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| From | Thomas Heger <ttt_heg@web.de> |
| Newsgroups | sci.physics.relativity |
| Subject | Re: Hidden dimensions could explain where mass comes from |
| Date | Tue, 6 Jan 2026 09:22:32 +0100 |
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Am Montag000005, 05.01.2026 um 16:59 schrieb Ross Finlayson: > On 01/05/2026 01:47 AM, Thomas 'PointedEars' Lahn wrote: >> Chris M. Thomasson wrote: >>> Every dimension has time, >> >> Scientifically that statement does not make sense. You appear to be >> referring to a definition of "dimension" that is used in science-fiction >> and fantasy instead. >> >> In mathematics, a dimension is basically an additional degree of >> freedom for >> choosing a coordinate in a space. In a different meaning, /the/ >> dimension >> of a vector space is the magnitude of its basis, the minimum number of >> basis >> vectors to represent an element (vector) of that space; since basis >> vectors >> have to be linearly independent, when they are written in components as >> column vectors, this is equal to the number of components per vector. >> For >> example, for 3-dimensional Euclidean space R^3 (by "R" I mean the set of >> real numbers; see below) one defines vectors of the form >> >> (x) >> (x, y, z)^T = (y), >> (z) >> >> where x, y, and z are coordinates, and the standard basis vectors >> >> (1) (0) (0) >> e_x := e_1 := (0), e_y := e_2 := (1), e_z := e_3 := (0). >> (0) (0) (1) >> >> These are linearly independent (the proof is an undergraduate mathematics >> exercise), and suffice to represent, by a linear combination of them, >> every >> vector in R^3; thus this set defines /a/ basis of R^3 (a vector space has >> potentially infinitely many different bases related by linear >> transformations; thus for every vector there is potentially an infinite >> number of representations, depending on the choice of basis -- notably, >> basis vector can, but do not have to be, unit vectors). >> >> [In physics, the term "dimension" also has another meaning with regard >> to physical quantities: apparently every physical quantity can be >> expressed as a product of integer powers of quantities of the types, >> called *dimensions*, length, time, and mass. For example, when we >> say that a quantity has (the) dimensions of a force, we mean that it >> can be written in terms of other quantities: >> >> [[force]] = [[mass]] * [[acceleration]] >> = [[mass]] * [[length]]/[[time]]^2. >> ] >> >>> therefore time is _not_ a "special dimension"? >> >> It *is*, and it is special at least in that its sign in a spacetime >> metric >> is the opposite of that of spatial dimensions. For example, the >> Minkowski >> metric can be written with in Euclidean spatial coordinates >> >> ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2. >> >> The peculiar (here: negative) sign for the temporal component of the >> metric >> can be understood (and in fact the Minkowski metric can be nicely >> derived) >> by considering two different ways to measure the straight-line spatial >> distance that a signal travels at a constant speed c: >> >> c^2 (∆t)^2 = (∆x)^2 + (∆y)^2 + (∆z)^2, >> >> where the left-hand side (LHS) is the square of the distance given by the >> time ∆t that it takes the signal to travel the distance, and the >> right-hand >> side (RHS) is the square of the Euclidean distance as given by the >> coordinates between the start point and the end point (the 3-dimensional >> version of the Pythagorean theorem). Then subtracting the LHS gives >> >> 0 = -c^2 (∆t)^2 + (∆x)^2 + (∆y)^2 + (∆z)^2, >> >> providing a *metric* for the separation of events: If the value RHS is >> equal >> to 0, then two events can be connected by a (light) signal, and the >> spacetime interval between them, their separation, is called >> *lightlike*; if >> it is negative, the two events can be connected by constant motion at a >> speed less than c, a *timelike* interval; and if it is positive, the >> motion >> would have to be faster than c which we assume is impossible, so the >> events >> cannot be causally connected, and the interval is called *spacelike*. >> >> For infinitesimally-separated events, one writes differentials instead of >> differences and drops the parentheses; so for lightlike-separated events, >> those on a lightlike worldline that is described by light in vacuum, >> >> ds^2 = 0 = -c^2 dt^2 + dx^2 + dy^2 + dz^2, >> >> and in general the infinitesimal spacetime interval in a flat >> (1+3)-dimensional spacetime called *Minkowski space* is