## Thursday, November 19, 2009

### 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.

Arun said...

Looks like all the comments posted so far are in a causally disconnected region. :)

Zephir said...

How Universe looks like...

http://infohost.nmt.edu/~imcgarve/perlin_noise/perlin_noise2.jpg

Kaleberg said...

"In reality, black holes are formed from collapsing matter and later evaporate."

This always gets me confused because stuff that falls into black holes appears red shifted so that an outside observer never sees a particle reach the singularity. Meanwhile, the black hole is evaporating because virtual particles created near the singularity lose their antiparticles and appear to that same viewer as radiation.

It always reminds me of traditional Christian eschatology in which one dies and goes to heaven or hell immediately, but one isn't judged until Judgment Day at some point in the distant future.

I never did understand either chronology. Is there a causal diagram that might explain things, at least about black holes? I doubt there is one for eschatology.

Bee said...

Dear Arun:
Yes! Some days ago I thought we're really getting a lot of comments these days, I should write something with content.

This post was actually a reply to a question by Phil, so where is Phil? Best,

B.

Bee said...

Hi Kaleberg,

the observer doesn't see the particle reach the horizon. There is no singularity at the horizon. (It was once thought to be a singularity, but it turned out to be due to the choice of coordinates and not a physical singularity.) You are right of course that there is a very high redshift at the horizon, and everything is very slowed down for the observer at infinity. It takes a very long time for the black hole to evaporate. This is why the static black hole solution is an excellent approximation in many cases, since the black hole is quasi-stable for a very long time. For the causal structure however, which is a global property, ignoring the formation and evaporation misses relevant features. Best,

B.

Phil Warnell said...
This comment has been removed by the author.
Phil Warnell said...

Hi Bee,

"This post was actually a reply to a question by Phil, so where is Phil?"

Like the cartoon character of your last post I was attempting to stretch my brain muscles after reading this one:-) Also, I do appreciate the post which I find to be excellent and can’t imagine what the follow up will be when you show what one would get with the universe having a beginning, Actually my question was in response of being aware you recently where working with such diagrams, which lead me to ask the following:

Are they causal based on what has happened or what is about to? That is are the drawings resultant of the causal past or future as being consequence or rather potential?

However, as you explain the diagram of a black hole between regions A and B are not causally connected at all, that is so far as it’s been defined as the transfer of information being restricted to light speed. Of course this is only true if one chooses not to consider perhaps the most ignored consequence of quantum mechanics, that does have things connected in their action(s) as being holistic and as such not limited from being time-like or space-like separated; which in essence forms the bases of my question.

That being this so called spooky action at a distance doesn’t appear to be so restricted, as it mandates the outcome of potential, which is what Bell was the first to demonstrate. This leaves us able to consider one of two conclusions drawn from his theorem, being that natures action in regards to information transfer at this level as mandated by QM, is either resultantly non-causal or the universe at its deepest level being necessarily holistic.

Personally I find that when having to choose between the two I am forced to except the latter as being true, rather than the former if for no other reason out of sheer logical necessity. The big question of course is how and not why, for the why being simply if it weren’t we wouldn’t have reality to be as it has demonstratedly presented itself, that is at least not as this one has.

Best,

Phil

Phil Warnell said...

Hi Bee,

Rather than rewrite my comment which is my usual terrible habit , I’ll instead just point out I meant to say “Personally I find that when having to choose between the two I am forced to ‘accept’ the latter as being true, rather than the former if for no other reason out of sheer logical necessity”. I had written excepted when I should have said accepted.

Best,

Phil

Andrew Thomas said...

Hi Bee, thanks for a really clear explanation of these diagrams. In the third paragraph you describe them as "intuitive". Do you really think so? I would instead describe them as completely counter-intuitive. The space they describe is not like any intuitive notion of space. Also, you provide some simple examples, but what the heck is someone supposed to make out of this horrible diagram?

I'm sure if you work in the field and you use them everyday, then you get used to them and they're very useful. But for everyone else I think they're just a nightmare! I've struggled with some for maybe 20 minutes before concluding that the idea they are trying to convey is actually pretty simple and could have been described better in a couple of sentences of words. I actually think you hit the nail on the head when you said "take time to see how pretty it is". I think that 50% of the attraction - they look cool.

Also, considering that last diagram of yours with the two diamonds next to each other, which appears to show a parallel universe next to ours (the other diamond). Some people seem to be saying "Oh, that's evidence of a parallel universe" (see here for example). Noooo! It's just shows how the diagram is not a good description of reality. Not the remotest evidence of a parallel universe.

Bee said...

Hi Andrew:

Well, what I've tried to say is that you can read off the diagram the causal properties very easily. Which is vastly more complicated if I just give you a metric. Sure, maybe there is an easier explanation in words, but then an explanation works well for somebody doesn't work well for somebody else, and I personally like figures better than essays. The diagram you link to is horrible indeed, but not because it is a causal diagram, but because the caption is missing. What does the dotted line represent, what the dashed-dotted, what the line labeled OH (or maybe AH, hard to read), what is M1, M2, N1 etc.

