From: ttt_heg@web.de   
      
   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   
      
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