A question I need answered.

[Reincarnate]

Re: A question I need answered.

This is an example of Zeno's Paradox. It's basically a way of asking "if we can divide some finite distance into infinitely many pieces, then how can we possible make it from one end to the other? There are INFINITELY many pieces to traverse!"

It's not really a paradox -- the question just sounds confusing because of the way it's phrased and because people tend to have a hard time grasping infinity. In this case, the fact that you're talking about a finite distance already solves the problem.

Say we drop the ball 10 feet. To traverse 10 feet, you have to first traverse 5. But to traverse that, you need to travel 2.5 feet, and so forth. Ultimately we can keep going until we ask ourselves the question, "How can anything move at all? To travel some distance X we have to travel some distance smaller than X first, but to travel that we need to travel something even shorter than that, etc."

The problem lies in the sort of assumption that adding up an infinite number of terms equals infinity. Some infinities are "larger" than others. There is technically no such thing as "the number infinity." Infinities only make sense when you speak of limits, and when we're talking about limits, we're really talking about rates.

Even with a fully continuous underlying space, Zeno's Paradox still fails. An infinitesimal amount of distance may be infinitely small, but it isn't nothing.

*****

I also want to point out to you guys: There's a ***HUGE*** difference between quantizing space and making space discrete. You don't need to invoke discrete space to solve this problem.

[Reach]

Re: A question I need answered.

I also want to point out to you guys: There's a ***HUGE*** difference between quantizing space and making space discrete. You don't need to invoke discrete space to solve this problem.

But space is not continuous, which in and of itself does solve the cognitive aspect of the problem. It might not be necessary, but it's an easy solution.

This is, as far as I know, also a very common solution to this problem, so I fail to see why you have an objection to this.

Even with a fully continuous underlying space, Zeno's Paradox still fails. An infinitesimal amount of distance may be infinitely small, but it isn't nothing.

Everything you've said up until here makes sense, but this doesn't. It's a contradiction. What do you mean?

Your second statement is exactly why Zeno's Paradox is so hard to wrap your head around; it's not a solution. Because infinitely small distances are still distances, and because there are an infinite number of them if you treat this problem as a series, you arrive at the paradox.

You haven't provided a solution to the paradox, other than " In this case, the fact that you're talking about a finite distance already solves the problem.", which is fine and obviously right LOL, but it doesn't 'explain' anything.

(Yes, everybody and their dog knows the sequence can be defined in a finite manner using a convergent series. At least, if you've taken Cal. It's a terrible dissatisfying answer though, IMO, because it doesn't even answer the fundamental question the paradox is asking in the first place.)

[qqwref]

Re: A question I need answered.

The problem lies in the sort of assumption that adding up an infinite number of terms equals infinity.

I wouldn't say this assumption is made in the paradox at all. In fact, it's implied (even if you know no math) that the sum of the infinite number of terms is finite. The thing that makes Zeno's Paradox confusing at all is the implication that, to move at all, you have to complete an infinite number of tasks - even though you might think you could complete them in a finite time, there are still too many to ever count.
of course, the Paradox is a load of bollocks anyway, but it does require some thinking/explanation the first time you see it
To me, the issue Zeno's Paradox brings up is actually quite distinct from real-world questions such as "does this ball bounce an infinite number of times".

[Reincarnate]

Re: A question I need answered.

Reach: Whoa, whoa, whoa there.

"Space is not continuous" has *definitely* not been proven one way or the other. One of the problems with combining GR and QM is that QM states that space must be quantized. Applying these quantization methods to gravity fails on a GR scale. To say that something is quantized doesn't mean that there are discrete quanta of space -- it just means the objects and operators you discuss need to have quantization methods applied to them.

So when we talk about things like Planck lengths, it's an assumption made by the model that you arrive at through dimensional analysis. It doesn't say that things can't be smaller than Planck length -- it just says that beyond this point, we can no longer probe them. To go beyond the dimensional analysis, we'd need to invoke a new physical theory.