given by the >> *line >> element* >> >> ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2. >> >> Finally, you can see that instead of subtracting the LHS we could also >> have >> subtracted the RHS, leading to >> >> 0 = c^2 (∆t)^2 - (∆x)^2 - (∆y)^2 - (∆z)^2, >> >> and therefore to >> >> ds^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2. >> >> Now for lightlike intervals we would still have ds^2 = 0, but timelike >> intervals would have ds^2 > 0, and spacelike intervals would have ds^2 >> < 0. >> >> So there is a *sign convention* that can (and has to) be chosen for the >> metric, but the temporal component must always have the opposite sign >> of the >> spatial ones (-+++, called "mostly plus"; or +---, called "mostly minus") >> for the physics to make sense. >> >> [Unless one gets clever and defines the *Euclidean time* >> x^4 := i x^0 = i c t. Then (dx^4)^2 = i^2 (dx^0)^2 = -c^2 dt^2, and >> the metric becomes Euclidean (now it looks like a 4-dimensional >> Pythagorean theorem; previously it, and the manifold it describes, >> was called *pseudo-Euclidean*): >> >> ds^2 = (dx^4)^2 + (dx^1)^2 + (dx^2)^2 + (dx^3)^2. >> >> This coordinate transformation is called Wick rotation¹ and becomes >> useful in quantum field theory. Stephen Hawking uses "imaginary >> time" >> in explanations in some of his popular-scientific books, even when >> only >> discussing general relativity, and I think he means Euclidean time >> (but >> IIRC he never explains it in terms of a Wick rotation).] >> >> Analogously to 3-dimensional Euclidean space, one defines *4-vectors* >> >> (c t, x, y, z)^T >> >> or in general >> >> (x^0, x^1, x^2, x^3)^T where x^0 = c t, >> >> or >> >> (x^4, x^1, x^2, x^3)^T where x^4 = i c t. >> >> For example, to describe spherically-symmetric situations, it is more >> convenient to use spherical coordinates: (c t, r, θ, φ)^T. Such is the >> case, for example, with the Schwarzschild and the FLRW metric. [For >> simplicity of notation and calculation, usually c is set equal to 1; we >> need to restore it when we want to compare theory and measurements.] >> >> So you can see that time really is a (colloquially: "the fourth") >> dimension >> of this mathematical space. >> >> See also: >> >> <https://www.britannica.com/topic/Albert-Einstein-on-Space-Time-1987141> >> >> Time is also special in that apparently, by contrast to the spatial >> dimensions, we do not have the freedom to move arbitrarily in time, >> but only in the positive direction, from the past to the future; and >> there are processes that are *irreversible*: there is an *arrow of time*. >> >> ____ >> ¹ after Gian Carlo Wick (1909–1992), Italian theoretical physicist who >> made >> important contributions to quantum field theory >> > > When there's mentioned Wick rotation it may be kept in mind that > it's after an account of a sort of "screw" arithmetic after what > is the Eulerian-Gaussian or as about the Eulerian identity after > de Moivre and Euler's account of the telescoping in infinite series, > and the Gaussian the complex analysis and as with regards to accounts > of the hypergeometric, of which Gauss gives an example, but about > that the regular singular points of the hypergeometric are 0, 1, > and infinity. When mentioning coordinates, then one might further > distinguish rectilinear and polar coordinates, since they have quite > different treatments while the one has unique and the other non-unique > representations in the space, a plain mathematical space vis-a-vis > a usual notion of a (linear) vector space. > > > A "Wick rotation" may be better called a "Wick screw rotation", > since for example that other formalisms that so establish the > "screw" arithmetic would fill the same role in a derivation. > > The "dimensions" as relating R^2 to C in the complex diagram, > or, the Eulerian-Gaussian vis-a-vis the Cartanian for Elie Cartan > and reflections and rotations then later the geometric algebras > of what's often called the "hypercomplex" numbers, has that the > Cartanian has ready representations in the complex number diagram, > yet reflections and rotations are also simply on their own sake > in the affine about the convolutive and symmetries and not so > necessarily, though readily, in the usual ideas of symmetry groups. I interpreted 'Wick rotation' as multiplication with the imaginary unit i. So: the axis of time is the result, if space itself is 'Wick rotated'. ... See https://docs.google.com/presentation/d/1Ur3_giuk2l439fxUa8QHX4wTDxBEaM6lOlgVUa0cFU4/edit?usp=sharing TH
