On Thursday, March 7, 2019 at 3:19:39 AM UTC-7, agrays...@gmail.com wrote: > > > > On Wednesday, March 6, 2019 at 11:42:33 AM UTC-7, Brent wrote: >> >> >> >> On 3/6/2019 1:27 AM, agrays...@gmail.com wrote: >> >> >> >> On Wednesday, March 6, 2019 at 1:03:16 AM UTC-7, Brent wrote: >>> >>> >>> >>> On 3/5/2019 10:02 PM, agrays...@gmail.com wrote: >>> >>> >>> >>> On Saturday, March 2, 2019 at 2:29:50 AM UTC-7, agrays...@gmail.com >>> wrote: >>>> >>>> >>>> >>>> On Friday, March 1, 2019 at 10:14:02 PM UTC-7, agray...@gmail.com >>>> wrote: >>>>> >>>>> >>>>> >>>>> On Thursday, February 28, 2019 at 12:09:27 PM UTC-7, Brent wrote: >>>>>> >>>>>> >>>>>> >>>>>> On 2/28/2019 4:07 AM, agrays...@gmail.com wrote: >>>>>> >>>>>> >>>>>> >>>>>> On Wednesday, February 27, 2019 at 8:10:16 PM UTC-7, Brent wrote: >>>>>>> >>>>>>> >>>>>>> >>>>>>> On 2/27/2019 4:58 PM, agrays...@gmail.com wrote: >>>>>>> >>>>>>> *Are you assuming uniqueness to tensors; that only tensors can >>>>>>> produce covariance in 4-space? Is that established or a mathematical >>>>>>> speculation? TIA, AG * >>>>>>> >>>>>>> >>>>>>> That's looking at it the wrong way around. Anything that transforms >>>>>>> as an object in space, must be representable by tensors. The informal >>>>>>> definition of a tensor is something that transforms like an object, >>>>>>> i.e. in >>>>>>> three space it's something that has a location and an orientation and >>>>>>> three >>>>>>> extensions. Something that doesn't transform as a tensor under >>>>>>> coordinate >>>>>>> system changes is something that depends on the arbitrary choice of >>>>>>> coordinate system and so cannot be a fundamental physical object. >>>>>>> >>>>>>> Brent >>>>>>> >>>>>> >>>>>> 1) Is it correct to say that tensors in E's field equations can be >>>>>> represented as 4x4 matrices which have different representations >>>>>> depending >>>>>> on the coordinate system being used, but represent the same object? >>>>>> >>>>>> >>>>>> That's right as far as it goes. Tensors can be of any order. The >>>>>> curvature tensor is 4x4x4x4. >>>>>> >>>>>> 2) In SR we use the LT to transform from one* non-accelerating* >>>>>> frame to another. In GR, what is the transformation for going from one >>>>>> *accelerating* frame to another? >>>>>> >>>>>> >>>>>> The Lorentz transform, but only in a local patch. >>>>>> >>>>> >>>>> *That's what I thought you would say. But how does this advance >>>>> Einstein's presumed project of finding how the laws of physics are >>>>> invariant for accelerating frames? How did it morph into a theory of >>>>> gravity? TIA, AG * >>>>> >>>> >>>> *Or suppose, using GR, that two frames are NOT within the same local >>>> patch. If we can't use the LT, how can we transform from one frame to the >>>> other? TIA, AG * >>>> >>>> *Or suppose we have two arbitrary accelerating frames, again NOT within >>>> the same local patch, is it true that Maxwell's Equations are covariant >>>> under some transformation, and what is that transformation? TIA, AG* >>>> >>> >>> >>> *I think I can simplify my issue here, if indeed there is an issue: did >>> Einstein, or anyone, ever prove what I will call the General Principle of >>> Relativity, namely that the laws of physics are invariant for accelerating >>> frames? If the answer is affirmative, is there a transformation equation >>> for Maxwell's Equations which leaves them unchanged for arbitrary >>> accelerating frames? TIA, AG * >>> >>> >>> Your question isn't clear. If you're simply asking about the equations >>> describing physics* as expressed* in an accelerating (e.g. rotating) >>> reference frame, that's pretty trivial. You write the equations in >>> whatever reference frame is convenient (usually an inertial one) and then >>> transform the coordinates to the accelerated frame coordinates. But if >>> you're asking about what equations describe some physical system while it >>> is being accelerated as compared to it not being accelerated, that's more >>> complicated. >>> >> >> *Thanks, but I wasn't referring to either of those cases; rather, the >> case of transforming from one accelerating frame to another accelerating >> frame, and whether the laws of physics are invariant. * >> >> >> For simplicity