How did Einstein interpret Euclidean geometry?

Eduardo Prachedes
5 min readOct 12, 2023

Everyone knows that one of the greatest theories proposed by Albert Einstein was the Theory of Relativity. You’ve certainly heard of it, whether through the famous formula E= MC² or even the concept of Space-Time.

What most people don’t know is that the theory of relativity is largely based on Geometry — a branch of mathematics that is dedicated to the study of the properties, relationships, and measurements of shapes, sizes, figures, and spaces. You’ve probably studied it in school, but what you may not know is that the classical geometry taught in schools is called Euclidean Geometry, proposed around 300 BC.

Given this, you might wonder, “What does the subject I learned in school have to do with the Theory of Relativity?” To answer you simply, dear reader: nothing.

What is Euclidean Geometry, really?

At this point, you may have more questions than answers. That’s normal; you probably didn’t expect this proposition to be contradicted. At first, it’s asserted that Einstein’s Theory is largely based on Geometry, and then it’s said that the Geometry you’ve known for years is not related to this idea.

To clarify things for my dear reader, I would like to first bring to light what someone who has never been exposed to the ideas of the German physicist understands as Euclidean Geometry.

Euclides

How is it commonly treated?

You’re planning to renovate your house, and suddenly, you’re measuring a wall to put up your beautiful wallpaper. Measuring from one point to another to determine the length of the roll, measuring two other points to find the wall’s height, and finally, relating these measurements to determine the total amount of fabric you’ll need.

What you’ve just used in this simple, everyday situation is Euclidean Geometry. Relating points, finding their distances, and proposing measurements between them. This is internalized in everyone; they all tend to think of a distance as two marked locations on a certain body, and because it’s so common, it becomes a “truth” even though it isn’t.

Is this geometry not true?

Euclid’s postulates are treated almost as nearly unquestionable truths because they start from simple propositions. Thus, all subsequent ideas end up having their validity based on these axioms, and the label of “true” becomes common in geometry.

With this in mind, when we delve into this perspective, we realize that it’s not possible, or even reasonable, to ask why Euclidean propositions are true.

“One cannot ask if it is true that only one straight line can pass through two points. It can only be stated that Euclidean geometry deals with objects called ‘lines’ that possess the property of being univocally determined by any two of their points.”

Therefore, it’s possible to affirm that geometry itself cannot be a “truth” simply because it’s not real.

Geometry and Physics

“Let us suppose that, following our customary way of thinking, we add to the propositions of Euclidean geometry a single additional one: to two given points of a practically rigid body, there always corresponds the same distance (a straight line segment), no matter how the body may be situated. In this case, the propositions of Euclidean geometry generate propositions about the possible relative positions of practically rigid bodies.”

Einstein’s quote suggests the need to abandon the idea of an absolute space, where geometry is invariant, and embraces the idea that geometry and physics must be consistent for all observers. This passage from his work serves as a groundwork for presenting his ideas on Geometry for the Theory of Relativity.

Einstein proposes that from the moment the invariance of the distance between two points in a body no longer fits Euclidean geometry, it’s necessary to view geometry as a branch of Physics. Once this is proposed, it becomes possible to question whether any geometric proposition is “true” since it finally applies to real objects.

Image for illustration purposes only

Non-Euclidean Geometry

After all these ideas are proposed, Einstein finally reaches the point and shows that it’s no longer possible to understand the geometry of Space-Time with Euclidean Geometry. To fully comprehend his Theory of Relativity, non-Euclidean geometry is required.

Non-Euclidean Geometry is essential for understanding the Theory of Relativity. It was what altered our understanding of gravity, space, and time and had a profound impact on theoretical physics and cosmology.

Einstein was brilliant in stating:

“Matter tells space-time how to curve, and space-time tells matter how to move.”

He demonstrates how a body can directly affect space-time with its mass, causing it to curve and form geometry, subsequently manifesting gravity and resulting acceleration.

Closing Thoughts

This was a short text to try to expose as much as I could of Einstein’s ideas about Euclidean Geometry and its impact on the Theory of Relativity. I relied heavily on Albert Einstein’s book, “The Theory of Relativity” especially in Part 1.

I sought additional information in research and in my notes from my studies of the Theory of Relativity to bring greater clarity and content to the subject. Clearly, I’m not immune to criticism or even corrections, so I ask that if someone more knowledgeable corrects me, I would be very grateful.

If you like the text, don’t forget to support me. I intend to put as much of my knowledge into this profile and expand my knowledge even further. Please get in touch with me if you have something to say about the subject; I’m always open to discussion.

With that, I extend my thanks to you for reading this far.

This text was translated from Portuguese to English, therefore it is subject to grammatical and translation mistakes. Report any mistake

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Eduardo Prachedes

Young individual with a deep interest in programming, Math, Science and everything.