On Tuesday, April 16, 2019 at 9:15:40 PM UTC-6, Brent wrote: > > > > On 4/16/2019 6:14 PM, [email protected] <javascript:> 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 >>> >> >> *Sorry; I was thinking about QM, where the state of the system is a >> linear combination of component states of the vector space representing it. >> In GR, since there is an infinite uncountable set of tangent spaces, how >> can we be sure that our test particle is in one of those subspaces, called >> tangent states? That would be the case, I surmise, if the tangent spaces >> spanned the manifold. I think they do so since there's a tangent space at >> every point in the manifold. AG * >> > > *The presumed test particle has a history, and each tangent space is a > proper subset of the manifold. So is there a guarantee that an arbitrary > test particle will have a history contained in a particular tangent space? > AG* > > > No. It's guaranteed that at every point on the particles world line there > is a tangent space. > > Brent >
*Agreed. For example, if we consider a sphere as a manifold, its tangent spaces are NOT proper subspaces of the underlying manifold, so a test particle's history (assuming we go to 4 dimensions) will generally lie outside the manifold except for history points contained in some tangent space. AG * -- 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.