Like I mentioned, Zeno's Paradox is confusing to people because they think that adding up infinitely many pieces means that you're somehow arriving at an unachievable infinity even though we're talking about a finite distance. Yes, we could think of chopping up distance (or time!) into smaller and smaller units -- and we could think about doing this forever -- this doesn't mean the act of traversing distance or time needs to also take forever (which is how we incorrectly arrive at the confused conclusion that we should be unable to move and that time should stand still).


qqwref: It's the assumption that confuses people. They think "in order to get from point 1 to point 2, I must first travel to 1.5, but then to get there I must travel through 1.25, etc" and we could do this forever. So it confuses people into thinking "If we can chop up this distance into infinitely small pieces, how can we possibly get anywhere if we always have to go to an intermediary first?" This is what I mean by "adding up an infinite number of terms equals infinity" as the confusing implicit assumption (it's the same thing as when you say "The implication that to move at all you have to complete an infinite number of tasks").

[Izzy]

Re: A question I need answered.

I'm still going with that the assumption of the paradox is flawed. It makes sense that once you get small enough there is no smaller unit of space which means that there is always a finite number of these units between 2 points.

Only conceptually can you continue to divide the measurement in half, but physically this will inevitably become impossible. But even if you could infinitely divide a measurement in half causing an infinite amount of points between 2 points I don't think that automatically means that everything is impossible due to some kind of limitation on time.

[Reach]

Re: A question I need answered.

whoa

I suppose it hasn't been objectively proven that space-time is absolutely discontinuous. For all practical purposes though, QM is pretty clear. Without getting into a heated semantic debate in physics, my point was more or less here:

It doesn't say that things can't be smaller than Planck length -- it just says that beyond this point, we can no longer probe them

Case in point. Reality is for all practical and material purposes discontinuous as we know it beyond this point. Therefore, thinking of points within points at this scale is, well, rather imaginative. Also, simply add what ledwix said to this and you have the point I am trying to make.

Like I mentioned, Zeno's Paradox is confusing to people because they think that adding up infinitely many pieces means that you're somehow arriving at an unachievable infinity even though we're talking about a finite distance. Yes, we could think of chopping up distance (or time!) into smaller and smaller units -- and we could think about doing this forever -- this doesn't mean the act of traversing distance or time needs to also take forever (which is how we incorrectly arrive at the confused conclusion that we should be unable to move and that time should stand still).

I completely get what you're saying. However, you're addressing the paradox and solving it by simply declaring it a non-paradox, which is...fine. It mean, it's right, because it's not a paradox if you treat it as a finite problem.

You just don't provide an answer to the fundamental cognitive problem the paradox provides, which is what qqwref is getting at.

[rushyrulz]

Re: A question I need answered.

can't help but think this is described by something like this:



but as a graph of force?

[Reincarnate]

Re: A question I need answered.

Technically qq and I said the same thing in different words. The problem of Zeno's Paradox is the notion of traveling to an infinite number of places in a finite time.

But no, the Planck length issue means that we simply can't probe it. It doesn't mean that space is discrete like pixels or something. Even when we talk about pixels on a screen, we can still describe aspects *about* them using continuous mathematical concepts, and likewise for elements that involve sub-Planck-lengths. When we talk about Zeno's Paradox, we would basically be saying that any movement that occurs could only be observed down to the Planck scale for certain units of time. But this is still not dt time and dx distance we're talking about it. We're talking about something larger, and so it just means that we can't *observe* anything smaller even if they take place and take some sort of measurement. Think of it like using a camera with a finite FPS counting objects whizzing by that eventually move faster and faster. The camera can only probe so much. It doesn't say anything about the stuff it can't detect.

And so assuming that spacetime is somehow discrete is a sort of handwaving solution that doesn't really address the reason this Paradox is tough: Calculus. To invoke discrete spacetime is to basically say "We'll remove the psychological problem we experience with this paradox by fixing the division of space with an eventual stopping point" when the real crux of the paradox is that the psychological problem need not exist in the first place. The approach you propose just fixes the symptom and misses what the calculus is actually saying.

