Edgar, you wanted me to address your examples so I will, although I thought it better to hold off on this until we settled the question of whether the basic assumption you seem to be making in case #1 leads to contradictions. Given your recent post at https://groups.google.com/d/msg/everything-list/jFX-wTm_E_Q/qurIcfr_qT4J I am not even sure if I am understanding the rule you use to derive your conclusions about case #1 correctly, though--I really think we should settle that before getting into an involved discussion of more complicated cases involving general relativity like your case #2 and case #3.
On Sun, Feb 9, 2014 at 2:44 PM, Edgar L. Owen <[email protected]> wrote: > Jesse, > > Let me try to clarify my response to your A, B, C past p-time simultaneity > example, because I think I misstated it in my previous post. > > Assume three observers A, B, C, with three clock times t, t' and t''. It > is important to specify these are the clock time readings of their OWN > clocks in their OWN frames. > Do you understand that "clock time readings" for an observer are what physicists mean by the "proper time" of that observer, and that these readings are completely frame-independent, so it doesn't even matter what coordinate times are assigned to these clock readings in "their own" frame? Also, do you understand that any observer is free to use any coordinate system they like in relativity, and that while it is a matter of convention that the inertial rest frame of an inertial observer in flat spacetime is often labeled "their own" frame, there is no such convention for accelerating observers or observers in gravitational fields (which in relativity requires the curved spacetime of general relativity)? > > > Case 1: Assume they are all initially in the same inertial frame with > synchronized clocks. In this same inertial frame they are in the same > current moment of p-time at every synchronized clock time tick of their > clocks. > See my comment at the top about my confusion resulting from your comment at https://groups.google.com/d/msg/everything-list/jFX-wTm_E_Q/qurIcfr_qT4J -- in this case, are you assuming a GENERAL RULE that says that in flat SR spacetime with no gravity, when you have some observers "initially in the same inertial frame" with clocks that are synchronized RELATIVE TO THAT FRAME (as opposed to assuming from the start that they are 'synchronized' in p-time without assuming this means the clocks are synchronized in their inertial rest frame), that this ALWAYS IMPLIES that the clocks are synchronized in p-time as well? If you aren't assuming such a general rule I don't really understand the basis for your conclusions above, while if you are assuming such a general rule, then I think my example with two pairs of observers shows that this rule leads to a contradiction. > > > Case 2: Assume A is in a gravity well that makes it's clock run at 1/2 the > rate of B's clock. > Not a well-defined assumption. When gravity is involved, relativity says we must describe this in terms of the curved spacetime of GR. But in GR there are only two meaningful ways to compare the rates of clocks at different points in space: 1. Use a particular spacetime coordinate system with a particular definition of simultaneity, and then you can talk about the rate of one clock ticking relative to the other *in that particular coordinate system*. However, different coordinate systems will disagree on the rate of ticking of one clock relative to the other, and all smooth coordinate systems are equally valid in general relativity. 2. Calculate the rate that one observer will *see* the other clock ticking in a purely visual sense--for example, if my clock elapses 2 seconds between moments when I receive light signals from successive one-second ticks of the other clock, I see it ticking half as slow as my own. The relationship between purely visual rates need not be symmetrical, though: If I see a distant clock ticking at half the rate of my own at a particular point on its worldline, at that point on its worldline it is *not* necessarily seeing my clock ticking twice as fast as its own. Which of these do you mean, or do you think there is some third alternative that can be defined in an empirical way? > > Case 3: Now C is in relative motion to B (and therefore to A as well). > We are still assuming that A and B are as described in case 2, correct? But then we must still be in the curved spacetime of general relativity, and in general relativity "relative motion" of distant observers has no well-defined coordinate-independent meaning. In SR this does have meaning, because you can "parallel transport" a velocity vector at one location in spacetime A to a different location B to compare it with a velocity vector at another point in spacetime B. But in general relativity, if you try to do parallel transport of a velocity vector from A to B, the final vector at B will be different depending on what path it was transported along. See the discussion at http://math.ucr.edu/home/baez/einstein/node2.html for a basic explanation, with mathematical details at http://math.ucr.edu/home/baez/einstein/node10.html So, all talk of whether two observers are "at rest relative to each other" or "in relative motion" can only make sense in the context of a particular coordinate system. But you haven't given any rule for choosing what coordinate system in general relativity should be used to decide facts about p-time. Without clarification on these points, the rest of your arguments are ill-defined. Jesse -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
