Tried three different browsers and no luck, but other links in other posts work fine. How could it be when quadratic dependence between stress and energy density is near universal, certainly in linear response range of medium. An entity cannot logically be proportional to it's own square!! First declare it ("energy density directly proportional to stress") to be meaningless (I just think it's wrong), then in the very next sentence declare it to be true (Stress is energy). If stress IS energy (I will properly add density) then the two must be directly proportional and differ at most by some proportionality constant, unity virtually by definition. This is just basic. Elastic energy is NOT 'stress energy'!! Any more than it is 'strain energy'. Sometimes it is loosely termed as either (mostly the latter), but that is loose language. Elastic energy requires the combination of both stress and strain acting together, is neither of those contributing inputs but may be expressed in terms of the two either in combination or separately. Together with some material dependent constitutive relationships that couple them. As per your formulas in #65! Good grief. Again - if as you claim stress IS energy then linear addition of non-orthogonal stresses must reflect in linear energy addition, but that's NOT how it works! And the stress-strain product terms there are ALL elastic energy density terms - all can be expressed as proportional to the square of stresses and/or strains, none proportional to simply stress itself. Which is my point exactly. So in effect you are imo conceding there is indeed NO 'stress energy' present, just the usual elastic energy. I rest my case.
I have tested them on a PC with IE, on a Mac with Safari, and even on my iPhone. They work fine in all instances. Whatever you do, you are doing something wrong. I don't believe I stated any such thing. Yes, that is exactly what I am trying to explain, and which is why they are combined into just one tensor in GR. It is what you dispute ! No - they way it works is that you add the stress-energy tensors of the various contributing factors to obtain the total stress-energy tensor of the system. It's simply tensor decomposition. See MTW chapter 5. ...which means that they are all simply energy, since their linear sum is energy. So what's your point ? Did you not tell me that opinions are irrelevant here, only facts count ? The facts, as given in step-by-step rigorous derivations, are given in the links I provided. Your statement that these derivations are wrong is clearly your personal opinion. Or can you back up your claim that stresses, strains and pressures are not energy, and thus have no place in the SEM tensor ( and, by extension, that GR is thus wrong ) with appropriate textbook references ? If your case is that our understanding of the stress-energy tensor is wrong, and ( as an example ) the entire chapter 5 of MTW is a big huge lie ( and hence all interior metrics of GR are unphysical ), then yes, I believe you better rest your case Please Register or Log in to view the hidden image!
Actually, I would be very interested to hear what your definition of the stress-energy tensor actually is...would you like to enlighten us ? The official definition is : T(u,..) = T(..,u) = 4-momentum per unit of 3-dimensional volume, as measured in observer's Lorentz frame where T is chosen T(u,n) = T(n,u) = Component of 4-momentum density along the n-direction, as measured in observer's frame T(u,u) = Mass-energy per unit volume as measured in frame with 4-velocity u. \(T_{jk}\) = j-component of force acting from side \(x^k-\varepsilon\) to side \(x^k+\varepsilon\), across a unit surface area with perpendicular direction \(e_{k}\). This ties up nicely with the fact that the SEM tensor is the conserved Noether current of space-time translation invariance. So now, what is your definition, and how does it fit in with Noether's theorem ? Also, is it correct or not that according to yourself two bodies of identical mass and dimensions, one of which experiences stress/strain/pressure and the other one not, have identical gravitational fields ? Is it correct that no energy is involved in inducing stress and pressure in a body, and that no work can be performed by pressures/stresses/strains ? Because that is pretty much what you are saying if these aren't forms of energy, according to you.
You imply it in stating, repeatedly, that stress IS energy(density). Well 1 joule/cc + 1 joule/cc = 2 joules/cc. Linear addition. You do NOT add 1kPa + 1kPa = 2kPa and get a resultant elastic or pneumatic energy density that adds linearly that way. The energies add quadratically. Ipso facto - stress/pressure thereby cannot be an energy density. Hence my statement you quote above. So make up your mind. Either stress/pressure IS energy density and both, being the same thing with a different name it seems, add linearly, or one - energy density - is QUADRATICALLY related to but most definitely NOT simply stress/pressure. Again - this is basic logic. No - you dispute it as per above and all previous posts - linearity vs quadratic dependency. I dispute functional relevance of stress as bona fide source of gravity yes - especially given the linear relationship assumed. But that is really outside this current discussion. Keep it simple here please - as in my example above re adding energy densities vs adding pressures. Yes legitimately they and their linear sum is properly energy(density) - but that is true because such terms are quadratic, not linear in stress (or equivalently in strain). Repeat, there are NO stress-only terms in that W expression. Strange omission, if stress IS energy(density). Yes, and your creative editing left out "just the usual elastic energy." - which makes all the difference to the meaning of above partial quote. He he. Please Register or Log in to view the hidden image! This is just going in circles. Signing off and good day/night to all.
