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Hungry4info
2010-Feb-14, 06:01 AM
Suppose I have a right triangle. Two of the legs are equal to planck length. Mathematically, the hypotenuse would be ~1.41421356 planck lengths. But what about physically?

Thanks.

hhEb09'1
2010-Feb-14, 06:17 AM
You mean, if three objects were separated by those pair of measurements? Would the third leg agree with the Pythagorean theorem? Or would the physical forces act, as if the third leg agreed with the Pythagorean theorem?

Hungry4info
2010-Feb-14, 06:20 AM
Both. They're both what I was trying to get at and something I would certainly like to know. Sorry for being a bit vague D:

I guess at the heart of this is, how long would it take a photon to traverse the hypotenuse? Planck time? Two planck times?

hhEb09'1
2010-Feb-14, 06:24 AM
Are they bigger, or smaller, than a Planck mass?

Hungry4info
2010-Feb-14, 06:29 AM
I don't understand how that's relevant.

I hadn't imagined the triangles legs to have mass.
What would happen if the legs are bigger than Planck mass?
What would happen if the leg's are equal to Planck mass?
What would happen if they were less than Planck mass? (am I correct in guessing that such isn't possible?)

hhEb09'1
2010-Feb-14, 06:37 AM
I don't understand how that's relevant.

I hadn't imagined the triangles legs to have mass.I meant the three objects.
What would happen if they were less than Planck mass? (am I correct in guessing that such isn't possible?)A Planck mass is about 22 micrograms. I think it might be possible to see with the naked eye something with a Planck mass.

sabianq
2010-Feb-14, 06:37 AM
well, the hypotenuse is found by taking the square root of x^2+y^2 (x squared plus y squared) where x and y are the lengths of the other sides of the triangle.

so your answer would be c (the speed of light) over the hypotenuse (the distance)..

is that right??

Hungry4info
2010-Feb-14, 06:41 AM
Ah. The objects. What about for all three options? (I didn't know planck mass was so large).

so your answer would be c (the speed of light) over the hypotenuse (the distance)..

is that right??

It makes sense but it seems naive. Planck length is the smallest meaningful length, so I'm not sure what a distance of a non-integer amount of planck lengths would physically mean.

Edit: Just found a reference saying that contrary to popular belief, space isn't quantized into planck lengths. I'm no longer convinced that my question is really as meaningful as I thought, but I'll stick through and see where this leads.

hhEb09'1
2010-Feb-14, 06:48 AM
It makes sense but it seems naive. Planck length is the smallest meaningful length, so I'm not sure what a distance of a non-integer amount of planck lengths would physically mean.

Edit: Just found a reference saying that contrary to popular belief, space isn't quantized into planck lengths. I'm no longer convinced that my question is really as meaningful as I thought, but I'll stick through and see where this leads.Maybe the hypotenuse would be both one and two? :)

Hungry4info
2010-Feb-14, 06:50 AM
Maybe the hypotenuse would be both one and two? :)

Really?

Anything sandwiched between planck units of something is considered to be both of them?

sabianq
2010-Feb-14, 06:51 AM

the square root of (1.61×10−35 meters)^2 + (1.61×10−35 meters)^2

2.5921e-70 + 2.5921e-70 = 5.18e-70

the sqr root of 5.18e-70 is
2.26x10-35 meters

so how long does it take light to travel 2.26e-35 meters?
299792458 m/s is c (the speed of light)

1.32 x 10 -43 seconds

which is almost as fast as a plank time..
(5.3x10-44 seconds)

Hungry4info
2010-Feb-14, 06:53 AM
Sabianq, I know how to do the math...

I'm asking for the physical interpretation

which is almost as fast as a plank time..

What does almost a planck time actually mean?

(if planck length is how far light moves in planck time, then the hypotenuse would be traversed by a photon in more than planck time since the hypotenuse is longer than the legs)

sabianq
2010-Feb-14, 06:57 AM
Sabianq, I know how to do the math...

I'm asking for the physical interpretation

What does almost a planck time actually mean?

It is the time required for light to travel, in a vacuum, a distance of 1 Planck length..

lol
http://en.wikipedia.org/wiki/Planck_time

sabianq
2010-Feb-14, 06:59 AM
hmmm...

Hungry4info
2010-Feb-14, 07:02 AM
sabianq, please stop toying with me.

If time is quantized into units of Planck time (as I've heard thrown about), then what physically would non-integer units of Planck time mean?

hhEb09'1
2010-Feb-14, 07:03 AM
Really?No, I said "maybe" :)

sabianq
2010-Feb-14, 07:05 AM
yes, it takes light a little longer to travel the distance equal to the hypotenuse of a right angle with sides equal to a plank length..

sabianq
2010-Feb-14, 07:08 AM
here:

http://en.wikipedia.org/wiki/Planck_units

sabianq
2010-Feb-14, 07:16 AM
I guess at the heart of this is, how long would it take a photon to traverse the hypotenuse? Planck time? Two planck times?

the answer is the square root of planck length squared plus planck length squared

what 1.414 planck time?

