Henceforth space by itself, and time by itself,
are doomed to fade away into mere shadows,
and only a kind of union of the two will preserve an independent reality.
~Hermann Minkowski

Since the mathematicians have invaded the theory of relativity,
I do not understand it myself anymore.
~Albert Einstein

In my previous discussions of how we know the speed of light is constant and how this results in special relativity, I used Albert Einstein’s thought experiments to derive the time-dilating, length-contracting results. There’s another way to describe special relativity, though, invented by the Polish mathematician Hermann Minkowski. It uses a view of the world we now call Minkowski Space, which elegantly describes special relativity as a consequence of the makeup and mixture of space and time.

Space, Time and Graph Paper

Minkowski treated space and time as two interlinked pieces of the same whole. In Minkowski space, we give each point (or event) a position in space and a position in time.

In Minkowski space, people and objects exist at all times, but move between places. The line representing someone or something’s path through space and time is called a worldline. If an object is stationary, the worldline is vertical. If an object is moving, the worldline is at an angle, and the slope of the line is based on the speed at which the object is moving.

When working in Minkowski space, it is customary to work in units where the speed of light is one. We do this so that we can convert between position and time, which we treat as two different types of distance. (For instance, a second is the amount of time it takes light to travel meters.)

Using such units, the worldline of a photon is a line forty-five degrees off of each axis–i.e., a line whose slope is one. The worldlines of light traveling away from a point in every direction thus form the light cone for that point.The light cones traveling into the future are future-directed and the light cones traveling into the past are  past-directed.

Because nothing can travel faster than light, the light cones determine what events in the past can affect current events and what events in the future can be affected by the present. As shown below, if event B is in the past-directed light cone of event A, it would be possible for event B to affect event A. However, since event C is outside of the light cone, it can’t possibly affect event A.

Similarly, if event B is in the future-directed light cone of event A, A can influence B. However, if C is not in the future-directed light cone, A can’t influence C.

Temporal Distance: Time is Different

Now that we know roughly how Minkowski space looks, it’s time to talk about the weird stuff. I mentioned that we treat time as a sort of distance, related to position by the speed of light. What I didn’t mention was Hermann Minkowski’s great innovation: the minus sign.

If we measure the distance between two events that both occur at a particular time, the distance is, of course, positive. If we square the distance (multiply it by itself), it is, of course, still positive. However, if we choose a place and measure the square of the distance between two events that both occur there, the square is negative. This is what separates time from space. This minus sign tells us that time is different.

If we want to measure the distance between two points that are separated in both space and time, we break it up into the bits separated by space and the bits separated by time, then use the Pythagorean Theorem:

   

Since light points exactly half in the time direction and half in the space direction, distances traveled by light have a square length of zero. Distances that have a positive square length are called spacelike. Distances with a negative square length are called timelike. And distances with zero square length are called lightlike, or null.

Mixing Space and Time

Einstein’s theory of special relativity tells us that weird things happen when we go fast. To observe the world as we move at high speed, we rotate our view of Minkowski space. As we accelerate, the time and space axes move. If we were in Euclidean space, where square distances are always positive, we’d rotate the space and time axes together. However–because Minkowski time has a minus sign–if the space axis moves clockwise, the time axis moves counter-clockwise.

When we reach the speed of light, the axes align with the light cones—which, in Minkowski space, always remain at forty-five degrees. (In general relativity, they can change shape because of gravity.) This means that the speed of light is constant, no matter how fast you’re going. It also means that photons don’t experience time at all; they experience space and time as a single phenomenon. However, as massive objects, we can never reach the speed of light, and thus our fragile minds are safe from whatever sanity-shattering horror this confluence might bring.

As the time and space axes move together, all non-lightlike events are smushed together by the axes. This smushing forces distances to shrink and time to stretch out–in fact, it perfectly enforces the length contraction and time dilation predicted by Einstein! This rotation in Minkowski space is called a Lorentz transformation, after Hendrik Lorentz. (For experts: Minkowski space is preserved by the symmetries in the Poincare group.)

It’s important to note that the universe itself remains unchanged as we speed up. However, because we must respect the speed of light as a constant, different observers will experience the universe in different ways. Speedier observers will measure distances to be shorter and times to be longer.

But… Why?

An important question to ask is “Why bother with Minkowski space at all?” Einstein developed special relativity without any of the mathematical formalism introduced by Minkowski. There are a couple reasons why physicists prefer Minkowski’s picture to Einstein’s.

The first, I think, is because it is more beautiful. In Einstein’s formulation, we had to do a lot of thought experiments involving trains. However, in Minkowski space, the rules of special relativity emerge organically from the way we look at space and time.

The second reason physicists prefer the Minkowski picture is because it is substantially easier to use. Einstein’s picture requires the user to memorize a large number of equations–one for length contraction, one for time dilation, one for velocity addition, etc. However, in Minkowski’s picture, these relations are all very easy to re-derive on the fly.

Similarly, Minkowski’s picture of special relativity is easy to generalize. It’s simple to take other theories of physics, such as electromagnetism, and incorporate them into special relativity. Einstein’s picture usually requires more thought experiments if we want to involve other physical ideas.

Finally, Einstein himself built on Minkowski’s work. In the absence of gravity, Einstein’s theory of general relativity simplifies into Minkowski’s description of special relativity.

Obituary

After he died of appendicitis in 1909, Hermann Minkowski’s obituary was written by David Hilbert, one of the greatest mathematicians in recent history. It’s not relevant to special relativity, but it is a testament to what an amazing person Minkowski was:

Since my student years[,] Minkowski was my best, most dependable friend who supported me with all the depth and loyalty that was so characteristic of him. Our science, which we loved above all else, brought us together; it seemed to us a garden full of flowers. In it, we enjoyed looking for hidden pathways and discovered many a new perspective that appealed to our sense of beauty, and when one of us showed it to the other and we marvelled over it together, our joy was complete. He was for me a rare gift from heaven and I must be grateful to have possessed that gift for so long. Now death has suddenly torn him from our midst. However, what death cannot take away is his noble image in our hearts and the knowledge that his spirit continues to be active in us. (source)

Further Reading

• A video game by MIT lets you experience what it’s like to go very close to the speed of light. Very worth playing.
• David Metzler on Youtube has a series of videos on relativity and geometry. The first one is here.
• Louis Del Monte has a series on this, too. The first one is here.
• Hamilton Carter has a great blog post on Minkowski space here.