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Hidden dimensions could explain where mass comes from Anthk NM <anthk@disroot.org> - 2026-01-04 14:47 +0000
Re: Hidden dimensions could explain where mass comes from ram@zedat.fu-berlin.de (Stefan Ram) - 2026-01-04 15:41 +0000
Re: Hidden dimensions could explain where mass comes from Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-04 17:06 +0100
Re: Hidden dimensions could explain where mass comes from "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2026-01-04 15:08 -0800
Re: Hidden dimensions could explain where mass comes from Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-05 10:47 +0100
Re: Hidden dimensions could explain where mass comes from Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-01-05 07:59 -0800
Re: Hidden dimensions could explain where mass comes from Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-01-05 08:22 -0800
Re: Hidden dimensions could explain where mass comes from Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-05 22:43 +0100
Re: Hidden dimensions could explain where mass comes from Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-01-06 00:00 -0800
Re: Hidden dimensions could explain where mass comes from Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-01-06 09:28 -0800
Re: Hidden dimensions could explain where mass comes from Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-01-06 10:04 -0800
Re: Hidden dimensions could explain where mass comes from nospam@de-ster.demon.nl (J. J. Lodder) - 2026-01-06 10:43 +0100
Re: Hidden dimensions could explain where mass comes from Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-01-06 06:51 -0800
Re: Hidden dimensions could explain where mass comes from Thomas Heger <ttt_heg@web.de> - 2026-01-06 09:22 +0100
Re: Hidden dimensions could explain where mass comes from "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2026-01-05 14:10 -0800
Re: Hidden dimensions could explain where mass comes from Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-05 23:36 +0100
Re: Hidden dimensions could explain where mass comes from Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-05 23:45 +0100
Re: Hidden dimensions could explain where mass comes from Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-05 23:49 +0100
Re: Hidden dimensions could explain where mass comes from "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2026-01-05 15:09 -0800
Re: Hidden dimensions could explain where mass comes from "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2026-01-05 15:14 -0800
Re: Hidden dimensions could explain where mass comes from Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-06 00:47 +0100
Re: Hidden dimensions could explain where mass comes from "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2026-01-05 19:40 -0800
Re: Hidden dimensions could explain where mass comes from Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-06 17:29 +0100
Re: Hidden dimensions could explain where mass comes from Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-01-06 08:48 -0800
Re: Hidden dimensions could explain where mass comes from "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2026-01-06 17:29 -0800
Re: Hidden dimensions could explain where mass comes from "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2026-01-06 19:20 -0800
Re: Hidden dimensions could explain where mass comes from "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2026-01-08 23:20 -0800
Re: Hidden dimensions could explain where mass comes from "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2026-01-09 00:03 -0800
Re: Hidden dimensions could explain where mass comes from Thomas Heger <ttt_heg@web.de> - 2026-01-07 08:47 +0100
Re: Hidden dimensions could explain where mass comes from "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2026-01-07 11:46 -0800
Re: Hidden dimensions could explain where mass comes from Thomas Heger <ttt_heg@web.de> - 2026-01-08 09:03 +0100
Re: Hidden dimensions could explain where mass comes from Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-01-08 08:16 -0800
Re: Hidden dimensions could explain where mass comes from Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-09 02:55 +0100
Re: Hidden dimensions could explain where mass comes from Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-01-08 19:55 -0800
Re: Hidden dimensions could explain where mass comes from Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-01-08 22:34 -0800
