From Pythagoras to Einstein

The Pythagorean theorem is one of the most famous theorems in all of mathematics. Even people who are not very familiar with much of mathematics are at least familiar with the Pythagorean theorem, especially since its grade school level stuff. It’s also one of the most ancient theorems, obviously known to the ancient Greeks, although it may not have been invented by Pythagoras himself – it may go back to an even earlier time in history.

We recall the statement of the Pythagorean theorem. Suppose we have a right triangle, a triangle where one of the three angles is a right angle. The side opposite the right angle is called the “hypotenuse”, and we will use the symbol c to signify its length. It is always the longest among the three sides. The other two sides are called the altitude, whose length we symbolize by a, and the base, whose length we symbolize by b. The Pythagorean theorem relates the length of these three sides, so that given the lengths of two sides we can calculate the length of the remaining side.

$\displaystyle a^2+b^2=c^2$

Later on, when we learn about Cartesian coordinates, the Pythagorean theorem is used to derive the so-called “distance formula”. Let’s say we have a point A with coordinates $\displaystyle (x_{1}, y_{1})$, and another point B with coordinates $\displaystyle (x_{2}, y_{2})$. The distance between point A and point B is given by

$\displaystyle \text{distance}=\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2}$.

There is a very important aspect of the distance formula that will play an important role in the rest of the things that we will discuss in this post. Namely, we can change the coordinate system, so that the points A and B have different coordinates, by translating the origin, or by rotating the coordinate axes. However, even though the coordinates of the two points will be different, the distance given by the distance formula will be the same.

This might be a little technical, so let’s have a more down-to-earth example. I live in Southeast Asia, so I will say, for example, that the Great Pyramid of Giza, in Egypt, is very far away. But someone who lives in Egypt will perhaps say that no, the Great Pyramid is just nearby. In that case, because we live in different places, we will disagree on how far away the Great Pyramid is. But if, instead, I ask for the distance between the Sphinx and the Great Pyramid, then that is something we can agree on even if we live in different places (Google tells me they are about a kilometer apart).

We disagree on how close or far away something is, because the answer to that question depends on where we are. I measure distance from myself based on my own location, and the same is true for the other person, and that is why we disagree. But the distance between the two objects in my example, the Sphinx and the Great Pyramid, is an invariant quantity. It does not depend on where we are. This invariance makes it a very important quantity.

We will rewrite the distance formula using the following symbols to simplify the notation:

$\displaystyle \Delta x=x_{2}-x_{1}$

$\displaystyle \Delta y=y_{2}-y_{1}$

Furthermore we will use the symbol $\Delta s$ to represent the distance. The distance formula is now written

$\displaystyle \Delta s=\sqrt{(\Delta x)^2+(\Delta y)^2}$

That giant square root over the right hand side does not look very nice, so we square both sides of the distance formula, giving us

$\displaystyle (\Delta s)^2=(\Delta x)^2+(\Delta y)^2$.

Finally, we switch the left hand side and the right hand side so that the analogy with the Pythagorean theorem becomes more visible. So our distance formula now becomes

$\displaystyle (\Delta x)^2+(\Delta y)^2=(\Delta s)^2$.

Again we did all of this so that we have the same form for the distance formula and the Pythagorean theorem. Here they are again, for comparison:

Pythagorean Theorem: $\displaystyle a^2+b^2=c^2$

Distance Formula: $\displaystyle (\Delta x)^2+(\Delta y)^2=(\Delta s)^2$.