The implications of what Zeno is saying is that there is a halfway point between where you are and where you want to go... as long as you're not yet at the finish line. The problem with infinity is that if you're going to talk about finite concepts you have to imagine that you're at the "end" of an infinite process. Zeno is trying to get you to think of it in terms of something you count through, much like how you might count through writing .3 repeating. If you fall into this trap, you'll never make it. No matter how many 3's you write, you'll never be at 1/3. No matter how much you subdivide Zeno's distance, you are by definition subdividing the distance and are logically leaving pieces behind that you aren't dealing with in your future calculus of further divisions.

But that's why the solution to the paradox "feels" tautological and not very satisfactory. It's all due to the nature of infinity. If we can travel distance x in time t, then we travel .5x in time .5t and distance dx in time dt. If we're talking about infinite divisions, then dx and dt are valid concepts to talk about. This means we ARE moving some distance in some time -- both nonzero. We can keep subdividing distance, but this means we're also subdividing time and we fall into this trap where we're "counting" things. But when we're talking about dx and dt, this also means we can *actually arrive* at the full-blown destination. That's the entire reason a limit works. We can keep subdividing, but if we subdivide to infinity, that means we can actually talk about MOVING that distance in that amount of time.

In other words, yes, to arrive somewhere you must arrive at its midpoint first. Assuming we can move at all, we'll get there eventually. The subdivision argument is a psychological/logical trick that will ensure you never solve the paradox if you attack it from that angle.

[Reach]

Re: A question I need answered.

As much as I didn't need it, thank you for the elaboration Rubix.

It was completely and utterly missing from your original blurb on the subject.


As an aside, discussing space continuity further will only break this thread, so I will only add this: How are you defining continuous? A mathematical space is only defined as continuous if it's metric can undergo infinitesimal subdivision.

When I talk about discontinuous space time, I'm talking about distance being undefined beyond a certain threshold, causing a breakdown of metric continuity. The plank length is exactly this scale.

I wasn't aware that there was any dispute over this. It's what I was taught in quantum physics.

[Reincarnate]

Re: A question I need answered.

It's not so much a dispute as it is a notoriously tricky concept to grasp. It's hard to distinguish between quantizing something and making something discrete because they both sound like they're saying the same thing. We can talk about discrete energy states and technically be referring to "quantized" states at the same time. When we start talking about space itself, though, "discrete" loses its meaning because we're not talking about something countable like energy states.

We just say that the Planck length is the smallest unit where things make any physical sense. At lengths/times less than one Planck unit, quantum theory no longer applies to that realm. We don't have a good, solid quantum understanding of GR yet so we try to determine realms of relevancy by combining GR constants (G and c) with QM constant (h) to result in the fundamental units.

So yes, Planck length is the smallest possible length we can talk about meaningfully, but really it's just that we don't have any reason to believe one way or the other than our current theories have any application below that scale.

[benguino]

Re: A question I need answered.

How about this:

f(x) = int[g(x) / p]
      -----------------
                p

Where:
x is time elasped
g(x) is a function someone would normally use to get distance in meters, assuming continuity in time and space.
p is the number of Planck lengths in a meter
f(x) is the actual distance in meters, assuming that space is divided into discrete units.
int is a function that truncates the decimal portion of any real number.

I don't know if "int" is the appropriate function, as opposed to "ceiling" or "round", so feel free to discuss. I was just thinking about the topic and I came up with the above formula.

[Reincarnate]

Re: A question I need answered.

A note on all this stuff because it pisses me off when I see people treating Planck stuff this way:

A Planck length is 1.61605*10^(-35) m, so that means there are 6.18792735 * 10^34 Planck lengths in a meter. That's all you need to convert.

You get Planck's constant from (hcross*G/c^3)^.5, where all the units cancel out to give meters.

Planck time is 5.39124*10^(-44) seconds, which is derived from (hcross*G/c^5), which is the amount of time it takes the speed of light to travel one Planck length.

You all know Planck speed, c -- an upper limit on speed, the speed of light.

Now for the interesting part:

Planck mass is (hcross*c/G)^.5, or about 22 micrograms, or the mass of a Planck particle which is a black hole with a Scharzschild radius (or event horizon) equal to Planck length. You'll notice it's actually quite large compared to everything else. This mass represents the smallest possible mass that can collapse into a black hole (which is good news for us, considering that most elementary particles are smaller than this).