I explicitly wrote that the diagram of the Schwarzschild-metric is not a description of reality. So what's your point? Best,

B.

Andrew Thomas said...

Hi Bee, my point is just that I find these diagrams very difficult to follow. Whereas spacetime diagrams are intuitive, I just find these a nightmare to follow.

My point about the parallel universe is that the resultant diamond shape of these diagrams almost seems to invite the drawing of these disconnected regions which are interpreted as parallel universes.

Like I say, I'm sure for people who work in the field they must be great. But I find them mind-bogglingly confusing. And I don't think I'm thick - I think they genuinely are confusing. Which I guess is why authors of popular science books never include them.

Not the slightest criticism of your article which was a very clear description. I just don't like these diagrams. I'll look forward to your next article on them. Maybe you'll end up convincing me!

Andrew Thomas said...

(I should add that that horrible diagram I posted was from Richard Gott's paper Can the universe create itself? and apparently shows the formation of a baby universe on the other side of a black hole (apparently!). It's just a diagram I have struggled with for ages.)

Andrew Thomas said...

I must say, I really like your idea of perspective on railway tracks. I'd never thought of it like that before, but I will from now on.

Bee said...

Hi Phil,

Sometimes I'm convinced there must be two of you, one who sleeps at night and one during the day ;-)

As to your question in which regard the diagrams are causal, I think you are looking for more in the name than there is. They are causal in the sense that they depict the causal properties: they show you which part of spacetime can influence another. You are instead asking a question about the evolution (which part of the diagram follows from what). The evolution is given by Einstein's field equations. To obtain a solution you need to specify initial conditions, and these you can evolve forwards or backwards. The diagram depicts the whole evolution, for all times. Also, as long as there's no quantum mechanics, either way is fine, and forwards is as good as backwards.

With regards to the spooky action, you certainly know that you cannot transfer information faster than the speed of light even in quantum mechanics. That's what I was referring to. Best,

B.

Bee said...

Hi Andrew:

Well, I did my best to explain the diagrams on a very basic level. I see no point in arguing with you about what you find confusing or intuitive. Just use what works best for you. The causal diagrams work fine for me, and as I said they are very widely used, so it seems to me I'm not the only one to find them useful.

My point about the parallel universe is that the resultant diamond shape of these diagrams almost seems to invite the drawing of these disconnected regions which are interpreted as parallel universes.

I wrote explicitly that the diagram is not a representation of a realistic situation, so in how far does that "invite" anything? Best,

B.

Bee said...

Oh, and regarding the horrible diagram, if you want to see easier to understand diagrams of baby universes, look at my and Lee's recent paper. The horrible diagram seems to me more to show the nucleation of Minkowski-spacetimes from deSitter, rather than "on the other side of a black hole" wherever that might be. But maybe I should look at the paper. Best,

B.

Andrew Thomas said...

Hi Bee, I see how you say that the "white hole" is a mathematical artifact. But is that the same as saying that region A (or B) is also a mathematical artifact? If so, then yes, you've made a good point there.

My reservation about "inviting parallel universes" is well-described in the page by John Horton you link to at the end of your article. He goes on to say: "If we keep extending the spacetime in that way, we end up with a new and interesting black hole. The great novelty of the new black hole is that it is twice the size of the old one. On the "other side" of the event horizon is a complete duplicate of the exterior of the black hole. Everything that happens in region I can happen in region III. Both can have planets and moons and space travelers." So it's like he's managed to find a parallel universe from somewhere, no? So OK, you did well by stressing the unreality of the situation, but this guy is in full flow by this stage: "What can come out of the singularity? Ejection from it is the reverse process in time of falling into the future singularity. So anything that can fall into the future singularity can be ejected from the past singularity. That means anything--dinosauars; the socks you lost; TVs showing re-runs of "I love Lucy"; and so on." It's this kind of stuff I'm not keen on.

Bee said...

Andrew: I have said the above depicted diagram represents a solution to Einstein's field equations, yet not a scenario that is realized in Nature. Yes, in the realistic scenario there is no region B, that will be content of another post. Best,

B.

Bee said...

Well, if you dislike what Horton wrote, why do you complain to me?

Andrew Thomas said...

Hi Bee, because I don't have his email!

But the point is, there's a lot of of this stuff out there. You did well to stress the point, but plenty of people are going to read Mr. Horton's piece and think "Causal diagrams indicate parallel universes". I think you're in the minority.

Phil Warnell said...

Hi Bee,

Thanks for the clarification. Also yes I know that what is referred to as being information is not to be found in the entanglement of quantum systems, as the consequence of quantum randomness precludes such correlations considered as being such. Yet this goes back to asking what information is? That is information what stands as being the actual state of something or is it only information if we are able to become aware of it, as to known for certain.

