Obelix

Schutz, D' Inverno, and Rindler above are very good introductory text books. Another one, quite recomendable could be: L. P. Hughston and K. P. Todd: "An Introduction to General Relativity", Cambridge University Press For more a more advanced study I would think (among MANY TITLES) of the following: 1) Hans Stephani: "Relativity", Cambridge University Press 2) Hobson, Efstathiou and Lasenby: "General Relativity", Cambridge University Press 3) Sean M. Caroll: "Spacetime Geometry", Addison Wesley Finally, for a high level study, the following are indicative titles: 1) Robert Wald: "General Relativity", The University of Chicago Press 2) F. De Felice and C. J. S. Clarke: "Relativity on Curved manifolds", Cambridge University Press 3) Jerzy Plebanski and Andrej Krasinski: "An Introduction to General Relativity and Cosmology", Cambridge University Press, and of course: 4) S. W. Hawking and G. F. R. Ellis: "The Large Scale Structure of Spacetime", Cambridge University Press There is a VAST number of books of all levevels, anyway. The above deal, more or less, with General Relativity in the overall. Nevertheless there are also books on specialized topics, e.g. Relativistic Astrophysics. Typical example is: Norbert Straumann: "General Realtivity with an application to Astrophysics", Springer - Verlag Many titles about Black Holes, e.g.: 1) Erric Poisson: "A Relativists's Toolkit", Cambridge University Press (introductory) 2) S. Chandrasekahr: "The Mathematical Theory of Black Holes", Oxford University Press (advanced) Many books on Relativistic Cosmology too, e.g.: Jayant Narlikar: "An Introduction to Cosmology", Cambridge University Press Finally two monumental books that need some special comments: 1) Stephen Weinberg: "Gravitation and Cosmology", Wiley. This book can serve almost as anything: Introductory, intermediate even advanced level (chapters 12 and 13). It is also very usefull as an introduction (or maybe even beyond that level!) to Relativistic Astrophysics and Cosmology. However care must be taken on the author's "heterodox" (in his own confession!) philosophy about spacetime: In contrast with Einstein (and, for that matter, with Relativity mainstream) Weinberg attempts to move the importance of the overall analysis from Spacetime Geometry to a non geometric theory of Quantum Gravity where geometric terms, although present, are in a sense "illusions", and attempts to present Relativity as compatible with that, thanks to the Equivalence Principle. Small wonder: The book was written in 1973, a time of triumph for Quantum Field Theory (Electroweak Unification, 1968, 1969) by one of the leading figures in that triumph - and Nobel prize winner for his work! 2) Missner, Thorne, and Wheeler: "Gravitation", Freeman. This is, in a sense, "the book about everything on Gravitation". Yet I have always found it cumbersome to use, because of its huge size and bulk of contents...Impossible - in my view - to use as a textbook; only as a detailed "Encyclopaedia of Gravitation"! Sean Caroll's lecture notes can be found here: http://xxx.lanl.gov/abs/gr-qc/9712019 See also: http://preposterousuniverse.com/grnotes/, http://search.yahoo.com/search?ei=utf-8&fr=slv8-&p=Relativity%3eSean%20Caroll&type=, and: http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=find+a+s.m.carroll&FORMAT=WWW&SEQUENCE=

It is reasonable to expect that gravitational waves and particles (gravitons) will display a wave - particle duality. Unfortunately, as yet, gravitational waves have not been detected (let alone studied experimentally) and there is no complete theory regarding gravitons. Hence, no definite conclusions can be drawn, so far.

Could you please suggest a link for the above?

What kind of energy? There maybe different kinetic energy, yet there will also be different potential energy. What about the TOTAL energy? Same question as above. How do you measure the equality of mass in all frames? Why does the question have to refer to an entire "Universe" containing only 3 balls? Why not simply an "isolated system" of three balls?

