Why does light have invarient speed?

[W.RonG]

But this problem is also simple to resolve in the case of continuous space and time, using calculus. Yes, you can divide the trip into an infinite series of smaller and smaller increments, but the time for the arrow to cross each successive increment will be also be getting smaller and smaller, and in calculus it is quite possible to have an infinite decreasing series which sums to a finite number, like 1/2 + 1/4 + 1/8 + 1/16 + ... = 1

If I recall, calculus is based on the assumption that the summation of a very large number of very small increments is for our purposes the equivalent of a continuous function. I'll wait while someone adds up all the above fractions. Oops, there's an infinite number of them so it would take (literally) forever and I don't have that long. In this case we can only have faith that the asymptote actually reaches the final value.

Why couldn't an object move more than one units of space in a single unit of time? After all, for slower-than-light objects, they'd have to move more than one unit of time for each unit of space along their path. ...

I see it as being very difficult to travel two Planck distances in space without first having moved one Planck distance. The only way to achieve more than one Planck distance of movement in one time increment is to traverse all Planck distances simultaneously. Not happening. So I see it as all movement is 1Lp/1Tp, 1Lp/2Tp, 1Lp/3Tp, etc. This implies (since we can't occupy space in an interval smaller or between Lp), that all movement is described by spending a number of Tp's at each location of Lp. Then on to the next Lp location, wait another number of Tp's. etc.
rg
(does this forum have a sig. function? mine is "I thought I was wrong once, but I was mistaken". Oh and smileys.)

 

[JesseM]

If I recall, calculus is based on the assumption that the summation of a very large number of very small increments is for our purposes the equivalent of a continuous function.

No it isn't. It's based on limits, and in the case of a continuous function, you're talking about the limit as the size of the increments goes to zero (and the number of increments in the sum goes to infinity).

I'll wait while someone adds up all the above fractions. Oops, there's an infinite number of them so it would take (literally) forever and I don't have that long. In this case we can only have faith that the asymptote actually reaches the final value.

Again, you seem not to understand the idea of limits which is the foundation of calculus. To say the sum of the infinite series is 1 means the limit as you increase the number of terms in the sum is 1, so that for any tiny number "delta" (say, delta = 0.0000000000000000000000000000000001), there's some rule that gives you a finite number N such that, if you add together the first N terms of the series, the sum will be larger than 1 - delta, but it's also possible to prove that no finite number of terms will ever give a sum larger than 1. This can all be proved in a rigorous way, you don't have to wait around to test every possible delta to make sure there are no exceptions to the rule.

I see it as being very difficult to travel two Planck distances in space without first having moved one Planck distance.

Why? If space is quantized, then things are always "jumping" discontinuously through space anyway. We could certainly program a computer so that simulated objects could hope more than one pixel in single unit of time, and I see no reason that the universe can't be following any conceivable algorithm that could be programmed into a computer.

 

[W.RonG]

Are the wheels spinning out there or did everyone else bail?
Hi JesseM. Since these posts are contiguous I won't re-quote; I believe we've said essentially the same thing regarding calculus, just that I've used more colloquial terms. Since we don't want to wait around to add an infinite number of infinitesimally small items, we just say "close enough" (within your delta) and project the result as if we had added everything up (invoking the limit). Kind of a to-may-to to-mah-to scenario. You can add 0.9 + 0.09 + 0.009 ... til you're exhausted and you won't (and never will) reach "1". When the result gets within "delta" of "1" we say "done" but you have to admit it's still not "1".
I'll stop retorting about calc - if you look at my bio you'll see that I'm more in line with the practical side of things and less of the theoretical (my 3-decade-old degrees are in applied science and engineering technology), plus the fact that I've spent your lifetime not using calculus since I last had it in class.
rg

 

[matheinste]

W.RonG.

I think the sum of the infinite series 0.9+0.09 and so on is 1. If it is not so can you give me a number which is between this sum and 1.

Matheinste.

 

[JesseM]

Since we don't want to wait around to add an infinite number of infinitesimally small items, we just say "close enough" (within your delta) and project the result as if we had added everything up (invoking the limit).

A limit is not just "close enough", it is the number that we can rigorously prove the sum would get arbitrarily close to if we kept adding terms forever.

Kind of a to-may-to to-mah-to scenario. You can add 0.9 + 0.09 + 0.009 ... til you're exhausted and you won't (and never will) reach "1". When the result gets within "delta" of "1" we say "done" but you have to admit it's still not "1".

