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. >>>>>>>> >>>>>>>> >>>>>>>>> >>>>>>>>> >>>>>>>>>> What is gets you is it enforces and explains the equivalence >>>>>>>>>> principle. And of course Einstein's theory also correctly predicted >>>>>>>>>> the >>>>>>>>>> bending of light, gravitational waves, time dilation and the >>>>>>>>>> precession of >>>>>>>>>> the perhelion of Mercury. >>>>>>>>>> >>>>>>>>> >>>>>>>>> I was referring earlier just to the transformation to the tangent >>>>>>>>> space; what specifically does it buy us; why would we want to execute >>>>>>>>> this >>>>>>>>> particular transformation? AG >>>>>>>>> >>>>>>>> >>>>>>>> For one thing, you know the acceleration due to non-gravitational >>>>>>>> forces in this frame. >>>>>>>> >>>>>>> >>>>>>> *IIUC, the tangent space is a vector space which has elements with >>>>>>> constant t. So its elements are linear combinations of t, x, y, and z. >>>>>>> How >>>>>>> do you get accelerations from such sums (even if t is not constant)? AG* >>>>>>> >>>>>>> So you can transform to it, put in the accelerations, and transform >>>>>>>> back. >>>>>>>> >>>>>>> >>>>>>> *I see no way to put the accelerations into the tangent space at any >>>>>>> point in spacetime. AG* >>>>>>> >>>>>>> >>>>>>> The tangent space is just a patch of Minkowski space. d/t(dx/dt) = >>>>>>> acceleration. >>>>>>> >>>>>>> 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.