In his famous gedankenexperiment, regarding the lightnings striking thw front and the rear of a rushing train (relativity of simultaneity) Einstein ponders momentarily over the question whether light has the same speed in two different directions. This was clearly the issue of the Michelson - mOrley experiment. Nevertheless, einstein dismisses the question, declaring that "the independence from direction of the speed of light is a POSTULATE we can make in adopting this method (seeing the lightnings simultaneously in a mirror) in order to DEFINE simultaneous events". In other words, when one DEFINES what is to be meant by "simultaneity" or by "time - " or "length - interval", details like uniformity of light speed, or the history of clocks and/or measuring rods may be irrelevant. E.g. the time interval for a certain event is DEFINED as the ticks of that specific standard clock (or clocks, if more than one are employed) which is (are) synchronized with the event under consideration". Such "tacit assumptions" are necessary in all theories, all branches of science. Consider for example a very major tacit assumption made for centuries in Newtonian mechanics: "The gravitational pull between two masses is independent of their history". Indeed one can ask the question: "We measure masses A and B, e.g. by the Cavendish device, and find them to satisfy the inverse square law. HOW DO WE KNOW, nevertheless, that, if we had measured the same masses A and B after they had been through an entirely different history, they would yield the same result?" Or, even worse: "If we measure any other pair of masses, would the result be the same (inverse square law)?" Or even: "If we measure the same pair of masses one more time (however many times we have already done so) how do we know the result would not differ, this one time?"

This has to be checked. Nevertheless it appears that if I held a gyro and rotated I would be a totally different reference frame compaired to the one in Tsolkas' Exp. 12: The latter is OUTSIDE gravitation, and the gyro would experience a fictitious gravitation due to inertia. In my case there would be Earth's gravitation - and the direction of each force (gravitational or inertial) in the above would be different altogether. So, the question stands: What torque would cause the gyro to move at all in Tsolkas' Exp. 12?

There are several attempts to interprete singularities in such a way as to include them in a physical scheme. I suggest yet one more.

Even as far as the analogy is concerned, 3D sun sits on 4D SPACETIME, not 3D space alone. Time is also curved. Appart from the analogy, Sun, Earth, or any mass, is not something independent of spacetime, seperate from it. The analogy with the rubber sheet is misleading. A massive object is not something "sitting on space or spacetime, causing it to curve": According to Einstein's gravitational field equations, it is spacetime itself, or, more specifically, a geometric feature of the latter. It is a curved neighbourhood of spacetime, a specific local geometry thereof. It is not General Relativity that is incomplete; it is the rubber sheet analogy that is oversimplified. I see nothing "incomplete" in the concept of a singularity, whether in a black hole or at the "beginning" of spacetime (nor do I see any "beginning" of spacetime for that matter). Singularities may or may not exist - till either is proved beyond doubt by solid observational facts - yet, as a theoretical concept, they are terribly misunderstood. I think they must not be viewed as something existing statically, with an "eventually achieved" infinite density, or as an area where time "has eventually come to a halt", but rather as some kind of "lower limit" ("infimum") to which spactime, matter, energy, along with their properties, converge dynamically, but never reach conclusively. That is, something like classical infinity of Newtonian Universe: Such an infinity didn't exist anywhere "as such". It was just an abstract global outcome of all endless dynamical processes in the Universe. There is nothing more or less metaphysical in Newtonian infinity compaired to singularities. A good analogy is that with the "absolute zero", as suggested by Malocolm Ludvigsen in his "General Relativity", Cambridge University Press.

Why would that be so? What momentum would cause the gyro to rotate at all?

I suppose you mean "A very good novel", or "A very well written novel". It's not only your knowledge on Physics that's questionable, as I see... Thanks anyway (NOT for the answers to my questions which I have long ceased expecting from you)!

Mr. Tsolkas can't get to his mind this: From the point of view of mathematical accuracy, Equivalence Principle is a MODEL. Like the principle of Inertia: The latter claims that any object, moving free of any force, moves with a constant velocity with respect to an inertial frame of reference. Yet this is an IDEAL situation. Inertial frames of reference do not exist in nature: They can only be APPROXIMATED. Sticking to Tsolkas' reasoning, the principle has to be wrong. Regarding the Principle of Equivalence, Mr. Tsolkas argues precisely this: There are no homogeneous gravitational fields in nature. In particular, his experiments - like the No. 13 one - are all about a two- or three- body problem, i.e. gravitaional fields which are ANYTHING BUT HOMOGENEOUS. Mr. Tsolkas says nothing that wasn't known to Newton, Einstein, and all physicists to the present day. In reality Equivalence Principle is about an idealized ONE BODY PROBLEM, in an ideal UNIFORM GRAVITATIONAL FIELD. A situation which, in real world, can only be a LIMIT CASE: Indeed, as HUNDREDS of experiments show, the more the mass ratio in a two body problem tends to zero and the gravitational field tends to a uniform one (which is adequately approximated, e.g, by the situation on Earth) the better experimental outcomes fit this very principle. That is the latter's confirmation! Regarding Exp. 12: WHY should the gyroscope's axis rotate under the circumstances outlined? What is the moment that would cause it to rotate till it became parallele to the axis of rotation of the frame of reference? Even as a thought experiment, it seems anything but convincing. Counter example: If a gyroscope behaves the way predicted on Exp. 12, why is it then that the spin axis of Uranus lies on the surface of its trajectory, pointing towards the Sun, for a few...billions of years now? Final remark: There are top men of Relativity, like Synge, who argue that Equivalence Principle, is, after all, INESSENTIAL for the validity of the theory. That it has served as "midwife for the theory's birth" but now it has only historical importance. So, even disproving E. P. (which has NOTHING to do with Mr. Tsolkas' "genius") there is no reason at all to reject Relativity Theory.

