On Saturday, July 28, 2018 at 6:42:22 AM UTC-7, Gregor Scholten wrote:   
   > 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.   
      
   "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   
    * Origin: you cannot sedate... all the things you hate (1:229/2)