> On 19 Apr 2019, at 04:08, [email protected] wrote: > > > > On Thursday, April 18, 2019 at 6:53:33 PM UTC-6, Brent wrote: > Sorry, I don't remember what, if anything, I intended to text. > > I'm not expert on how Einstein arrived at his famous field equations. I know > that he insisted on them being tensor equations so that they would have the > same form in all coordinate systems. That may sound like a mathematical > technicality, but it is really to ensure that the things in the equation, the > tensors, could have a physical interpretation. He also limited himself to > second order differentials, probably as a matter of simplicity. And he > excluded torsion, but I don't know why. And of course he knew it had to > reproduce Newtonian gravity in the weak/slow limit. > > Brent > > Here's a link which might help; > > https://arxiv.org/pdf/1608.05752.pdf <https://arxiv.org/pdf/1608.05752.pdf>
Yes. That is helpful. The following (long!) video can also help (well, it did help me) https://www.youtube.com/watch?v=foRPKAKZWx8 <https://www.youtube.com/watch?v=foRPKAKZWx8> Bruno > > AG > > On 4/18/2019 7:59 AM, [email protected] <javascript:> wrote: >> >> >> On Wednesday, April 17, 2019 at 7:16:45 PM UTC-6, [email protected] <> >> wrote: >> I see no new text in this message. AG >> >> Brent; if you have time, please reproduce the text you intended. >> >> I recall reading that before Einstein published his GR paper, he used a >> trial and error method to determine the final field equations (as he raced >> for the correct ones in competition with Hilbert, who may have arrived at >> them first). So it's hard to imagine a mathematical methodology which >> produces them. If you have any articles that attempt to explain how the >> field equations are derived, I'd really like to explore this aspect of GR >> and get some "satisfaction". I can see how he arrived at some principles, >> such as geodesic motion, by applying the Least Action Principle, or how he >> might have intuited that matter/energy effects the geometry of spacetime, >> but from these principles it's baffling how he arrived at the field >> equations. >> >> AG >> >> >> On Wednesday, April 17, 2019 at 7:00:55 PM UTC-6, Brent wrote: >> >> >> On 4/17/2019 5:20 PM, [email protected] <> wrote: >>> >>> >>> On Wednesday, April 17, 2019 at 5:11:55 PM UTC-6, Brent wrote: >>> >>> >>> On 4/17/2019 12:36 PM, [email protected] <> wrote: >>>> >>>> >>>> On Wednesday, April 17, 2019 at 1:02:09 PM UTC-6, Brent wrote: >>>> >>>> >>>> On 4/17/2019 7:37 AM, [email protected] <> wrote: >>>>> >>>>> >>>>> On Tuesday, April 16, 2019 at 9:15:40 PM UTC-6, Brent wrote: >>>>> >>>>> >>>>> On 4/16/2019 6:14 PM, [email protected] <> wrote: >>>>>> >>>>>> >>>>>> On Tuesday, April 16, 2019 at 6:39:11 PM UTC-6, [email protected] <> >>>>>> wrote: >>>>>> >>>>>> >>>>>> On Tuesday, April 16, 2019 at 6:10:16 PM UTC-6, Brent wrote: >>>>>> >>>>>> >>>>>> On 4/16/2019 11:41 AM, [email protected] <> wrote: >>>>>>> >>>>>>> >>>>>>> On Monday, April 15, 2019 at 9:26:59 PM UTC-6, Brent wrote: >>>>>>> >>>>>>> >>>>>>> On 4/15/2019 7:14 PM, [email protected] <> wrote: >>>>>>>> >>>>>>>> >>>>>>>> On Friday, April 12, 2019 at 5:48:23 AM UTC-6, [email protected] >>>>>>>> <>wrote: >>>>>>>> >>>>>>>> >>>>>>>> On Thursday, April 11, 2019 at 10:56:08 PM UTC-6, Brent wrote: >>>>>>>> >>>>>>>> >>>>>>>> On 4/11/2019 9:33 PM, [email protected] <> wrote: >>>>>>>>> >>>>>>>>> >>>>>>>>> On Thursday, April 11, 2019 at 7:12:17 PM UTC-6, Brent wrote: >>>>>>>>> >>>>>>>>> >>>>>>>>> On 4/11/2019 4:53 PM, [email protected] <> wrote: >>>>>>>>>> >>>>>>>>>> >>>>>>>>>> On Thursday, April 11, 2019 at 4:37:39 PM UTC-6, Brent wrote: >>>>>>>>>> >>>>>>>>>> >>>>>>>>>> On 4/11/2019 1:58 PM, [email protected] <> wrote: >>>>>>>>>>>> >>>>>>>>>>>> >>>>>>>>>>>> He might have been referring to a transformation to a tangent >>>>>>>>>>>> space where the metric tensor is diagonalized and its derivative >>>>>>>>>>>> at that point in spacetime is zero. Does this