Tom Mattson

No' date=' it isn't. Newtonian mechanics fails to predict the deflection of light in the presence of a massive body, and it fails to predict the lifetime of a hydrogen atom. The former is a manifestation of the relativistic world, and the latter a manifestation of the quantum world. You're right about Newton working at high speeds though. But the definition of momentum has to change from p=mv to p=gmv in order for Newton's 2nd law to hold.

The summation of the series does involve taking a limit as n-->∞, but that does not mean that we can not say what the sum is. The sum of the series literally equals 1, despite the fact that we can not manually perform all the steps in the addition. No, the formula holds even with the correct definition of infinity. I am not making a scientific prediction with this series, I am telling you that the finite sum of the series is the reason that Zeno's paradox is not a paradox. In the argument leading up to the paradox, he assumes that the sum is infinite, when it is really finite. OK, so we agree that Zeno was wrong.

No, it shouldn't. First, there is the fact that the gravitational force of the planets is position-dependent, and becomes weaker with increasing distance. Second, there is the fact that once the probe has passed the planet, the planet attracts the probe, thus working against the acceleration away from the solar system! Third, what you have got to understand is that the space probe is not even close to approaching relativistic speeds, and so there is no apparent difference between the Galilean and Lorentz velocity addition law. But when the speed of the probe approaches the speed of light, it is found that a constant force does not produce a constant acceleration. There is a "diminishing returns" effect by which it costs more energy to attain that extra bit of speed, such that it takes an infinite amount of energy to attain light speed.

No, Zeno's series was the geometric series with (1/2)n as the summand. Since subsequent terms share a common ratio of 1/2 (which is less than 1), the series converges.

Yeah, either that or Math, because Goedel is taught in both departments.

Right. The infinite series that I presented is the one that is used in the argument for the infinity paradox. The "1" is dividable--that's why I can write it in terms of a series! The point is that all the terms in the series add up to 1, and that series represents the total time it would take to traverse the distance. When Zeno formulated his argument, he assumed that the series of terms would have to add up to infinity, simply because there were an infinite number of them. But what he didn't know was that an infinite series can add up to a finite number, and this one does. I agree: If the series actually diverges, then we cannot have continuous motion. I don't understand your puzzlement. Logically, it is possible to start with position and momentum operators with continuous spectra, construct a Hamiltonian, and get bound states with quantized energies and angular momenta. The program that accomplishes precisely that is quantum mechanics. But your question seems to be from a point of view that quantized angular momentum implies quantized linear momentum, which is not true.

Presently, all working physical models contain continuous variables to represent spacetime coordinates, and there is no problem with it. But it iis thought that, to be consistent, and to carry the quantum revolution "all the way", that spactime itself must be quantized. Only time will tell whether or not this is experimentally justified. Currently in quantum theory, dynamical observables of bound states are quantized. These include energy and angular momentum. Also, in the standard model of particle physics, charges (I mean that in the general sense to include EM, Weak and Strong charge) are quantized. All of this can take place in a continuous spacetime.

But quantized space does not appear in any working model that been tested by experiment.

There is no "after this 1". Actually' date=' I made a typo, the series should have read: T=L/v=(1/v)Σ[sub']n[/sub]L(1/2)n (I left out the L in the summand last time). Anyway, the sum of the series is the total time that it would take to cover the entire distance. Zeno assumed that the sum of an infinite number of terms is necessarily infinite, and it is not.

No, it hasn't. The only scientific journals in which you can find discussion on quantized spacetime are the ones which discuss quantum gravity. All such theories are far from being finished, let alone tested.

An axiom of a theory is something that is not proven, but rather taken as self evident. Every theory has to have axioms because, due to the finiteness of our language and concepts, any theorist will eventually run out of words and ideas to define his axioms in terms of. A synonym is "postulate", such as the postulates of Euclidean geometry. Axioms must also be independent to be considered "axiomatic". That is, it must not be possible to derive an axiom from among the set of axioms. There was a time when challengers of Euclid charged that his 5th postulate was not really a postulate, but actually derivable from the others. The charge was eventually found to be false, however. Another interesting point is the work of Goedel, who proved that every formal system that is at least as powerful as arithmetic is either incomplete or inconsistent. That is, it either contains true statements that can't be proven from within the systems (incomplete) or, for some statement X, it is possible to derive both X and ~X from within the system. Euclidean geometry happens to be an example of a formal system that is both consistent and complete.