I think perhaps MD is implying that, a mass moves through space due to the net-force(unbalanced force) F=ma. The rest of the forces(balanced force) being applied on the mass causes only pressure or stress and strain to the mass. These balanced forces does not equate to 'ma'.
So you are saying that GR is wrong. I suspected that all along. Perhaps you would like to suggest an alternative which is still in accordance with experiment and observation ? And along the way you might also explain why GR gives us correct numbers in the first place, if it is so wrong... I am talking tensors, which should be quite clear from my post 65. And as you can see there, these tensors do add linearly. If you follow the links I gave you will find that very relation there. Yes, that is precisely because stress and energy are equivalent, hence you will not find any "stress only" terms. I have no idea what you are doing here ! You are effectively showing yourself wrong...? I should hope you will be back to answer post 84 - that's the interesting bit. I'd really like to see your definition of the SEM tensor, and how you arrive at it !
Nothing moves through space faster than c. There is no closing speed faster than c. The equation for the relativistic addition of velocity is V[sub]1[/sub]+V[sub]2[/sub]/1+(V[sub]1[/sub]V[sub]2[/sub]/ c[sup]2[/sup]). This will always yield a closing speed less than c or equal to c.
"Closing" speed (the speed of two objects covering the distance between them wrt one reference frame) can be \(\le 2c\)
The constant speed of light is a component of SR, and it depends on inertial frames of reference. GR doesn't have inertial frames, simultaneity is only defined in GR for objects whose worldlines intersect. That isn't really an accurate picture. If an object isn't accelerating it has no net force acting on it. That means all the forces cancel, that means forces are additive, hence linear. It does NOT mean that the forces don't have units. In physics, everything has units, even some constant where all the units cancel has units (or how would you cancel them all and get a 'dimensionless' number?)
Howdy.....Hello.....the dark matter of the star is connected to each other. as the dark matter evolved, the star etc. became extant. As the dark matter evolves through mechanical duration the strengths within the layering of the solar system evolves and different 'properties per layering' evolve. The dark matter is compressed compression pressure density as formed from force pressure density as structures within the laws of collision, friction, and motion. Ex. clap your hands. F from right hand attempts to enter the force from left hand attempting to enter the right hand......Among the different actions occurring, layering within, as part of the collision, is but one. Angular momentum collides with angular momentum, mass collides with mass, curvi-linear momentum collides with curvi-linear momentum. Each produces geometries and are each a part of the whole. Each with a given mechanical duration,i.e. strengths and geometries. As one changes each will react in it's own way. When a layer's density varies, other stronger layers will take it's place, but not it's strengths. The dark matter moves inward to the stars core until resistance occurs and a new series of layer evolves.So keeping it very simple Fpp' creates compressed compression pressure density during it's mechanical process of geometrical evolution. Mechanical duration of compression strength and geometry allows for a time line for/of layering.////as per initiation process, i.e. angular, mass, curvi. each with it's own mechanical duration, but since it was formed from the same strengths at the same time (or close enough to) each stays within a given strength-mechanical duration.......Most everything observed astronomically has been, is being formed, held in form-mechanical duration of and by Fpp' evolution. .
Closing speed refers to the rate that two objects approach each other, when viewed from a third frame of reference. An example is two photons coming at you from opposite directions. They close the distance between them at 2c. I did add theoretically, because there is some question whether any complex matter, meaning whole atoms including the associated electrons, could ever, in practice, be accelerated to relativistic velocities and remain intact. Once whole atoms are dismissed, closing speeds nearing 2c are achieved in the LHC, with protons and heavy ions.., each beam with an independent velocity greater than 0.9c.
The apparent rate. But nothing is moving through space at faster than c. Measure the speed of one beam, you get .9c. Measure the speed of the other beam, you get .9c. Measure the speed of one beam from the other and you get .9944c. Again, there's nothing moving through space at greater than c.
Is the speed of light really constant, or only in SR? Is GR a working model if it includes a variable speed of light? Apparently we can't tell, is the first answer, and the second is yes, or it's maybe depending on the cosmological constant.
