XPost: rec.aviation.military   
   From: PointedEars@web.de   
      
   Stephen Harding wrote:   
   > I recall reading "somewhere"   
      
   "The Science of 'Interstellar'"? 8-)   
      
   > that space travel in the future (very, very, very* future) won't involve   
   > actually traveling gazillions of light years away but instead, "punching   
   > through" the space-time fabric as a sort of short cut.   
      
   _Spacetime_ is not a fabric, but a way to assign coordinates to events.   
      
   You mean a wormhole, an additional path between events that is spatially   
   shorter than the one along the spacetime manifold.   
      
   > I suppose it would be like a great circle route on a   
   > globe is shorter than the apparent straight line route   
      
   It is the other way around, of course.  With a positively curved spatial   
   manifold, the Euclidean (straight-line) distance between points on it is the   
   shortest.  This can be seen by considering the corresponding metrics.   
      
   Euclidean metric:   
      
     ds^2 = dx^2 + dy^2 + dz^2.   
      
   Spherical coordinates:   
      
     x = r sin(theta) cos(phi)   
     y = r sin(theta) sin(phi)   
     z = r cos(theta)   
      
   ==>   
      
     dx = dr sin(theta) cos(phi)   
          + r cos(theta) (d theta) cos(phi)   
          - r sin(theta) sin(phi) (d phi)   
      
     dy = dr sin(theta) sin(phi)   
          + r cos(theta) (d theta) sin(phi)   
          + r sin(theta) cos(phi) (d phi)   
      
     dz = dr cos(theta) - r sin(theta) (d theta)   
      
   ==>   
      
     ds^2 = dr^2 + r^2 (d theta)^2 + r^2 sin^2(theta) (d phi)^2.   
      
   A path P along the equator, dr = 0, theta = pi/2 ==> (d theta) = 0:   
      
     ds = r (d phi)  ==> s = int_P ds = r int_0^phi d phi' = r (Delta phi).   
      
     [Due to the spherical symmetry, we can rotate the coordinate system   
      such that any geodesic lies on the equator.]   
      
   Suppose, for example, the first point on a sphere with radius r has coordinates   
      
     (r, theta = pi/2, phi = 0)    <--> (x = r, y = 0, z = 0),   
      
   and the second point has coordinates   
      
     (r, theta = pi/2, phi = pi/2) <--> (x = 0, y = r, z = 0)   
      
   then the Euclidean distance between them is   
      
     s_2 = sqrt[(0 - r)^2 + (r - 0)^2 + (0 - 0)^2]   
         = sqrt(2 r^2)   
         = r sqrt(2) =~ r * 1.41   
      
   but the geodesic distance (along the equator) is   
      
     s = r (pi/2 - 0) = r pi/2 =~ r * 1.57 > s.   
      
   You are maybe confusing this with the *apparent* straight-line path if the   
   curved surface is being projected on a flat surface.   
      
   > or as we think of SciFi wormholes and such.   
   >   
   > But without that sort of mechanism, there is no way we'll see other life   
   > forms,   
      
   Not true.  We might see lifeforms on Mars, and possibly Europa and Enceladus   
   (there are oceans under the frozen surfaces of those moons).   
      
   > let alone intelligent ones.   
      
   There are intelligent lifeforms on Earth, not counting humans :-p   
      
   > As far as we know, even with exoplanet systems that we have now found,   
   > the odds are scarce that a planet will have the requirements for life   
   > (as we know it) let alone the roll of the dice evolutionary process   
   > to produce intelligence.   
      
   Humans know too little about the conditions for life to make even educated   
   guesses.  Currently they are in a position similar to that of person who   
   knows a hammer as the only tool.   
      
   > That's a primary reason I don't believe in UFOs and extraterrestrial   
   > incidents.  Some weird stuff no doubt, but explainable at some point,   
   > now or in the future with more knowledge.   
      
   There are much better reasons for not believing that UFOs are of   
   extra-terrestrial origin.   
      
   --   
   PointedEars   
      
   Twitter: @PointedEars2   
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