Tom Mattson

Ack, I was too sloppy. I should have said that the bullet doesn't turn around in the Earth frame only if the recoil speed of the bullet is less than the speed of the train. In that case the bullet will be seen by the Earth based observer to collide with the wall and then continue moving forward, but slower. At no time will the speed of the bullet be zero in the Earth frame. So in that case, it is not true that the bullet turns around in all frames. But now let's look at what happens when the recoil speed of the bullet is greater than that of the train. Let [imath]v_B[/imath] be the initial speed of the bullet as measured from the train, [imath]v_T[/imath] be the (constant) speed of the train as measured from the Earth, and let [imath]v_B>v_T[/imath]. Further, let the bullet hit the front wall of the train at time [imath]t=0[/imath] and let the deformation and recoil take place from [imath]t=0[/imath] to [imath]t=T[/imath]. Let the acceleration of the bullet be constant during the entire collision-deformation-recoil process (this is for simplicity). We can write down the following function for the velocity of the bullet with respect to the train. (Sorry for how it looks; I tried to define a piecewise function as an array, but the system said I had a syntax error in my LaTeX. Couldn't find it though.). [imath]v_{BT}(t)=v_B[/imath] for [imath]t \leq 0[/imath] [imath]v_{BT}(t)=v_B-\frac{2v_B}{T}t[/imath] for [imath]0 \leq t \leq T[/imath] [imath]v_{BT}(t)=-v_B[/imath] for [imath]t \geq T[/imath] Note that I assumed that this collision is perfectly elastic in the frame of the train. Now the question is, at what time [imath]t_{stop,BT}[math] is the speed zero? It can only be zero during the deceleration, so we'll set that piece of [imath]v_{BT}[/imath] equal to zero. Solving the equation yields [imath]t_{stop,BT}=\frac{T}{2}[/imath], which is exactly the midpoint of the deceleration process (makes sense). Now let's transform to the Earth frame with the Galilean transformation. If we call the velocity of the bullet as measured from the Earth frame [imath]v_{BE}[/imath] and the velocity of the train as measured from the Earth frame [imath]v_{TE}[/imath] then the Galilean transformation reads as follows: [imath]v_{BE}=v_{BT}+v_{TE}[/imath], which leads to: [imath]v_{BE}(t)=v_B+v_T[/imath] for [imath]t \leq 0[/imath] [imath]v_{BE}(t)=v_B+v_T-\frac{2v_B}{T}t[/imath] for [imath]0 \leq t \leq T[/imath] [imath]v_{BE}(t)=-v_B+v_T[/imath] for [imath]t \geq T[/imath] Now let's ask the same question that we asked for the train's point of view: When is the speed of the bullet equal to zero in the Earth frame? Set the middle piece of [imath]v_{BE}(t)[/imath] equal to zero and solve. You will get [imath]t_{stop,BE}=\frac{v_B+v_T}{2v_B}T[/imath]. This resolves the apparent paradox. Yes, the bullet turns around in the Earth frame, provided that it recoils with a speed greater than that of the train. But it doesn't turn around at the same time in both frames. When the bullet turns around in the Earth frame, is moving with respect to the train, and thus there is no reason to suppose that the train must be stopped at that time. Now if you want to see this done with the Lorentz transformation, you're going to have to start paying buddy.

I don't think that anyone has tried to do anything with it. The reason would be that state variables that appear in the laws of physics arise from the analysis of a physical system, as opposed to being forced into them. The dynamics of a mechanical system requires that [imath]x[/imath] and [imath]\dot{x}[/imath] be used as state variables, because potential energy depends on the former and kinetic energy depends on the latter. The reason [itex]\int xdt[/itex] isn't used is because there is no form of energy that depends on it. Of course, there is nothing stopping you from re-writing [imath]x[/imath] as [imath]\frac{d}{dt}\int xdt[imath], but why would you want to do that?

No. Zeno of Elea has 4 paradoxes of motion attributed to him (by Aristotle), but this isn't one of them. http://plato.stanford.edu/entries/paradox-zeno/

Wowee, good show guys!

Where have you ever seen position integrated with respect to time? Not that you can't do it, but there's no physical reason to do it.

Speed is still s/t, but in the so-called "natural units" of SR speed becomes a dimensionless number between 0 and 1.

That's fine, but it's not a convenient point of view when doing modern physics. The distinction I've been describing is something we want to be rid of, because it arises from nothing more than a choice of units and is therefore artificial. But there is still a very important distinction that remains. The basic invariant of SR is [imath]x^2+y^2+z^2-t^2[/imath]. The difference between the spatial coordinates and the time coordinate is that minus sign. That distinction between space and time is preserved in the metric, and it cannot be dispensed with.

Easily fixed: Just let the initial momentum be zero, and the integral will give you momentum.