Note that GR has no such restriction on minimum sizes of black holes. All it predicts is that if a mass is squeezed smaller than its Scharzschild radius, it collapses into a black hole (the radius here is Gm/c^2). But on the quantum level, we talk about Compton length, where quantum effects become dominant (given by h/mc). These two figures are inversely proportional to each other, which is why we typically treat GR and QM as mutually exclusive theoretical frameworks. However, when you thrust these units together you find that QM and GR are both dominant at these levels where the Schwarzschild and Compton radii are equal. Planck density (specifically, Planck mass/(Planck length)^3 or c^5/(hcross*G^2)) gives you the largest meaningful density -- which also happens to be the exact same as the density of the universe one Planck time after the Big Bang (Planck temperature is the highest possible temperature -- a radiation with a Planck-length wavelength).

These lengths, masses, and times are just constraints derived from the frameworks of the underlying theories. These units say nothing about what may or may not be actually happening at levels beyond these. At any rate, we haven't even begun to plunge into some of these levels yet. We're still multiple orders of magnitude away from a Planck second or Planck length. These units are limits on quantum field theory and classical gravity. Here you start getting into quantum gravity stuff (otherwise, trying to marry GR and QM by just hard-fusing shit from both sides at this level results in some LOLworthy mistakes where vacuum energy is over 100 magnitudes off).

To give you an idea of the magnitudes, we all know that if you were to blow up an atom to the size of a football stadium, the nucleus might be the equivalent size of a pin placed in the center of that stadium. But say you blew that atom up to the size of the whole goddamned observable universe. A Planck length might be the equivalent of a typical Earth tree.

Basically:

"Substances are the smallest units! Grains of sand! Dust! We're right!"
"No, no, atoms are the smallest, duh-doi. We're right!"
"JK protons and electrons fo shizzle. We're right!"
"WAIT! Stop the presses! QUARKS! We're right!"
"Planck lengths, dude! We're right!"

I think you see where this is going. All these limits are derived from the basis of the models in which they're derived. Planck mass alone should give you a pretty big clue. We derive it with the same sort of methods that we derive Planck time and Planck length, and yet we experimentally know that there are things larger and smaller than Planck mass. Planck mass is a limit *within the confines of the frameworks you're discussing*. Planck length and Planck time just happen to be smaller and faster than anything we can probe. But that says nothing about potentially new, future physical theories that speak about things even more extreme.

Regarding problems like Zeno's Paradox, this is a problem of calculus. This assumes that you're capable of observing something with infinite precision. Invoking discrete space is just a handwaving tactic to get out of trying to understand why the paradox itself is so tricky to understand within the arena it was built on (calculus). Even if you do invoke discrete spacetime, you're not really solving the paradox. All you're saying is that you can't infinitely divide space and time and therefore the entire problem itself is bunk to begin with. Even if you take this route, you're not explaining how the reality is actually working. It's like people think of discrete spacetime as pixels where, at the lowest possible level, you have one Planck-length pixel "lighting up" every Planck-time. But then you have to answer "what does it mean to light up every Planck time?" We're still talking about something finite -- something continuous, but subdivided up into a discrete scale. All this is doing is solving the initial psychological boundary and not the problem of infinity itself.

Sorry for the long rant but holy shit I get fumed when I see math and physics so blatantly pillaged.

[qqwref]

Re: A question I need answered.

hcross

lol

[Reincarnate]

Re: A question I need answered.

That's how it's pronounced. You can also call it hbar or reduced h.

[qqwref]

Re: A question I need answered.

I'm pretty sure h-bar is at least 100 times more popular. Compare:

http://www.google.com/search?hl=en&q=physics+%22h-cross%22&aq=f&aqi=g-b4&aql=&oq=

http://www.google.com/search?hl=en&q=physics+%22h-bar%22&aq=f&aqi=&aql=&oq=

(Don't look at the number of results, look at the top results - "h cross" brings up a lot of fake ones so the number's off)