A Space-Time Cocktail: Minkowski Space and Special Relativity

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 Hermann Minkowski's description of the theory of special relativity, space and time mix together to form a new, strange whole. (Artist: frankenstijin at worth100.com).
In Hermann Minkowski’s description of the theory of special relativity, space and time mix together to form a new, strange whole. (Artist: frankenstijin at worth100.com).

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.

Two events in Minkowski Space
Two events in Minkowski space. Event B happens after event A, but both happen at different places.

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.

Worldlines in Minkowski Space
Worldlines in Minkowski space. Because Jack isn’t moving, his worldline is vertical. Jill, on the other hand, is moving away from Jack at a constant rate, so her worldline is angled at a constant slope.

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 3\times 10^8 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.

Light cones in Minkowski Space
Future- and past-directed light cones emanating from event A.

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.

Past-directed causality
Because event B is in the past-directed light cone of event A, it can affect event A. However, because event C is outside the light cone, it cannot 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.

Future-directed causality
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.

Distance in Minkowski space
The distance squared between events A and B is positive. However, the distance squared between points C and D is negative.

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:

    \[a^2 + b^2 = c^2.\]

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.

Pythagorean Theorem in Minkowski Space
To calculate the distance between two events on Jill’s worldline, we look at the part pointing only in space and the part pointing only in time, then use the Pythagorean Theorem.

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.

Rotations in Minkowski space as opposed to in Euclidean space
In Euclidean space (left), where time has a positive sign, we rotate or speed up just by rotating the time and space axes in the same direction, so that they have an angle of ninety degrees between them. However, in Minkowski space, if we rotate in the time direction, the space and time axes move towards each other (right).

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

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Questions? Comments? Hatemail?

This turned out a bit denser than I had hoped. If anything at all isn’t clear, please don’t hesitate to ask about it! And, as always, if you have any questions, comments, or insults, please let us all know in the comments!

25 thoughts on “A Space-Time Cocktail: Minkowski Space and Special Relativity

  1. Very good indeed! easy to follow and thorough. Thank you.

    Just one question to me a bit unclear: why is the time length negative? is it because time slows down at nearly light speeds therefore the time interval is longer? I am not clear as to why Minkowski uses a negative sign. Can you please clarify this point a bit further?

    Thank you again. Luis Bonorino

    1. Hi Luis. I’m glad you liked it!

      A small correction that I should have made more clear. The _square length_ of time is negative. In other words, if a spacelike length is a, then a^2 > 0. But, if a timelike length is b, then b^2 < 0. What this actually means is... well, there's a blunt but meaningless answer, a technical answer, and a hand-wavey answer. I'll give you all three. The blunt but meaningless answer is "because it works." I wrote two articles on special relativity a while ago, where I explained how Einstein derived it. (Number 1: http://www.thephysicsmill.com/2012/11/19/the-speed-of-light-is-constan/, number 2: http://www.thephysicsmill.com/2012/11/25/the-universal-speed-limit/). Einstein’s derivation doesn’t use Minkowski space, but Minkowski space and Einstein’s method produce the same results. Since the Minkowski space method is easier to use and more generalizable, people prefer it… and they don’t really care where the minus sign comes from.

      The technical answer is that the timelike lengths are actually _imaginary_. The square root of a negative number is imaginary, so you can think of the time axis as an imaginary one. (I’ll write a post about imaginary numbers soon.) When you make the time direction imaginary, you can look at how the equations for rotating the axes change, and they become the equations for a Lorentz transformation, which is the physics law that explains time dilation and length contraction.

      The hand-wavey answer is that Einstein realized time and space have to be unified together in a _spacetime_. BUT, somehow we have to distinguish time from space… if time had positive length, it would behave the same as space, which physicists know creates the wrong equations of motion. Minkowski didn’t know this, but we do now from more modern studies of quantum gravity. So you can think of the minus sign as a marker. It says “Time is different. Time is special.”

      Does this help?

  2. From experience of real and imagenary quantities, the time length seems to be a real physical quantity. So I could not figure out why the time axis need to be taken imagenary or square time length as negative. Is it something to do with the fact that world line of any object can bot be a circle in Minkowski space.

    1. Thanks for reading, Amit! You shouldn’t think about imaginary numbers as “non-physical.” They’re very much physical things! Check out my article on imaginary numbers for more of an intuition about this: http://www.thephysicsmill.com/2013/09/23/between-being-and-non-being-imaginary-numbers/

      The reason we take the time axis to be imaginary is to mark that time can only travel forwards, not backwards. This isn’t the only way to mark time as special! It’s just a very convenient one because the mathematics of the Lorentz transformations works out.

      And as you point out, no object can have a circular worldline in MInkowski space. This is a direct consequence of time only traveling forwards. You can think of the minus sign in the square length as coming from this. The reason is that a circle becomes a ellipse:
      x^2 + t^2 = r^2
      becomes
      x^2 – t^2 = r^2
      And this is why a rotation in Minkowski space is a Lorentz transformation.

  3. I think it is much easier to simply analyze motion. This analysis leads you to all of the equations found within Albert Einstein’s theory Special Relativity. It also exposes the absolute foundation that creates the relativistic outcome. The idea is to figure motion out in your mind first, and then convert your understanding into equations as step two, rather than try it the other way around.

    After all, it’s just like Einstein said, “Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore.”. Mathematics, if placed first in line, do not reveal a complete understanding. In fact they can be quite confusing and misleading.

    Videos found at http://www.youtube.com/watch?v=KKAwpEetJ-Q&list=PL3zkZRUI2IyBFAowlUivFbeBh-Mq7HdoQ provide that step by step, and somewhat humorous, analysis of motion and final production of the equations.

    1. Thanks for reading, Sean. I actually wrote about the motion. You can find those articles here:
      http://www.thephysicsmill.com/2012/11/19/the-speed-of-light-is-constan/
      http://www.thephysicsmill.com/2012/11/25/the-universal-speed-limit/

      The Minkowski space picture is more complicated than Einstein’s original description. However, it’s absolutely required to understand general relativity. I discuss for that reason and because multiple perspectives on the same physics are helpful.

      Also, your videos are wrong. You say in video 8 that an absolute frame of reference must exist and that anyone who thinks differently must be pretty stupid. Given that Liebnitz, Einstein, Macht, and many others all felt that there is no such thing as an absolute frame of reference, you must think the greatest physicists in history are pretty dumb, huh?

      The point of rulers and clocks in Einstein’s description is to remove the notion of an absolute frame of reference. We cannot measure the length of an object with respect to the universe, but we can measure the length of an object with respect to a long stick.

      1. Actually, it was simply stated by the big boys, that an absolute frame of reference can NOT be detected, rather than say it does not exist. The universe itself absolutely exists. It does not just exist relativistically.

        1. The universe exists but an absolute frame of reference to describe it does not. There is no preferred frame of reference in general relativity.

          This is an important idea, and one Einstein emphasized. If we can’t DETECT a frame of reference, then we might as well work under the assumption that one doesn’t exist.

  4. Can I first thank you for this exceptional contribution.
    Where can I learn more about-
    we rotate or speed up just by rotating the time and space axes in the same direction
    Many thanks

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