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.
So 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. 
You 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. 
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.]
-cvj
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- 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]
- Now the opposite situation to footnote 1 is relevant: [tex]R<2M[/tex]. [return]
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I have to defend Plato. Light appears to be the motion of waves of action that exist in an the extreme variations of the dimensions of space-time. The space-time subgroup of dimensions has the dimensions of area or length*length, time, and momentum (or energy, its product with the speed of light c). That kind of approach, so fundamental in dimensions the dimensions of action, also is an important factor of general relativity particularly in the light cone. Action can exist in many other forms than the motion of light, though it is in the motion of light that the unit of action called the quantum appears. The quantum appears to be independent in its magnitude regardless of what units time and distance are observed. We could use fortnights as time and furlongs as distance the quantum would be the same size physically (of course with much smaller numbers). Feet and inches are sometimes used as an exercise. But in science, meters and kilograms are used and one of those is based on the size of the Earth. The other is a dimension characteristic of planets, moons, stars and all other mass.
Action’s dimensions mean it can exist in one form as motion of momentum or energy waves in space-time, which in quantum relativistic physics is the ds space-time path unit defined. For instance, the Schwarzschild equation describes the way the mass of the Sun and Earth stress space-time into a slightly strained form in which time and distance are slightly different from unstressed space-time or just space. Corrections for that were necessary in designing and understanding the resulting trajectories for spacecraft traveling to the outer parts of the solar system and beyond, as in the Pioneer and Voyager extrasolar space probes.
These corrections were also vital for the Global Positioning System, where calculations indicated they would be necessary though it had not been proven. Because GPS launches were so expensive, the first probes were designed with switchable electronics so that either of the nonrelativistic and relativistic physical theories could be put into service. The other was turned off of course. The result was that the nonrelativistic (Newtonian) model proved false. The relativistic (Einstein and more modern) models proved true, and provided accurate results in GPS units on the ground.
Very good art, Tamsen van Essen’s ‘Black Holes’. Best image I’ve ever seen, even in my own eye, of what seems to be the double-sided whirlpools near the centers of galaxies, into which stars fall and
explode because all of a sudden their gyroscopic inertia is in a distant cosmological (firmament) that appears upside down. Spin makes their charges go different directions, possibly out the poles. That’s only my guess. Whatever is true, van Essen’s art is sure to present a path to eventual discovery of the truth of black holes.
No. Not at all. This is a common misunderstanding. Quantum teleportation still requires information to be send by regular (light or sublight) channels in order to work. So it does not violate the light cone. Sorry to disappoint.
-cvj
Hello quantum teleportation!!!
We can send the photons at 4c down coax cable
That ought to mess with your cones!!!
Light cones and what is outside of a light cone was explained to me in this way.
What is within a light cone is where we are causally connected and what is outside of our light cone we are not causally connected. That outside is called the “elsewhere”. It takes about 11 light minutes for a message to get from a rover on Mars to Houston. If a landslide is about to come down on the rover 5 minutes in the rover’s future, there is nothing Houston can do to signal back to the rover to move out of the way in time. That event is in the “elsewhere” and the rover is doomed, assuming it has no circuitry with a program that allows survival in such a situation.
Anyway, that is my two cents. The tipping of the light cone is something I am waiting for. It appears that the future and past cannot be distinguished but I am no expert.
In the lab?
So.. continuing a bit*… one can ask the question in the full interior geometry (i.e. not schwarzschild, but the interior schwarschild)… the mass function varies with r and so forth, and so the proper radius vs the coordinate radius is a more subtle issue, depending upon the hydrodynamics… But the upshot is, as far as I know, entirely consistent – there’s no tardis behaviour (way bigger inside than outside) going on and so forth! This is not the place for a lecture on this (and frankly, I’ve not looked at this for a while so would have to review), but it should be in Wald, or MTW.
-cvj
(*I can see that your question did not necessarily apply to the exterior geometry, so sorry if you were correctly thinking in terms of the appropriate interior geometry.)
What I was pointing out was that there is nothing wrong with saying that the radius r=2M is where the star’s radius must fall for it to form a black hole. Both sides are measured in Schwarzshild coordinates, and therefore it is unambiguously correct.