Of course, real life does not exist on a plane, and if we wanted a distance formula with three dimensions, as may be quite useful for applications in engineering where we work with three-dimensional objects, we have the three-dimensional distance formula:

$\displaystyle (\Delta x)^2+(\Delta y)^2+(\Delta z)^2=(\Delta s)^2$

Following now the pattern, if we wanted to extend this to some sort of a space with four dimensions, for whatever reason, we just need to add another variable w, as follows:

$\displaystyle (\Delta w)^2+(\Delta x)^2+(\Delta y)^2+(\Delta z)^2=(\Delta s)^2$

As far as we know, in real life space only has three dimensions. However, we do live in something four-dimensional; not a four-dimensional space, but a four-dimensional “spacetime”. The originator of the idea of spacetime was a certain Hermann Minkowski,  a mathematician who specialized in number theory but made this gigantic contribution to physics before he tragically died of appendicitis at the age of 44 years old. But Minkowski’s legacy was passed on to his good friend, a rising physicist who was working on a theory unifying two apparently conflicting areas of physics at the time, classical mechanics and electromagnetism. This young physicist’s name was Albert Einstein, and the theory he was working on was called “relativity”. And Minkowski’s idea of spacetime would play a central role in it.

People have been putting space and time together since ancient times. When we set an event, for example, like a meeting or a party, we need to specify a place and a time. But Minkowski’s idea was far more radical. He wanted to think of space and time as parts of a single entity called spacetime, in the same way that the x-axis and the y-axis are parts of a single entity called the x-y plane. If this was true, then just as there was an invariant “distance” between two points in the x-y plane, then there should be an invariant “interval” between two events in spacetime. This would simplify and explain many phenomena already suggested by the work of Einstein and his predecessors such as the measurement of lengths and the passing of time being different for different observers.

However, the formula for this interval was different from the distance in that there was a minus sign for the one coordinate which was different from the rest, time. This was needed for the theory to agree with the electromagnetic phenomena that we observe in everyday life.

$\displaystyle -(\Delta t)^2+(\Delta x)^2+(\Delta y)^2+(\Delta z)^2=(\Delta s)^2$

There’s still a little problem with this formula, however. We measure time and distance using different units. For time we usually use seconds, minutes, hours, days, and so on. For distance we use centimeters, meters, kilometers, and so on. When we add or subtract quantities they need to have the same units. But we already have experience in adding or subtracting quantities with different units – for example, let’s stick with distance and imagine that we need to add two different lengths; however, one is measured in meters while another is measured in centimeters. All we need to do is to “convert” one of them so that they have the same units; a hundred centimeters make one meter, so we can use this fact to convert to either centimeters or meters, and then we can add the two lengths. More technically, we say that we need a conversion factor, a constant quantity of 100 centimeters per meter.

The same goes for our problem in calculating the spacetime interval. We need to convert some of the quantities so that they will have the same units. What we need is a conversion factor, a constant quantity measured in units that involve a ratio of units of time and distance, or vice-versa. Such a quantity was found suitable, and it is the speed of light in a vacuum c, which has a value of around 300,000,000 meters per second. This allows us to write

$\displaystyle -(c\Delta t)^2+(\Delta x)^2+(\Delta y)^2+(\Delta z)^2=(\Delta s)^2$

This formula is at the heart of the theory of relativity. For those who have seen the 2014 movie “Interstellar”, one may recall (spoiler alert) how the main character aged so much more slowly than his daughter, because of the effects of the geometry of spacetime he experienced during his mission, and when he met up with her again she was already so much older than him. All of this can really be traced back to the idea of a single unified spacetime with an invariant interval as shown above. If space and time were two separate entities instead of being parts of a single spacetime, there would be no such effects. But if they form a single spacetime, then neither time nor distance are invariant; the invariant quantity is the spacetime interval. Time and distance are relative. Hence, “relativity”. Hence, contraction of length and dilation of time. Such effects in real life are already being observed in the GPS satellites that orbit our planet.

The theory of relativity is by no means a “complete” theory, because there are still so many questions, involving black holes for example. Like most of science, there’s always room for improvement. But what we currently have is a very beautiful, very elegant theory that explains many phenomena we would otherwise be unable to explain, and all of it comes back to some very ancient mathematics we are all familiar with from grade school.

References:

Theory of Relativity on Wikipedia

Spacetime on Wikipedia

Spacetime Physics by Edwin F. Taylor and John Archibald Wheeler

Spacetime and Geometry by Sean Carroll