DrRocket

No In fact no simple expression that produces only prime numbers is known, let alone one that would list all of them in increasing order.

The integral of the derivative of a periodic function over a period will always be zero. But as Timo observes, you have imposed additional conditions that make the function non-differentiable, in contrtadiction to your problem statement. You have something messed up in the problem statement.

What is "expanding" are spacelike hypersurfaces corresponding to a one-parameter foliation of spacetime by a timelike parameter. This requires homogeneity and isotropy. See the thread on cosmo basics. Calabi-yau manifolds are irrelevant to this. Inflation is one explanation for the horizon and flatness problem, but the post to which I responded said nothing about those problems. Greene's books are very good, very entertaining, and very speculative. He tends to put the caveats in inconspicuous places, but at least he puts them in. You should read all books on string theory with a grain of salt. If it is a Kaku or Susskind book, a train car load is needed.

Sure it is. A worldline is just a curve in spacetime. The velocity vector at a point on a parameterized curve is a perfectly well-defined object. If you parameterize a world line by arc length divided by c then the tangent vector is just velocity with respect to proper time. The magnitude is always c.

4-velocity is particularly simple, since the speed of anything, not just light, is c.

This appears to be homework. What do you know about cardinality ? This is a relatively simple exercise in the theory of cardinal numbers.

Right. And this does not require inflation. But inflation does help to explain why regions that might not otherwise be expected to have been causally connected in the distant past actually were in communication -- the "horizon problem". Right. "objects" are not events. Spacetime points are events. Spacelike and timelike separation applies to events. Events don't move so neither do the intervals that connect them. Note the correction added in edit to that earlier post of mine. I hope I did not confuse you too much.

"The spacetime manifold is fixed" means just that. It embodies ALL of "time" and ALL of "space". Nothing changes it. That is one reason for the giant incompatibility between general relativity and quantum theory. No single light cone can contain any two spacelike separated points. This has nothing to do with inflation. So, there are lots of such points -- the tip of your nose and the tip of your right index finger at any single moment in time for instance. Edit: This is not correct. What is true is that neither are in the forward light cone of the other. If you go far enough back in time you may and probably will find a light cone that contains both points. In special relativity you will be able to do this. When you throw in the big bang then a lot depends on the nature of the associated singularity. The big bang can be considered to be any Cauchy surface (if you don't understand this Google "Cauchy surface") near t=0. But it is not a single event. VERY imprecisely you can think of it as a point with the entire universe in its light cone, but this is so imprecise that you should not try to draw conclusions from this picture. The big bang itself is not well understood. We have theories that do a pretty good job from say t+10^-33 sec (this perhaps optimistic), but t=0 is essentially a complete mystery. Despite the impression that you may get from popularizations, expressions such as "the moment of the big bang" are not well-defined. Moreover the singularity associated with the big bang is not a point or even a set of points in spacetime itself. By definition, there are no singular points in the spacetme manifold. Singularities in general relativity are very subtle and technical things.

This amounts to speculation about speculation.

yes No Events are spacetime points vand are neither timelike nor spacelike. Vectors are timelike or bspacelike and smooth curves are timelike or spacelike if their tangent vectors are everywhere timelike orb spacelike. Moreover, in general relativity, the spacetime manifold is fixed an neither inflation nor anything changes it. This makes no sense. "Locations" would usually be interpreted as spacelike related points (not joinable by any timelike curve) and hence even in the flat case no two "locations would be in any single light cone. neither This site is interesting, but in trying to express everything without using mathematics, also introduces misconceptions and distortions.

This makes no sense. pi is a dimensionless pure number. The speed of light has dimension distance/time and the numerical value varies with the arbitrary choice of units for distance and time.

http://onnes.ph.man.ac.uk/nano/Publications/Naturemat_2007Review.pdf

ajb and Schrodinger's hat each provided explanations accessible to beginning physics students, and in the case of ajb an explanation devoid of calculus that could be understood by most high school students. The path to education involves some work on one's part to elevate one's understanding to the level required by clear concise explanations, such as those provided. Expecting answers to be dumbed down to one's current level of ignorance only prolongs that undesirable state.

You now have the unenviable task of formulating a precise definition of "matter'. Good luck.

1/2

Your example fails to exhibit the fact that there are in fact many infinities. But if you would care to research the topic of cardinal numbers you will find that there are infinitely many infinite cardinals, which measure the size of infinite sets. The cardinality of the real numbers, for instance, is strictly larger than the cardinality of the natural numbers, though both are infinite sets. However 2 x any infinite cardinal is just that same infinite cardinal again.

Nothing is ill-defined at the event horizon. In fact if you were crossing the event horizon you would note nothing remarkable. All sorts of odd things may appear to happen in some coordinate system, including the event horizon appearing to be singular (as in Schwarzchild coordinates) but that is illusory, an artifact of an arbitrary choice of coordinates. The singular nature in the interior, as proved by Penrose, is independent of any coordinate system. Note: The whole point of "general covariance" in general relativity is the formulation of physics without reference to any coordinate system. That is how Riemannian geometry works.

That is to say that [math] \lim_{n \to \infty} P_n = \infty [/math] so that to think that the sum might actually converge is absurd. You are considering an "infinite sum" of positive integers, with the individual terms growing rapidly -- which is obvious from the fact that there are infinitely many primes and more so from the prime number theorem.

rapidly

Your solution is correct, but your logic is invalid in solution 1. In solution 1 you merely showed that no x>0 can satisfy the inequality. You then need to consider separately the case x<0. In solution 2 both the logic and solution are correct. You reduced the problem to showing that a certain fraction which has a negative numerator must be positive, and hence that the denominator, x, must be negative. Since all the operations performed were invertible, the solution set of your final inequality coincides with the solution set of the initial inequality.

As far as I can tell this subject is just Tom Bearden bull. If you want bto pay for the paper with his nonsense, here is the site where you can do it. http://iopscience.iop.org/1402-4896/61/5/001

It depends on what it is that you might be doing.

I have managed several science and engineering organizations. In my experience, on average, the female engineers were better than the male engineers. I suspect that this is because social factors mitigate against mediocre females entering the field.

One possible alternative is Einstein-Cartan theory, in which one does not make the a priori assumption that spacetime is torsion-free. This results in a more complicated mathematical theory, but one in which the singularites may either disappear or take on a different character from GR. Unfortunately that is about the limit of my knowledge. In any case I think we both agree that focusing on the singularities in GR is not productive from a physical perspective. What makes a black hole is the event horizon, not the singularity.

Power flow from an electromagnetic field is described by the pointing vector whicfh is just [math] \frac{1}{\mu_0}E \times B [/math] The only references that I could find to the "Heaviside curled EM energy flow component" are in web sites of well-known wackos like Tom Bearden. There is a good reason why such a thing would be omitted from mainstream electrodynamics books.