If you are Tsolkas of the Greek Forum, then you have ignored my questions again and again, sticking to your own "speak speak speak"... I need more data to criticize the experiment: Why would the gyroscope move that way if the chamber rotated? Could you please give a diagram with momentum, moment of inertia, angular momentum etc? Why do you demand "proof" for exp. 12 only? What about the other 11 experiments?

I happen to bew Greek and have encountered repeatedly Dr. Tolkas' "scientific authority". He has been around for years (first publication was in 1986) and has gone beserk during the past few years. He is very active in Greek fora like this (e.g http://www.phorum.gr/viewtopic.php?f=39&t=104203&start=270 - for those who can read Greek). It worths mentioning that, for all I know, by proffession he is a topographer, not a physicist. His main reasoning is approximately as follows: 1) He declares he HAS disproved both the idea that Ether does not exist, and the Equivalence Principle. More precisely, he claims to have proved that the speed of a freely falling body depends on its mass, by using high school level mathematics of Newtonian Mechanics. (Example: Use the principle of Conservation of Energy and Conservation of Momentum to show that!) 2) His conclusion is that, since the Equivalence Principle does not hold WITH A MATHEMATICAL ACCURACY in Nature (e.g. the gravitational field of a planet is, eventually, inhomogeneous, which clearly violates this principle) then it does not hold at all. As a rule (if not always) his "experiments" are characterized by erroneous reasoning and stupenduous mistakes even in his simple calculations. His general attitude is indicative of a person in need of medical assistance: I have remarked MORE THAN FIVE TIMES his mistakes and inaccuracies to him, in that other forum, and instead of replying to me about that, he calls me an "Einstein's Taliban" while repeating again and again that he has "uncovered that big fraud and charlatan (Einstein)". His only argument is: "Read my experiments!" When comments are made on his experiments, he simply ignores them. I strongly suggest that anything related to him is placed in the section of "Alternative Science".

The worldline is likely to be given as a function of proper time: [math]\gamma(\tau)=(ct(\tau),x(\tau),y(\tau),z(\tau))[/math]. Likewise with 4 - velocity: [math]V(\tau)=(c\frac{dt}{d\tau},\frac{dx}{d\tau},\frac{dy}{d\tau},\frac{dz}{d\tau})=(c,\frac{dx}{dt},\frac{dy}{dt},\frac{dz}{dt})\frac{dt}{d\tau}=(c,\vec{\upsilon})\sqrt{1-\frac{\upsilon^2(t)}{c^2}},[/math] we throughout assume the coordinates and their derivatives as functions of proper time, i.e. as evolving along the specific world line - otherwise differentiating with respect to [math]\tau[/math] would be meaningless! The specific form of [math]\vec{\upsilon}(t)[/math] and [math]\upsilon^2(t),[/math] above, result accordingly from this specific functional form of [math]t(\tau),x(\tau), y(\tau), z(\tau)[/math] Working with the Minkowski metric we obtain: [math]d\tau^2=dt^2-\frac{1}{c^2}(dx^2+dy^2+dz^2)\Rightarrow\frac{d\tau}{dt}=\sqrt{1-\frac{\upsilon^2(t)}{c^2}}\Leftrightarrow\tau=\int\sqrt{1-\frac{\upsilon^2(t)}{c^2}}dt[/math] Integration with respct to [math]\tau[/math] is thus performed along this specific world line, since [math]\upsilon(t)[/math] has resulted as above. D' Inverno's book is good, and was recomended as an introductory text by prof. David Robinson of King's College/London, in his General Relativity class, spring semester, 1993. (Can you guess when I was there?) I haven't read the other two. A good introductory text with a lot of examples could also be Wolfgang Rindler's "Relativity: Special General and Cosmological", Oxford University Press, 2nd. edition (2006), as well as his "Introduction to Special Relativity", Oxford University Press, 1991 (not sure if available now, but a good part of it is contained in the former). For a more detailed study one can go on with D. F. Lawden's "An introduction to Tensor Calculus, Relativity and Cosmology", John Wiley (I have the 1986 reprinting, but there has been a quite recent edition) L. P. Hughston's and K. P. Tod's "An introduction to General Relativity", Cambridge University Press (1990 - not sure if available) and eventually, for an acquaintance with higher formalism: Hans Stephani's "Relativity", Cambridge University Press, 3rd edition (2004). Finally, Robert Wald's "General Relativity", University of Chicago Press, 1984 (an excellent book, still widely availbale) can serve as an introduction to the highest mathematical formalism of General Relativity, like rigorous Differential Geometry and Topology. I believe you can order any of that online, with a credit card (350 km from nearest Uni! Where are you man, middle of the desert???) or get information about titles in general. Of course all the above titles are only indicative. The bibliography is VAST (I suppose there must be Australian editions too, unknown in Europe or America, just like there are some Greek books on the subject, this end of the line) and it is much better if you do your own digging, gaining experience! Most authors include their institute, so you can contact them via e - mail in case of unanswered questions and/or doubts. Do not hesitate to do so: Many of them are happy to offer help. Not all though, and least of all the famous ones: It's practically impossible to get a reply, e.g., from Stephen Hawking or Roger Penrose. Quite understandable: They receive thousands of e - mails each day...