Just as an example related to what I referred to earlier, let’s set up a thought experiment with having two entangled particles separate, with one heading off to be taken in by a super massive black hole, with the other being taken in by another. Now as we know that nothing unusual is suppose to happen while passing through event horizons of such black holes resultant of tidal forces and such, then so let’s also suppose an observer who entered along with the particle takes a spin measurement of the particle of the nature described in the Bell (EPR) scenario.

This then would have any spin measurement performed on the particle to directly reflect as to what the spin state would be of the corresponding particle that went into the other black hole. So that by the rules as mandated and demonstrated by QM the state of one particle would have one know the state of the other. So despite the two particles being as physically and temporally separated as you can have them, the quantum mechanical correlation between the two particles remain. The question then to ask is, does this correlation represent as being information in relation to the shared reality of the two particles and if not why not?

Best,

Phil

Bee said...

Hi Phil,

No, it's no information because you don't learn from it what's going on in the disconnected region. The measurement doesn't tell you what quantum numbers are inside the black hole or anything. See, the problem with the information loss is that you need to get the information about the stuff that collapsed previously back out of the black hole. Throwing additional things in that are correlated with the outside doesn't help to that end. (This btw was also already discussed in an earlier post. I might be mistaken, but I believe Andrew brought it up. It's the same reason why the entanglement in the Hawking radiation doesn't help you to get information out.) Best,

B.

Bee said...

Hi Andrew:

Yes, that might be. Well, I'll make sure to stress this point even more in the next post then. Best,

B.

Shivang said...

Shame on me!

Phil Warnell said...

Hi Bee,

I think I understand your point, yet for me this still has information being contingent upon the ability to have it be known, which of course in many respects is reasonable simply by way of definition. However, I have to at the same time wonder if such considerations in the end boils down to being any different than the question about the tree which falls in the forest with no one to hear. I quess what I’m actually saying is I’ve always struggled with the correlations that are made as to entropy and information as being one in the same , yet that is of course is to have it as being my problem and certainly not yours:-)

Best,

Phil

Giotis said...

"spacelike curves always have angles less than 45°, and timelike curves on which particles can move have angles more than 45° (in every point)"

I think is the other way around.

Bee said...

I was talking about angles between the curve and the horizontal axis. This is unclear though, thanks for pointing out.

Arrow said...

Since you invoke spherical symmetry your particles and photons are actually giant spherical shells descending upon or receding from the observer ;)

Bee said...

Yes, that's correct.

Andrew Thomas said...

One of the things I find hard to understand is why the singularity is always shown as a horizontal line. Isn't a black hole singularity basically a point in space, and so should be a vertical line (would surely be that on a spacetime diagram)?

Bee said...

Exactly how the line looks like depends on the coordinates, thus it's not necessarily exactly horizontal, it's just spacelike, but that doesn't matter for the causal structure. You will indeed sometimes find the singularity depicted as being slightly hyperbolic. It is however indeed not a point in the diagram. The easiest way to see this is to realize that the singularity is where all light rays end, and light rays can't move towards each other. Best,

B.

Andrew Thomas said...

Thanks.

Arun said...

QFT vacuum in Minkowski space is a nice coherent, zero-entropy thing; yet (Unruh effect) an accelerated observer sees thermal radiation in that vacuum. I'm assuming it can't be zero-entropy thermal radiation. Which implies that entropy/"information" is in part dependent on the state of motion of the observer. But then information loss may also be in part dependent on the state of motion of the observer (even if it is an inertial observer in GR).

We equate "information-loss" with loss of unitarity in QM. But if unitarity itself is observer-dependent, then what?

---
If we use what Penrose calls the U (unitary evolution) and R (reduction, measurement) operators, R also apparently leads to information loss, but we don't have a problem with that. Certainly nobody thinks QM is threatened by R. So stuff falling in a blackhole somehow undergoes an R-like process, perhaps; and while the physical mechanism is mysterious, it is not more or less mysterious than any ordinary QM measurement.

Arun said...

A different thought - in the EPR-like experiments, where a two-particle state is prepared and the particles are then launched to distant observers who make various measurements, depending on whether the first observer makes a measurement or not, the second observer is supposed to represent her situation by a pure-state or mixed-state density matrix. The key emphasis is however that the result of a measurement is the same for the second observer no matter which description applies. There may be a mysterious correlation with the measurement made by the first observer, but there can be no faster-than-light information exchange precisely because the measurement results are the same for the pure- and mixed- state descriptions.

If we are comfortable with that, the analogous situation for the supposed blackhole information loss is to be considered, and if you can show (in a thought experiment maybe) that the result of a measurement will differ if the observer external to the blackhole uses a mixed state versus a pure state (which would be available if the blackhole gave up its information), then there is a problem with information loss.
If the results of measurement are the same, worrying about information loss in a blackhole is just so much theoretical mental exercises.

Uncle Al said...

A stereogram Schlegel diagram is elegant: 4-D into 3-D, then pick your perspective.

Does reality reduce cuddly flat everywhere, or is it a K_5 graph (Kuratowski's theorem)? If reality is K_5, physical theory has problems. If reality is chiral and K_5, physical theory has bigger problems.

How is quantized gravitation doing?

