From: jthorn@astro.indiana.edu   
      
   J.B. Wood  wrote:   
   > Hello, all.  One concept that I can seem to get my thoughts around is   
   > that arguably gravitational force fits in to the Newtonian concept of a   
   > force that can act on a mass.  But general relativity informs us that   
   > gravity isn't a force at all but a phenomenon due to the curvature of   
   > space-time.  How can merely the curvature of space give rise to the fact   
   > that two untethered masses will tend to be attracted to and move toward   
   > one another?  We've seen those rubber-sheet demos but they rely on   
   > earth's gravity and we end up with a cause-effect-cause situation?   
   > Thanks for your time and comment.  Sincerely   
      
   and later   
   > Why   
   > does the curvature of space-time give rise to a force/acceleration?   
      
   As you point out, the rubber-sheet analogy is not very useful.   
      
   In my opinion a better analogy is to consider horizontal motion on the   
   Earth's surface (which we can treat as being spherical for present   
   purposes).   
      
   Suppose you have two nearby ships on the Earth's (ocean) surface, whose   
   courses are initially parallel (i.e., d/dt of the distance between them   
   is zero), and that both ships move along "locally straight" lines, i.e.,   
   symmetrical hulls and rudders not deflected left or right.  Then each   
   ship will travel along a great circle on the Earth's surface.  This   
   implies that the ships paths will *converge* (i.e., d^2/dt^2 of the   
   distance between them must be negative).   
      
   You could attribute this convergence-of-paths to a mysterious   
   "Newtonian gravity" ship-to-ship attraction, but it's informative to   
   instead regard it as a manifestation of the Earth's surface having   
   "intrinsic curvature", i.e., *not* satisfying the axioms of Euclidean   
   geometry.   
   	[Here the adjective "intrinsic" means that we're treating   
   	the curvature as an attribute of the 2-dimensional surface   
   	itself; not as an artifact of an embedding in a 3-dimensional   
   	world.]   
      
   As supporting evidence for this interpretation, note that the   
   ship-to-ship acceleration (i.e., d^2/dt^2 of the distance between   
   nearby ships at the moment they're some standard distance apart) is   
   universal, independent of the properties of this ship.  [This is   
   analogous to the universality (at a given place and time) of the   
   free-fall acceleration of test masses in Newtonian gravitation,   
   regardless of the properties of the free-falling test mass.]   
      
   Moreover, we observe that a "locally straight" route [= that taken   
   by ships & aircraft] from (say) Paris (France) to Vancouver (Canada)   
   curves far to the North of the origin & destination cities, typically   
   passing over central to northern Greenland.  If we didn't know about   
   the curvature of the Earth's surface, we might explain this by saying   
   that there is an equator-attracting "action at a distance" force on   
   the Earth's surfce, causing ship and aircraft trajectories to be   
   concave towards the equator for the same reason that a thrown ball's   
   path is concave towards the ground.   
      
   >From a "Newtonian" perspective, this equator-attracting force can   
   acclerate ships and airplanes "down" towards the equator, and thus   
   is a real physical force (it can do work in the Newtonian sense).   
   >From a curved-space perspective, we instead see that these ship and   
   aircraft simply follow great circles (geodesics).   
      
   Ok, that's enough of the analogy.  I can think of 2 major places where   
   falls doen:   
   1. Gravitation is actually a phenomenon of curved *spacetime*, not just   
      of curved *space*.  It's this that allows a very weak gravitational   
      field like that of the Earth, to curve a thrown ball's path so   
      strongly (from 45-degrees up to 45-degrees down in just a second   
      or so).   
           [One of Wheeler's books has a nice sketch showing a   
           ball's path in 3-D, with axes x, y, and c*time, making   
           it clear that in a 1-second flight, the thrown ball   
           has travelled some small number of meters in x and y,   
           but also 300,000 km in c*time, so the actual *curvature*   
           of its path in spacetime is very small.]   
   2. In our analogy, the Earth's surface has *extrinsic* curvature: it's   
      curved (does not satisfy Euclid's axioms of plane geometry) due to   
      the way it is embedded as a 2-dimensional surface in a 3-dimensional   
      world.  In contrast, we do not believe that the curvature of   
      4-dimensional spacetime is due to it's actually being embedded in   
      some 5-dimensional "hyper-spacetime"; rather, spacetime curvature   
      is seen as *intrinsic* to spacetime itself.   
      
   --   
   -- "Jonathan Thornburg [remove -animal to reply]"    
      Dept of Astronomy & IUCSS, Indiana University, Bloomington, Indiana, USA   
      "There was of course no way of knowing whether you were being watched   
       at any given moment.  How often, or on what system, the Thought Police   
       plugged in on any individual wire was guesswork.  It was even conceivable   
       that they watched everybody all the time."  -- George Orwell, "1984"   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)