consider just flat Minkowski space time. If you know the >> motion of a particle in reference frame, whether the reference frame is >> accelerated or not, you can determine its motion in any other reference >> frame. As for the particle path through spacetime, that's just some >> geometric path and you're changing from describing it in one coordinate >> system to describing it in another system...no physics is changing, just >> the description. If the reference frames are accelerated you get extra >> terms in this description, like "centrifugal acceleration" which are just >> artifacts of the frame choice. This is the same as in Newtonian mechanics. >> >> But if the particle is actually accelerated, then there may be more to >> the problem than just it's world line through spacetime. For example, if >> the particle has an electric charge, then it will radiate when accelerated >> and there will be a back reaction. >> >> *Here the "laws" could be ME or Mechanics. It seem as if GR is a special >> case for gravity, but I was asking whether invariance, or covariance, has >> been generally established. * >> >> >> Einstein's equations are written in a covariant form, so they look the >> same for all (smooth) coordinate systems. >> > > *I know this is pretty basic, but with ME's we can apply the LT and find > that the equations are invariant, or covariant (which I think means the > same as invariant). But in GR, there's no general transformation from one > accelerating frame to another accelerating frame to verify that the field > equations are covariant, yet you say they are. Could you elaborate a bit on > this? AG* >
*Does GR require or imply that the laws of gravity are invariant, or covariant, for all accelerating frames? If so, how can that be the case if there's no general transformation from one accelerating frame to any other? Recall, that the independence of coordinate systems doesn't imply that invariance, or covariance, among accelerating frames AFAICT. TIA, AG * > > But the problem arises on the right hand side, the stress-energy tensor. >> If you are considering the motion of a charged particle then the >> stress-energy tensor has to include the EM field of the particle and the >> interaction of the particle with that field. This requires a global >> spacetime solution since, due to the curvature of spacetime, the particle >> can emit EM radiation at one event and then run into that same radiation at >> a later event. The solution may even include singularities; which we know >> are unphysical. So there is no simple transformation between frames like >> the Lorentz transform between inertial frames in flat spacetime. The >> classical theory is probably not even self-consistent when applied >> globally. Here's a paper that addresses a simple form of the problem: >> >> https://arxiv.org/pdf/1509.08757.pdf >> >> *Also, if the LT works locally in GR, how do we transform between >> non-local frames?* >> >> >> There can be no general answer to that. In curved spacetime, you would >> have to solve the problem in a global frame to determine the relation >> between two local frames. The LT can only be relied on within a local >> patch where spacetime is approximately Minkowski. >> >> Brent >> >> * TIA, AG* >> >> >>> Maxwell's equations apply to the description of the EM field of an >>> accelerating charged particle and show that the particle loses energy to an >>> EM wave, but how the particle interacts with it's own field when >>> accelerated produces unrealistic results which were superceded by quantum >>> field theory. Bill Unruh showed that the accelerated system interacts with >>> the vacuum as though the vacuum is hot. >>> >>> Brent >>> >> -- >> You received this message because you are subscribed to the Google Groups >> "Everything List" group. >> To unsubscribe from this group and stop receiving emails from it, send an >> email to everything-li...@googlegroups.com. >> To post to this group, send email to everyth...@googlegroups.com. >> Visit this group at https://groups.google.com/group/everything-list. >> For more options, visit https://groups.google.com/d/optout. >> >> >> -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
