Heisenberg Uncertainty principle:
Dp*Dx>= k
Where D is delta, which I can’t seem to type in this editor, p is momentum, x is position, and k is a constant. “>=” means greater than or equal to, which I also can’t type properly in this editor.
The error in momentum times the error in position is a constant.
Imagine we measure position with arbitrary accuracy; as we do so the range of error in momentum gets arbitrarily large. The velocity of the object is bounded, however, by the speed of light. Since momentum is mass times velocity, by measuring the position accurately enough we can cause the range of possible masses to be arbitrarily large.
This results in a paradox.
The Scwartzchild radius is the radius within which, for a given point mass, the escape velocity exceeds that of light, creating a black hole. It is given by 2GM/c^2 where M is the mass, c is the speed of light, and G is the gravitation constant. Obviously, the particle we are measuring the position of cannot have a mass so large that it would create a black hole larger than the detector (if it did we would be getting this position info out of a black hole - an impossibility). Thus we have an upper limitation on the mass, and with the speed of light being an upper limit on the velocity we have an upper limit on the momentum, and hence the error in the momentum, Dp.
No matter how accurately we measure position, the error in momentum cannot exceed a certain amount, contradicting Heisenberg.
The “certain amount” is dependent on the size of the detector, and to calculate the theoretical limit we have to consider that. Since this is a gedanken detector, take the limiting case and make the detector infinitessimally larger than the particle itself. We can now put the Schwartzchild radius formula together with the Heisenberg uncertainty formula and calculate the smallest possible Dx. I don’t have my calculator handy (I came up with this awhile ago) but it comes out to something like 1.6 x 10^-34 meters IIRC. I call this the APB9999 radius. It is meaningless to speak of a smaller division of space than this since no measurement could be made more precise.
[I’m simplifying a little, you actually get several different cases when you plug the formulas together. But the end result is the same: Either you have a black hole of radius greater than or equal to the APB9999 radius, or you have a simple Dx greater than or equal to this distance.]
The smallest possible division of space: Cool, no?
Interestingly enough, you may have gotten the right number there, but for the wrong reason. The length scale you derived is what’s known as the Planck distance, and we honestly don’t know what goes on at scales that low. if there is, in fact, a smallest possible division of space, then that’s probably it. However, your reasoning for deriving it is a bit flawed. First of all, when you’re dealing with relativistic speeds, momentum isn’t just mv, it’s gammam*v, where gamma is the relativistic dilation factor. This means that the momentum can get arbitrarily large, even with the speed of light limitation.
Secondly, you assume that the object has to have a Schwartschild radius smaller than the size of the measurement device. This would be true, if the object had to be inside of the device, but it doesn’t have to be. Consider the Sun: It’s got a Schwartzchild radius of approximately 3 kilometers, but you can detect and measure the Sun with a device much smaller than that. of course, even if we were somehow inside of the sun, and within 3 km of the center, there still wouldn’t be a problem, because most of the mass of the Sun is outside of that radius, so there’s no black hole.
Another, subtler problem is that we’re not even sure that black holes can exist at the Planck scale… Like I said, it’s not known what happens down there. In order to say anything, we’d need a quantum theory of gravity, which we do not yet have.
That’s great reasoning on your part, though… Keep it up!
I love how these threads actually show the intelligence of Straight Dopers. Are you guys physicists or just hobby astronomers? I’m mainly talking about Chronos, but there are some others that haven’t posted in this thread. (I’m not going to give the names, as I don’t care to put anyone on an ego trip). I can follow along with the threads, but can’t add anything to them unless I’m the very first person to respond. I guess I could quote something from my textbooks, but that’s too much trouble just be outshined by the next guy. I find comfort in believing that the quasi-geniuses are homely fat guys that can’t find dates. Whether it’s true or not, I like to think that.
Well, I’m a biologist, actually, but I’m not very good at it.
Chronos, I don’t understand your first point. Can you elaborate and put the contraction in the formula where it goes? Momentum getting arbitrarily large is part of the paradox, so I don’t see why this would invalidate it. It’s just that since the velocity can’t exceed c, we can interpret the minimum mass as p/c, can’t we? For a gamma to change this interpretation, wouldn’t it have to have units? If we’re still talking about a mass, the paradox should hold up. Maybe I’ll see how it works when you put in the contraction.
