DrRocket

Good point. I stand corrected.

Let's be clear here. You are the one asking for help. You can receive help by showng your rationale and work. We will not do your work for you. What you are "allowed" to do is your problem. If you want help from me, I require that you provide your work. I don't need the practice of solving freshman problems.

No. It merely shows that the proposed counter example fails.

Why don't you show us your reasoning and your answer, with appropriate units ?

That would be an insistence on accuracy, evidence and scientific rigor no doubt.

The stress-energy tensor, which determines spacetime curvature, includes mass/energy as well as momentum flux and pressure.

For [math]x_1 = x_2 = 1[/math] we have [math]x_k = 1 \ \forall k[/math] which most certainly converges to 1.

There is no global notion of time in general relativity, so there is no "moment of time" that applies to the entire universe. Conservation of energy in general relativity holds at a point, but not over a finite volume.

Nope I have advanced degrees in both EE and mathematics. Neither was overly difficult, largely because both are logical with objective critreria. That is not to sat that either did not require study and diligence.

Because velocities do not add linearly in special relativity. http://math.ucr.edu/home/baez/physics/Relativity/SR/velocity.html

It gets worse. http://einsteinsintuition.com/who-is-thad-roberts/about-thad/

"I was proceeding down the road. The trees on the right were passing me in orderly fashion at 60 miles per hour. Suddenly one of them stepped in my path." -- John von Neumann

Pick up a brick. It gains potential energy. Drop it. The potential energy is converted to kinetic energy. Let it land on your foot, where the kinetic energy is absorbed. Is that real or imagined pain ?

The best available definition, which is none too satisfactory, is that energy is the conserved quantity associated with the time translation symmetry (Google Neother's theorem). Unfortunately this is all but useless in general relativity. J.C. McSwell is right.

Show us your reasoning. You might start with why you have eliminated 1 and 3. Then think about what B Λ C Λ D --> E means and why that would result from the original sentence. It might help to think about this in terms of set theory and draw yourself a Venn diagram.

Don't worry about speed. That will come naturally. Engineers not only must calculate, they must present their calculations to others with clarity. Focus on writing out your problem and solutions so as to be able to present to someone eldse, the problem, the approach to a solution, the logic of that approach and finally the actual solution. By doing that you will force yourself to understand the underlying principles which is the ultimate source of accuracy. Speed will come with expeerience and practice.

Current cosmology is based on general relativity. Hawking and Penrose showed that, given general relativity, the observed expansion of the universe, and a minimal amount of matter consistent with observation that the universe began in a very dense state and that spacetime is singular. The nature of that singularity is that timelike geodesics cannot be extended indefinitely backwards -- that things do not extend indefinitely backwards in time. Thus, within the context of general relativity there is no such thing as "before the big bang ". We have no means to physically travel back in time to observe or explore the big bang. We are forced to rely on what we can observe and what we can infer from well-supported theoretical models. To sensibly discuss "before the big bang" requires theories with which such a question can be modeled. There is ongoing research that may or may not produce such a theory sometime in the future. There is speculation, but only speculation now. Any sensible answer will have to await a real theory. Until then we have only general relativity, and in that context the question of "before the big bang" is meaningless.

Singularities in GR are rather subtle. First, by definition the curvature tensor exists and is finite at all points of spacetime. There is no singular point in the spacetime manifold. A region of spacetime is said to be singular if timelike geodesics canot be extended in thev forward or backward direction. In the case of the big bang singularity it is the backward direction that fails. In te case of black holes it the forward direction that fails. It is not necessarily the case that curvature becomes arbitrarily large in a singular region. You can find singularities discussed in greater detail in The large scale structure of space-time by Hawking and Ellis, Spacetime and Singularities, an Introduction by Naber and The Analysis of Space-Time Singularities by Clarke.

