativity needed to account for gravity if it was to properly represent the physical universe.

For roughly two millennia before the 20th century, geometry was built upon the work of Euclid, the preeminent Greek mathematician. His Elements is a compendium of geometrical axioms, self-evident truths about the structure of the universe. Until the late 19th and early 20th century, Euclid’s assertions were the bedrock of geometry, physics, and other mathematical disciplines. Upon these axioms were the prevailing assumptions in physics regarding time, space, and the mathematical coordinate system which could disclose them.

As general relativity stepped to the foreground of physics, the traditional assumptions of Euclidean geometry were unable to account for relativity’s new insights into the fabric of the universe. Conundrums abounded not only for the older notions of space and time, but also about the curvature, warp, and expansion of space. The new cosmology was altering our perception of geometry as well as physics.

Imagine being in front of a marble table, looking at its surface. Any one point on this area is connected to every other point, such that one could measure, point to point, across the entire table. With a large number of equally-sized metal rods, one could connect the points on the table with these rods like lines in between them. Small squares could be formed, each leading to more lines, more squares, and ultimately an even grid is mapped onto the table surface. This grid could then be used to map out distances between points, perfectly even and approximated by numbers. This exercise relies on Euclidean principles that govern the way points and lines are understood and applied to the universe. 

The table itself and anything that occurs on the table could be measured in this way because doing so would reflect the simple order governed by traditional geometry. Now consider a portion of the table being heated to such a point that the metal rods are affected. Since metal expands when heated, the rods on that part of the table would expand, thus distorting the Euclidean surface that was constructed by rods touching at simple points.

This is about as far as the analogy goes before it breaks down. One could substitute some non-metallic rods that wouldn’t expand, but that misses the point. Instead of expanding metal, consider that space itself—the points and lines represented above by marble and metal—is expansive, curving and warping according to the objects within it. 

Euclidean geometry, dealing with simple, regular objects and planes, was unable to account for space and time themselves being a part of the increasingly complex makeup of the universe. In its wake, a new geometry was needed to account for the universe general relativity disclosed.

The introduction of gravity into the now general theory of relativity had numerous consequences on our model of the universe. It seems as though the universe is infinite in all directions. Stars populate every corner of space, and though they differ in size and makeup, one can conclude that the universe is evenly populated with these lights. Since the time of Newton, however, this view of the universe was challenged by his findings on gravity. Because of the gravitational force exerted on and by stars, it makes more sense to think of the universe as having a center. In this center, stars are more densely packed together, which lends to our perspective of them filling up the whole of space.

This view of the cosmos as an island of stars in a sea of empty space had two issues which required resolution. The first was motivated by a distaste for the idea that the universe has a center. Such an assumption seemed like a bias toward our earthly perspective, a bias that lacks warrant in the face of the grander cosmos. Second, and more significantly, if this clustered universe was surrounded by empty space, then this means that the light of the stars is spilling out into an infinite void, with no recourse to the universe which spawned it. Taking into account entropy, this meant that the universe was rapidly diminishing in its overall energy as light was being radiated out toward nothing.

Advancements in the study of gravity and light provided a solution. Now that gravity is understood as a field, physicists proposed that an increased distance between two objects significantly diminishes the potency of their gravitational force. With this understanding of gravity as a field around celestial bodies, it became possible to speak of the universe not as a cluster, but again as a broadly distributed cosmos. 

In summation, the theory of relativity not only unsettles traditional assumptions in mechanics, Newtonian physics, and mathematics, but it also affirms that the universe is finite and irregular. Recall that space is not an independent concept, but rather is dependent on matter. Matter exhibits a gravitational field, which affects the very fabric of space, even if it goes unnoticed by the naked eye. This entails that for geometry, understanding the structure of the universe is dependent on seeing the state of a region of matter. Space, in this sense, is imposed upon the moving universe as a tool of approximation.

As previously mentioned, this invalidates Euclidean geometry, which is dependent on fixed, independent points of reference. Given that matter is constantly interacting with and imposing upon other matter, it is impossible to assert a fixed point of reference such that Euclidean geometry can seamlessly measure the universe. Of course, this does not totally dispense with Euclid’s work. Euclidean geometry allows us to make plenty of geometric approximations, which is still useful. 

What matters here is that these approximations can only handle some, not all, of the data that the universe gives. The universe is not as simple as a flat marble table. Much of these normally unaccounted-for influences operate below the level of notice. The exceedingly smaller effects of various gravitational fields complicate the picture of a simple, structured universe.

The universe is like the rippled surface of a lake, ebbing continually but with appreciable order and structure. To that point, relativity has revealed that the universe is not an infinite empty expanse beyond a small island of stars, but rather a region that is like a sphere. The shape of the universe is not uniform, which problematizes many oversimplified images of it. Though its shape is irregular and dependent on its matter, this spherical image of the universe confirms that the universe has boundaries, a finite size. Knowing these facts gives shape to further experimentation, which will disclose more of the grandeur of our universe.