make any sense? >>>>>>>>>>> >>>>>>>>>>> Sort of. >>>>>>>>>>> >>>>>>>>>>> >>>>>>>>>>> Yeah, that's what he's doing. He's assuming a given coordinate >>>>>>>>>>> system and some arbitrary point in a non-empty spacetime. So >>>>>>>>>>> spacetime has a non zero curvature and the derivative of the metric >>>>>>>>>>> tensor is generally non-zero at that arbitrary point, however small >>>>>>>>>>> we assume the region around that point. But applying the EEP, we >>>>>>>>>>> can transform to the tangent space at that point to diagonalize >>>>>>>>>>> the metric >>>>>>>>>>> tensor and have its derivative as zero at that point. Does THIS >>>>>>>>>>> make sense? AG >>>>>>>>>> >>>>>>>>>> Yep. That's pretty much the defining characteristic of a Riemannian >>>>>>>>>> space. >>>>>>>>>> >>>>>>>>>> Brent >>>>>>>>>> >>>>>>>>>> But isn't it weird that changing labels on spacetime points by >>>>>>>>>> transforming coordinates has the result of putting the test particle >>>>>>>>>> in local free fall, when it wasn't prior to the transformation? AG >>>>>>>>>> >>>>>>>>> It doesn't put it in free-fall. If the particle has EM forces on it, >>>>>>>>> it will deviate from the geodesic in the tangent space coordinates. >>>>>>>>> The >>>>>>>>> transformation is just adapting the coordinates to the local >>>>>>>>> free-fall which removes gravity as a force...but not other forces. >>>>>>>>> >>>>>>>>> Brent >>>>>>>>> >>>>>>>>> In both cases, with and without non-gravitational forces acting on >>>>>>>>> test particle, I assume the trajectory appears identical to an >>>>>>>>> external observer, before and after coordinate transformation to the >>>>>>>>> tangent plane at some point; all that's changed are the labels of >>>>>>>>> spacetime points. If this is true, it's still hard to see why >>>>>>>>> changing labels can remove the gravitational forces. And what does >>>>>>>>> this buy us? AG >>>>>>>> >>>>>>>> You're looking at it the wrong way around. There never were any >>>>>>>> gravitational forces, just your choice of coordinate system made >>>>>>>> fictitious forces appear; just like when you use a merry-go-round as >>>>>>>> your reference frame you get coriolis forces. >>>>>>>> >>>>>>>> If gravity is a fictitious force produced by the choice of coordinate >>>>>>>> system, in its absence (due to a change in coordinate system) how does >>>>>>>> GR explain motion? Test particles move on geodesics in the absence of >>>>>>>> non-gravitational forces, but why do they move at all? AG >>>>>>>> >>>>>>>> Maybe GR assumes motion but doesn't explain it. AG >>>>>>> >>>>>>> The sciences do not try to explain, they hardly even try to interpret, >>>>>>> they mainly make models. By a model is meant a mathematical construct >>>>>>> which, with the addition of certain verbal interpretations, describes >>>>>>> observed phenomena. The justification of such a mathematical construct >>>>>>> is solely and precisely that it is expected to work. >>>>>>> --—John von Neumann >>>>>>> >>>>>>>> >>>>>>>> Another problem is the inconsistency of the fictitious gravitational >>>>>>>> force, and how the other forces function; EM, Strong, and Weak, which >>>>>>>> apparently can't be removed by changes in coordinates systems. AG >>>>>>>> >>>>>>>> It's said that consistency is the hobgoblin of small minds. I am >>>>>>>> merely pointing out the inconsistency of the gravitational force with >>>>>>>> the other forces. Maybe gravity is just different. AG >>>>>>> >>>>>>> That's one possibility, e.g entropic gravity. >>>>>>> > > > -- > 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] > <mailto:[email protected]>. > To post to this group, send email to [email protected] > <mailto:[email protected]>. > Visit this group at https://groups.google.com/group/everything-list > <https://groups.google.com/group/everything-list>. > For more options, visit https://groups.google.com/d/optout > <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. 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