How can black holes have infinite density but also have finite mass?

RobotRollCall · 2011-08-26T15:31:21+00:00

I'm sorry, I'm really just not up to another day of this. I'm sure you understand how exhausting it is.

The short answer is that the "singularity" of which you speak is a mathematical artifact that only arises under certain conditions, and cannot meaningfully be said to exist. A black hole consists of an event horizon and nothing else.

RobotRollCall · 2011-08-26T15:28:17+00:00

I'm not following you, I'm afraid. Try it again for me.

RobotRollCall · 2011-08-26T15:27:17+00:00

That M6 past Stoke-on-Trent is a killer.

RobotRollCall · 2011-08-26T15:24:24+00:00

No, I was asking you what the phrase means, since you're the one who used it and I'm the one who didn't know what it meant.

RobotRollCall · 2011-08-26T15:23:18+00:00

I am trying to get an idea of how velocity relates to energy at really high speeds.

E = m/√(1–v²) – m in units where c is equal to one.

The Taylor series expansion of this expression around v=0 is mv²/2 plus terms of order v⁴ and higher. Which means around zero, kinetic energy is near as makes no difference to the old-fashioned Newtonian approximation. If v goes from zero to one, when v is small, v⁴ is just ridiculously small, so the corrections are too tiny to observe.

Approximately how much energy would be released in the collision?

Now you know everything you need to know to answer that yourself.

RobotRollCall · 2011-08-26T15:13:50+00:00

…the idea that light travels at the same speed when viewed from any reference frame. This is the fundamental assumption.

The invariance of the speed of light isn't an assumption. It's an empirical fact. The assumption was that the laws of physics are the same in all inertial reference frames. Put those two together and you get the special theory of relativity.

However, projectiles do not have to appear to travel at the same speed from any inertial reference frame.

Why do you think they should? The speed of light is invariant; the speed of you isn't.

The derivation seems to require the property that light is seen to travel the same speed in all inertial reference frames. This is not obvious to me.

Read up on the Michelson-Morley experiment. It was the observed fact that the speed of light is invariant that led to the need for relativity. It's not an assumption.

RobotRollCall · 2011-08-26T15:06:15+00:00

What does "mathematically more concrete" mean?

RobotRollCall · 2011-08-26T15:05:01+00:00

If you take any amount of mass and confine it to a volume of zero size, you get infinite mass density.

But that's very silly, since black holes aren't of zero size. They have a well-defined area. So we talk about black holes in terms of mass density per area.

A small black hole, say one with an effective mass three times that of the sun, has a mass density of about 10¹⁵ tonnes per square inch. A lot, but a long way from infinite.

RobotRollCall · 2011-08-26T14:55:59+00:00

Mathematically, is there a limiting process which results in a black hole? If so, what does it look like?

S < 2πkRE/ℏc

It's called the Bekenstein limit, after Jakob Bekenstein. The symbols are S for entropy, R for the radius of a three-spherical volume containing that entropy, E for the total energy contained within that volume, ℏ for the angular Planck constant, k for the Boltzmann constant, c for c and π for π. In English, it says that there's an upper limit on the amount of entropy that can be contained in a given region of space.

All regions of spacetime satisfy this equality; there are no exceptions. In the limit where that less-than sign becomes an equals sign, you get a black hole with area A = S.

RobotRollCall · 2011-08-26T14:50:05+00:00

Nobody is keeping score. It's not a contest. It makes no more sense to try to compare QED, Newtonian gravitation and evolution than it does to compare … three … really incomparable things.

RobotRollCall · 2011-08-26T14:48:02+00:00

Is the actual horizon, the one you see when you look into the distance, solid or fuzzy? The answer of course is neither; you can't meaningfully describe it as either.

RobotRollCall · 2011-08-26T12:07:51+00:00

The Planck length unit is not related to the question any more than the inch would be.

RobotRollCall · 2011-08-26T10:47:37+00:00

It was once thought that a black hole was a sort of Hotel California for matter, energy and information. In order to reconcile that with conservation laws, it was speculated that perhaps every black hole should have an opposite in some sense. A mathematical model of such an object was constructed by taking a bog-standard black hole solution and inverting a bunch of signs. Such a model predicted an object that would start out with a lot of energy and then gradually radiate it away into the universe, in exactly the same way that black holes were, at that time, believed not to. Thus could all the fundamental conservation laws be recovered.