Your second point doesn’t change the paradox, I think. We can’t be sure where in a black hole the mass is (although of course we assume it is in the center). If the detector is outside the object and the object has enough mass to be a black hole, we cannot possibly measure its location with arbitrary accuracy. That would be getting information out of a black hole, in defiance of relativity.
You are right in that I am assuming that the particle is contained within its Schwarzchild radius, since we are trying to measure a position to greater accuracy than that distance. If the particle is bigger than that then it’s a different problem (i.e., in getting more accurate position readings we end up measuring where the surface of the particle is, or its center, or something like that).
Do we really not know if black holes can exist at small scales? I don’t know much about this. Aren’t physical laws assumed to be invariant wrt scale? Why should the escape velocity formula vary?
This is neat stuff! I look forward to your answer.
Yes, I am a physicist, and for what it’s worth, I can’t get dates, either… But I’m not fat.
APB9999, the reason that we can’t constrain the mass of the particle is that the gamma part all by itself can account for the momentum getting very large. It’s the rest mass that counts for black hole formation, not the relative mass, otherwise anything would be a black hole, to the point of view of someone moving fast enough.
With respect to a black hole, we can’t say much about where the mass is inside it, but we can say that it’s spherically symetric about the center of the event horizon. In fact, that’s the only thing that we can say about the mass distribution, because if it were anything else, we would be able to determine that, and get illegal information out of the hole.
Also, it’s possible to determine the position of the center of an object (what we probably mean when we say the position of the object) just by observing the surface. The size of the object does not put a lower limit on the uncertainty in its position.
As to physical laws and scale: They aren’t really invarient, depending on what you mean by that. The very notion of a smallest possible division of space should illustrate that-- Physics would certainly be different below that scale, and above it. Further, some physical effects can be more or less significant at different scales: Quantum mechanics is irellevant for a stellar mass black hole (yes, there’s Hawking radiation, but only an insignificant amount of it), which is a good thing, because we wouldn’t know how to deal with the quantum effects if they were there. On the other hand, a Planck black hole would be small enough that the quantum effects would be significant. To give you an idea of the problem, massive black holes radiate less than low-mass ones. By the time that you get down to the Planck scale, if we just extrapolate the formulae for Hawking radiation in a normal hole, you find that a Planck hole would be expected to evaporate away completely with the emission of a single particle. However, that particle would still have that much mass, and it would still be contained in a comparable volume, so it’d be a hole, itself: This probably points to some flaws in just blindly extrapolating down to those levels.
Okay, I’ll bow to your expertise as a physicist. I don’t really understand some of your post, though. Maybe you could explain.
First, I get this part:
because when you replace p=mv with p=mv/(1-(v/c)^2)^1/2 then p can get arbitrarily large as v gets closer to c without m changing. [I’ve really got to learn how to do super and subscripts!] So that makes sense, and invalidates my paradox.
But this raises an interesting secondary issue: I’m sitting in space and a body goes by at some fairly slow speed. I measure the gravitational pull and thereby calculate the mass. Now the same body comes past again, this time going at a high (relativistically relevant) speed. Will I measure the same gravitational pull? The answer, based on what you’ve said, must be yes, right? Because otherwise the paradox still works - an accurate enough position measurement will create a range of momentum that includes a gravitational pull strong enough to produce a black hole (as opposed to saying a “mass large enough”). The constancy of gravitational pull when two objects are moving at different, even relativistic, speeds relative to each other is an interesting thing to make note of. I’ll have to think about that. Cool, thanks!
This part:
leaves me a little confused but I think it’s probably just semantics. If I talk about the position of a basketball on an Angstrom scale, what do I mean? I could mean the center of volume, the center of mass, or a surface in space that corresponds to the basketball’s surface. Usually it’s the COM that’s of interest, as in my defunct paradox, but you can’t determine that from a knowledge of where the surface is alone, unless you make additional assumptions about the distribution of mass. At any rate, the paradox works with a point particle. Maybe it’d be simpler to just specify that from the outset.
