From: PointedEars@web.de   
      
   Mikko wrote:   
   ^^^^^   
   Please post to Usenet using your full name.  It is considered polite and   
   avoids ambiguities.   
      
   > On 20/01/2026 15:54, Paul B. Andersen wrote:   
   >> You are standing on the floor in a small room   
   >> without windows with an accelerometer fixed to the wall.   
   >> The accelerometer shows that it is accelerating   
   >> in the direction which to you is upwards.   
   >>   
   >> If the accelerometer shows that the acceleration   
   >> is constant 1g, then there is no way you can   
   >> decide whether you are stationary at the ground,   
   >> or if you are accelerating at 1g in a spaceship.   
   >>   
   >> If the accelerometer shows that the acceleration   
   >> is changing with time, then you know that you are   
   >> in a spaceship.   
   >   
   > Depends on how much it changes. On Earth you have small variations   
   > from the gravity of Moon and Sun. Sometimes you may have even large   
   > variations (of short duration) from earthquakes.   
      
   The gravitational field of a celestial body varies with location -- altitude   
   and surface coordinates --, and it may vary with time.   
      
   Your description considers the variation with surface coordinates and time   
   only to some extent.  The primary variation with surface coordinates and   
   altitude is obvious from Newton's Law of Universal Gravitation (which --   
   because it describes gravitation so well -- has to be a limit of any theory   
   of gravitation, and actually is a limit of general relativity):   
      
          F(R) = m_i A(R) = m_g g(R) = -m_g G M/r^2 R/r   
      
     <==>                       g(R) = -G M/r^2 R/r,   
      
   where F(R), A(R), g(R) are vector fields, R is the radius vector, and   
   r = ||R||_2, and we have assumed (as Einstein did, from observations)   
   the equivalence of inertial mass m_i and gravitational mass m_g.   
      
   One can see that the magnitude of the gravitational field varies with r, and   
   as the law is assuming a spherically-symmetric object, this means that its   
   strength varies with altitude h:   
      
     ||g(h)||_2 = G M/(R + h)^2,   
      
   where now R is the radius of the object; but also   
      
     R = r [sin(θ) cos(φ), sin(θ) sin(φ), cos(θ)]^T,   
      
   so the direction of the field, always pointing towards the center of mass,   
   varies with latitude (90° - θ) and longitude φ.  (Notwithstanding that there   
   probably is no celestial object with uniform mass density; certainy Terra is   
   not; and that a rotating celestial object is not spherical; certainly Terra   
   is not.)   
      
   Finally we have to consider variations of the gravitational field with   
   surface coordinates and time which you partially alluded to: There are other   
   celestial objects by whose existence the gravitational field is modified,   
   dependent on surface coordinates, altitude (some celestial objects, like   
   Sol, are farther away, but may have a larger mass; other celestial objects,   
   like Luna, may have a smaller mass but are closer to the object where we are   
   discussing the gravitational field), and time (the position-relation between   
   those objects changes with time, and so may the mass/energy distribution of   
   any of them).   
      
   That is why the Einstein Equivalence Principle -- "physics in a frame of   
   reference at rest under the influence of gravitation is experimentally   
   indistinguishable from physics in a constantly accelerating frame that is   
   not under the influence of gravitation" -- only applies under the assumption   
   of a *uniform* gravitational field, i.e. if the considered spacetime volume   
   is small enough that the variations mentioned before can be neglected.   
      
   Remarkably, though, it sufficed/suffices to formulate general relativity   
   which has been observationally confirmed to high precision.   
      
   --   
   PointedEars   
      
   Twitter: @PointedEars2   
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