Albert Einstein is widely regarded as one of the most renowned and influential scientists in human history. To this day, Einstein’s works serve as a guide for physicists, mathematicians, and astronomers studying everything from gravitational waves to Mercury’s orbit.
He is also thought of as one of the most intellectual people of our time, having proposed things previously unheard of in many fields, including math.
While most people, even young children, are aware of Einstein’s contributions to physics, few are aware of his astounding contributions to mathematics. Though Einstein did not make direct contributions to mathematics in the same way that Isaac Newton or Carl Friedrich Gauss did— by developing calculus and number theory, respectively—he made many indirect contributions to mathematics, including the most notable theory of relativity.
Einstein’s equation E = mc^{2} is a well-known formula, even among children, and it helps to understand “special relativity.” His theory of “general relativity” that explains gravity is also well known, and his work explaining the behavior of electrons under certain circumstances, that is, the photoelectric effect, earned him the Nobel Prize in physics in 1921. At the time of his death, he was also working on a theory that would unify all the forces of the universe into a single theory, or the theory of everything.^{1}
The foundation of Einstein’s contributions was in fact his mathematical creativity. Although he is recognized as the greatest physician of the 20th century, his mathematical prowess is not as popularly known, because he did not directly create any theory in math per se, but rather used mathematics in an essential way to come up with remarkable ideas that have sparked great advances in geometry and parts of math that are apparently far removed from physics.
Because math and physics are inextricably linked and cannot be separated, we can emphasize how Einstein’s monumental contributions, such as replacing the “Newtonian theory of gravity” with what is now known as the “general relativity theory,” are critical to many fields of mathematics, not just geometry.
This theory enabled the transition from three-dimensional geometry to four-dimensional geometry by incorporating time as the fourth variable, and the interpretation of gravitation at the curvature of the fourth-dimensional space geometry.
Some of Einstein’s most-notable contributions to the world of mathematics are:
- He discovered the Einsteinian tensor^{2} and, through his application of tensors in general relativity theory,^{3} he urged mathematicians to develop multidimensional geometries.
- He gave the first correct mathematical expression of Galilean relativity,^{4} which was a problem that had defeated both Newton and Leibniz at that time.
- He not only derived the Lorentz transformation,^{5} establishing relativity as a mathematical theory of physics, but also adapted Differential Geometry to use the Lorentzian metric to show that it gave the correct description of gravity.
- He came up with the “Einstein summation convention.”^{6}
- Einstein demonstrated the bending of light^{7} and the advancement of the perihelion.
- It was Einstein who first recognized that the phenomenon of entanglement was implicit in the mathematical structure of quantum mechanics.^{8}
- Apart from these, many interesting mathematical problems arose as a result of his discoveries in physics. For example, his paper on Brownian motion^{9} in 1905 led to Wiener’s development of the Wiener process, a key concept in probability and stochastic processes.
- He also published many papers on important concepts like photoelectric effect, brownian motion, special relativity, and energy equivalence.
- Among the many math articles that he published, the two articles on differential geometry and field equations^{10} have had the greatest impact on contemporary mathematics.
- In addition to introducing index notation,^{11} he made significant contributions to the field of index notations.
Albert Einstein was a man of extraordinary intelligence and a brilliant scientific mind, who, by combining his love for knowledge, intuition, and mathematics, created extraordinary theories in physics that no other scientist has yet matched. It is because of his contributions that we have the kind of technology we have at our disposal today; many technological advancements would not have been possible without him. Thanks to him, the world has a better understanding of physics today, and he has provided us with the tools for the future of math, science, and technology.
References:
- This Month in Physics History: Einstein’s quest for a unified theory. (n.d.). Retrieved April 1, 2022, from https://www.aps.org/publications/apsnews/200512/history.cfm
- Einstein Tensor – an overview | ScienceDirect Topics. (n.d.). Retrieved March 24, 2022, from https://www.sciencedirect.com/topics/mathematics/einstein-tensor
- Einstein’s theory of general relativity | Space. (n.d.). Retrieved March 24, 2022, from https://www.space.com/17661-theory-general-relativity.html
- Galilean relativity | physics | Britannica. (n.d.). Retrieved March 24, 2022, from https://www.britannica.com/science/Galilean-relativity
- (PDF) Analysis of Einstein’s derivation of the Lorentz Transformation. (n.d.). Retrieved March 24, 2022, from https://www.researchgate.net/publication/338774760_Analysis_of_Einstein’s_derivation_of_the_Lorentz_Transformation
- Einstein Summation Convention. (n.d.).
- Einstein’s Light-Bending Concept | Exploratorium Video. (n.d.). Retrieved March 24, 2022, from https://www.exploratorium.edu/eclipse/video/einsteins-light-bending-concept
- Quantum Theory: The Einstein/Bohr Debate of 1927 | AMNH. (n.d.). Retrieved March 24, 2022, from https://www.amnh.org/exhibitions/einstein/legacy/quantum-theory
- This Month in Physics History. (n.d.). Retrieved March 24, 2022, from https://www.aps.org/publications/apsnews/200502/history.cfm
- The Einstein Field Equations. (n.d.).
- BME 332: Mathematical Preliminaries. (n.d.). Retrieved March 24, 2022, from http://websites.umich.edu/~bme332/ch1mathprelim/bme332mathprelim.htm