From: g.scholten@gmx.de   
      
   jmreno wrote:   
      
   >> Can the fact that galaxies are receding from each other faster   
   >>    than light violate causality?   
   >>   
   >   
   > The universe is expanding because space itself is expanding.   
   >   
   > And since space and time are intrinsically linked, maybe it is spacetime   
   > that is expanding.   
   >   
   > That would mean that time is expanding along with space.   
   >   
   > Presumably, that means that time is slowing down.   
      
   If we follow General Relativity (GR), there is no such thing like   
   expansion of spacetime or expansion of time. There is time dilation, but   
   that is something different. Let's consider more in detail what gives   
   movitation for the statement "space is expanding":   
      
   In cosmological solutions of GR, one usually assumes that on large   
   scales, the curvature of spacetime is described by the   
   Friedmann-Lemaitre-Roberson-Walker (FLRW) metric. This can be expressed as   
      
   ds^2 = -(c dt)^2 + S(t)^2 [dr^2 / (1 - k r^2) +   
                                r^2 (dtheta + sin(theta)^2 dphi^2]   
      
   where (t,r,theta,phi) are so-called co-moving coordinates. S(t) is the   
   scale factor, k is the so-called curvvature index (for curvature of   
   space, not of spacetime), that can be -1, 0 or +1. Now consider two   
   galaxies, one located at r = 0, the other at r = r0. The spatial   
   distance between both galaxies, i.e. the distance on a spacelike   
   hypersurface defined by t = const (<=> dt = 0), is given by   
      
   D = \int_0^r0 [S(t) / (1 - k r^2)^(1/2)] dr   
      
   what equals D = S(t) r0 for k = 0. Now, if the universe is expanding,   
   the scale factor S(t) is increasing by time: dS/dt > 0, so that the   
   spatial distance between both galaxies increases by time, too. In other   
   words: the two galaxies move apart.   
      
   However, this movement does not yield special-relativistic effects like   
   time-dilation. For the proper time dtau = 1/c (-ds^2)^(1/2) that elapses   
   for each galaxy, we find, with ds = dtheta = dphi = 0 since each galaxy   
   has a fixed spatial position in the co-moving coordinates:   
      
   dtau = 1/c (-ds^2)^(1/2) = dt   
      
   So, the proper time tau matches the coordinate time t.   
      
   For comparison: in Special Relativity (SR), if we consider a body moving   
   with velocity v = dx/dt in x-direction with respect tome some inertial   
   frame (t,x,y,z) in which the (Minkowskian) metric is   
      
   ds^2 = -(cdt)^2 + dx^2 + dy^2 + dz^2   
      
   we get   
      
   dtau = 1/c [(c dt)^2 - dx^2]^(1/2)   
         = (dt^2 - dx^2/c^2)^(1/2)   
         = (dt^2 - v^2 dt^2 /c^2)^(1/2)   
         = (1 - v^2/c^2)^(1/2) dt   
      
   what is the well-known formula for time-dilation.   
      
   That's why we say that in SR, bodies are moving on their own, yielding   
   time dilation, whereas when galaxies move apart due to the expansion of   
   the universe, there is no time dilation because the galaxies do not move   
   on their own, but space is expanding between them.   
      
   --- SoupGate-Win32 v1.05   
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