DrRocket

You can say the temperature is the average kinetic energy of the molecules of the substance in question. At that point, to become more fundamenta,l you are stuck with explaining what energy is and that is just as mysterious as what time is. But wait, you say, KE=1/2 mv^2. But when you think about it, to measure v you need to know length and lenght is "what rulers measure". Not to mention that pesky "mass" thing, which becomes even more sticky.

Let the capital letters represent the areas in which you are interested. If I=O-I then you should be able to solve for I in terms of O. You should also be able to represent I and O in terms of the radii of the associated circles. From that the answer ought to become apparent.

yes

Time is what clocks measure. In general relativity what clocks measure is the proper time of the world line of the clock.

Neg rep countered. You don't deserve a neg rep, as you are sincere. (And those rep points don't mean much anyway). But you are still wrong. I know what a frame is, but your are a bit confused as to what that means in the context of a manifold and hence in the context of general relativity. A coordinate system in general relativity and a chart in the mathematical theory of manifolds are the same thing. There is no such thing as a global chart or a global coordinate system, and there is no meaning to a comparison of charts based at different points -- there is no meaning to "time here" vs "time there". That applies as well to so-called "gravitational time dilation", which is measured in a single local coordinate system and is a coordinate effect, which locally approximates a difference in world liines (.ie. proper time). Strictly speaking the only time in general relativity is proper time. Proper time is (in units in which c=1) just the length of the (time-like) world line of a physical body and for the twin paradox the issue at hand is the proper time of the two twins measured between two spacetime points at which their world lines intercept. I have not read the Wiki article in detail but it seems to be dealing with a calculation of proper time. You can make that calculation as difficult and obtuse as you want, but the beauty of general relativity is that the metric and arc length are completely independent of even local choices of coordinates. The "twin paradox" is stated qualitatively (no definte trip length, no velocity vs time history, etc,) and it can be solved, as I showed you, completely and rigorously by means of merely comparing a geodesic world line with any other world line in terms of the spacetime metric. No laborious calculations are needed.

Use the fundamental definition.

I don't follow you, but all possible situations are summarized in a tuth table. The truth table for A--->B was my first post. The entries for the truth table are intuitively clear if you read A--->B as "If A then B" withe perhaps the value when A is false being a bit mysterious until one realizes that a false hypothesis can imply literally anything. The fact that a false hypothesis implies anything is important mainly in the implications for an inconsistent set of axioms. Given an inconsistent set of axioms, literally any sentence formulated using those axioms is both true and false, hence inconsistent sets of axioms are both uninteresting and to be avoided. Thus "If pigs could fly then I woudl be rich" is both a true implication and an information-free statement. I don't follow you, but all possible situations are summarized in a tuth table. The truth table for A--->B was my first post. The entries for the truth table are intuitively clear if you read A--->B as "If A then B" withe perhaps the value when A is false being a bit mysterious until one realizes that a false hypothesis can imply literally anything. The fact that a false hypothesis implies anything is important mainly in the implications for an inconsistent set of axioms. Given an inconsistent set of axioms, literally any sentence formulated using those axioms is both true and false, hence inconsistent sets of axioms are both uninteresting and to be avoided. Thus "If pigs could fly then I woudl be rich" is both a true implication and an information-free statement.

Bet large. You can't lose.

Go read the link. The center of mass of any closed gravitational system does not move relative to the inertial reference frame in which the initial total momentum of the system is zero, and is in uniform motion in any other inertial reference frame. That fact follows from a simple application of Newton's third law, so it is not "ever-changing". (See Classical Mechanics by Goldstein, page 5) Gravitational motion need not be anything that anyone in their right mind would call an "orbit" except in the most abstract mathematical sense. In fact (again read the link) there are systems that are characterized as chaotic. It doesn't matter whether anyone here agrees or not. Physics wins. The point is that gravitational systems have been studied for quite a long time and it is well-known that systems with 3 or more bodies can have motions that are not describable in closed form and that those motions need not look anything like "orbits around the common center of mass." If you prefer books, here are some useful references: Classical Mechanics -- Goldstein Classical Dynamics of Particles and Systems -- Marion (see page 279 of 2nd edition for a nice overview of the 3-body problem) A Treatise on the Analytical Dynamics of Particles and Rigid Bodies -- Whittaker An Introduction to the Mathematics and Methods of Astrodynamics -- Battin New Methods of Celestial Mechanics -- Poincare