Re: Hidden dimensions could explain where mass comes from Thomas Heger <ttt_heg@web.de> - 2026-01-10 09:24 +0100
Lagrangian mechanics (was: Hidden dimensions could explain where mass comes from) Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-09 12:13 +0100
Re: Lagrangian mechanics Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-01-09 08:37 -0800
Re: Lagrangian mechanics Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-01-09 09:03 -0800
Re: Lagrangian mechanics Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-01-12 07:58 -0800
Re: Lagrangian mechanics nospam@de-ster.demon.nl (J. J. Lodder) - 2026-01-09 21:17 +0100
Re: Lagrangian mechanics Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-01-09 13:23 -0800
Re: Hidden dimensions could explain where mass comes from Thomas Heger <ttt_heg@web.de> - 2026-01-10 09:13 +0100
Re: Hidden dimensions could explain where mass comes from "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2026-01-08 20:13 -0800
Re: Hidden dimensions could explain where mass comes from Thomas Heger <ttt_heg@web.de> - 2026-01-10 09:19 +0100
Re: Hidden dimensions could explain where mass comes from "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2026-01-10 13:08 -0800
Re: Hidden dimensions could explain where mass comes from Thomas Heger <ttt_heg@web.de> - 2026-01-11 10:13 +0100
Re: Hidden dimensions could explain where mass comes from Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-06 00:45 +0100
Re: Hidden dimensions could explain where mass comes from "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2026-01-10 13:12 -0800
Re: Hidden dimensions could explain where mass comes from ram@zedat.fu-berlin.de (Stefan Ram) - 2026-01-04 17:07 +0000
Re: Hidden dimensions could explain where mass comes from Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-04 18:39 +0100
Re: Hidden dimensions could explain where mass comes from Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-04 18:41 +0100
Re: Hidden dimensions could explain where mass comes from Anthk NM <anthk@disroot.org> - 2026-01-05 20:11 +0000
Re: Hidden dimensions could explain where mass comes from ram@zedat.fu-berlin.de (Stefan Ram) - 2026-01-05 20:52 +0000
Re: Hidden dimensions could explain where mass comes from Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-05 22:36 +0100
Re: Hidden dimensions could explain where mass comes from nospam@de-ster.demon.nl (J. J. Lodder) - 2026-01-05 22:38 +0100
Re: Hidden dimensions could explain where mass comes from nospam@de-ster.demon.nl (J. J. Lodder) - 2026-01-05 22:03 +0100
Re: Hidden dimensions could explain where mass comes from Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-05 22:26 +0100
Re: Hidden dimensions could explain where mass comes from nospam@de-ster.demon.nl (J. J. Lodder) - 2026-01-05 22:55 +0100
Re: Hidden dimensions could explain where mass comes from ram@zedat.fu-berlin.de (Stefan Ram) - 2026-01-09 14:06 +0000
Re: Hidden dimensions could explain where mass comes from nospam@de-ster.demon.nl (J. J. Lodder) - 2026-01-09 21:31 +0100
Re: Hidden dimensions could explain where mass comes from nospam@de-ster.demon.nl (J. J. Lodder) - 2026-01-04 20:46 +0100
Re: Hidden dimensions could explain where mass comes from Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-01-04 13:54 -0800
Re: Hidden dimensions could explain where mass comes from Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-01-04 14:45 -0800
Re: Hidden dimensions could explain where mass comes from athel.cb@gmail.com <user12588@newsgrouper.org.invalid> - 2026-01-05 10:18 +0000
Re: Hidden dimensions could explain where mass comes from nospam@de-ster.demon.nl (J. J. Lodder) - 2026-01-05 18:32 +0100
Re: Hidden dimensions could explain where mass comes from Thomas Heger <ttt_heg@web.de> - 2026-01-06 09:06 +0100
Re: Hidden dimensions could explain where mass comes from nospam@de-ster.demon.nl (J. J. Lodder) - 2026-01-06 10:43 +0100
Re: Hidden dimensions could explain where mass comes from Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-01-06 06:50 -0800
Re: Hidden dimensions could explain where mass comes from nospam@de-ster.demon.nl (J. J. Lodder) - 2026-01-06 16:12 +0100
Re: Hidden dimensions could explain where mass comes from Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-01-06 07:47 -0800
Re: Hidden dimensions could explain where mass comes from ram@zedat.fu-berlin.de (Stefan Ram) - 2026-01-06 18:00 +0000
Re: Hidden dimensions could explain where mass comes from Richard Hachel <rh@tiscali.fr> - 2026-01-11 22:32 +0000
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