DrRocket

In general relativity there is no such think as a global time. Time is a local notion only. The time measured by clocks is proper time, and there is a proper time associated with each world line. You cannot compare "time here" with "time there" except as a local, coordinate-dependent approximation. In this regard the special theory of relativity is misleading until one recognizes that special relativity is simply the local approximation to general rrelativity, special relativity really applies on the tangent space to the spacetime manifold and not to the manifold itself. It works very well on small and modest scales, but not at all in the current context. So, there is no such thing as a "universal present" which makes "presentism" rubbish.. You can, and apparently are, confusing yourself by using global models from cosmology that use a universal notion of time. This is justified by assuming that the universe is homogeneous and isotropic, in which case spacetime decomposes as a one-parameter foliation of spacelike hypersurfaces ("space") by a timelike parameter ("time"). This decomposition is not unique, and in fact exists only as an approximation on the largest scales. The universe is manifestly neither truly homogenous nor isotropic, though the contrary assumption is a useful fiction in cosmology. So, you can stop your bluff and your typical attempt to baffle with bullshit. It is quite easy to see through your arguments, which have no content -- just a lot of buzz words that you fail to truly understand. The Wheeler De Witt equation and all manner of failed attempts at quantum gravity are quite irrelevant.

If a=b and c=d then a-c=b-d. QED

What we know of the big bang is based on general relativity. The singularity theorems of Hawking and Penrose show that it is impossible to indefinitely extend timelike geodesics into the past. That means that, within the context of general relativity, there is no such thing as "before the big bang". An no, this is not "just a theory". It is a logical consequence of what is actually observed (that the universe is expanding and has at least a minimal matter content) and the general theory of relativity, which is also supported by a great deal of evedence. What is a theory (not "just a theory") is the general theory of relativity. To call something a theory is to say that it has proven predictive power and is consistent with what is actually observed. A theory is NOT a conjecture. It is not gospel, but it is much more than mere musings.

The short answer is that polynomial exponents have nothing whatever to do with dimension.

Has anyone noticed that while uncool's observation is correct -- you do get the expression quoted from the book for the sum of the areas of the rectangles associated with the infinite geometric series if you take into acount the bases of those rectangles -- the result is a lousy approximation to the area under the curve. In fact it is high by a factor of approximately n+1 for small values of r ? You can see this from doing the integral. But if you simply consider the case n=a=1 (a triangle over the unit interval) for which any school boy can see that the area is 1/2 , while Ar is about 1 for r close to 0.

If the speed is faster than c, then the existing theory fails and all bets are off. Special relativity does not allow such speeds. If you mindlessly continue to apply Lorentz transformations you wind up with things like imaginary mass which no one knows how to interpret sensibly.

Yes. His name was Albert Einstein. It that work that was cited when he received the Nobel Prize. Google "photoelectric effect".

If this is homework, you need to show some thought or preliminary work on the problems. If not this is not the right forum. If this is homework it would be interesting to know the nature of the class. The problems do not seem to have much in common. That said, in #1 I see nothing obviating the possibility that all ropes smolder on the lit end for a period arbitrarily close to one hour then burn rapidly to meet the 1 hour period for complete burning -- in essence a simple time delay signal of one hour with nothing happening qalong the length of the rope during that period. In that case no matter how you configure the ropes, from any single ignition you get only signals spaced at one hour intervals. Lighting multiple ropes would only introduce sets of signals with one hour intervals interspersed depending on the unknown ignition spacing. In this case I don't see how the specified task can be completed. You might as well have bells that ring at one hour intervals. If you could use the known interval to calibrate a linear device then you could do something. But no other tools are allowed. #2 is possible with a judicious choice of division of the stone plus the knowledge that only integer weights between 1 and 40 kg are possible. A little thought and some trial and error will get you to a solution. #3 is, as you guessed, impossible. Google "Euler circuit" -- this is a well known class of problems in graph theory. The most well known similar problem is the "Konigsberg bridge problem."

This is just <j |Qi> or in terms of inner products b<j, Qi>. And there is a way to insert latex code: [math]\langle j |Q| i \rangle = \langle j, Q i \rangle [/math] (if you reply the codes are revealed iin the reply box). I am a bit unsure of what it is that is troubling you. Perhaps this Wiki article can help. http://en.wikipedia....ra-ket_notation It relates the Dirac notation to more conventional notation for operators on a Hilbert space, and to the relationship between a Hilbert space and its dual, reflected by bra and ket vectors.

What is thermal energy ? It is juat the kinetic energy of the random motion of molecules. What is sound ? It is the effect of motion of gas molecules in a somewhat organized linear oscillatory motion. So as a sound wave is continually reflected and refracted by bouncing around and contacting solid objects and as the molecules collide with other molecules, the motion becomes more and more random until it can finally be classified as thermal energy.

But [math]E=ms^2[/math] would imply that [math]E \rightarrow 0[/math] as [math]s \rightarrow 0[/math] which we know is grossly wrong (or nuclear weapons and nuclear power plants would not work). This also flies in the face of Einstein's recognition of the equivalence of mass and energy, so it is a long way from a mere tweak -- it is just plain wrong. And THAT is why swansont's challenge to you to derive your equation from some set of basic principles is important. You don't revise a theory that is supported by a mountain of empirical data and developed logically from simple principles by simply pulling an equation out of ................ the air perhaps.