What ajb is alluding to are difficulties with both general relativity and quantum field theories. In general relativity the usual notion of conservation of energy has problems. It can be shown that mass/energy is conserved at a point, but unlike the classical Newtonian case the reasoning does not extend to a volume of finite size. The conservation of energy that one gets in classical and quantum theories from Noether's theorem from time translation symmetry does not work in GR since there is no meaning to time translation. Spacetime is not a vector space. On the other hand quantum field theories on curved spacetime have all sorts of problems. Among them, particles tend to lose individual identity. So you are trying to describe vacuum energy using a theory that is ill-formulated and even more poorly understood. It is a halting and imperfect step toward a theory of quantum gravity, which as ajb notes, is what is really needed. Hawking radiation, BTW, is a result of predictions made using quantum field theory on curved spacetime. It is therefore on somewhat shaky ground (though no one has a better approach at the moment and the result is probably correct), hence the issue over the potential loss of information, or lack of unitarity in the evolution of the state function. The concession of Hawking to Preskill, that information is not lost, is based in large part on an application of the AdS/CFT correspondence from string theory. Unfortunately the Ads/CFT correspondence is itself an unproved conjecture of Maldecena, dating from 1997. Thus there is doubt remaining, and Kip Thorne, also a party to the bet, has not conceded. The message here is that what has been elucidated is not a theorem, but rather an opinion, and there continue to exist differing opinions held by well-qualified people.

One sign of genius is that when you try something and it doesn't work, the next time you try something else. -- paraphrase of a statement by Eugene Wigner

Special relativity was developed before general relativity. It was developed to explain effects that are primarily electrodynamic and specifically ignores effects due to gravity. SR was announced in 1905. Einstein spent the next 10 years trying to extend the special theory of relativity to include gravity. The result was general relativity. General relativity explains gravity in terms of the curvature of spacetime viewed as a Lorentzian manifold. A manifold is an object that "locally looks like" ordinary euclidean space. A simple example is the surface of a sphere, like the Earth, which can be described in small patches as a flat plane. Similarly the curved Lorentzian manifold of general relativity is described in small patches Minkowski space. The geometry of Minkowski space is mathematically just special relativity. You can think of it as general relativity on flat space, which since gravity is a manifestation of curvature is just general relativity without gravity. Alternately you can think of SR as the local version of GR, and just realize that all manifolds are "almost flat" in a sufficiently small patch. So SR provides the local, linearized picture, and GR puts the small patches together and "fits it (gravity) back in" just as you suggest. Where you find a situation where gravity is left out is in freefall -- as with astronauts in orbit experiencing "weightlessness". A local reference frame in freefall is Lorentzian and is a convenient local frame/patch for use in general relativity. In this frame the equations of special relativity hold locally and this is the physical reflection of the statement that "SR is the localization of GR".

In addition to what ajb told you, you might consider that, given that matter is composed of atoms and molecules that are themselves composed of elementary particles, ALL forces are force at a distance at a sufficiently small scale. Fields are the means by which force at a distance is understood. Also the fact that the resolution of the electromagnetic field into magnetic and electric components is dependent on the reference frame of the observer was one of the things that guided Einstein in his development of the theory of special relativity. You can find a good treatment of this aspect of electrodynamics in any good text on the subject. Classical Electrodynamics by J.D. Jackson is a good source.

You will get an oblique projection of a circle, which is a section of a cone at an oblique bangle to the axis, otherwise known as an ellipse. Unfortunately that is not what you want.

The market went down about 1000 pts in the last several days, based largely on hysteria and the unsurprising idiotic behavior of Congress. The value of my stocks dropped a small fortune. I can either sell, cementing the loss, and put the money in cash accounts yielding about 0 and holding dollars of questionable long-term value, or leave it invested in corporations that have strong balance sheets,produce useful products, pay dividends and invest in growing their business. I think I can figure this out.

It is only right if one in addition to the assumption that all elementary events are equally likely one makes the unconventional interpretation of "outcomes" as "possible outcomes" rather than "observed occuences of A". That may be what Khaled means, but it is contrary to normal useage. If one uses the conventional interpretation of "outcomes" then his equation would result in the probability of A being inversely proportional to the relative frequency of occurrence of A, which is backwards.