DrRocket

Define "something". "Something" is not a scientific term. You are going down a path to philosophy, and you find no answers on that path.

There is some good information on the internet. There is also a lot of utter nonsense. Unless you have enough knowledge of the field of interest to be able to tell the difference you would be better off sticking to information from known sources.

Before critiquing general relativity it would behoove you to invest the time and intellectual capital to understand it.

You have it backwards. Starting from the natural numbers, in the form of the Peano Axioms, or more rigorously the Zermelo Fraenkel axioms you can construct (not derive but construct) the integers, rational numbers and complex numbers. From that you can construct Euclidean geometry. Now throw in the axiom of choice and you can construct essentially all of the remainder of mathematics. The important thing is that the only assumptions are really the existence the natural numbers and the axiom of choice. Everything else is a logical consequence. If you would like to see the construction through the complex numbers you can find it done in Rudin's Principles of Mathematical Analysis or Landau's Foundations of Analysis. It is not difficult. I have taught it to sophomores and juniors. It is, however, a bit tedious.

Books are your best source. Books by people who have demonstrated an ability in the subject are usually a good bet. Physics: The Feynman Lectures on Physics by Feynman, Leighton and Sands Chemistry: General Chemistry by Linus Pauling

Take the log of both sides of the solution in the book.

Particularly those authored by Kaku.

I'll take a chocolate [math]W^-[/math] please.

Your logic is flawed (non-existent) and your conclusion is false. The term "clock" in relativity, and in physics in general, does not refer to any specific device, or in fact to any device at all, but rather any means, real or idealized, for measuring time. "Time is what clocks measure". So, yes, if you have a physical device that is your "clock" then the elementary particles of which it is composed will each be in some quantum state and that state will not be one of 0 energy even if the clock is at 0K -- not all (in fact no two) fermionic particles in a system, according the Pauli exclusion principle, can be in the same quantum state, so they cannot all have 0 energy. But they can be in the lowest allowable set of states and that is what is meant by absolute zero.

A useful idealization.

Certainly a clock can be at rest. Just pick the rest frame of the clock. There is no such thing as absolute motion. The fact that something moving is most emphatically NOT universal.

incorrect In fact in special relativity a clock at rest with respect to a given observer, the clock which measures time in that observer's rest frame, has zero kinetic energy.

The situation is this: There is nothing in general relativity per se that prohibits a time machine. Nevertheless no one believes that such a thing is possible. Stephen Hawking has shown that under general relativity, with the additional assumption of the "weak energy condition" that no time machine is possible. He has also offered the "chronology protection conjecture" which posits that no time machine (closed timelike curve) is possible on the macroscopic scale. The nature of quantum field theories is such that closed timelike curves might be possible at the level of elementary particles -- Feynman diagrams include such possibilities but they represent vanishingly small probabilities. So, as it stands now, the chronology protection conjecture remains an open question. General relativity by itself will not answer the question. Quantum field theories indicate that a proof will likely involve considerations at the macroscopic level as distinct from the quantum level -- but there is no clear dividing line, so it will take greater understanding of physics than what is now available to answer the question with finality. Nevertheless there are all sorts of paradoxes associated with the existence of time travel (closed timelike curves) that result in the nearly universal belief that time travel is not possible. There are still people who profess otherwise, mostly crackpots. I can guarantee that any representation of the possibility time travel on YouTube will fall into the crackpot category. YouTube, except for lectures by recognized scientists from recognized institutions, is not a good source for scientific information.

Google (or Bing) is your friend. http://www.lifesci.ucsb.edu/eemb/programs/graduate/research/freshwater_biology/freshwater_biology.html

If I understand your starting point, you are not making any assumptions abuot the specirfic mechanism for photosynthesis, even though you apparently do understand it and used that understanding a bit in formulating your experiment. Assuming that my understanding of your starting point is correct, then it seems to me that you need more controls for your experiment. So, for instance it might be that the effect is not one of simply "more CO2 results in more sugar" but it might be that "less oxygen or less nitrogen results in more sugar". If this makes sense to you the you might want to to consider adding some controls to your experiment -- say try having some sugar beets exposed to a pure nitrogen atmosphere (no oxygen), a pure oxygen atmosphere (less nitrogen) and a near-vacuum (less everything). With regard to the kinetics this Wiki article might help. There are several reactions involved, all fairly fast, but the mass-rate at which end products is produced will depend on the rate at which the necessary reactants are transpported to the reaction site, which is an entirely different matter. http://en.wikipedia.org/wiki/Photosynthesis

His ruler is broken ?

