Einstein’s most revolutionary idea is that gravity is geometry. The theorems of geometry are not eternal truths, but can be changed by mass and energy.

This is my first longer explanation of ideas that are well established in science, but not well understood by the public. To keep this from becoming too long and overwhelming, I have split the six parts of this explanation into three blog posts. (I&II, III&IV, V&VI)

The two parts for today are the introduction and an explanation of how different geometries have different versions of the Pythagorean Theorem.

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Prerequisites: The Pythagorean Theorem. I am serious about explaining the key ideas of general relativity without using calculus or anything more difficult. If you find something hard to understand, please ask me for more clarification.

Originally Written: October 2016.

Confidence Level: Established science since the early 1900s.

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I. Introduction

Einstein’s most famous contribution to science is $$ E = m c^2 , $$ which boldly declares that energy and mass somehow consist of the same stuff.

Despite its fame, this is not the most revolutionary idea of Einstein. That honor goes to: $$ \textbf{Gravity is Geometry} $$ This idea is completely non-obvious to the uninitiated. I hope to explain it to you using nothing more than high school geometry and trigonometry.

Most people think of the laws of geometry as though they were given to us by God (or Euclid). Every high school student is taught these laws, from the rules for determining if two triangles are equivalent to the areas of various shapes to the Pythagorean Theorem.^[1]Notice that this is called the Pythagorean Theorem, not Pythagoras’s Theorem. Pythagoras not only was a mathematician, he also founded a cult. The Pythagoreans were a group of mathematicians … Continue reading We like to think of these theorems as eternal truths. Regardless of what’s happening in the foreground, geometry always exists in the background. Matter and energy move and change, passing through the static laws of geometry.

This way of thinking conforms nicely with our intuition and was the dominant – if not exclusive – view of scientists up to 1900. Unfortunately, it is wrong.

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II. Multiple Geometries

The first cracks in this logic came from the realization that the laws of geometry are not unique. Euclid’s Elements laid out a fully self-consistent description of two-dimensional objects. It is not the only possible description. By changing the underlying assumptions Euclid puts forward at the beginning of his work, you can get other fully self-consistent descriptions of two-dimensional objects. Some of these other geometries have intuitive pictures associated with them. For example, elliptical geometry describes objects on the surface of a sphere. Others, like hyperbolic geometry, are much less intuitive. Regardless of whether these geometries are intuitive, there is nothing wrong about any of them. Each is based on slightly different assumptions, which are then used to prove dramatically different results.

If you have ever looked into non-Euclidean geometries before, you might be expecting me to state how the assumptions are modified and then follow through Euclid’s proofs to see what changes. Although this is necessary for any mathematicians who want to construct proofs using the results of non-Euclidean geometry, it is difficult to build intuition this way without becoming a mathematician yourself.

Instead, I will focus on one theorem: the Pythagorean theorem.^[2]Other theorems in geometry also change. The Pythagorean Theorem is the easiest to explain, but not the only difference. As another example, the circumference of a circle in elliptic geometric is $C … Continue reading A right triangle with sides of length $dx$ and $dy$ and hypotenuse of length $ds$ will satisfy: $$ ds^2 = dx^2 + dy^2 $$ Once Cartesian coordinates (think graph paper) were developed, this expression took on a slightly new meaning. The length of any line segment $s$ can be determined by its lengths along the $x$ and $y$ axes. This explains my notation: the distance along $s$ is determined by the distances along the $x$ and $y$ coordinates. ^[3]The $d$ actually means differential, not distance, but that’s not important right now. The expression extends naturally into higher dimensions. The length of a line segment can be expressed in terms of the lengths of its projections onto each of the Cartesian axes. $$ ds^2 = dx^2 + dy^2 + dz^2 $$

One important feature of this equation is that it is independent of rotations. Take a line segment of length $ds$ on a piece of graph paper. The line segment has a length $dx$ in the $x$ direction and a length $dy$ in the $y$ direction. Rotate the graph paper while holding the line segment fixed or rotate the line segment while holding the graph paper fixed. The lengths in the $x$ and $y$ directions will change: $dx$ and $dy$ do not remain the same after a rotation. However, the total length of the line segment $ds$ must stay the same, so $dx^2 +dy^2$ always has the same value, regardless of how you rotate the graph paper.

If you were to use a non-Euclidean geometry, the Pythagorean theorem would change. This does not mean that the Pythagorean theorem is wrong. It just means that the assumptions used to prove it are not fulfilled. In particular, the Pythagorean theorem assumes that you are doing geometry on a perfectly flat plane. If you were doing geometry on a curved surface, you would get a different result.

As an example, I will state the Pythagorean theorem in elliptical geometry. Suppose you are on a sphere of radius $R$ (this shouldn’t be too hard for any Earthlings to imagine). On this sphere, the latitude will be denoted by $\phi$ and the longitude will be denoted by $\theta$.

Consider a right triangle drawn on the surface of the sphere which is small compared to the radius of the sphere. The triangle is located close to a specifc latitude $\phi$ and longitude $\theta$. One side is aligned along east-west and corresponds to a change in longitude of $d\theta$. The other side is aligned along north-south and corresponds to a change in latitude of $d\phi$. The length of the hypotenuse of this triangle is $ds$. The Pythagorean theorem in this case reads $$ ds^2 = R^2 \ d\phi^2 + R^2 \cos^2\phi \ d\theta^2 . $$

Notice that near the poles, $ \phi \approx 90^\circ $, so $ \cos \phi $  is close to zero. This means that changes in longitude count for less distance the closer you get to the poles. Traveling a quarter of the way around the world in terms of longitude is much shorter if you are closer to the poles than if you at the equator. If you are standing exactly at either the north or south pole, then you could change your longitude without moving at all. If $ \phi = 90^\circ $ (you are at the north pole), then $ \cos^2 \phi = 0 $, so you could have nonzero $ d\phi $ (change in longitude) and still have zero $ ds $ (the distance that you traveled).

You can find a Pythagorean theorem for larger triangles on the surface of the sphere using the Pythagorean theorem for small triangles. You break up the triangle into a bunch of little triangles and them sum over all of the little ones. I will not go through the details of it because it involves an integral and I promised not to use calculus.

In Euclidean geometry, straight lines are extremely important objects. They are also important in physics. Newton’s first law of motion states than an object at rest will stay at rest and an object in motion will stay in motion on a straight line at constant speed unless acted on by an outside force. There are no straight lines on the surface of a sphere – every path has to be curved just to stay on the surface. There are other curves that take the role of straight lines in non-Euclidean geometry. These curves are called geodesics. In Euclidean geometry, a straight line is the shortest distance between two points. Similarly, in non-Euclidean geometry, a geodesic is the shortest distance between two points. On the surface a sphere, geodesics are circles which go around the circumference of the sphere. The equator and all of the lines of longitude are geodesics. The other lines of latitude are not.

You might be wondering at this point if we can describe geometry on other curved surfaces. A sphere has an extremely simple surface compared to a raspberry or a doughnut. It is possible to describe geometry on any curved surface. Edges and points sometimes cause difficulties because crossing them leads to an abrupt change in direction. The geometry on any surface without any edges or points can be described using a modified version of the Pythagorean theorem. Near a place on the surface which looks like a saddle, the geometry is hyperbolic. Near a place on the surface which looks like a dome, the geometry is elliptic.

Next: Parts III & IV.