Pope Maledict XVI said...

Andrew Said:

"One of the things I find hard to understand is why the singularity is always shown as a horizontal line. Isn't a black hole singularity basically a point in space, and so should be a vertical line (would surely be that on a spacetime diagram)?"

Ha, that's funny, I was trying to think of an example to convince you how wonderful these diagrams are, and you hit the nail on the head! I mean: the way many people, including lots of people who should know better, talk about the black hole singularity, you would think that it is timelike, as you thought. But it isn't: it's spacelike! And this diagram is the easiest way to prove that. The singularity is *not* a point in space, it is a moment of time in the future [of anyone who goes inside the black hole]. The idea that when you fall inside, the singularity is in front of you, pulling you in, is completely *wrong*! Instead it is in your future, and you are "being pulled into it" in precisely the same sense that you are currently being "pulled into" 2010. So you see that black holes are actually a lot more interesting than you thought, and the Penrose diagram is the best way to appreciate that.

Similarly the Penrose diagram of de Sitter spacetime is the best way to understand why, in an accelerating universe, you lose contact with distant objects.

Pope Maledict XVI said...

By the way, Bee, Penrose diagrams should really be called "Hong Kong and Shanghai Banking Corporation Diagrams", as you can clearly see at the top left corner of this page:

http://www.hsbc.com.hk/1/2/home

Perhaps it's a subtle message about the likely fate of one's savings these days.

Giotis said...

Why whenever I hear about the information loss paradox, I picture an advanced extraterrestrial life form that really knows physics and monitors science progress on earth, laughing at us while eating popcorn in front of a monitor?

Andrew Thomas said...

Pope Maledict said: The way many people, including lots of people who should know better, talk about the black hole singularity, you would think that it is timelike, as you thought. But it isn't: it's spacelike! And this diagram is the easiest way to prove that. The singularity is *not* a point in space, it is a moment of time in the future.

Thanks a lot, that's a very interesting way of looking at it and I'll try to get my head round it. I'm sure you're right.

But, at the end of the day, a black hole singularity is a point in space, it has no spatial extent, but that is not how it is shown on these diagrams. And I don't think that's great.

Here's a couple of diagrams taken from the same chapter of Roger Penrose's book "The Road to Reality". The diagram on the left is a nice, clear, intuitive spacetime diagram with a timelike singularity. Now a few pages later on Roger Penrose draws the causal diagram on the right with a spacelike singularity:

Roger Penrose diagram

But the reason for this fundamental shift in how we illustrate a black hole singularity is not explained. Penrose just says: This is somewhat counter-intuitive, since we tend to think of a singularity as a point". Yeah, that's because it is a point, Roger.

Anyway, I'll have to go away and have a think. Thanks for all your help.

Bee said...

Hi Giotis:
Yes, I have this thought very frequently when I sit in seminars...
Best,
B.

Bee said...

Robert: I just deleted your last comment. You seem to be a bit slow in understanding, so let me repeat it one more time. We will not allow links to your site that you dump into the comments unless they are of relevance to the topic. This means you either scrape the URL from your signature or your comments will continue to be deleted. Best,
B.

Bee said...

Hi Andrew, Pope:

"The singularity is *not* a point in space,"

It is a point in space, but not a point in space-time. Though that depends on what you mean with "space," whether you attach it to the radial coordinate (the singularity is at r=0) or whether you demand the coordinate to be "spacelike."

I think the information Andrew is missing is that inside the horizon time- and spacelike coordinates are interchanged. The horizon is the boundary between these domains. It is at a fixed radius r=2M, as you probably know, but this surface is null, not spacelike.

It might be useful in this context to look at the un-squeezed diagram. This is more details than I meant to go into, but you find it in my PhD thesis (PS file, needs Ghostview). Ignore all the German words, just go to page 37 (45 in the PS file), and look at Figure 3.3. There you see curves of constant t and constant r. See how the curves inside the horizon are spacelike? Now if you make a conformal transformation of that diagram, you get the causal diagram shown above. (In my thesis it's on the next page.) If you can't open the file, let me know and I'll make a screenshot of the figure and upload it.

Best,

B.

Bee said...

Andrew:

"a black hole singularity is a point in space, it has no spatial extent, but that is not how it is shown on these diagrams. And I don't think that's great"

It's a space-time diagram, points are events, not fixed values of a spacelike coordinate. I don't see why that is confusing. If you draw a particle sitting at a fixed value of the coordinate in an ordinary (x,t) plane, it will be a curve, the "worldline" not a point. Best,

B.

Phil Warnell said...

Hi Bee,

It is interesting to imagine the universe being what exists within an event horizon, as to wonder where the entropy is coming from, which forces it to expand and to further wonder if the information will ever be returned to wherever it came. That is to say if the point of its creation was a white hole as to consider it sort of a anti black hole, what then constitutes to be an anti event horizon? That would seem to me to be a place where although there is a continuance of time, it marks the end of space, which at least from our perspective it indeed appears to be.

Best,

Phil

Bee said...