Yes, the principle of diminishing returns does apply. So?

Once you understand relativity, you turn that question on its head: Why not treat time in the same way as distance? Using different units for time and distance obscures the symmetry of relativity, which puts space and time on the same footing. In that book I told you about it is explained that the factor [imath]c[/imath] is just a conversion factor between space units and time units and that it is no more significant than the conversion factor "5280 feet = 1 mile".

Neutrinos do interact with matter' date=' but they only do it via the weak interaction (I'm neglecting gravity because it doesn't fit into particle physics yet). That's why they are so difficult to detect. That is true, but I just want to point out that the inverse isn't true. That is even if the neutrino did not have mass, it would still have a momentum.

This problem is treated in every calculus book. You'll find it in the index under "arc length".

The reason that c isn't carried around in the upper level textbooks is that there is no reason that you couldn't think of time as a distance. 1 second of time is about 3*108 m of distance traveled for a light pulse. This is explained in Taylor and Wheeler's Spacetime Physics, which is meant for sophomores or juniors (post Halliday and Resnick).

Of course they are. Fractions are elements of the rationals, which are a subset of the reals.

Higher level books on relativity do use t for the 4th dimension. ct is used in elementary treatments because students are still used to old systems of units (SI, English, what-have-you) that necessitate a conversion factor c with the dimensions of velocity.

I would be wary of any wikipedia article on Bell's inequality or the foundations of quantum mechanics. There is a battle raging behind the scenes among the editors, one of whom is the famous (infamous?) local realist crackpot named Caroline Thompson. She is constantly reverting sections of these articles towards her biased POV, and the others constantly revert them back to a more neutral, encyclopedic nature. They watch her pretty carefully but you never know what these articles are going to say on any given day.

No you don't. Electrons do not and can not observe anything. Only sentient beings can observe things. And even if you were correct, your statements in no way imply that energy is an entity unto itself. It is blatantly obvious that you came here with a preconception that energy exists objectively, and it is equally obvious that you ignored all the responses that contradict your preconception. Of course, you reserve the right to hold whatever cockamamie view of energy that you please, but let's not pretend that your view is supported by any actual evidence. The fact of the matter is that energy (as discussed by physicists) has no reality of its own. It has no reality apart from its functional dependence on state variables which are real (distance, velocity, etc.). If you think that energy possesses existential qualities, then you are not talking about the same "energy" that one finds in physics literature. You have yet to make a deductively sound inference about energy in this entire thread. "How and when energy sums up to become matter?" Of course there isn't a mathematical treatment of that, because that is nothing other than word salad. It means nothing and refers to nothing that any physicist actually works on. That is a bit hollow, in view of the fact that you ignored every input you received on this issue.

Buddy' date=' the phenomenon of radioactivity is a [b']prediction[/b] of QFT, which is manifestly relativistic. Nonrelativistic theories do not and can not account for radioactivity because they do not acknowledge the existence of energy due to mass ([imath]E=mc^2[/imath]). You will find comprehensive, noncontradictory relativistic descriptions of strong, weak, and EM decay processes in any number of modern books on QFT, if you take the trouble to look.

First of all the Bohr model is internally inconsistent. It relies on a particular solution of Maxwell's equations (Coulomb's law)' date=' and said equations imply that accelerated charges must radiate. But Bohr answered this by postulating that there were certain special nonradiating orbits, and these orbits are precisely those that can contain an integral number of deBroglie wavelengths around the circumference. The people at the Stanford Encyclopedia of Philosophy cited the Bohr model as an example in their entry on Paraconsistent Logic. Second, while the Bohr model does quantize orbital angular momentum, it gives the wrong quantization rule. Bohr says that [imath]L=n\hbar[/imath], whereas QM says that [imath]L=\sqrt{l(l+1)}\hbar[/imath]. And third, observables in the Bohr model are not operators but rather real valued algebraic expressions. Since they are just elements of [imath]\mathbb{R}[/imath] they always commute, which implies no fundamental restrictions on the compatibility of observables. For instance in Bohr's theory there is no reason that you cannot measure [imath]L_x[/imath] and [imath]L_y[/imath] simultaneously to any desired accuracy. That's just what I can think of off the top of my head.

[imath]e^2[/imath] is certainly not 1/137 in any system of units. In SI units [imath]e^2=2.56 \times 10^{-38} C^2[/imath]. Your statement is true in natural units in which [imath]\hbar=c=1[/imath].

It's already known that what you propose is not possible. For example electric charge is not derivable from mass, distance, and/or time.

Come on people, if future humans are smart enough to dig a hole clean through the Earth then they'll be smart enough to evacuate the center, cool the core, stop the rotation, and turn our planet into a perfect homogeneous sphere. Stop nitpicking!