So if one wants to define the tangent plane on a manifold, and that manifold is spacetime, one includes vectors corresponding to v > c even though they're physically disallowed? This seems wrong since the tangent space would be defined by violating a key postulate of relativity. What about the "vectors" on the light cone you referred to earlier? What exactly are they? They seem different from those used to define the tangent plane. TY, AG
On Sunday, August 25, 2024 at 12:25:03 AM UTC-6 Brent Meeker wrote: > > > > On 8/24/2024 8:54 PM, Alan Grayson wrote: > > The vectors you refer to; are they the velocity vectors on a tangent space > of a manifold? > > I'm confused about the definition of a tangent space on a manifold, say at > point P. If the manifold is *spacetime *and we must exclude all test > particles moving at velocities > c. > > You don't exclude vectors. You just realize that some vectors don't > represent velocities. > > Brent > > Does the set of tangent vectors at P form a vector space? I don't see how > it can, since under ordinary addition parallel vectors can sum to > velocities > c, *negating* the closure property for a vector space. If > they *don't* form a vector space, this compromises the definition of the > metric tensor which is defined on a *vector space *on the tangent space > at each point P. TY, AG > > On Saturday, August 24, 2024 at 2:37:42 PM UTC-6 Brent Meeker wrote: > >> The difference of two time-like vectors can be space-like. That's why >> the CMB at opposite points on the celestial globe have not interacted. >> >> Brent >> >> >> >> On 8/24/2024 10:36 AM, Alan Grayson wrote: >> >> TY, but if you use time-like velocity vectors on spacetime as the >> manifold to construct a tangent plane, that set *won't* form a vector >> space (on the tangent plane) on which the metric tensor is allegedly >> defined, because a vector space is defined using ordinary addition of >> vectors, and the only way to preserve time-likeness is to use the Lorentz >> transformation, but a frame transformation is not the same as ordinary >> addition. AG >> >> On Saturday, August 24, 2024 at 8:09:24 AM UTC-6 smitra wrote: >> >>> You only need 4 linearly independent vectors; orthogonality is not >>> needed. Once you have this linear space you can construct an orthogonal >>> basis, but you can't directly probe spacelike vectors. >>> >>> Saibal >>> >>> On 24-08-2024 04:32, Alan Grayson wrote: >>> > If calculating the metric tensor is ill-defined, it seem impossible to >>> > use Einstein's Field Equations to get any useful prediction, but yet >>> > that's the claim. Baffling, to say the least. AG >>> > >>> > On Thursday, August 22, 2024 at 2:32:51 AM UTC-6 Alan Grayson wrote: >>> > >>> >> A related question is this; if the manifold is spacetime, and at >>> >> some point P we consider all test particles moving at velocities >>> >> LESS THAN C, containing point P, is the set of these tangent vectors >>> >> sufficient to define the tangent plane at P? IOW, can we OMIT all >>> >> test particles moving at velocities GREATER THAN C moving through >>> >> point P, and still have a well defined tangent plane at point P? If >>> >> so, the problem I am struggling with is whether omitting such test >>> >> particles on the GR manifold, as we must, will the result be a >>> >> tangent plane whose tangent vectors at P do NOT form a vector space, >>> >> since it isn't closed, and therefore have a problem with defining >>> >> the metric tensor at P which is defined on a vector space. TIA, AG >>> >> >>> >> On Tuesday, August 20, 2024 at 2:26:16 AM UTC-6 Alan Grayson >>> >> wrote: >>> >> >>> >>> One way to do it, say at point P on the spacetime manifold, is to >>> >>> imagine test particles of all velocities LESS THAN C, passing >>> >>> through point P. Then, presumably, these vectors define the >>> >>> tangent space at P. But for these vectors to form a vector space, >>> >>> ON WHICH THE METRIC TENSOR IS DEFINED, we have to include vectors >>> >>> with velocities GREATER THAN C. So, the original set of velocity >>> >>> vectors are NOT closed under usual addition. We must expand the >>> >>> original set of velocity vectors moving at velocities less than c, >>> >>> to presumably close that set in order to construct the vector >>> >>> space, without which the metric tensor is undefined. >>> >>> >>> >>> What bothers me about this procedure is that applying it to the >>> >>> closed interval, say [0.1], and asking whether it forms a vector >>> >>> space under addition (with the real numbers forming the scalar >>> >>> field), it's claimed this interval is NOT a vector space since it >>> >>> isn't CLOSED under addition. Well, if we can add elements to the >>> >>> tangent vectors with velocities less than c, with those having >>> >>> velocities greater than c, doesn't the original set of vectors >>> >>> ALSO fail the closure test, and therefore fails to make the total >>> >>> set a VECTOR SPACE, necessary for defining the metric tensor? >>> >>> >>> >>> TY, 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 everything-li...@googlegroups.com. >>> > To view this discussion on the web visit >>> > >>> https://groups.google.com/d/msgid/everything-list/6a7411bd-94cc-4687-9cee-269d495497dfn%40googlegroups.com >>> >>> > [1]. >>> > >>> > >>> > Links: >>> > ------ >>> > [1] >>> > >>> https://groups.google.com/d/msgid/everything-list/6a7411bd-94cc-4687-9cee-269d495497dfn%40googlegroups.com?utm_medium=email&utm_source=footer >>> >>> >> -- >> 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 everything-li...@googlegroups.com. >> >> To view this discussion on the web visit >> https://groups.google.com/d/msgid/everything-list/d6ff681e-ca0f-468a-b2fb-55d759ba7e7an%40googlegroups.com >> >> <https://groups.google.com/d/msgid/everything-list/d6ff681e-ca0f-468a-b2fb-55d759ba7e7an%40googlegroups.com?utm_medium=email&utm_source=footer> >> . >> >> >> -- > 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 everything-li...@googlegroups.com. > > To view this discussion on the web visit > https://groups.google.com/d/msgid/everything-list/af749fac-b531-466c-ad16-76a64a7e4a71n%40googlegroups.com > > <https://groups.google.com/d/msgid/everything-list/af749fac-b531-466c-ad16-76a64a7e4a71n%40googlegroups.com?utm_medium=email&utm_source=footer> > . > > > -- You received this message because you are subscribed to the Google Groups "Everything List" group. 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