Swansont is right, this is equivalent to Zeno's paradox of motion which states that no destination can ever be reached, because, at any point in the motion, you always have at least half the remaining distance in front of you. The reason Zeno's paradox fails is that, in his argument, he assumes that the infinite series of powers of 1/2 diverges to infinity, when in fact it converges to 1. It goes like this: Assume you are traveling at speed v and must cross a distance L in time T. Before you can get to L, you must get to L/2. Once you have gotten to L/2, you must then get to 3L/4, etc. This continuous halving of distances can be represented by the series: Distance=L/2+L/4+L/8+... Note that it takes an infinite number of steps to travel the distance. Since it is not possible to perform an infinite number of steps in a finite time, it is not possible to cross any distance L in time T at speed V. Zeno's mistake is in assuming that an infinite number of steps cannot be done in a finite amount of time. In other words, he tacitly assumes that it would require an infinite amount of time, and so could never be done. Is that true? Let's see. T=L/v=(1/v)Σn=1oo(1/2)n The series on the right is a geometric series. In order for Zeno's argument to hold up, it would have to blow up to infinity--but it actually converges to 1. edit: typo

volleyball ("playa" is Spanish for "beach" )

<--Tom

I didn't say it was "29" (what unit is that in, by the way?). I said that the Schwarzchild radius of an atom is 29 orders of magnitude smaller than the Planck length. As far as the e-p distance in an atom goes, it is on the order of 10-11m (roughly the Bohr radius). And the p-p distance in the nucleus is somewhere in the ballpark of 10-15m, or 1 Fermi. That is precisely my point! The matter in an atom--or even in a nucleus--is not confined to anywhere near small enough a space to form a black hole. Got it now? edit: typo

Yes, there is space, but is there precisely the right amount of space? This is a pretty simple problem to figure out. You take the mass of the object and calculate its Schwarzchild radius. If the object is smaller than that, then it is a black hole. Let's estimate the Schwarzchild radius for an atom. G~10-10Nm2/kg2 M~10-27kg c2~1017m2/s2 Using those estimates, I get something in the ballpark of: R=2GM/c2~2*10-54m, which is 29 orders of magnitude smaller than the Planck length. So no, there is no black hole in an atom. Let us never speak of it again. :nono:

No, that is still wrong. Read what I wrote again: the tensor Tmn is the stress-energy tensor, which contains all nonzero contributions to the 4-momentum density (that includes energy). (emphasis added) When I say "all nonzero contributions", that is precisely what I mean. All rest mass energy, all electromagnetic energy, all kinetic energy, all of it goes into the stress-energy tensor, and all of it works to curve spacetime to give rise to the effect called "gravity".

Yes, it does. The field equations of GR are: Gmn=(8pG/c4)Tmn The tensor Gmn contains all the information about the curvature of spacetime (the effect we call "gravity"), and the tensor Tmn is the stress-energy tensor, which contains all nonzero contributions to the 4-momentum density (that includes energy).

I think all you guys are missing the point of the original post. You are talking about thermodynamics when you should be talking about Doubly Special Relativity. carluk is asking if there is an upper limit to energy in the same sense that there is an upper limit to speed. The answer is, "that seems to be the case". That is, some recent observations of cosmic rays seem to indicate an upper limit on energy, and this has led to the postulation of the theory of Doubly Special Relativity by Giovanni Amelino-Camelia. Here's an introduction, straight from Amelino-Camelia himself. Doubly Special Relativity edit: fixed url tags

How about this one? Standard Model: An Introduction

A good place to start would be looking at a college catalog, and noting what the prerequisite courses are for the various physics courses. For instance, if you want to study quantum mechanics you need to know multivariable calculus, ordinary and partial differential equations, Fourier analysis, and linear algebra. As to linear algebra, you want a course that emphasizes a rigorous treatment of vector spaces.

In what experiment has, for instance, a photon with a nonzero invariant mass ever been detected?

Not at all. Real photons are detected all the time, be it with a photographic plate or a photoelectric cell or. It is quite impossible to detect virtual photons in that way.