Tensor
2010-Feb-14, 04:13 PM
sabianq, please stop toying with me.

If time is quantized into units of Planck time (as I've heard thrown about), then what physically would non-integer units of Planck time mean?

That's a new one on me. I've heard that the Planck Time and Planck length are the smallest meaningful distances and times. But, I've never heard it claimed that they are quantized at that level.

If, however, they are quantized at that level, then there wouldn't be any meaning to non-integer units, of those types. It would be like having non-integer units of spin or non-integer units of h-bar.

Paul Beardsley
2010-Feb-14, 04:21 PM
If we're considering a triangle whose sides are measured in (single) planck lengths, then surely it would be impossible to say that two of the sides are at right angles? One cannot use a planck-sized protractor!

Andrew D
2010-Feb-14, 04:51 PM
If we're considering a triangle whose sides are measured in (single) planck lengths, then surely it would be impossible to say that two of the sides are at right angles? One cannot use a planck-sized protractor!

True. do we know how the curvature of spacetime affects measurements at that scale? Can we even be sure a planck-sized triangle has 180 degrees? Or is area a limiting factor? What would the lenths of the side of a trinagle with an area of 1 planck-length(sqrd) be?

Tensor
2010-Feb-14, 05:04 PM
True. do we know how the curvature of spacetime affects measurements at that scale? Can we even be sure a planck-sized triangle has 180 degrees? Or is area a limiting factor? What would the lenths of the side of a trinagle with an area of 1 planck-length(sqrd) be?

It would really depend on the curvature of space-time. Actually, since space-time probably wouldn't be Euclidean, it would be possible to have three right angles and have not only two, but all three legs be one Planck length long.

Andrew D
2010-Feb-14, 07:02 PM
It would really depend on the curvature of space-time. Actually, since space-time probably wouldn't be Euclidean, it would be possible to have three right angles and have not only two, but all three legs be one Planck length long.

That's what I was getting at. In this case, you don't have to define (sqrt)2 h.

Strange
2010-Feb-14, 07:29 PM
That's a new one on me. I've heard that the Planck Time and Planck length are the smallest meaningful distances and times. But, I've never heard it claimed that they are quantized at that level.

What does "meaningful" mean in this context? (if not quantized)

ETA: I have seen it implied, if not explicitly stated, in popular science articles that they are quantized.

Hungry4info
2010-Feb-14, 07:35 PM
Interesting idea about non-Euclidean geometry in the case of a triangle. But it would seem to only work for the case of a triangle. What if we have a pentagon with lengths of planck length, how long would it take for a photon to go from one vertex to any other non-adjacent vertex?

Andrew D
2010-Feb-14, 08:59 PM
Interesting idea about non-Euclidean geometry in the case of a triangle. But it would seem to only work for the case of a triangle. What if we have a pentagon with lengths of planck length, how long would it take for a photon to go from one vertex to any other non-adjacent vertex?

I assume you mean from one vertex to a non adjactent vertex, not travelling along the edges.

If it can't travel in non-integer amount planck time or planck distance, I would say that it follows a curve (on a plane prepandicular to the plane with the pentagon, for example) such that the total distance travelled and travel time (both in planck units) are integers. This might be the reason that extra degrees of freedom are necessary, and why they are so small: they're not observable in scales that don't vary at that degree. Or, it might just be nonsense.

Tensor
2010-Feb-15, 02:49 PM
What does "meaningful" mean in this context? (if not quantized)

Meaningful means, that, at present, our equations don't really work at distances smaller than a Planck length, nor is there any reality to anything smaller. Look at size of the Planck length ~1.6 x10[dup]-35[/sup]m. Electrons and quarks are thought to be point particles. But in actual measurements, electrons have been found to have a maximum quantum radius of ~10-22 m. Even if the electron's size is off by a couple of order of magnitude, you'll notice that this is still huge compared to the planck length. If the smallest thing we can measure is orders of magnitudes larger than the Planck length, then how meaningful is it?

NorthernBoy
2010-Feb-15, 03:29 PM
It makes sense but it seems naive. Planck length is the smallest meaningful length, so I'm not sure what a distance of a non-integer amount of planck lengths would physically mean.

I'd agree with that. It maybe makes sense to think of it like the resolving limit for a telescope. There is a smallest possible size of feature that we can make out, but that does not mean that bigger objects have to look like integer multiples of this limit. We don't assume that the universe is a grid marked out in Planck lengths, with particles destined to sit only at the nodes, but, even if we did, that would still not mean that every separation was a multiple of the grid size.

Andrew D
2010-Feb-16, 04:08 PM
Ah. The objects. What about for all three options? (I didn't know planck mass was so large).

It makes sense but it seems naive. Planck length is the smallest meaningful length, so I'm not sure what a distance of a non-integer amount of planck lengths would physically mean.

Edit: Just found a reference saying that contrary to popular belief, space isn't quantized into planck lengths. I'm no longer convinced that my question is really as meaningful as I thought, but I'll stick through and see where this leads.

I just skipped to the end here, I apologize if this has been addressed.