The body's proper time is given by Minkowski's metric: [math]d\tau^2 = dt^2 - \frac{1}{c^2}(dx^2+dy^2+dz^2)[/math] The latter yields: [math]d\tau = \sqrt{1 - \frac{1}{c^2}(\frac{dx^2}{dt^2}+\frac{dy^2}{dt^2}+\frac{dz^2}{dt^2})}dt=\sqrt{1 - \frac{\upsilon^2}{c^2}}dt[/math] I understand you already have [math]\upsilon[/math] as a function of [math]t[/math], so all you need to do is integrate the above expression. What book are you using? Who is the author?

In his 1920 paper Einstein wrote that "light cannot propagate without an Ether". He also wrote that "no standards of length and/or time can be given without an Ether", and that "the fact that matter interracts with space means that there is an Ether". However that Ether was not the one assumed by Physicists of 19th. century. The latter was a "quasi - solid" medium such that light propagated in the form of a MECHANICAL VIBRATION along it. It had to be ("quasi - ") SOLID(!) as well, since light displays properties of a TRANSVERSE wave (polarization). All that was in accordance with the spirit of the 19th century Physics. During that time period Mechanics was the only complete (and very successful) branch, whence there was a tendency to interprete everything in terms thereof. Electromagnetic waves were interpreted as a mechanical vibration... The "Ether" Einstein had in his mind in 1920 was something different altogether: It was the FIELD: Gravitational, electromagnetic, etc. It is quite clear that without some kind of "Ether" (electromagnetic field in this case) light cannot possibly propagate. It is clear that, according to General Relativity, without another form of "Ether" (gravitational field and a metric tensor thereof) nothing can be defined as regards length, time, etc. Einstein explained that light, geometry, etc. cannot propagate through, or be properties of an "empty space". In other words, what was thought of as "vaccuum" was nothing of the kind: It was "something" - something DYNAMICAL too, interracting with matter. It also clear that General Relativity identifies field with spacetime, since gravitation is a property of the latter's geometry. Moreover, Einstein had in mind to identify ALL fields (= all forms of "Ether") with spacetime and spacetime geometry via his planned "Unified Field Theory", which, however, was not achieved. In another paper of 1950, Einstein drops the term "Ether" and adopts openly the term "Field". In 1920 "Ether" was still in use, however in 1950 that had changed. More details can be found in in pp. 235 - 246, and pp. 414 - 418 of: "A stubbornly persistent illusion", ed.: Running Press, forwarded by Stephen Hawking.

Matter DOES interact with Spacetime in General Relativity, according to Einstein's Field Equations: [math]R_{\alpha\beta}-\frac{1}{2}g_{\alpha\beta}R=-\kappa T_{\alpha\beta}[/math] Matter/Energy defines the geometry of Spacetime, and is itself defined, in turn, as a geometrical feature of the latter. This is what is meant by "interraction: Not the classical interraction by mutual exertion of "forces".