There is no such thing as a reference frame in general relativitiy. There is no such thing as a gravitational field in general relativity. I showed you how to resolve the twin paradox in terms of general relativity. I am not conflating anything. You are not only conflating, but confusing, mixing, dicing and chopping the entire theory. The Schwzrzchild metric applies to a radially symmetric spacetime. It has NOTHING to do with the twin paradox. Go read the book that I suggested, a real book, not some popularization, not some simplificatioin, but the real theory. I give up.

If you need someone to hand you a coordinate system to notice that when that dragster takes off you are pushed back into the seat, then I can't help you. The whole point of general relativity is what Einstein called "general covariance", which in modern parlance means that the equations are formulated without reference to any coordinate system (physicists embody this notion in the terminology of tensors). Coordinates are only invoked for specific calculations, but the physics is invariant. It is time that you actually studied general relativity. Gravitation by Misner, Thorne and Wheeler would be the place to start. There you will find real not only general relativity but an introduction to differential geometry and the theory of manifolds. Those subjects are necessary to accurately explain general relativity. What you are missing is that in general relativity spacetime is a manifold, not an affine space and that there is no such thing as a reference frame. The coordinate systems that you see in general relativity are "charts" in the language of manifold theory and there is no meaning whatever to a comparison of charts based at two different spacetime points. There is no global notion of space. There is no global notion of time. "Time here" vs "time there" makes no sense. This is a major difference between general relativity and special relativity. It is more profound than the difference between special relativity and Newtonian mechanics.

Proofs are known. http://www.archive.org/details/diophantusofalex00heatiala http://en.wikipedia.org/wiki/Fermat_polygonal_number_theorem#CITEREFHeath1910

As a bed bug.

wrong Your argument relies on your misconceptions of general relativity being valid. It also relies on rejection of a mountain of experimental evidence. You are essentially arguing that when a car in which you are riding accelerates that you do not notice that fact.

no

What makes you think that that they are ? Have you solved the general n-body problem that has eluded scientists and mathematicians for the past couple of centuries ? http://en.wikipedia.org/wiki/N-body_problem

Fundamentals of Diophantine Geometry by Serge Lang would be a good place to start.

Unfortunately the notion of a "field" is a mathematical construct that has been shown to make physical predictions that are consistent with experimental results. A field is not "made" of anything. It is not true that a pair of force fields are needed to create a photon (there is no such thing as a non-propagating photon). A photon in quantum theory is a disturbance in the electromagnetic field. Viewed classically a large number of photons comprise a classical electromagnetic field. That classical field can be resolved into two components -- the electric component and the magnetic component -- but the resolution is dependent on the reference frame of the observer. The electromagnetic field itself, however, is a single invariant entity. Gravity is not explained by "distorted space". It is the result of curvature (you can call that distortion) of spacetime which is a lot more than space. Modern physical theories are expressed in terms of mathematics. There is no avoiding that fact. Those theories have been successful in extending into realms in which everyday language simply fails. Thus there are no answers to many questions posed in that everyday language, such as "What are fields made of ?".

You are the who is not listening. I am indeed addressing what you are saying, even if you do not recognize that fact. Any two global inertial reference frames must be in uniform motion with respect to one another. It is impossible for both of the frames in the twin paradox to be inertial. The term inertial reference frame in special relativity refers to a global inertial reference frame. I have explained to you how the twin paradox is addressed in general relativity. The mathematics of general relativity is quite different and more involved than the simple mathemtics of special relativity. You are trying to mix the two in a way that cannot be done.

This is quite clearly a homework problem. You need to show a reasonable attempt before we will help you.

Notice the smiley. The article speaks of a somewhat statistically preferred direction for the net angular momentum of rotating galaxies. I don't think it has anything to do with Godel's rotating universe model. The title of the article seems to have been selected to titillate.

Right. One might suspect that superluminal neutrinos are at the heart of this phenomena.

The right title is in there somewhere.

Just noticed my typo. I trust that your copy is Quanatum Field Theory on Curved Spacetime and Black Hole Thermodynamics.