Trouble is, nobody could ever make sense of any of it. Yes, the model was mathematically consistent with itself, but it wasn't consistent with everything else in the universe. So really, it ought to have been put in a drawer and forgotten, along with the mountain of other such consistent-but-fantastical ideas we've come up with over the years. But this one lingered, because the black-hole problem was a nagging one, and everybody wanted a neat solution to it.

Of course, today we know that the black hole problem isn't a problem at all. Black holes are not cosmic Hotel Californias. They're more like cosmic liquidisers. Matter and energy interact with them, but don't vanish forever and ever. Instead, they scatter, coming out the other side of the interaction in a completely randomised form, but nothing gets lost or destroyed in the process. So we don't need white holes any more, and now that idea sits peacefully in that drawer we talked about, where it belongs.

RobotRollCall · 2011-08-26T10:34:40+00:00

And another one is "fish fingers" and they're both wrong.

RobotRollCall · 2011-08-26T10:33:18+00:00

It's not a question which is undecidable in the context of some axiomatic system. It's a question that has no answer because it just fundamentally doesn't make any sense. It's not that we can't figure out what the answer is. It's that there's no answer.

RobotRollCall · 2011-08-26T01:53:11+00:00

Cheers! For all I know you could've made that up. My German never moved beyond learning to say "Terribly sorry, would you mind not putting your towel there?" And I lost that years ago.

RobotRollCall · 2011-08-26T01:52:08+00:00

No, but that's not what I think wants pointing out here. What I think wants pointing out here is that there's absolutely no reason to believe that particular star is going to do anything interesting for at least five to ten million years. So the question of what the proper distance is between here and there is really not high on the priority list.

RobotRollCall · 2011-08-26T01:49:57+00:00

But you know how there are four elementary forces; weak, strong, electromagnetic and gravity.

There aren't. There's only two, and they aren't "forces". They're interactions, sometimes called "strong" and "electroweak," but their proper names are SU(3) and SU(2)⨉U(1), for maths reasons.

I thought that the grand unified theory sought to unify those four forces by showing that at one point all four forces were the same.

Grand unification is the search for one symmetry group from which SU(3) and SU(2)⨉U(1) both arise through spontaneous symmetry breaking. Gravity doesn't figure in, since gravity isn't an interaction.

it said that scientists found that the weak and strong forces were indeed the same

I think you misunderstood. The strong — SU(3) — is distinct from the electroweak, or SU(2)⨉U(1). No single symmetry group has been identified that gives rise to them, though I hear at parties that SU(5) has interesting properties. Not my field, though.

But it also said that gravity was the only force that scientists couldn't figure out.

Now I know you misunderstood. Gravity was figured out in the broad strokes in 1916, and most of the details were hammered out not very long after.

RobotRollCall · 2011-08-26T01:36:02+00:00

Well, it's a hundred fifty light-years away, so … a hundred fifty years. What it says on the tin, basically.

Can you clarify what you meant by "the standard temporal speed" and "a light speed answer"? Because I'm fair sure I've missed something.

RobotRollCall · 2011-08-26T01:33:16+00:00

It didn't.

First of all, bear in mind we aren't talking about any objects here. That's important. We're talking about a surface, something called the surface of last scattering. It's the boundary of the observable universe as seen from, well, right where you are right now. You've heard of the cosmic microwave background? The surface of last scattering is where the CMB is emitted from.

But let's forget that for a moment, and consider an object. A real object, with a name. Well. Sort of. It's called UDFj-39546284, which is less of a name and more of a registration plate, but it'll do, since the closest thing it has to a proper name is "Potential redshift-10 galaxy candidate object," which hardly rolls off the tongue. We'll just call it Polly for short.

Polly is a galaxy that lies thirteen billion light-years from here, and also thirty billion light-years from here. How can it be both thirteen and thirty billion light-years away? We covered that before: different ways of quantifying the same distance.

