Causal Diagrams

I once witnessed a physicist explain the universe to an artist. The artist had approached the physicist to learn how to understand extra dimensions, a concept, so he explained, that would undoubtedly enhance the depth of his artwork, and be of great inspirational value for his quest to capture the contextuality of essence. Or maybe essence of contextuality. Or something like that. Either way, the physicist took a piece of chalk and drew a line on the blackboard. "That is our universe," he said. It took several minutes before the artist stopped laughing and said "Now THAT is what I'd call an abstraction."

You see, the fact that our universe is at least 4-dimensional and infinitely large (or damned close to that) creates some problem with visualization. The average blackboard is 2-dimensional, somewhat smaller than infinite, and my female brain already finds 3d plots messy and confusing. Add to this that most physicists aren't particularly great in drawing the universe.

Thus arises the need to picture 4 dimensions in an intuitive and illuminating way. Penrose-Carter diagrams, also called "causal diagrams," do exactly that. Though they do not work for the most general space-times, but only when additional symmetries simplify the scenario, they capture the essence of a 4-dimensional space-time. Or maybe the essence of 4-dimensional contextuality.

Understanding causal diagrams is one of the most basic skills you need if you want to work in General Relativity.

It works like this.

First, we have the problem of getting 4 dimensions down to 2, where one of the 4 dimensions is time. That's not so complicated. We will assume that space is spherically symmetric, such that when you sit in one point, all directions from that point look similar. This would be the case for example if you sat in the middle of a ball or, to reasonably good precision, if you sat in the middle of the Earth. The only interesting information is then in the change of scenery as a function of the distance from you, who you are sitting in the center of symmetry. We can thus capture the full 3 space dimensions by just considering what happens with the distance to the center of symmetry. This distance is of course just the radial coordinate r. Besides that, we will draw the time-coordinate t, which is usually depicted vertically, whereas r is horizontally. This is shown in the picture below, left. You've seen that before.

An infinitely flat 4-dimensional space-time is then just a half-plane. Note that a flat space is spherically symmetric around every point. (If you want to nitpick, what I mean with "flat" is that the curvature tensor identically vanishes.)

Next thing we do is to notice that if we had a particle moving towards the center of symmetry at r=0, passing through it, and moving away from it again, it would look on the half-plane like a reflection instead. Sometimes we thus mirror the half-plane to the other side, such that the curves of particles just go through. Keep in mind though that r increases in both directions. The world-lines of particles with a fixed velocity move on straight lines in that plane. Don't try to draw curves for particles that do not approach the center radially because the symmetry doesn't allow it. We now adopt the first convention for causal diagrams:

Light moves on 45° angles.

Curves on which light moves are called "lightlike," or, due to their property of having zero length in a Minkowski-metric, "null curves."

The next step is more tricky, because now we have to deal with the infinitely large space. How do we get it to fit on a blackboard? If you have ever done perspective drawing, you know the answer already. The "horizon line" and the "vanishing points" depict the infinite distance on a finite sheet of paper. The price to pay is that what is equally spaced far away, moves closer and closer together on the 2-dimensional picture. An example is the photo with railroad tracks to the right.

To draw a picture of an infinite space-time, we do exactly the same: we make infinity finite by squeezing together what is far away. Since the space-time is infinite in more than one direction an additional assumption is that we

Squeeze infinity equally in all directions.

The resulting squeeze is also called a "conformal transformation," and has the merit of preserving angles, such that most importantly null curves still move on 45°, no matter which such transformation you used. There are many different squeezes, though qualitatively they look all similar. An example for an often used squeeze is the tangent function in the interval [-π/2,π/2], shown to the left. If you take equal spaces on the vertical axis, the corresponding values on the horizontal axis produce a no longer evenly spaced representation of that infinite vertical axis.

If we now go and squeeze our flat space-time what we get is a diamond.


In this diagram, spacelike curves always have angles less than 45°, and timelike curves on which particles can move have angles more than 45° (in every point). All spacelike curves come from and end in the side corners, called "spacelike infinity," whereas timelike curves all come from the bottom corner and end in the upper corner, called "past timelime infinity" and "future timelike infinity," rspt. Light comes from the lower V-shaped boundary and end at the upper Λ-shaped boundary, called "past null infinity" and "future null infinity." The null infinities are usually denoted with an I in a script font, and are thus for short often called "scri minus" for past null infinity and "scri plus" for future null infinity.

So far so good, but flat Minkowski space is admittedly somewhat boring. Let us thus look at something more interesting. The causal diagram of the maximally analytically extended Schwarzschild-solution, describing a static black hole. You have seen it thousands of times in the header of this website.


It is futile trying to explain how to obtain the diagram without telling you what a metric is and what to do with it, but the big advantage of these diagrams is exactly that you can learn something about the space-time properties without bothering with tensor equations, so let's see.

The first thing you will notice is that the diagram contains regions (A and B) that cannot be connected by any lightlike or timelike curve. This means there is no way to send information from one to the other, and A and B are thus causally disconnected. You will also see that there are two spacelike boundaries on the bottom and top where time- and lightlike curves end without having reached infinity. The spacetime is thus geodesically incomplete or, equivalently, has singularities. The maybe most important property to identify is the boundary of the region from which lightlike curves can reach future null infinity. If not at an infinite distance, this boundary it is called a future event horizon. Similarly, the boundary of the region to which light can be send from past null infinity is a past event horizon. These horizons are always lightlike surfaces.

When you're done thinking, take time to see how pretty it is.

This Schwarzschild-metric does not only depict a black hole in the upper part, which contains a region where no information can ever come out to future infinity, but also a region in the lower part where no information can ever get in from past infinity. That second region is called a white hole. It is however a mathematical artifact since this diagram describes an unrealistic situation: a black hole that has been there since forever and will be there until eternity. In reality, black holes are formed from collapsing matter and later evaporate. We will discuss the more realistic diagram in another post, so stay tuned.

Finally, upon Googling for images I found that somebody else had used the same motivation from perspective drawing that I came up with. Well. If one thousand monkeys hit they keyboard for long enough, they will eventually type the complete Misner, Thorne, Wheeler. Not only once, but an infinite amount of time.

If you arrived here by just scrolling down, shame on you. The minimum amount of information you should take home is that Penrose-Carter diagrams, aka "causal diagrams," are used to depict the causal properties of 4-dimensional space-times with additional symmetries.