Concave and convex, with respect to curvature are terms that one typically encounters in calculus. They are not generally used to describe manifolds of dimension >1. One needs to be a bit careful and precise when using the term curvature in general relativity, and with respect to manifolds in general. It does not necessarily mean what you think it means. The notion of curvature is actually rather sophisticated, and not easily described. First, Riemannian curvature is a tensor and the Einstein tensor of general relativity is derived from it. It has no simple explanation. There are actually several different notions of curvature tensors. ALL curves are "flat" in the terminology of differential geometry. The usual right circular cylinder is also flat. There is such a thing as a flat torus, and a flat torus is one potential model for the curvature of space in cosmological models. Also, you are correct in that energy, as well as mass (and in fact pressure and momentum flux) are all part of the stress-energy wich determines the Einstein curvature tensor of general relativity. You are also correct in that curvature from different particles does not cancel -- and since curvature is a tensor field there is no reason to think that such cancellation is even reasonable, since (remember that general relativity is not a quantum theory) two particles cannot be in the same place. As a particle and its anti-particle have the same rest mass/energy they enter into the stress -energy tensor in the same way. Also note that the curvature tensor is an invariant, independent of any local reference frame, and hence it is the invariant mass/energy and not the relativistic mass/enrgy that is important in the stress-energy tensor. So, before some asks, no you cannot increase the gravitational pull of something and create a black hole simply by viewing it from a reference frame in which it is moving very fast. Bottom line -- general relativity relies on differential geometry and other sophisticated mathematical descriptions, that may not conform to one's naive intuitive notions. The only way to really understand GR is to invest the necessary time and intellectual capital to understand it in the mathematical framework in which it is formulated. Just Euclid tole told King Ptolemy with respect to geometry, "There is no royal road to general relativity."

Geometry IS mathematics.

no Only if it is transported by a jolly fat man in sleigh pulled by reindeer.

You need to show what you have done and some reasonable attempt. We won't do your homework for you.

If they didn't have charge they wouldn't be an electron or a positron. See swansont's post. Apparently they can. One cannot say with certainty exactly what particles are produced in a given interaction. That is the nature of quantum theory. It only predicts probabilities. While one outcome may predominate, other outcomes may occur with less frequency. I am not sure of all the possibilities of a gamma-gamma interaction. But unless the energies are quite high, nothing is produced since one must conserve a number of quantities including mass/energy, charge and momentum. Electrons and positrons have the attractive feature that the mass/energy requirement is relatively small among particles with positive rest mass -- and it takes a LOT of energy to produce much rest mass.

To quote the dean of my graduate school, "The only requirement for a PhD is an acceptable dissertation." Everything else can be waived (course work, residency requuirement, etc.) However, the purpose of the qualifying and general exams is to determine if you are ready to procede further in course work or research. If you fail one of those you might be given the opportunity to take them again, but unless you eventually pass you are out of the program. However, no one who is doing significant original research is likely to fail (and no one who fails is likely to do significant original research). Courses are just there to get you ready to do research and to take those two exams. Not every course even has associated tests.

You can't, nor does the problem ask you to. You are given two vectors in the kernel of T. That does not define T uniquely. You are only asked to find some transformation with the specified kernel, not any specific one. There are, in fact, infinitely many solutions to your problem.

There are intellectually honest theists, many of them. There are also intellectually honest atheists. But as you have so clearly demonstrated, not all atheists are intellectually honest.

Relativity shows that there is no preferred reference frame with respect to the formulation of the laws of mechanics. But there is a reference frame with respect to which it can be reasonably said to be "at rest with respect to the universe". That (local) reference frame is the frame in which the cosmic background radiation (CMBR) is isotropic. The observed CMBR is red-shifted in one direction and blue-shifted in the opposite direction, revealing a motion relative to that reference frame. This reference frame is of utility in cosmology and is commonly used in that discipline. Note that this frame is local and appies to the special theory of relativity. There are no global frames in general relativity in the presence of gravity. Relative to this reference frame the galaxy is moving towards the Hydra constellation at about 300 km/sec. http://hypertextbook...riciaKong.shtml http://en.wikipedia....round_radiation