Jesse, Here is a clearer, unambiguous and more general way to define p-time simultaneity in terms of proper times. Let me know what you think. I'll also address your latest questions in separate replies.......
Drop an arbitrary coordinate system onto an arbitrary space. Place a clock at each grid intersection. I don't think we even have to worry about those clocks being synchronized initially. (We do assume only that physical processes, including the rate of time, follow the same relativistic laws at all locations.) Place a stationary observer with each clock just for terminological convenience. We don't really need this coordinate clock system but I include it to address your concerns. Each clock will display the coordinate time of its grid intersection, which will also be the proper time of the stationary observer at that location. These grid clocks will run at different rates depending on the gravitational potentials of their grid locations. Do you agree? Now also introduce an arbitrary number of observers either stationary, or moving relative to this grid, each with its own proper time clock, some accelerating, some with just constant relative motion. This model covers all possible types of relativistic time effects (disregarding black holes and other types of horizons for the moment). Do you agree? It is possible for all observers in this space to have knowledge of the relativistic conditions of all other observers as well as themselves. In other words they can know the equations governing how any observer would view any other observer. Do you agree? Thus it is possible for all observers to know the RATES of all proper clocks in this system, and all observers will agree on all those proper clock rates. E.g. all observers would agree that the proper clock in a certain gravity would be running at 1/2 the rate as clocks in no gravity. All observers would agree that the proper clock rates of all observers in inertial motion would be running at the same rate. And all observers would agree that the proper clock of an observer with a specific acceleration close to the speed of light would have a proper clock rate half that of a non-accelerating observer. Do you agree? So all observers in this space can agree on all proper time RATES. But to tie p-time simultaneity to proper times we need to establish a notion of proper time SIMULTANEITY as well as proper time RATES. So how do observers know the actual proper time clock reading of any other observer in the general case without having to initially get together and synchronize their clocks? (They of course can do this if they wish but they don't actually have to.) Well, it's actually easy. They simply tell each other what their proper clock time readings are and compare them. Observer A sends a light message to any observer B saying "when I sent this message my proper time was t". Observer B knows how long it takes to receive that message by both A's and his own proper clock RATES. As soon as he receives the message he sends a light reply telling A that when he received the message his proper time was t'. In this way it is simple for both A and B to calculate what the other's proper time was at every past moment of their own proper time, and what it will be for every future moment of their own proper time, and what it is now for their current proper times. Do you agree? And this proper time relationship is transitive among all observers. In other words, by all observers establishing a proper time correlation with some other observer(s) until all observers are included in this proper time network, every observer will agree on how his proper time correlates 1:1 with the proper times of all other observers in the space. In this way we establish a 1:1 proper time correlation among all observers which all observers agree upon. Every observer knows exactly what the proper time t' of any OTHER observer IS, WAS, OR WILL BE that corresponds to any proper time t of his OWN. Note that this is NOT a proper time plane of simultaneity. The proper times of various observers can be different, but always in a 1:1 knowable way that all observers agree upon. Do you agree? I know you said you had a counter example. If so please present it. Now the current proper time of any observer always tells the current p-time (the current present moment). Thus we have now established both the FACT of a universal p-time common to all observers, AND an operational method to unambiguously determine that in terms of proper time readings for all relativistic observers. Do you agree? We have thus established a current P-time plane of simultaneity in terms of the (differing) proper times of all observers in our test space, and a method to determine all previous (and theoretically future) P-time planes of simultaneity in terms of the proper times of all observers in our test space. Do you agree? Edgar On Monday, February 24, 2014 9:23:20 PM UTC-5, jessem wrote: > > > > On Mon, Feb 24, 2014 at 6:53 PM, Edgar L. Owen <edga...@att.net<javascript:> > > wrote: > >> Jesse, >> >> Well, I thought I was expressing your own model, but apparently not. >> >> However IF, and a big if, I understand you correctly then I do agree that >> "if two events have the same space and time coordinates in a single >> inertial frame, they must also satisfy the operational definition of "same >> point in spacetime" I gave earlier? And I would agree this means that the >> two events happened at the same p-time?" >> >> I'm assuming this means we agree that the meeting twins do meet in the >> same space and time coordinates of the inertial frame in which they meet, >> though obviously NOT in the same time coordinates of their own proper >> comoving frames? >> > > Depends what you mean by that. Say that in the original inertial frame we > first use to analyze the problem (which may not be the rest frame of either > Alice or Bob), the event of Alice turning 30 has the same space and time > coordinates as the event of Bob turning 40, i.e. these two events happen at > the same point in spacetime. Then the event of Alice turning 30 could be at > a time coordinate of t=30 in her own comoving rest frame, but in her > comoving frame the event of Bob turning 40 would ALSO be at t=30 (and both > events would have identical space coordinates in this frame). And the event > of Bob turning 40 could be at a time coordinate of t'=40 in his own > comoving rest frame, but in his comoving frame the event of Alice turning > 30 would ALSO be at t'=40 (and again the space coordinates would be the > identical). So no matter what frame we use, these two events--Alice turning > 30, and Bob turning 40--are assigned the same time-coordinates AS ONE > ANOTHER in that specific frame, but the actual time coordinate common to > both events can differ from one frame to another (in Alice's frame they had > a common time coordinate of t=30, while in Bob's frame they had a common > time coordinate of t'=40). Is the latter all you meant by "NOT in the same > time coordinates of their own proper comoving frames", or would you > actually disagree with my claim that if these two events have the same > space and time coordinates as one another in some frame, they must still > have the same space and time coordinates as one another in any other frame > as well? > > Also, would you agree that crossing through identical space and time > coordinates implies satisfying the operational definitions I gave even if > they don't actually stop and come to rest relative to each other, but just > cross paths briefly while moving at a large relative velocity? That they > would still satisfy the operational definition of crossing through the > "same point in spacetime" in the sense that if they were sending continuous > signals to one another, the time for the signal to be reflected and return > would approach zero as they approached the space and time coordinate that > both their paths cross through? I can give an example if this scenario > isn't clear. > > 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 everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