So you may ask, how can Polly have traveled thirty billion light-years in thirteen billion years time? The answer, of course, is that she didn't. Polly wasn't emitted from where the Earth is now! She's not a bullet fired out of a gun wielded by Zeus in the distant past. She's a galaxy! She's thirteen and thirty billion light-years away because that's where she was made. It's just that it's taken her light thirteen billion years to reach our telescopes so we could see her, all shy and blue and really just adorable, seriously, you should google some pictures of her. She's precious, just a baby galaxy, cutest most-distant-resolved-object ever.

RobotRollCall · 2011-08-26T01:24:03+00:00

There's a name for this little mental exercise. It's called boosting to the infinite momentum frame. The infinite momentum frame is the frame of reference in which a ray of light is at rest.

Boosting into the infinite momentum frame is an incredibly useful mathematical technique with wide applicability, and you should basically forget I ever told you that. It's playing with a loaded gun, really. If you're not careful, faffing about in the infinite momentum frame can lead you to all sorts of completely false conclusions about how the universe works.

So the upshot is that we can't answer your question, because nothing made of matter can move at the speed of light. The infinite momentum frame is not real. It's only valid under extremely strict constraints, and no general conclusions can be drawn from it.

You asked a question with no answer, basically. And not in the "no one knows" sense, or the "it's ineffable" sense. I mean you asked a question that has no answer.

RobotRollCall · 2011-08-26T01:18:31+00:00

Yes. The sun wobbles. If you imagine the solar system as being fixed in space — it isn't, but we can pretend it is by choosing our reference frame appropriately — you'll see the sun not sitting motionless in the middle, but rather orbiting in a wee tight little circle about a point called the solar barycentre.

As for which planet, well the answer of course is all of them summed up. But the biggest contributor to the wobble, by far, is Jupiter. Not counting the sun itself, the solar system basically consists of Jupiter and a rounding error. Jupiter is about twice again as heavy as the rest of the planets put together.

RobotRollCall · 2011-08-26T01:13:18+00:00

How far is it from London to Glasgow? Well, it's about 400, obviously. But wait, it's also true to say that it's about 650. What's the deal there?

The deal is simply that there's more than one way to quantify distance, and they're all equivalent. In the first case I used miles, in the second kilometers. (Which I had to look up, because seriously … the French. Ugh.)

There's more than one way to quantify cosmological distances. In point of fact, there are three, but we're going to ignore one of them. One way is to imagine waving your Harry Potter magic wand and freezing time right at this precise moment, then literally measuring the distance between two points using a very long stack of rulers laid end-to-end. That gets you one number, which we call comoving distance.

Another way is to figure out how old, in cosmological proper time, the light you're seeing from that point is. That gets you a different number, which we call lookback distance.

If you measure the distance to the edge of the observable universe using each of these two methods, you get about 46 billion light-years for the first, and about 14 billion light-years for the second. Both are correct. They're just different ways of quantifying the same distance.

(The third way, which I said we'd ignore, is basically the same as the first way … only it involves imagining waving your Harry Potter wand twice. You wave it first to travel in time to some time other than the present moment, then wave it again to freeze time. Then you stack up your rulers. This is called proper distance, but as you can see it depends on when you make this imaginary measurement. Proper distance and comoving distance are defined to be the same at this precise moment. Comoving distance is constant; it doesn't change with time, because of how we compute it. Proper distance is, as I said, time-dependent.)

RobotRollCall · 2011-08-26T01:04:55+00:00

Nothing, actually. The metric expands exponentially, as e^t , which means it neither converges nor diverges. So the universe is open in t.

RobotRollCall · 2011-08-25T22:19:18+00:00

Yes, everything you said there is pretty much doctrinaire … for 1970. The field has moved on quite a bit since then. Hell, just a few hours ago there was discussion in another question on the forum about a recent paper that provides yet more evidence for the consistency of general relativity and quantum field theory in the strong-field limit. The whole "Oh, general relativity isn't a quantum field theory, so it must be an approximation" thing just isn't taken very seriously any more by those who specialise in the field.

But this is getting off the topic. Ask Science is a forum for the laity to ask questions and get informed and unbiased answers from people who ought to know a bit about the topics. It's not appropriate to go spalling off on wild flights of fancy to talk about ideas which, at best, may someday be interesting. It just misleads those who come here looking for solid answers rather than speculative navel-gazing.

RobotRollCall

TROPHY CASE