> On 20 Apr 2019, at 23:14, [email protected] wrote: > > > > On Friday, April 19, 2019 at 2:53:00 AM UTC-6, Bruno Marchal wrote: > >> On 19 Apr 2019, at 04:08, [email protected] <javascript:> 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 > > I've been viewing this video. I don't see how he established that the metric > tensor is a correction for curved spacetime. AG
ds^2 = dx^2 + dy^2 is Pythagorus theorem, in the plane. The “g_mu,nu” are the coefficients needed to ensure un non-planner (curved) metric, and they can be use to define the curvature. Bruno > > > >> >> AG >> >> On 4/18/2019 7:59 AM, [email protected] <> 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. >>>>>>>> >> >> <br class="webkit > > > -- > 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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