On Tuesday, March 12, 2019 at 4:05:04 PM UTC-6, [email protected] wrote: > > > > On Thursday, March 7, 2019 at 3:19:39 AM UTC-7, [email protected] wrote: >> >> >> >> On Wednesday, March 6, 2019 at 11:42:33 AM UTC-7, Brent wrote: >>> >>> >>> >>> On 3/6/2019 1:27 AM, [email protected] wrote: >>> >>> >>> >>> On Wednesday, March 6, 2019 at 1:03:16 AM UTC-7, Brent wrote: >>>> >>>> >>>> >>>> On 3/5/2019 10:02 PM, [email protected] wrote: >>>> >>>> >>>> >>>> On Saturday, March 2, 2019 at 2:29:50 AM UTC-7, [email protected] >>>> wrote: >>>>> >>>>> >>>>> >>>>> On Friday, March 1, 2019 at 10:14:02 PM UTC-7, [email protected] >>>>> wrote: >>>>>> >>>>>> >>>>>> >>>>>> On Thursday, February 28, 2019 at 12:09:27 PM UTC-7, Brent wrote: >>>>>>> >>>>>>> >>>>>>> >>>>>>> On 2/28/2019 4:07 AM, [email protected] wrote: >>>>>>> >>>>>>> >>>>>>> >>>>>>> On Wednesday, February 27, 2019 at 8:10:16 PM UTC-7, Brent wrote: >>>>>>>> >>>>>>>> >>>>>>>> >>>>>>>> On 2/27/2019 4:58 PM, [email protected] 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 * >
*I suppose the foregoing is another dumb question. I think the answer has something to do with how tensors transform. Further, I suppose Einstein started with the motivation of finding a general transformation from one accelerating frame to another, and later gave up on this project and settled for a theory of gravity. Is this true? 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 [email protected]. >>> To post to this group, send email to [email protected]. >>> 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 [email protected]. To post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
