Tipping the Light Cone: Black Holes

Black Holes by Tamsin Van Essen: http://www.vanessendesign.com/

Black Holes, by Tamsin Van Essen. Part of a series of lovely ceramics with a physics theme. For more, visit the websites here and here.

As you may recall from the post I did some time ago, the “Light Cone” is a rather important concept in physics, and keeping track of it in a given physical scenario is an extremely important tool and technique for understanding many physical situations. (I urge you to review that post before continuing reading this one.)

One way to understand a most important concept – the event horizon – is by keeping track of lightcones, and so let’s go ahead and explore that here. The outcome is that you’ll end up with an understanding of one the the most striking, simple, and beautiful objects in physics – the black hole.

So let’s just study what called the “future light cone”, which is that divider between events that are in your future (if it is your lightcone) and events that are not. The point (review the post) is that events are only in your future if you can reach them by moving no faster than the speed of light. By “in your future” I mean that you can be connected to them physically in some way. And remember that an event is a time and place together. Yankee stadium (or Wembley Stadium, for our readers in the UK) are not examples of events. Three o’clock in the afternoon on Thursday is not an event. Yankee stadium at three o’clock on Thursday afternoon is a event. Given that it take light about 8 minutes to move between the earth and the sun, the event “three minutes from now at the sun” is an event, but it is not inside your light cone. “9 minutes from now at the sun” is, however. Events inside the future light cone are in your future, ones outside are not. The surface of the cone itself is the borderline case: events connected by motion at the speed of light.

To understand the structure of a region of space and time (“spacetime” is the term we should and will use), it is important to understand the light cones for every point in that spacetime. To find out how the landscape of events are connected. The key point here is that we’ve learned from Einstein’s General Relativity that certain situations (often, but not always, involving gravity) can radically change the structure of spacetime so that it’s lightcone landscape gives rise to rather striking physics. Light cones are usually drawn on spacetime diagrams in the way that I showed earlier. Rather than try to draw all the four dimensions of our spacetime, it is usually enough to draw two of them, time pointing up, and one spatial coordinate pointing along the horizontal axis. A future light cone would strictly be an infinite triangle in this diagram, but we take poetic license and do two extra things. We shorten it, for a start, and then draw a little ellipse to complete the top of it, to remind us that there are more spatial directions (you’d have a cone for real if we had two spatial coordinates, but we actually have three, and so the cone is not really the cone you eat ice-cream from -fixed time cross sections of the cone are spheres, not circles.) Have a glance at the sketch to the right and refer to the earlier post.

light cones in flat spacetimeSo regular flat (or only gently curved) spacetime that we are used to from everyday physics has a simple light cone structure. The cones all are the same for every point. If I were to use a radial coordinate [tex]r[/tex] to denote how far away I am from a given point in spacetime, for example, the spacetime diagram would be like the one I’ve drawn to the right.

Well, when spacetime is curved by the presence of mass-energy, the lightcone structure gets distorted. For a mass that is significant (so that we need to worry about its effects) but at the same time is not too compact, such as for our earth and our sun1 the light cone structure looks like the diagram on the left. light cones in  spacetime with not too compact massYou can see that the cones get a bit bent towards the mass! One result of this is that light has to make more of an effort (as it were) to get away from the mass. In fact, it has to give up energy to do this, and so as a result gets its frequency reduced (its wavelength lengthened) so that it appears “red-shifted” to observers further away. This structure will also end up allowing you to deduce that the mass bends light that passes near it (see an earlier post) and several other important effects, such as time slowing down as you get nearer the mass, and so forth. These are all well known measurable features of gravity as predicted by General Relativity.

Now when the mass is too compact2, something marvellous happens. Look at the sketch to the right. light cones in  spacetime with a compact mass - a black hole After a certain point of closest approach ([tex]r=2M[/tex], see the footnotes 1 and 2), the light cones tip over completely! What does this mean? Well, it means that anything back beyond that point (at larger radius) is no longer in the future light cone of any object that has gone past it. Check: If the cone has tipped so that the right side is past vertical, then anything emitted, including light, will only go to the left. In other words, nothing can get out of the region inside that radius. That radius is called, understandably I hope, an event horizon, and the object we’re talking about – that compact mass – is a black hole. The mass has all disappeared to [tex]r=0[/tex], leaving this rather simple, pure geometry. (The details of what happens at or near [tex]r=0[/tex], the famous “singularity”, needs a better theory of spacetime physics. See my post from last week about these and related issues in string theory.)

You might be thinking that this is all weird, and that it’s more science fiction dreamed up by those theoretical physicists, but think again. Black holes, and the sort of physics they exhibit, seem to be an extremely natural part of our universe. A situation where some mass is compact enough (squeezed into a small enough space) in order to give rise to such a situation is not rare, it seems. The end-state of a large enough star’s life is inevitably going to end in a collapse which leads to a black hole. (Not our own star, since it is a bit too small. Stars from about 20 times the size of our sun are expected to be able to collapse to black holes.) These stellar black holes are quite reasonably sized – ranging from a few times the mass of our sun to about ten times. Even more dramatically, the centers of most galaxies contain a lot of mass that is compact enough, making supermassive black holes that are millions and even billions of times as massive as our sun. (There are also expected to be (and evidence has been found of) intermediate mass black holes too. See a post of mine on Correlations here.) There’s also the possibility of microscopic black holes having formed in the very early (hot, dense, violent) universe, but we’re some way from putting that to the test (both theoretically and observationally), but there’s probably more to come in that area of research.

[Up next on this topic: “Tipping the Light Cone: Event Horizons in the Lab”. See you then.]



  1. Imagine that the mass is roughly a spherical blob. Then the radius, [tex]R[/tex] of this blob is larger than [tex]2GM/c^2[/tex], or [tex]2M[/tex] if I measure things in units such that Newton’s constant [tex]G[/tex] and the speed of light [tex]c[/tex] are unity. [return]
  2. Now the opposite situation to footnote 1 is relevant: [tex]R<2M[/tex]. [return]


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