From: hees@itp.uni-frankfurt.de   
      
   On 22/12/2024 09:57, Luigi Fortunati wrote:   
   > Jonathan Thornburg [remove -color to reply] il 21/12/2024 09:27:44 ha   
   > scritto:   
   >> ...   
   >> Now let's look at the same system from a GR perspective, i.e., from a   
   >> perspective that gravity isn't a force, but rather a manifestation of   
   >> spacetime curvature.  In this perspective it's most natural to measure   
   >> accelerations relative to *free-fall*, or more precisely with respect   
   >> to a *freely-falling local inertial reference frame* (FFLIRF).  An   
   >> FFLIRF is just a Newtonian IRF in which a fixed coordinate position   
   >> (e.g., x=y=z=0) is freely falling.   
   >   
   > Can we define the interior space of the elevator as "local" or is it   
   > too big?   
   >   
   > If it is too big, how big must it be to be considered "local"?   
   >   
   > If it is shown that there are real forces inside the free-falling   
   > elevator, can we still consider this reference system inertial?   
   >   
   > Are tidal forces real?   
   >   
   > Do we mean by "freely falling bodies" only those that fall in the very   
   > weak gravitational field of the Earth or also those that fall in any   
   > other gravitational field, such as that of Jupiter or a black hole?   
   >   
   > Luigi Fortunati.   
      
   This problems in understanding GR is, in my opinion, due to too much   
   emphasis on the geometrical point of view. Of course, geometry is the   
   theoretical foundation of all of modern physics, i.e., a full   
   theoretical understanding of physics is most elegantly achieved by   
   taking the geometric point of view of the underlying mathematical   
   models. However, there's also a need for a more physical, i.e.,   
   instrumental formulation of its contents.   
      
   Now indeed, from an instrumental point of view, the gravitational   
   interaction is distinguished from the other interactions by the validity   
   of the equivalence principle, i.e., "locally" you cannot distinguish   
   between a gravitational force on a test body due to the presence of a   
   gravitational field due to some body. In our example we can take as a   
   test body a "point mass" inside the elevator, with the elevator walls   
   defining a local spatial reference frame. The corresponding time is   
   defined by a clock at rest relative to this frame at the origin of the   
   frame (say, one of the edges of the elevator). Now, the equivalence   
   principle says that it is impossible for you to distinguish by any   
   physics experiment or measurement inside the elevator, whether you are   
   in a gavitational field (in our case due to the Earth), which can be   
   considered homogeneous (!!!), for all relevant (small!) distances and   
   times around the origin of our elevator reference frame or whether the   
   elevator is accelerating in empty space. A consequence is also that if   
   you let the elevator freely fall in the gravitational field of the   
   Earth, you don't find any homogeneous gravitational field, i.e., free   
   bodies move like free particles locally, and thus the free-falling   
   elevator defines a local inertial frame of reference.   
      
   Translated to the "geometrical point of view" that means that you   
   describe space and time in general relativity as a differentiable   
   spacetime manifold. The equivalence principle means that at any   
   space-time point you can define a local inertial frame, where the   
   pseudometric of Minkowski space (special relativity) defines a   
   Lorentzian spacetime geometry.   
      
   If you now look at larger-scale physics around the origin of the   
   freely-falling-elevator restframe, where the inhomogeneity of the   
   Earth's gravitational field become important, there are "true forces"   
   due to gravity. In the local inertial frame these are pure tidal forces,   
   named because they are responsible for the tides on the Earth-moon   
   system freely falling in the gravitational field of the Sun.   
      
   So it's important to keep in mind that the equivalence between   
   gravitational fields and accelerated reference frames in Minkowski space   
   holds only locally, i.e., in small space-time regions around the origin   
   of your coordinate system, in which external gravitational fields can be   
   considered as homogeneous (and static). T   
      
   he physically interpretible geometrical quantities are tensor (fields),   
   and the general-relativistic spacetime at the presence of relevant true   
   gravitational fields due to the presence of bodies (e.g., the Sun in the   
   solar system) is distinguished from Minkowski space by the non-vanishing   
   curvature tensor, and this is a property independent of the choice of   
   reference frames and (local) coordinates, i.e., you can distinguish from   
   being in an accelerated reference frame in Minkowski space (no   
   gravitational field present) and being under the influence of a true   
   gravitational field due to some "heavy bodies" around you, by measuring   
   whether there are tidal forces, i.e., whether the curvature tensor of   
   the spacetime vanishes (no gravitational interaction at work, i.e.,   
   spacetime is described as a Minkowski spacetime) or not (gravitational   
   interaction with other bodies present, and you have to describe the   
   spacetime by some other pseudo-Riemannian spacetime manifold, which you   
   can figure out by solving Einstein's field equations, given the   
   energy-momentum-stress tensor of the matter causing this gravitational   
   field, e.g., the Schwarzschild solution for a spherically symmetric mass   
   distribution).   
      
   --   
   Hendrik van Hees   
   Goethe University (Institute for Theoretical Physics)   
   D-60438 Frankfurt am Main   
   http://itp.uni-frankfurt.de/~hees/   
      
   --- SoupGate-DOS v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)