DrRocket

Energy dissipated in a resistor is [math]I^2R[/math] The maximum energy transfer to a resistive load by a voltage source occurs when the load resistance is equal to the internal resistance of the source.

"There was a time when the newspapers said that only twelve men understood the theory of relativity. I do not believe that there ever was such a time. There might have been a time when only one man did, because he was the only guy who caught on, before he wrote his paper. But after people read the paper, a lot of people understood the theory of relativity in some way or other, certainly more than twelve. On the other hand, I can safely say that nobody understands quantum mechanics." – Richard P. Feynman in The Character of Physical Law

You seem to think that you are going to re-discover and understand general relativity by introspection. That was only marginally successful for Einstein, who needed a lot of outside expert help. You are not Einstein. Read a book.

Blindingly, utterly wrong. What I said is absolutely true in general relativity, precisely as stated. If you have some other view, that view is most certainly not general relativity. Yep, not even in the ballpark. Talking to yourself.

You are wrong all over the place, Pay attention to what swansont, Sysiphus and Janus are telling you. Correcting your mistakes is beginning to resemble "wack a mole". You really do need to read a book that will give you a complete and consistent treatment of relativity. Rindler's Essential Relativiity, Special, General and Cosmological might be appropriate. His Introduction to Special Relativity is also pretty good. I like Naber's The Geometry of Minkowski Spacetime but you might find it a bit too heavy on mathematical abstraction. Someone else might have better recommendations.

k In physics a "tensor" is usually really a tensor field on some manifold. A covariant tensor is a tensor field, a field of multilinear forms, on the tangent bundle. A contravariant tensor field acts on the cotangent bundle. A tensor of order, or type, (q,p) or (q+p) acts on T x T x ... x T x T* x T* x ... x T* with q copies of T and p copies of T*. People who fool around with this stuff regularly do this without thinking. I have to stop and remember what the notation means. If you haven't become confused by this yet, you will be sometime.

Yes. The point is that you can't do that by adopting a reference frame in which an object is moving. The curvature will be the same as in the rest frame. Motion won't help. Counter-curvature ?

This is not correct. The stress energy tensor, like the curvature tensor to which it is equated by the Einstein field equations is invariant -- independent of the observer. Motion is not an invariant. You cannot increase curvature by adopting a reference frame in motion with respect to your initial frame. If what you are suggesting were true, light would see anything with non-zero rest mass as the source of a black hole. It would be really dark around here.

A nitroglycerine storeage tank .

It is because you are using the term "observed" in terms of what would be recorded by a camera held by A and not in terms of what is actually occuring relative to the reference frame of A. What you "see", optically in special relativity is not what one would measure. This is due to the finite speed of light and relativity of simultaneity. If you consider a circular hoop, circular in the rest frame of the hoop, passing an observer at relativistic speed then length contraction in the direction of motion would present that hoop to the observer as an ellipse. That is correct. But when light transit times are included in the calculation, the observer, and his camera, would "see" a circle. Roger Penrose was the first to recognize this. The "paradox" that you have noted is the result of failing to discriminate between "what you see" and "what you get". WYSIWYG does not hold in relativity.

Summary: Iggy and Michael are talking past one another. Iggy is correct, but one has to think in the abstract setting of Minkowski space to see why. Such thinking is necessary in special relativity. Michael123456 is trying in good faith, but can't seem to grasp the difference between Minkowski space with the Minkowski metric as opposed to Euclidean space with the Euclidean metric. Michael is out somewhere in left field. He should listen to Iggy. Owl is not even in the ballpark. He is talking to himself.

Scenario 1 A mathematician and an engineer are in a cabin by a stream, which they can see through a curtained window. There is an empty bucket on the floor. The curtains are on fire. The engineer takes the bucket to the stream, fills it with water, returns to the cabin and uses the water to put out the fire. The mathematician does likewise. Scenario 2 Identical to scenario 1, except that the bucket on the floor is full of water. The engineer picks up the bucket of water and puts out the fire. The mathematician picks up the bucket and pours the water on the floor, reducing the problem to that of scenario 1, which he has already solved.

If you assume global hyperbolicity the question of existence of CTCs is settled immediately -- they don't.