Hi Phil,

A white hole doesn't have an "anti event horizon," but it has a past event horizon, in contrast to there being a future event horizon for the black hole. A future event horizon is the boundary of a region where nothing can come out, a past event horizon is the boundary of a region where things can only come out. You see that in the causal diagram. Best,

B.

Robert L. Oldershaw said...

Well, you are are quite right about one thing: I am a very slow learner. But I make up for that with persistence and tenacity.

"How can physics live up to its true greatness except by a new revolution which dwarfs all its past revolutions?
And when it comes, will we not say to each other, 'Oh, how beautiful and simple it is!
How could we have missed it for so long!'."

John Archibald Wheeler, 2000

Amen, brother
RLO

Andrew Thomas said...

Bee said: "I think the information Andrew is missing is that inside the horizon time- and spacelike coordinates are interchanged."

Oh thanks Bee! Yes, that was absolutely the information I was missing. I had no idea. I'll definitely have a look at that part of your thesis.

I've just done my bit for the internet community by updating the Wikipedia page for Penrose diagrams with that little gem of information, so other people don't get as confused as I have been.

Plato said...

Again, one is looking for "some phenomenological" relation.

Veneziano knew well the limits to which cosmologists like to think in terms of the box, yet pondered a greater question of what could exist outside that box?

It is the characteristics of the QGP( in Navier Stokes thinking) that allows you to move forward in this thinking, and is not just confined to these "arbitrary diagrams?":)

While you might get some who see the microcosm demonstrated in the macrocosm as orbitals, it is not so far fetched to see this relation as well in cosmological analogies of the process of "rotation in blackholes and their jets?"

Best,

Andrew Thomas said...

Regarding this switch between timelike and spacelike coordinates inside the event horizon, I think it makes sense if you imagine that once you pass the horizon there's no turning back: your progress in space is uni-directional, just like your progress in time is uni-directional outside the horizon.

If the singularity was drawn as a timelike boundary on these diagrams, it would be possible for an object which has fallen into the black hole to avoid the singularity by taking evasive action. So the singularity is drawn as a spacelike boundary to make it clear that once an object has fallen into the black hole it will inevitably hit the singularity even if it takes evasive action. I've done a litle diagram to illustrate this:

Why the singularity is depicted as a spacelike boundary in these diagrams

I think the Pope was right when he said that understanding why the singularity is drawn the way it is is a key step in understanding these diagrams.

Bee said: "It's a space-time diagram, points are events, not fixed values of a spacelike coordinate. I don't see why that is confusing. If you draw a particle sitting at a fixed value of the coordinate in an ordinary (x,t) plane, it will be a curve, the "worldline" not a point."

Yes, you're right. The thing is, we all tend to think of singularities as points. But of course a point becomes a line (or boundary?) in a spacetime digram. And it is the switch of that boundary to a spacelike boundary which is shown in these causal diagrams. It's important to get away from the "singularity = point" way of thinking. This is opening my mind.

I think I'm getting the hang of them. Thanks to Bee and the Pope for your help. (I still have my reservations about certain aspects of them, but I'll keep quiet about that).

Bee said...

Hi Andrew,

Yes, your explanation is correct. Let me just clarify that this is not just a matter of how one likes to draw things, the diagram does actually follow from the metric which follows from Einstein's field equations. Best,

B.

Phil Warnell said...

Hi Bee,

“A future event horizon is the boundary of a region where nothing can come out, a past event horizon is the boundary of a region where things can only come out. You see that in the causal diagram. “

Yes we do and that effectively is what I meant by an anti horizon, where past is exchanged for future in respect to them. So in your opinion does this have our universe to be the possible product of a white hole or not, or will that be addressed when the other shoe drops in your follow up post?

Best,

Phil

Bee said...

Hi Phil,

The above diagram shows a time-independent vacuum solution. It does not describe the evolution of our universe. Best,

B.

Phil Warnell said...

Hi Bee,

Yes you have been quite clear on this and so I will be interested in the diagram you have which you hold as being relevant to our universe.

Best,

Phil

Bee said...

The diagram I had in mind does not describe our universe, it describes a black hole in an otherwise entirely empty universe. But yes, we'll come to more realistic cases.

Giotis said...

Andrew maybe the confusion arises because you have in mind the typical Schwarzschild metric. Of course you can't use the Schwarzschild metric to describe the region inside the event horizon. If you switch to Kruskal coordinates these things become apparent. Kruskal are ideal in that respect. As for the singularity my opinion is that you shouldn't think of it as anything. GR breaks down there and we simply can't describe it. The singularity is not part of space-time in the conventional sense.

Phil Warnell said...

Hi Bee,

I guess the tricky point will be to define what other then the black whole an empty universe could possibly be.

Just as a follow up of my own and in relation to something you said earlier in regards to the old analogy which relates to things coming to be out of chance being the monkeys with typewriters having infinite time. I would agree for instance that in such circumstances that they eventually would write all of Shakespeare’s works or anything else you could imagine. However I would ask if this serves to have it as being (meaningful) information without those who could interpret it as being such.