That is one reason why the UNIT circle is used for convenience.

They are in separate frames of reference by virtue of thev fact that are each in uniform motion relative to the other. How they came to be in that situation is outside the special theory of relativity. Special relativity is really a theory about space and time and the nature of space and time which is revealed by the fact that observers in relative motion perceive space and time differently. The real key, though a bit abstract, is that what is the same for all observers is a unified entity called spacetime, and a quantity, the spacetime interval. The spacetime interval combines into a "length" both spatial and temporal elements, which vary among observers. Those variations are what are called "time dilation" and "length contraction". But when all elements are considered simultaneously, the interval itself ([math] \Delta s^2 = c\Delta t^2 - \Delta x^2 -\Delta y^2 - \Delta z^2 [/math]) is the same for all observers. When you throw in questions involving acceleration you either have to find a single inertial frame in which to do all of your analysis or, better yet, go to the general theory of relativity. In general relativity you find that space and time are only local concepts (you cannot really talk about "time here" vs "time there" for instance) and that special relativity is just a local approximation. The critical thing to recognize is that relativity is really a radically new theory about the very nature of space and time. The resolution is the emergence of spacetime, and the recognition that it is spacetime, and neither space nor time, that is fundamental. One observer's space includes elements of another observer's time, and vice versa.

The unit circle is there and has the same significance whether you are measuring angles in degrees or radians. The utility of radians is that they are mathematically "natural" because the measure of an angle in radians is precisely the arc length of the subtended arc on the unit circle. It is also natural to think of the unit circle in the complex plane, which is a group under the operation of multiplication. The utility of degrees is that the angle is easily represented to an accurate approximation in terms of an integer number of degrees, minutes and seconds (you don't have to deal directly with the transcendental number pi). Mathematicians tend to use radians. Engineers and physicists use degrees or radians as convenient to a specific application.

That is somewhat a matter of taste and definitions (in particular what is meant by "mass"). [math]E^2 = m^2 c^4 + p^2 c^2[/math], applies with [math]m[/math] being rest mass ([math]m_0[/math]), which is currently in fashion. [math]E = m c^2 [/math], applies if m is taken to be relativistic mass ([math] \gamma m_0[/math]) and if the rest mass is non-zero. It is equivalent to [math]E^2 = m_0^2 c^4 + p^2 c^2[/math] in that case. So for massive neutrinos [math]E = m c^2[/math] is fully accurate if properly interpreted

No. Quadratic forms are functions defined on vectors from an n-dimensional vector space. I have no idea what an "expression of space" could possibly mean.

That's what the technical language is for. There is so much BS on the board surrounding the word "infinity" that God only knows how some people might interpret your statement.

This statement relies rather heavily on "see" implying literal detection of an electromagnetic signal, and that being a pure signal -- i.e. not the result of some event subsequent to the one of interest such as passing through a medium as you mentioned. Given those restrictions and idealizations the statement is essentially a trivial application of the definition of a light cone and the rather obvious fact that light travels at the speed of light. Logic trumps Wiki for understanding every time.. Note that you can easily be affected by events in the interior of the past light cone. If fact the set of events that can affect you is precisely your past light cone. That is whole reason that the past light cone is of interest.

Since "see" implies light reaching us at a point in spacetime, this is indeed true, essentially by definition. Causality rules out the light approaching from the future.

About the only thing sensible to me in the OP is the connection between a conservative vector field and the existence of a a potential function, which is true in a simply-connected domain. I think it was some sort of free association without much real content, and very little appreciation for the underlying mathematics. Mathematical modeling can be as much art as science, depending on the specific problem. Much of the value comes from understanding the nature of the idealizations and approximations that are made, knwing the limitations of them, and in interpreting the results of the model. As always, both with the model itself and with the interpretive capabilities of the modeler, GIGO.

The rationals, and the set of algebraic numbers are both countable, hence of Lebesgue measure 0. The transcendentals, being the complement of the algebraic numbers in are uncountable and of full measure -- you can do this in either the real or complex numbers as you please. Not sure exactly what you mean by "infinitely more numerous", but the above ought to cover it.

What you have is the restriction of a quadratic form on 3-space to the two-dimensional subspace determined by the variables x and y. This is a rather unusual notion. More commonly one studies quadratic forms of some given dimension which are determined by a symmetric matrix, often positive-definite, and almost always non-degenerate (determinant not 0). There is a large literature on quadratic forms -- see for instance the books by O'Meara or Lam.

Actually, you can go in reverse. In which case it is rather easy to rule out gravity, the strong and weak interactions. That leaves only the electromagnetic interaction (other cues such as sound or visual sigtnals are ruled out by simple isolation). One can then test for electromagnetic signals by several methods, and I think that this has been done. Result: Telepathy has no basis in physics. So we can rule out telepathy. This does not mean that some people do not "hear voices". But hallucinations are an entirely different matter.

Thank you. That makes the situation quite clear. The ample evidence supporting relativity show that "presentism" is utter rubbish, with no basis whatever in science.

Because that is NOT a vector equation. In fact the right hand side includes nothing that could be interpreted as a vector. The vector form would be [math] \vec F = G \frac {Mm \vec r}{||\vec r||^3}[/math] where [math] \vec F[/math] is the force exerted on [math]m[/math] by [math]M[/math] and [math]\vec r[/math] is the vector from [math]m[/math] to [math]M[/math].