For this discussion, you have to separate natural and observational. The shortest distance in nature is not a Planck length. There surely exist 'things' smaller than 1 planck length (be they particles, strings, lengths of space, etc.), or the 1 planck length 'grid' would be well known, and scientists would rejoice in having found the lower limit of the universe. This, however is not the case. The reason the planck length is the smallest meaningful measurement is due to our shortcomings as observers. The problem is that once you get down to distances shorter than 1 planck length, the energies required to make measurements are so great that they have a significant effect on the quantities measured. Because of this, the unceartainty of the measurements is greater than the values measured. So, quite literally, measurements at smaller than planck scale are meaningless, and although the planck length may not be the smallest 'natural' distance (if such a thing even exists) it is the smallest observational distance.

In fact, many theorists believe that this is why physics has hit a dead end. We have come to the point where we can continue making discoveries through pure mathematics, thought experiments, and the study of contradictions and paradoxes, but it may not be possible to learn anymore through physical experimentation and direct observation.

Ken G
2010-Feb-16, 04:32 PM
I think there may be some confusion about the difference between a mathematical theorem and a physical truth. The mathematical theorem requires assumptions and logic, and if we use the normal ones, we get the Pythagorean theorem at all scales, with no Planck length involved. If we ask about the physical truth on that scale, it is no longer clear that the assumptions that went into the mathematical proof will serve. The expectation is, they will not.

As for the issue of quantization, yes I think the Planck length is sometimes pinned as a possible unit for a quantum. But we have no idea, if there is such a quantum it might easily appear on a scale a thousand times that! All we know is that our physics breaks down at that scale, the rest is pure speculation. But I think the answer to the OP is basically that if distances are quantized at any scale, then the mathematical assumptions that go into the Pythagorean theorem will not hold well on that scale.

Dragonchild
2010-Feb-16, 06:50 PM
OK, it seems some have figured out that Hungry4info's initial confusion came from the assumption that space is quantized at the Planck length level. FWIW, there's no physical evidence of that (although, to be fair, IIRC there's no physical evidence against such an assumption largely because, up to this point and for the foreseeable future, the distance is indeed physically meaningless).

FWIW, distances shorter than the Planck length would indeed be theoretically meaningful if space was quantized, because if there was any sort of conservation law at work, any interaction that would intuitively involve a length shorter than the Planck length would (again, theoretically) result in behavior that is both observable and testable. In fact, that's how we'd discover such a phenomenon. The Ultraviolet Catastrophe of the Rayleigh-Jeans Law comes to mind.

Either way, it remains "meaningless" because with our current understanding of physics, measuring lengths on that scale is like trying to measure the width of a virus with a straightedge. It's not a matter of refining current technology; we'd need another paradigm shift. . . but considering the Planck length compares to an electron like how a millimeter would compare to the Earth, what's the incentive in finding a way to measure it? Quantization alone isn't a practical reason; even if it was true (and thus a theoretical curiosity), we know of nothing that would give a rat's *** about space quantization.

Hungry4info
2010-Feb-19, 09:36 AM
But I think the answer to the OP is basically that if distances are quantized at any scale, then the mathematical assumptions that go into the Pythagorean theorem will not hold well on that scale.
I would have guessed that much, though I am curious as to what would happen?

Hungry4info's initial confusion came from the assumption that space is quantized at the Planck length level.
Yeah. So if space isn't quantized, and we're permitted any arbitrary length, then mathematical assumptions should work at arbitrarily small scales.

FWIW, distances shorter than the Planck length would indeed be theoretically meaningful if space was quantized, because if there was any sort of conservation law at work, any interaction that would intuitively involve a length shorter than the Planck length would (again, theoretically) result in behavior that is both observable and testable. In fact, that's how we'd discover such a phenomenon. The Ultraviolet Catastrophe of the Rayleigh-Jeans Law comes to mind.
So, in essence, the conservation laws would not permit anything to be gained or lost due to something not falling on an integer planck unit? And thus, it would not be nonsense to speak of sub-planck-length scales? By nonsense, I mean, the math breaks and cries, as opposed to it being unobservable due to technological reasons, or no existing processes occuring at such scales. Of course, one could easily ask "if there are no existing physics at these scales, why does it matter?" And for that I don't have an answer other than curiosity.

Tensor
2010-Feb-19, 01:03 PM
Of course, one could easily ask "if there are no existing physics at these scales, why does it matter?" And for that I don't have an answer other than curiosity.

The real frustration may be that there may not be an answer for that question.

gzhpcu
2010-Feb-19, 04:48 PM
By nonsense, I mean, the math breaks and cries, as opposed to it being unobservable due to technological reasons, or no existing processes occuring at such scales. Of course, one could easily ask "if there are no existing physics at these scales, why does it matter?" And for that I don't have an answer other than curiosity.
Our physics may break down at that scale, because it is an approximation (just like Newton's model was a good approximation) I assume when say "no existing physics at these scales", you mean that we have not defined any physics applicable at this scale. Our limitations in defining physics are set by the scale of the observeable by us (directly or indirectly).