I guess this is what has me confused by the whole information question being tied to entropy to begin with, for like they say in the computer business if what most if not all that goes in being garbage why should it be so surprising what comes out wouldn’t be the same. I’m thus reminded of the speculation of a photon horizon or a blue sheet that rests outside of the actual event horizon, which would effectively act as a paper shredder for anything then passing through to the event horizon. I suspect this has been covered off before yet not in any of the explanations I’ve read.

Best,

Phil

Steven Colyer said...

Phew! I'm glad Bee and Andrew Thomas made peace with each other as they're two of my favorite people in MathPhys. Glad I didn't jump in, and yes I'm skilled at moderating disagreements, but I'm happy these two intelligent people made peace, good.

Speaking of which, once this learner is up to speed, two debates I look forward to moderating one day (or would pay money to witness before then) are Peter Woit v Lubos Motl in the string theory wars, and David Gross v Leonard Susskind re the Amthropic Landscape within String Theory.

Back on-topic, I hope you re-read your thread Bee and expand it someday, with more expository detail. This is a really good subject, and make no mistake, I love it.

However if you wish to keep it as is and to help others gain a better base knowledge to understand this important issue, I would recommend reading, for background: Roger Penrose's wonderful book "The Road to Reality," particularly section 30.4 (Hawking's black-hole temperature) beginning on page 823 and figure 30.4 on page 827, and sections 30.5 and 30.6 as well, where you'll see similar diagrams to ones uniting Stefan and Bee. :-)

Andrew Thomas said...

Hi Steven, Bee is great - respect is due. Lively discussion is productive though - that's how progress is made. It's totally not an argument. Bee gets in right old arguments with some people, which is naughty of those people. This is an area which she understands probably as well as anyone in the world. People should show respect.

Her article here and this page (including the discussions) is the best description of these causal diagrams on the web. These diagrams are very poorly explained generally, which is bad news as Roger Penrose admitted in that quote of his I posted - parts are "counter-intuitive", so explanation is vital. This idea of a singularity actually being a boundary rather than a point is brain-bending.

Yeah, I read those relevant sections of Roger Penrose's book and he doesn't explain it well at all. Which is bad news as he thought of these diagrams in the first place. He totally skips over the question of how to denote singularities. And the Wikipedia page was useless as well (it's hopefully looking a bit better now).

I think these diagrams are cool now. Very clever.

I still don't like the way they are so frequently used to describe parallel universe hell!

Phil Warnell said...

The way I look at it a singularity could be considered as either a boundary or a point and relates to perspective. For instance to say something diminishes to zero doesn`t force it to come to be nothing, yet rather from the Cartesian perspective marks merely a location, although it being a unique one as marking the location between the positive and negative. One could say also that a line is to two dimensions as a point is to one in acting as a boundary.

Phil Warnell said...
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Phil Warnell said...

Hi Steven & Andrew,

My way of distinguishing a discussion from an argument is the first is meant to seek the truth through resolution facilitated by way of listening and reason, while the latter only serves to convince one’s self they are right. The first is the essence of science while, the second only righteousness. The way I see it with a discussion between Woit and Lubos we should expect a lot the latter while little of the former:-)

I`d rather see a panel discussion between t`Hooft, Witten, Smolin and Penrose. One problem being I couldn`t imagine who could serve as a moderator; at least not being someone currently alive. The other problem of course is that much of the discussion I would have trouble following. However the one thing I`ve learned in all of this is there is a distinct difference between causal and casual considerations:-)

Best,

Phil

Andrew Thomas said...

Hi Phil,
Nicely put. I'll be honest, sometimes I feel like I'd really like to post something on Lubos's blog as some of his articles are interesting, and some I don't agree with. But I just know that if he disagrees with me then the whole thing will just grind to a halt and no progress will be made.

In the case of these diagrams I had read what Roger Penrose had to say, but it was a comment by Bee that finally made sense of them for me.

--------------------
Phil said: "The way I look at it a singularity could be considered as either a boundary or a point and relates to perspective." Yeah, I just think it shows how weird life would be like inside a black hole.
---------------------
Giotis said: "Andrew maybe the confusion arises because you have in mind the typical Schwarzschild metric. If you switch to Kruskal coordinates these things become apparent." Thanks for that, Giotis.

Andrew Thomas said...

Phil said: "The way I see it with a discussion between Woit and Lubos we should expect a lot the latter while little of the former."

I actually wouldn't mind watching a panel discussion between those two. You could probably sell it to a sports crowd on pay-per-view.

Phil Warnell said...

Hi Steven,

Yes but first they need to find a referee, instead of a moderator. In the end it would turn out to be nothing more than a reality show, which of course never truly represents reality, yet rather serves to justify our faults. I’m afraid satire is a concept lost on many these days, as rather it serving to be introspective, it serves only to reinforce what it’s suppose to have eradicated.

