From: tjroberts137@sbcglobal.net   
      
   On 5/25/20 8:38 AM, Nicolaas Vroom wrote:   
   > On Friday, 22 May 2020 21:15:39 UTC+2, Tom Roberts  wrote:   
   >> On 5/21/20 2:36 AM, Luigi Fortunati wrote:   
   >>> [...] Where is the conceptual difference that should make the   
   >>> time of a twin different from that of the other if we are   
   >>> talking about Special Relativity ONLY?   
   >>   
   >> The difference between the twins comes from the simplicity that   
   >> applies only to inertial frames.   
   >   
   > What is the reason that you cannot use one reference frame for the   
   > whole experiment from start to finish i.e. when both clocks again   
   > can be compared at point O?   
      
   There is no such reason, and my discussion explicitly showed using the   
   inertial frame of A for the entire analysis.   
      
   >> If twin A remains at rest in an inertial frame, it is easy to   
   >> calculate the age (elapsed proper time) of each twin: simply   
   >> integrate   
   >>   
   >> T = \integral sqrt(1-v^2/c^2) dt   
   >>   
   >> where T is the elapsed proper time, the integral is taken over the   
   >> path of the twin relative to A's rest frame, v is the speed of the   
   >> twin relative to that frame (as a function of t), and t is the time   
   >> coordinate of the frame.   
   >   
   > My understanding is that t represents the time (clock count) of a   
   > clock at rest and T the (time) clock count of a moving clock.   
      
   Not quite. t is the time coordinate of the inertial frame, and T is the   
   elapsed proper time of the clock whose velocity profile is v(t). The   
   equation applies to EVERY clock, and can be used for multiple clocks   
   with different velocity profiles.   
      
   [[Mod. note -- Note that each clock has its own individual velocity   
   profile  v(t)  and hence its own individual elapsed proper time  T .   
   -- jt]]   
      
   One can apply the equation to a clock at rest in the inertial frame: v=0   
   so the integral is trivial: T=t. So it is clear that the time coordinate   
   of the inertial frame is the same as the elapsed proper time of a clock   
   at rest in it (reset the clock to zero at t=0). They are conceptually   
   different but numerically equal.   
      
   	A clock can only indicate its proper time, and only   
   	along its worldline; the time coordinate of an inertial   
   	frame can indicate the frame's coordinate time anywhere   
   	the frame is valid. Of course such coordinates are   
   	models, and to implement them requires physical clocks   
   	located where events of interest occur; all such clocks   
   	must of course be synchronized to the coordinate time.   
      
   	In practice, large experiments place detectors where   
   	events of interest might occur; they connect the   
   	detectors with cables to a common place where timing can   
   	be applied (e.g. a data-acquisition computer); of course   
   	they must correct for the time delays in the cables,   
   	each of which must be measured.   
      
   > The question is how is v (the speed of the clock) measured?   
      
   In the usual way: v=dx/dt, where x(t) is the position of the object at   
   time t, and t is the time coordinate of the inertial frame.   
      
   > IMO in order to do that, you need a reference rod in the frame at   
   > rest with two clocks at backend x1 and the frontend x2 of this rod,   
   > also at rest. When you do that you can calculate the speed v by   
   > applying the following formula: v = (x2-x1) / (t2-t1) where t1 is   
   > the time that the moving clock coincides with the backend x1 and t2   
   > the time that the clock coincides with the front end x2 of the   
   > clock.   
      
   That works, too.   
      
   Note this is a gedanken, and we implicitly presume that when we use a   
   given inertial frame, we can identify its coordinate values for all   
   events of interest. This, of course, includes the events occupied by the   
   clock as it moves around.   
      
   >> Note that neither position nor acceleration appear in this   
   >> equation; all that matters is the speed of the twin relative to   
   >> the inertial frame being used to calculate.   
   >   
   > That is correct. But if the speed varies along this path you need   
   > more clocks to calculate the time T correctly.   
      
   Hmmm. I repeat: this is a gedanken, and we implicitly presume that when   
   we use a given inertial frame, we can identify its coordinate values for   
   all events of interest. This, of course, includes the events occupied by   
   the clock as it moves around.   
      
   If you want to place clocks at rest in the inertial frame, located along   
   the path of the moving clock, that is fine, but it provides no   
   additional information than is contained in the coordinates themselves.   
      
   > But there is another issue: How do you know that the equation to   
   > calculate T is correct?   
      
   This is NOT an issue. We are using SR to model this gedanken, and that   
   is the correct equation of SR. We know this because it has been derived   
   many times in many textbooks.   
      
   > The only way is when the calculated T at the end can be verified by   
   > means of an actual experiment.   
      
   Hmmmm. The model itself has no need of experiments, because it is merely   
   a set of postulates and theorems, plus definitions of the symbols that   
   appear in them. Testing the model and validating it certainly does   
   require experiments. For this equation of SR, that has been done many   
   times, for many different types of clocks and many different physical   
   situations.   
      
   > IMO experiments are the only way to derive this formula.   
      
   No. The equation is mathematically derived from the postulates of the   
   theory. Experiments are used to test and verify that the resulting   
   theory is in agreement with the world we inhabit. These are quite   
   different -- the first just uses math, while the second compares that   
   math to experimental results.   
      
   > Using different experiments you can also test if all clocks with a   
   > different internal physical construction behave the same i.e. show   
   > the same number of clock counts.   
      
   Hmmm. If you can apply different types of clocks in the same physical   
   situation, yes. But in general that is not practical, and what is   
   required is to show agreement with the theory; after all, it is the   
   theory we are testing.   
      
   Tom Roberts   
      
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