But what if the terms themselves represent actual physical times? In other words, you know the arrow takes 0.9 seconds to traverse the first interval, 0.09 to traverse the second, and so forth. Then how long it would take you to add them in your head is irrelevant, unless you can add each new term in the exact amount of time it is supposed to represent (and if you could, then you'd add an infinite number of terms in 1 second total).

 

[W.RonG]

W.RonG.

I think the sum of the infinite series 0.9+0.09 and so on is 1. If it is not so can you give me a number which is between this sum and 1.

Matheinste.

My point is in the real world there is not an infinite number of infinitesimally small items to add together, and if one really attempted to do so the effort would (a) never achieve an end and (2) never reach the ultimate result. Mathematics allows for a conceptual model of a real thing, it is not the real thing itself. Delta is an arbitrary number that is by definition not zero. Oops I retorted. no more. I promise.
rg

 

[matheinste]

In the real world the arrow reaches the target.

Matheinste.

 

[W.RonG]

...what if the terms themselves represent actual physical times? In other words, you know the arrow takes 0.9 seconds to traverse the first interval, 0.09 to traverse the second, and so forth. Then how long it would take you to add them in your head is irrelevant, unless you can add each new term in the exact amount of time it is supposed to represent (and if you could, then you'd add an infinite number of terms in 1 second total).

Maybe this is what got Max Planck started on his way to figuring out what the limits are for time and distance in the real world.
rg

 

[W.RonG]

Mdeng, I hope you don't feel left out. There were some good questions in #95 and some very insightful statements in #105. I think we should get back to the main question and continue the progress made up to that point. I'm going to brave the blizzard and head home now so I'll be checking in later.
rg

 

[rbj]

The inclusion of quantization was meant to describe the speed of light based on the nature of space-time and local interaction which propagates energy. It can only be 1/1=1. Everything else is that or a lower ratio; the only way to propagate faster would be 1/0=?

Why couldn't an object move more than one units of space in a single unit of time? After all, for slower-than-light objects, they'd have to move more than one unit of time for each unit of space along their path. In any case, if you think the notion of quantized space and time is established physics you're wrong; it's a speculation that emerges out of some approaches to quantum gravity,

i thought that it was a speculation that reality might be quantized in time and space, similar to cellular automa, where something around the Planck Time and Planck Length are the units of quantization which, since they're so damn small, all of these differential equations for EM, QM, and GR get turned into difference equations via Euler's method and these difference equations have no constants of proportionality (except maybe an occasional 2 or 1/2) in them since the quantities are in Planck Units. as cellar automa, some "action" can only propagate to adjacent discrete spatial cells in one discrete time unit, thus imposing a speed limit of 1 Planck Length per Planck Time.

but, of course it's not anywhere near established physics. just speculation and probably full of holes. but as a practitioner of Discrete-Time Signal Processing (sometimes called "DSP"), it's sort of gratifying to think about reality possibly as a sampled data system also (with a sampling frequency of about 10⁴⁴ Hz, the reciprocal of the Planck Time).

 

[JesseM]

Maybe this is what got Max Planck started on his way to figuring out what the limits are for time and distance in the real world.
rg

Someone correct me if I'm wrong, but I don't think Planck himself made any claims about "limits for time and distance", he just came up with "Planck units" as a convenient system of units for physicists to use, ideas about the physical significance of the Planck length and Planck time are modern speculations which emerge out of quantum gravity, which suggests that quantum gravitational effects should become significant at this scale.

 

[mdeng]

Quantized space and time answers the ancient conundrum of the arrow shot at a target. In half of the travel time it goes halfway to the target. Half again it is closer and if this is repeated ad infinitum the arrow will never reach the target. But we know it does so there must be a minimum distance unit and a minimum time unit and all speeds are integer ratios thereof.

TBH, I never understood why this is a conundrum. If time/distance are not infinitely divisible, the answer is obvious, no conundrum. Assume that time is infinitely divisible. Now, if we keep looking at only half of the remaining distance as the puzzle requires, we are just fooling ourselves by not looking beyond (and including) the target where the arrow is. The truth is, the arrow does not stop just because we willingly chose not to look beyond the time it takes the arrow to fly over the distance, or beyond the distance that the arrow will need to cover to reach the target.

If the conundrum is about "if we can move with an arbitrarily small step that can take infinitely small amount of time, can we ever cover a given finite distance?" Then the answer will depend on whether distance and time can be infinitely divided.

Thus I fail to see why the arrow's reaching of the target proves anything about whether time and distance must have a smallest unit or not (or whether they must be discrete).

 

[pervect]

This thread has nothing to do anymore with relativity,and has other severe problems as well. I'm locking it.