By the way two people I have in mind to moderate the discussion I proposed, would be Einstein of a mid forties vintage or J.S. Bell just before his untimely departure. That is given proper time to familiar themselves with the subject matter, I think either could keep the groups eye on the ball.

Best,

Phil

Phil Warnell said...

My humble apologizes as a meant to address Andrew and not Steven.

Giotis said...

Phil, this is indeed true and it is a clever way to bypass the problem but you must do it in a mathematical rigorous way. As far as I know there is no general solution for the identification of that boundary on a manifold with lorentzian metric (Bee may have more info on this). Singularities exist of course but you cannot think of them as points on the space-time manifold because the notion of space-time breaks there. Within GR literally they are out of place and time.

Phil Warnell said...

Hi Giotis,

Yes it’s true that my argument is not mathematically rigorous and yet I would say it might not to be able to, since not only does GR break down at the point of a singularity, yet more specifically the math which forms and supports its foundations. What I was actually attempting to convey is while as being a boundary can be taken as something with nothing beyond,it can also be perceived as a threshold, which in essence is what is being attempted to be decided with the use of such diagrams.

Best,

Phil

Andrew Thomas said...
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Andrew Thomas said...
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Pope Maledict XVI said...

Hi Andrew, glad to be of help. You in turn have treated us to the rare spectacle of somebody actually changing his mind!

This is a bit misleading too! Inside the black hole you are just as free to move around in space as you are outside. Again, inside the bh you don't *fall* into the singularity --- you *age* into it!

The whole idea that the singularity "pulls you in" is wrong. Do you feel yourself being "pulled" into the year 2010? What happens inside the bh is exactly the same. To speak of taking evasive action is like trying to take evasive action to avoid hitting your 50th birthday. I suppose botox might be described in that way....

The key thing to understand here is this. Inside the bh, the spatial geometry is a function of time. [Technically, the timelike Killing vector outside cannot be extended inside.] It's as if Pythagoras' theorem had functions of time built into it. Furthermore there are different functions of time attached to different directions in Pythagoras' theorem, so as time goes by your shape changes. These changes are fatal. That's why you die if you go inside a bh: your spatial geometry changes in painful ways as time goes by. The scary thing is not the singularity itself, it's the fact that the rules of geometry become time-dependent.

Hope this helps.

Pope Maledict XVI said...

I can't resist the temptation to point out that the penrose diagram of the black hole helps to dispel another very common misconception: the idea that inside the bh you are trapped there because "the gravitational pull is so strong". You know the routine: "a bh is black because gravity is so strong that even light can't escape." This is not true.

Consider the event: "where you are, right now". Now consider a sphere of light emanating from that event. In a spacetime diagram, it is represented by a V-shaped line, called the future light cone of that event. Now there is nothing you can do to get out of that light cone; you are trapped inside it forever. The interior of a bh is exactly like that: you are trapped there in exactly the same way that you, right now, are trapped inside your light cone. There is no escape for *you* either! You are trapped! Oh no! :-)

The only difference between your current imprisonment and the imprisonment of someone inside the bh is that the spatial geometry around you is [hardly] changing, whereas that poor guy is surrounded by a nasty time-dependent geometry. But his predicament has nothing to do with "being pulled inexorably inward... blah blah blah".

All this is immediately clear from the Penrose diagram.

Andrew Thomas said...

Hi the Pope,
Yes, considering the geometry is the right way to do it.

The Pope said: "I'm slightly concerned about your statement: 'your progress in space is uni-directional, just like your progress in time is uni-directional outside the horizon.' To speak of taking evasive action is like trying to take evasive action to avoid hitting your 50th birthday."

I actually think there are interesting parallels between your inexorable progress in space inside the black hole, and your inexorable progress in time outside the hole. And I think, yes, it's possible to take a form of evasive action in time as well!

As you said yourself, "Inside the black hole you are just as free to move around in space as you are outside." So although you cannot move backwards (in a direction pointing out of the hole) you are free to move laterally within your light cone to an extent, which represents what I called "evasive action". However, this will not prevent you reaching your inevitable doom when you hit the singularity. But evasive action inside the hole can result in the length of your path (your distance) increasing.

Similarly, in normal spacetime outside the hole, we are travelling towards our inexorable doom in time (rather than space) - we're all going to grow old and die. However, we can still take a form of evasive action by "moving laterally". We can get into our spaceship and travelling close to the speed of light. Hence, we take a curved path in spacetime (a form of lateral evasive action):

As a result, the space traveller with his "evasive action" ages less than the person who travels the direct route through spacetime, as in the twin paradox. So the effect of evasive action inside the hole is to make our distance longer, whereas the effect of evasive action outside the hole is to make our time longer.

But in neither case will the evasive action save us from our inevitable doom, either in space (inside the hole) or in time (outside the hole). Inside the hole, we can't break outside of our lightcone to leave the hole (to become safe), and outside the hole we can't break out of our lightcone to travel back in time (to become younger, and hence become safe from death).

I definitely think there are parallels between the two situations of space inside the hole and time outside the hole. And this switching of spacelike coordinates to timelike coordinates inside the hole just seems to reinforce these parallels.

