From: tjroberts137@sbcglobal.net   
      
   On 3/20/22 1:30 PM, Richard Livingston wrote:   
   > On Saturday, March 19, 2022 at 1:27:38 PM UTC-5, Tom Roberts wrote:   
   >> A major lesson of modern physics is to discuss only measurable   
   >> (observable) quantities [#]. Both "everyday lives" and   
   >> "wavefunction collapse" violate that dictum (in very different   
   >> ways).   
   >> [#] Interestingly, this applies to both QM and GR (for very   
   >> different reasons).   
   >   
   > Tom, I would be very interested in your expanding a bit on how this   
   > all applies to GR.  -Rich L.   
      
   Note that coordinates are arbitrary human constructs, which we use to   
   simplify, codify, and quantify our observations and descriptions. Nature   
   clearly uses no coordinates, so the choice of coordinates used to   
   describe some natural phenomenon cannot possibly affect that phenomenon.   
   That is the importance of coordinate independence in GR (and in all of   
   theoretical physics) -- for a quantity to correspond to some physical   
   phenomenon, it must be independent of coordinates (aka invariant).   
      
   For instance, each and every measurement is a definite value for   
   whatever physical phenomenon is being measured, and is inherently   
   invariant. So observer A can construct a locally inertial frame and use   
   it to measure the kinetic energy of a baseball, and all other observers   
   will agree that is the value A measures in that frame, even though they   
   themselves use other frames to measure a different value.   
      
   	[This has been called a "coordinate-dependent invariant   
   	 quantity -- the value depends on which coordinates are   
   	 used, but the result is invariant because it is   
   	 inextricably bound to the coordinates used.]   
      
   Take careful note of the wording: kinetic energy is NOT invariant,   
   but the kinetic energy of a designated object relative to a specified   
   inertial frame is indeed invariant. So observers using other frames can   
   make measurements of the designated object, transform them to the   
   specified frame, and agree on the value obtained in the specified frame.   
      
   > Mod. note -- I too would be interested in what Tom says.   
   > My take would be that in GR, there is no preferred coordinate system   
   > and all physical quantities (= those that are measurable, at least in   
   > a gedanken sense) should be independent of the coordinate system in   
   > use.   
      
   Yes, "should be" => "are".   
      
   Also beware of "preferred coordinate system", because those words are   
   ambiguous -- physicists use that phrase in the sense of a coordinate   
   system that appears explicitly in the equations of the dynamics; but in   
   many/most cases there is a particular choice of coordinates relative to   
   which the calculations are simplified, and we invariably prefer to use   
   them. The invariance of physical quantities ensures we can do so.   
      
   > Notably:   
   > * the coordinate "time" of an event, or the difference between the   
   >   coordinate times of two events) is merely a coordinate; it has no   
   >   inherent physical meaning and can be changed arbitrarily by changing   
   >   our coordinate system   
   > * *proper* time along some (timelike) worldline is measurable (it's   
   >   what an (ideal) clock moving along that worldline would measure),   
   >   can be said to have an inherent physical meaning (as the observable   
   >   result of that measurement), and *doesn't* change when we change   
   >   coordinates   
   > * similarly, the coordinate position of an object, or the (coordinate)   
   >   distance between two objects, is also a coordinate, has no inherent   
   >   physical meaning, and can be changed arbitrarily by changing our   
   >   coordinate system   
   > * the *proper* distance along a given path is measurable (at least in   
   >   a gedanken sense: one can imagine laying down a sequence of standard   
   >   rulers end-to-end along the path), and doesn't change when we change   
   >   coordinates;   
   > * coordinate singularities (and the set of events where they occur)   
   >   have no inherent physical meaning, and a change in coordinates can   
   >   change the set of events where there is a coordinate singularity;   
   >   only singularities in observable quantities like curvature invariants,   
   >   proper times/distances, etc, are physically meaningful   
   Yes to all. The Moderator and I agree, except for details in wording.   
      
   Tom Roberts   
      
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