Now what you’re pointing out is the result of a common mistake. The geometry inside the star is not the famous Schwarzschild geometry, and so the question (if that was Yonggary’s question) is problematic.
cheers,
-cvj
Clifford,
I’m guessing Yonggary was pointing out the difference between the coordinate r in Schwarzschild coordinates, and the proper distance from the center of the star to its edge, as measured along a geodesic of the three metric on some spacelike hypersurface. Perhaps a related issue is the question of what notions like length, area and volume mean for an object whose self-gravity cannot be ignored, and for which the space-time region that it occupies is not approximately Minkowski.
Well, let’s stop right there, ‘cos I’m confused by those last seven words. Why did you declare that? Your two statements are equivalent, since the line element for the angular coordinates is the same as it would be for flat space. There is no ambiguity that I can see by saying that in Schwarzschild coordinates, when r falls through 2M, it is a black hole. It’s perfectly clear – there’s no confusion there.
Best,
-cvj
Esprit d’escalier: when I say 2 pi I always mean 4 pi of course…..
Hi, I have often wondered about the following. We often say that a star will collapse to a black hole if its radius falls to 2M, but that isn’t really true of course — the correct statement is that this will happen if its *surface area* falls to 2 pi times [2M]^2, but radial distance is not given by r in the Schwarzschild metric. So in the case of *realistic* stellar collapse, what happens to the physical radius of the star? Could it be that the radius of a star which is on the brink of collapse is actually some huge number? A sort of Tardis Star? 🙂
Hi Plato,
Very happy to hear you like it. Please don’t be scared – it is just that the long rambling unstructured comments that you sometimes post can simply be discouraging of other readers or derail potential discussion, so I try to deter those a bit. Comments are quite welcome, in general.
Lt.milo,
Well, what can I say? That’s why I’m here (in part).
Cheers,
-cvj
“Depends upon the coordinates. In Eddington-Finkelstein coordinates (what I am more or less using in the post), you see them tip over a lot, making it explicit that the familiar radial coordinate actually becomes a time coordinate once you’re inside the horizon. It is another way of seeing that the singularity at smaller r is inevitably going to be visited by the traveller – it is no longer a place, but a time in their future.”
thank you Clifford! It is strange, I am just in 11th grade, but have read numerous physics books, nothing with extreme mathematics, mostly popular works (and everything Feynman). You would think that an example as clear and visual as a cone for descibing such concepts would be more widely implemented in popular physics literture.
Sorry Clifford.
Clifford:Sorry, I’m being a bit harsh, but I so often find your physics-related comments totally confusing.
It’s really been a struggle Clifford, and in no way meant to impinge on your presentations. You’ve been more then patient.
At the same time, I am always wary of the discipline to which you handle yourself and others. It is always with great trepidation that I wonder if I should or shouldn’t post, while I am a fan and whether my vote is worthy or not, I truly appreciate the efforts you take to blog.
Well, Plato, the whole point of the lightcone post was to stay away from this sort of thing and stick with the visuals of the lightcones, and have no equations. So no, not “by definition”.
-cvj
I thought this “by definition” should be along with tipping lightcones?
Escape Energy for Photon
If the gravitational potential energy of the photon is exactly equal to the photon energy then
[tex]\normal hv_o=\frac{GM}{rc^2}{v_o} \hspace9 \text or r=\frac {GM}{c^2}\\ \text so if Mass M collapses to radius r a photon will be redshifted to zero frequency[/tex]
Note that this condition is independent of the frequency, and for a given mass M establishes a critical radius. Actually, Schwarzchilds’s calculated gravitational radius differs from this result by a factor of 2 and is coincidently equal to the non-relativistic escape velocity expression
[tex]v_e_s_c_a_p_e_ = \sqrt {\frac{2GM}{r}} \hspace9 \\ \text which if V is set equal\\to c gives a radius r=\frac {2GM}{c^2}\hspace9 \text Schwarzchild Radius[/tex]
This equivalence is used as a mnenomic, but does not imply this is a valid way to derive the Schwarzchild Radius
My preference would be to delete it, since, with respect, I see no point to your placing of a standard textbook derivation of the redshift at this point.
[Update: Sorry, I’m being a bit harsh, but I so often find your physics-related comments totally confusing. Sorry.]
Are you trying to make some particular point …or practice [tex]\LaTeX ?[/tex] Let me see…. you want to show that the Schwarzschild radius can be derived as the surface of infinite red-shift? That is true. Yes.
Best,
-cvj
um Clifford, might you correct above? Sorry!