The Pope said: "The whole idea that the singularity 'pulls you in' is wrong. Do you feel yourself being "pulled" into the year 2010?" No indeed. The motion in space in the hole is just like motion in time outside the hole - it doesn't seem like we're being "pulled" in either case. It's just the same.

Bee might well get back and say this is all a complete load of twaddle. But it seems to me like there's a pattern there. Makes we wonder what implications this has for the arrow of time? Are we all falling into a form of black hole in time? :)

As regarding these causal diagrams, yes, I think they're very clever now. It was never the diagrams which were at fault - it was the explanations which were poor.

Bee said...

No, it's not a load of twaddle. However, Pope's argument about no escaping the lightcone entirely neglects to mention that in contrast to you heading towards the year 2010 in a basically entirely flat space-time, the curvature near the singularity diverges. If you calculate tidal forces, you get indeed pulled and stretched. Best,
B.

Andrew Thomas said...
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Andrew Thomas said...

I dunno. Maybe it's because there's three dimensions of space but only one dimension of time. You can't get those geometrical deformations in only one dimension.

Zephir said...

Casual space-time diagrams have good meaning in AWT, which describes motion of matter through space-time along geodesics by spreading of energy through density gradient simmilar to water surface, i.e. by Hamiltonian flow.

After then the time dimension is direction parallel to density gradient, i.e. perpendicular to watter surface and we can model the matter motion by path of rays through density gradient. Which is possible to use for prediction of various unknown yet phenomena, like the bouncing of matter from density gradient by total reflection mechanism, and so on.

The practical usability of these models is limited by fact, massive objects are collapsing or expanding during travel through time dimension, i.e. evaporate into radiation - so they've physical meaning only when the gradient of space-time remains sufficiently low.

Zephir said...

This is how "causual diagram" could appear in aether theory.

http://tinyurl.com/ybpafxe

Plato said...

Of course not to make "light of":) but understanding critical density associated with our current universe, or, relating to a "three body problem" helps to orientate the mind too, "changes going on in a region of space," much as you see, changes using assumption of non-euclidean perspectives according too, an Gaussian assumption of, while understanding well it's phenomenological relation. You see?:)

So you are actually working on two levels here. A introspective relation to data accentuation that ignites different regions of the brain:) One might say here that Dirac was working in two ways, one geometrically, while describing a phenomenological relation to the matrices in time? So Feynman saids hypothetically, I need a better way to map this thinking.

Ta da!:)

Best,

Plato said...

Oh and one more thing,

Under the Relativity banner it is only about gravity?

IN microscopic universe, it is a different story, that has been mapped detailing, to lead one to suppose phenomenological testing carefully marks the pathway?

Theoretical postulations as well.

These have to be grounded in fact for assumption to move theorization position forward.:)One indeed might not have all the facts.:) Changes of course do happen when one realizes.

Best,

Plato said...

Bee,

One question.

What is "the information" that dissipates?

Best,

Zephir said...

Information is just a an postmodern euphemism in this context, as no information can spread without energy.

Anonymous said...

If these are casual diagrams, I wonder what diagrams following the dress code look like?

Plato said...

Hi Zephir,

Bee:The mere existence of the paradox tells us that there has to be a solution (another green apple) somewhere. The question is not whether there is such a solution but what it is (where the apple is). Alice might come and say there's apples in the kitchen, but Carl might say, no, the apples are in the garden and Doug says they're from the market, etc. The task of science is to determine which is indeed the case. If you go and say the solution might be xyz (apple in the kitchen), that's not sufficient. You'll have to show it is THE solution and why. And in particular that it doesn't create any other problems along the way.

So, we arrive at this point.

If one looks at the cosmological events in locations throughout the universe, are we indeed seeing fragmentary schematics geometrically designed as to totality of the evolution of, and formation of the blackhole.

The diagrams lead us to this point, yet schematically do not reveal the geometrical expression that energy has taken, to help us see the end results, of jets emerging from. But Dirac may say, it is anti-matter?

At best, reference to decay elements of, are measured aspects of that energy as mentioned previous. Yet, do we see on a parallel level, a part of the geometrical expression of the supernova, as a part of the evolution of the blackhole?

So energy then, is information?

Best,

Andrew Thomas said...

I have a bit of a discussion about information here.

Zephir said...

/*..so energy then, is information?..*/
Longitudinal wave is spreading energy, but no information.

Anonymous said...

Hermann Weyl once commented:

'While topology has succeeded fairly well in mastering continuity, we do not yet understand the inner meaning of the restriction to differentiable manifolds. Perhaps one day physics will be able to discard it. '

We are now ready to do that. In fact it has been done and the results are so amazing that most theoretical physicists cannot even recognize or understand the successful completion of Einstein's 3-part relativity project: [1] Special Relativity (relativity of S-T for inertial frames); [2] General Relativity (relativity for inertial + accelerated frames); [3] discrete conformal relativity (discrete relativity of scale).

Welcome to the 21st century,
RLO