I have no argument with the formalities of that definition per se, merely arguing here that you have sought to obfuscate contradictory and absurd claims(s) - viz - that stress IS energy(I will keep adding - density), and momentum IS exactly energy, by entangling such notions within tensor expressions. Such 'clarifications' which you surely realize something like 99% of the demographic here would not understand in the least, are anything but. Once again, σ ≠ constant x σ^2, i.e. a stress component entity cannot conceptually be in character a scalar energy density given by it's own squared value - ever. And similarly once again, just on the basis of vector vs scalar alone, mv ≠ 0.5mv^2 - ever. Let's get real. That passage is breathtaking. Many times in various posts have made it crystal clear my position and how you could still be confused by now is staggering. We take the case of static mass distributions. My constant and unvarying position has been that the gravitational mass will be a function of the actual energies (elastic/hydraulic/pneumatic) associated with any stresses involved (that W you gave in #65, a QUADRATIC function of stress components and appearing exclusively in the Too term) - but not, as GR claims, owing also and generally overwhelmingly to, LINEAR in the stresses themselves (Tii terms chiefly). Hence my example of the 'disk brake' that you never tackled head on. Well actually you did make one half-baked statement that amounted to admitting that the thing would function as an overbalancing wheel, but immediately claimed 'there is no problem'. Sure.
Merely suspected? After so long? The acid test is still to come - actual proof/disproof of black holes, and that will be very hard to unequivocally establish. I consider Yilmaz gravity to be conceptually superior, though it may not thus be THE correct classical theory. Maybe you should check out this ppt article: http://www.powershow.com/view/1bbc8-ZjhlZ/P1246341516SeoJH_flash_ppt_presentation Hit the full screen button for legible read (Downloadable version is corrupted). With your superior grasp of all the math, perhaps you can point out all the 'inevitable errors' therein. And btw, watch out for a deceptive Wikipedia entry table that 'disproves' Yilmaz theory: https://en.wikipedia.org/wiki/Alternatives_to_general_relativity#Results_of_testing_theories Note carefully the dates: 1958, 1962 - early versions that unlike 1975 unlisted version, only considered static situation. A bit like presenting the 1912 version of GR as THE GR, and then shooting it down. Like I say, there may be some other better theory again, but just the fact it takes seriously the idea that 'gravity really does gravitate' impresses me for starters. You did ask.
What I presented is pretty much a word-by-word quotation from MTW's "Gravitation", which still is one of the most respected textbooks on the subject matter. If this, in your opinion, counts as an attempt at obfuscation, then I don't know how anyone is supposed to have a discussion with you on this topic. What do you mean by "first order", and "overwhelmingly" ? You do understand that the SEM tensor appears in the field equations not just because someone thought it's a good idea to put it there, but as the result of extremizing the Hilbert action, don't you ? In fact, all of GR can be derived via the principle of least action, quite analogous to classical mechanics. Have you considered that ? I did not 'claim' anything, I merely stated established facts. You on the other hand have yet to formally prove that energy in your example is not conserved. I am sorry that you are unable to follow tensor calculus, but, quite frankly, that is not my problem. It is the language GR is formulated in, after all, and there are good reasons for that. So what ? I also don't understand organic chemistry, along with 99% of the population - that doesn't make (CH2)4O as being the formula for Tetrahydrofuran any less right. If you come into the physics sections of a science forum, be prepared to see the generally accepted maths for the area you are discussing. What's that supposed to proof ? You can also have BHs in Newtonian gravity, and in most other alternatives to GR. This in itself would not establish very much, I think. GR is not the final word on gravitation either; we already know and understand this. As for Yilmaz theory, I have yet to find someone who can offer a satisfactory solution to the problem of disagreements with Newtonian gravity in the interior metrics of fluid bodies. But perhaps you can enlighten us a bit on the matter. The inevitable error is that due to the addition of a pseudo-tensor element to the SEM tensor, the usual Euler HD equations cannot be recovered from this theory. It can also be shown that Yilmaz theory allows for a solution to the field equations where two bodies ( e.g. sun and planet ) are permanently at rest with respect to each other without any gravitational forces acting between them, which is obviously unphysical. Btw, I never knew that there is a Wiki article on this ( http://en.wikipedia.org/wiki/Yilmaz_theory_of_gravitation ). Thanks for pointing it out. I have never heard the event horizon argument before, though; not sure what that is about. In standard GR the field equations are also self-interacting, so in that sense "gravity does gravitate", even in normal GR. That's why you can develop non-trivial topological constructs ( like geons ) even in the absence of any other gravitational sources. There definitely will be, but at this point in time GR is the best we have.
In "closing speed" you do not "measure the speed of one beam from the other". Look up the definition of closing speed.