Gravity and the Photon
The relativistic energy expression attributes a mass to any
energetic particle, and for the photon
[tex}E=mc^2=hv{/tex}
The gravitational potential energy is then
[tex]\LARGE U=\frac{-GMm}r=\frac{-GMh}{rc^2}{vo}[/tex]
When the photon escapes the gravity field, it will have a different
frequency
[tex]\large hv=hv_o[{1-}\frac{GM}{rc^2}] \hspace9 v=v_o[{1-}\frac{GM}{rc^2}] \hspace9 \frac{\bigtriangledown v} {v_o}={-}\frac{GM}{rc^2}[/tex]
Since it is reduced in frequency, this is called the gravitational
red shift or the Einstein red shift.
will continue later with regards to….
Escape energy for the photon/b>
If the gravitational potential energy of the photon is exactly equal to the photon energy then..
Depends upon the coordinates. In Eddington-Finkelstein coordinates (what I am more or less using in the post), you see them tip over a lot, making it explicit that the familiar radial coordinate actually becomes a time coordinate once you’re inside the horizon. It is another way of seeing that the singularity at smaller r is inevitably going to be visited by the traveller – it is no longer a place, but a time in their future.
-cvj
This “light cones” present a very interesting and nice way to visualize these concepts.
Possible novice question: How far can a “light cone” be “tipped over”? Or is the point at which the right side is past verticle, the limit, aka a black hole?
Hi David,
Thanks for the comment – long time since we’ve been in touch, so good to hear from you. Yes, what C.O. is trying to say is that – as far as we know, and it is borne out by everything we’ve done experimentally so far – the speed limit on light is an intrinsic property of space and time, or “spacetime” – the two thought of as making the physical whole. It is not that light is just the fastest thing and we’ve yet to find something faster…. it is that that particular speed itself is fundamental to the local structure of space and time. Some people like to claim that it is somehow not fundamental, and that it is a variable, and they are simply wrong, in that such a claim does not fit with what has been observed and tested in so many ways.
Best,
-cvj
David, the reason nothing can move faster than light is that the speed of light isn’t just “the speed of light”. It’s actually a conversion factor between time and space, and light simply happens to be the first thing mankind encountered (or studied) that travels at that magic speed.
Depends upon where the observer is. Are they on earth? If so, then the sun three minutes from now is not inside the future light cone. Simply because you can’t get there or send any signal that can get there in three minutes. However, the sun 20 minutes from now is inside. Similarly, the sun three minutes ago is also not inside the observer’s past light cone. Nothing on the sun could have sent you a signal that could have got here in three minutes. The sun 20 minutes ago is inside, however.
There’s no need for negative time and so forth to confuse the issue, and “observable past” is a bit confusing. See the light cone post for a re-read.
It is pretty simple. I know that the temptation is to want it to seem complicated… but it is not.
Cheers,
-cvj
Or, perhaps, I should use “observable past” for the past light cone, and “causal future” for the future light cone?
Thanks again.
Thanks for a helpful post for understanding some effects of black holes.
I am somewhat overwhelmed by the simplicity of the seemingly (to me, of course) usage of the jargon “future” and implied, I guess, “past” events. I have been under the impression that future and past referred to the positive and negative time intervals respectively measured in a particular frame, and that the causally past events of an observer are those which can influence him in the present as represented by distances not greater than that traversed by light during the time interval elapsed between them, i.e. events inside the past light cone. The corresponding causally future events are those which the observer might influence by the same criterion, and therefore lying in the future light cone. Then, at the sun three minutes from now is a future event which will have been in observer’s causal past after five minutes, and at the sun nine minutes from now is a future event in his causal future. Would this be a fair statement?
Would a better measure be toward the “evidence of the horizon” when the determinate is, “the escape velocity of the photon?” My dial-up is terrible right now. I will offer the equation later(for scrutiny) as it was presented while doing research.
This got me thinking about Susskind’s “thought experiment” as reported by Geftner.
Oh, and I second the lessons.:)
Mind blowing.
Thanks for the USC level lesson without the USC level tuition!
I’ve always been a bit confused as to why scientists are so insistent that nobody could ever be faster than light. Is it because nothing has ever been observed that is faster than light? And couldn’t light be sped up? If a black hole can bend light, it would lead me to believe that it is possible that light COULD move even faster if something wasn’t restricting it… but whats restricting it? Anyway, I’m a total noob on all this. Appreciate these online lessons!