From: PointedEars@web.de   
      
   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:   
      
      
      
   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*.   
      
      
   [continued in next message]   
      
   --- SoupGate-DOS v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)