Look at [math] \int_{a}^{b} \frac{dx}{dt}\frac{d^2x}{dt^2}dt[/math] and integrate by parts. Or let [math]u=\frac {dx}{dt}[/math] [math] \int_{a}^{b} \frac{dx}{dt}\frac{d^2x}{dt^2}dt = \int_{x'(a)}^{x'(b)} u \frac{du}{dt} dt[/math] [math] = \int_{x'(a)}^{x'(b)} u du = \frac{1}{2}x'(b)^2 - \frac{1}{2}x'(a)^2 [/math]

No, the "x" in front takes care of that, but adds an "N+1" term that is subtracted out in the last term. right

Suppose [math] x(t)=t^2[/math] [math] \frac {dx}{dt} = 2t [/math] [math]\frac{d^2x}{dt^2} = 2 [/math] [math] \int_{a}^{b} \frac{dx}{dt}\frac{d^2x}{dt^2}\ dt = \int_a^b 4t \ dt[/math] [math]= 2b^2-2a^2[/math] [math] \ne\frac{1}{2}\frac{dx}{dt} - \frac{1}{2}\frac{dx}{dt} [/math] [math]=0[/math] You have mis-stated something.

[math]\sum_{s=0}^{N} x^s = 1 +x \sum_{s=0}^{N} x^s - x^{N+1}[/math] [math] (1-x) \sum_{s=0}^{N} x^s = 1-x^{N+1}[/math] [math]\sum_{s=0}^{N} x^s = \dfrac {1-x^{N+1}}{1-x}[/math]

The solutions of the Einstein field equations depend on the distribution of matter/energy in the universe. Exact solutions, and Godel's spacetime is an exact solution, are known for only a handful of assumed mass/energy distributions. Godel assumed a homogeneous distribution of swirling dust (particles that interact only through gravity. This solution does not exhibit Hubble expansion, unlike the "real" universe. http://en.wikipedia.org/wiki/G%C3%B6del_metric

Absolutely. A conjecture should never be taken as a fact. I tend to think the conjecture is true, but that simply means that I think an attempt to prove the conjecture has a better chance of success than a project to build a time machine. If I were looking for a counter-example I would start by learning about Kerr black holes.

There is no question that GR, without some sort of additional constraint like an energy condition, admits CTCs. Several solutions are known. A prohibition of physical CTCs, i.e. CTCs in the real universe, in the context of general relativity is the "Chronology Protection Conjecture". That conjecture remains a conjecture, but I think it fair to say that it would be a major surprise if it were shown to be false. We have been surprised before. There are certainly legitimate physicists who have worked on or are currently working on the problem. Hawking and Thorne leap to mind. Mallet doesn't. Mallet's approach has apparently been looked at in detail and errors identified -- see comments in the wiki article noted in an earlier post. You might find this piece by Thorne interesting. http://www.its.caltech.edu/~kip/scripts/ClosedTimelikeCurves-II121.pdf One question that I have, and have no clue to the answer, is whether CTCs are allowed in Einstein-Cartan theory. I know no one who knows much about EC theory.

I don't have access to the whole paper, but I read the abstract a bit differently. 1. GR plus the weak energy condition disallows CTCs 2. Quantum theory might allow some violation of the weak energy condition, but (the expected value of) the stress energy tensor would be very large (indicating large local curvature). 3. Quantum gravity (whatever that is!) will have the final say on CTCs. Note: 1. GR by itself is known to admit spacetimes with CTCs. 2. Mallet is still a nut.

The explanation in terms of special relativity is quite simple" -- Special relativity is explicitly a theory regarding physics as described in inertial reference frames -- In the twin paradox only the reference frame of the non-traveling twin is inertial. Ergo analysis in the reference frame of the non-traveling twin is the correct analysis. Note: Thw traveling twin experiences measurable acceleration at take-off, and at return, and similarly at the distant star, and therefore his reference frame is not inertial. The problem is not symmetric.

In general relativity, space is NOT a 3-D volume and time is NOT duration between two chosen events. In fact, in general relativity, there is no such thing as space and no such thing as time. What there is is a 4-dimensional Lorentzian manifold, spacetime. The usual concepts of "time" and "space" the concepts that you clearly consider as absolute, are not absolute, but in fact are quantities that are applicable to the tangent space at a point, and not to the spacetime manifold itself. The study of special relativity alone gives rise to confusion because a global decomposition, though not a unique such decomposition, into "space" and "time' is possible and is central to the presentation of special relativity in introductory textbooks. The perspective of general relativity is needed to understand the real nature of spacetime. This has nothing to do with any ridiculous philosophical pigeon-holing of thought (e.g. subjective realism), but rather with a theory the validity of which is based on real measurements -- and a great many such measurements. The problem is quite clearly that, despite your protests to the contrary, you do not understand relativity. You don't begin to understand it. That is a condition that is relatively easily remedied. What is not so easily remedied is that you don't understand that you don't understand.

Paul Cohen proved that the Continuum Hypothesis is independet of the Zermelo-Fraenkel axioms plus the Axiom of Choice.

Nicely weasel worded. http://prd.aps.org/abstract/PRD/v46/i2/p603_1