Also, I may be wrong about this, but to measure where the center of a surface is to a particular accuracy, don’t you have the same problems as when you try to measure the position of a point particle? For example, to measure the location of a black hole’s center, you have to measure the precise location of the event horizon. How, even in principle, could you do this except by shooting a particle of some sort at it and seeing where, precisely, it vanished? And of course, the “precisely” means you are reduced to measuring the position of a particle with some given accuracy again, and this error will carry through to your knowledge of where the center is.
Finally, the business of scale is tricky, since the laws of physics are NOT written in a form that considers scale. So if scale matters, they must be wrong! Or at least limited in the way Newtonian physics was limited before the addition of the gamma you mentioned above. I know there’s some problems reconciling qm with relativity, but I won’t pretend to understand why. This is unsettling for my faith in the laws of physics as currently written, though.
Finally, this part:
This sounds like the problem with electrons spiralling down into a nucleus: they can’t, according to qm, because by the time they reach the s shell they would have to radiate more energy than they have. So atoms are stable. If a black hole could only evaporate by radiating a particle of it’s own mass, which would then be a black hole, doesn’t that mean that a black hole of that size would be stable?
Summary:
1)My paradox is wrong.
Two questions.
2)Does a particle A have the same gravitational pull on a particle B no matter what their relative speeds?
3)Would a black hole be stable wrt Hawking radiation (I guess - that may be the wrong term) if it were small enough?
You don’t have to find a black hole’s event horizon to measure it’s center. By measuring it’s gravitational attraction you can compute it’s center of mass. Since, as Chronos said, the mass of a black hole is “spherically symetric about the center of the event horizon”, once you know the center of mass from a few gravitational experiments (such as dropping plumb bob ), you know where the event horizon is.
Well, you’ve proposed a way that doesn’t involve measuring exactly where a particle vanishes directly. But I don’t know that it solves the problem.
Your plumb bob experiment would give you a vector along which the COM must lie, but to measure the POSITION of the center of mass in space to arbitrary accuracy requires more. Either:
A) You have to know the EXACT mass, then you can find the position from the gravitational force. But how do you find this out? Actually propose an experiment that would give the mass to arbitrary accuracy that DOESN’T depend on knowing the position to arbitrary accuracy. I can’t think of one; maybe you can.
B) Find a second vector along which the COM must lie by moving the plumb bob. The COM is at the intersection of the two vectors. But then the question comes up of how precisely we can determine vectors from the plumb bob. To get a single unit vector to arbitrary accuracy, we need to measure at least TWO points to arbitrary accuracy: The point at which the bob is fixed, and the point of the weight at the end of the bob. Thus, to use an intersection of vectors, we need to measure FOUR positions to arbitrary accuracy.
The gist of what I’m trying to say is that measuring the position of a point particle is the simplest, limiting case of measuring the position of anything. I don’t think that giving a particle volume in any way can make determining its position any more accurate. So it doesn’t matter whether a detector contains the object or not in my “paradox” (which is now moot anyway), because it doesn’t make the particle’s position any easier to measure if it’s bigger than the detector. And if the object is a black hole of arbitrary radius (as it was in my original argument) we always know that black hole can’t contain the detector, thus restricting the momentum.
I wasn’t really trying to solve the problem, I was just nit-picking with the statement “to measure the location of a black hole’s center, you have to measure the precise location of the event horizon” . My way isn’t any better or worse, fundamentally.
I agree with you on this.
I’m still not conviced that the detector can’t be in the black hole. Can’t I be inside the event horizon, along with my plumb bobs? (Of course, I might feel a little uncomfortable …)
With the plumb bob example, the key is that while there are limitations on the accuracy with which we can measure the plumb bobs, those inaccuracies are not related to the black hole. I can use my bobs with the same accuracy measuring a black hole as I can measuring the Earth.
As to the stability of Planck black holes: They might be stable, and in fact, that’s currently considered the most likely possibility. Unfortunately, though, we currently have neither the theoretical framework nor the the experimental evidence to say with certainty one way or the other.
), you know where the event horizon is.