This, however, must be combined with the fact that the other possibility - Earth and planets dragging the Ether along with them - has been excluded by astronomical observations (by Huygens, if my memory serves me right). By the context of his speech, it appears that Einstein says the following two things: 1) Special Relativity does not disprove the existence of an Ether. That was always known: Ether is just simply NO LONGER NECESSARY in the formulation of Electromagnetism, according to SR. 2) In General Relativity, spacetime interacts with matter. And it is this concept of spacetime that Einstein names "Ether" in his 1920 text.

Dear SFN: I want to delete my profile. I would have already done so if I knew how. There seems to be no button of "Delete profile" or so in the "Edit" options (none I could notice, anyway). This attitude of CENSORSHIP even as regards the FONTS I use seems IRREASONABLE and HUMILIATING. Not to mention that it is reminscent of a weird MILITARY DISCIPLINE. Reminiscent even of some totalitarian states where all people had to dress in the same uniform. I only used that font because I find it more attractive, and that brought forth even speculation regardng my "insecurity" about what I write. Isn't that reminiscent of countries where any off - mainstream behavior is connected with mental or emotional problems? I am a member in many fora and nowhere before have I met with a similar situation. Altogether I cannot stay and write under such circumstances. So please delete my profile for me, or let me know how to do it. Thanks, Obelix

Please let me get something straight: Is this font considered too large for this forum? It's only size 3. It doesn't look larger to me, compaired to the other fonts here. I understand that this font (Size 4) is big - and I've given it up! But THIS size too? As for this size, I think it is this one that's annoying: Too small to read easily. As for me, when something is written I pay attention to the meaning - not the font! There is a saying in my country, about a finger that was pointing at the moon..and someone was taking a good look at the finger!

I'm afraid that yours is a tautology, after all: "If one adds kinetic energy to a body, one will only increase its speed." This is what you seem to say. The question is: 1) Is speed increase the only way to increase a body's kinetic energy? Mass increase would do the same job. 2) Is addition of kinetic energy, in the way of increasing speed, possible to a body moving close to the speed of light? One can add energy, yes, but can one add that energy in the form of kinetic energy, in the above way? You seem to take for granted that the answer is "Yes!" But this is the very thing to be proved! If you take for granted that: "One can add kinetic energy to any body in any condition of motion but NOT in the way of increasing its mass" this means only one thing: "One can increase the body's speed". Altogether, your reasoning seems to go as follows: "If you increase the body's speed, this will only result in increasing its speed"! What does "Mass" mean? If it is Inertial Mass, it means a factor of proportion in the measure of Momentum and Force. There is no other way for it to manifest itself, either in relativity or Newtonian Mechanics. So what you ask above seems to be: "Is this the only way to understand mass?" I think the answer is yes. An increase in inertial properties (e.g. momentum) is the only way for mass itself to increase, as there is no other way for it to have a physical meaning. It would be an interesting experimental test to check if relativistic inertial - mass increase results in an increase of gravitational mass by an equal amount. A confirmation, that is, of the Principle of Equivalence ([math]m_{Inertial} = m_{Gravitational}[/math]): "If adding energy to a body results in an increase of its inertial mass, is the additional mass equal to another additional quantity of gravitational mass?" Anybody knows whether such an experiment has been carried out? I.e., have bodies or particles moving close to the speed of light ever been "weighed" in some way? Is it known whether beams of particles accelerated to relativistic speeds in accelarators (such as CERN) have ever been observed to curve as a result of an equal increase in their gravitational properties?

Thanks!

Could you please be more specific? Are there any papers about that?

I mean one can increase its energy without any limits. I mean mass as in mass of a particle. This is NOT NECESSARILLY its rest mass! Why do you say "it is wrong in the given scenario"? Do I need to remind you the relativistic mass increase? [math]m=\frac{m_{0}}{\sqrt{1-\frac{\upsilon^{2}}{c^{2}}}}[/math] You seem to take into account the non - relativistic laws about kinetic energy, speed, and mass. Special Relativity definetely reveals that a continuous increase in energy does not add to the speed - in other words, the added energy does not come in the form of kinetic energy, necessarily. It does so in the low speed limit of Newtonian Mechanics and everyday experience, but not in realtivistic speeds (i.e. close to that of light).

One can give an infinite amount of energy, but this will add to the MASS of the moving object, not to its SPEED.