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? In any case they clearly meet in the same point of spacetime by your operational definition, and thus clearly in the same p-time. Nevertheless by their own agreed upon different ages they meet at different times on their own proper comoving clocks, and meeting at a single coordinate clock doesn't change their real age clock differences. So after all this genuflection I don't see we are anywhere different than where we started though you may claim we are? And two caveats: 1. My agreement is subject to withdrawal if it turns out I didn't understand you correctly. 2. Those two events can still be the shaking hands and comparing clocks of twins with two actual DIFFERENT CLOK TIME AGES. In other words the fact that there can be an arbitrary clock time at the meeting point that both twins agree to use as the 'real' time of that meeting is completely arbitrary. It is no different operationally that either twin resetting his clock to the clock of the other and both twins agreeing that's the time they will use from then on. It makes no difference whatsoever to the actual age clocks of the twins which remain different permanently and agreed upon by both twins. Coordinate time does not explain the age differences of the twins. It ignores them. But it does provide a completely arbitrary 'SAME' time for the meeting, and that does correspond to a single agreed upon P-time. However that single agreed clock/coordinate time has no relevance to determining the clock times that correspond to the same P-time of the twins DURING the trip, because it does not refer to the trip in any way whatsoever. Edgar On Monday, February 24, 2014 2:35:07 PM UTC-5, jessem wrote: > > > > On Mon, Feb 24, 2014 at 7:24 AM, Edgar L. Owen <[email protected]<javascript:> > > wrote: > > Jesse, > > Let me make sure I understand what you are saying. > > You say we can drop an arbitrary coordinate system onto spacetime, and > then we can place an originally synchronized clock at every grid > intersection. Is that correct? > > > It depends whether we are talking about inertial frames or arbitrary > non-inertial coordinate systems. In non-inertial coordinate systems, the > only requirement is that the coordinate be "smooth"--no sudden > discontinuities in the coordinates assigned to infinitesimally-close points > in spacetime. Beyond that, not only are you free to drop an > arbitrarily-shaped rubbery coordinate "grid" with clocks at each > intersection, but you're also free to define "synchronization" any way you > want, you don't need to follow any standard procedure for deciding what > point on each clock's worldline is the one where it be set to read zero, > you can do this any way you like (again provided that the resulting > simultaneity surfaces are smooth, with no discontinuous "jumps"). And > there's also no requirement that the coordinate clock times actually > correspond to the proper times along their worldline--you could have a > coordinate clock that was designed to alternately run faster or slower than > a normal clock moving right alongside them, for example. > > But the example I gave with Alice/Bob/Arlene/Bart involved an inertial > coordinate system, not any non-inertial ones. In this case the rules for > constructing a coordinate system are more strict--you have to use a > Cartesian grid of straight rulers that are all inertial and at rest > relative to one another, and then you have to use the "Einstein > synchronization convention" to define what it means for clocks at different > grid intersections to be synchronized with one another--the most common > definition of this convention is that if you send a light signal from clock > A when it reads tA1, it reflects off clock B when it reads tB, and the > reflected light returns back to clock A when it reads tA2, then tB should > be exactly halfway between tA1 and tA2 (i.e. tB = (tA2 - tA1)/2 ). Another > equivalent definition is that if you set off a flash of light from a ruler > marking that's exactly halfway between the markings that A and B are > attached to, then both clocks should show the same reading when the light > from the flash reaches them. The Einstein synchronization convention > ensures that each inertial frame will measure the speed of light to be the > same in all directions. > > > > > And that those clocks read what is called the coordinate times of those > grid intersections, and this gives us in some sense a measure of the actual > time coordinate of that spatial coordinate? > > > Yes, or more specifically they give a time coordinate for any EVENT that > happens at a given spatial coordinate. For example, if a firework goes off > at position x,y,z, then the time coordinate of the firework exploding would > be defined by the reading t on the coordinate clock at x,y,z as the > firework was exploding right next to it (so a photo of this location at > that moment would show both the exploding firework and the clock there > reading t). > > > > One clarification before I agree. The clocks on this grid that are in > gravitational fields will be running slower than the clocks that are not? > And we can compare the clocks across the grid to determine which are > running slower and which faster? Is that correctly part of the model? > > > > In the case of inertial frames, these spacetimes are defined only in the > flat spacetime of special relativity, where there is no gravity (since > gravity involves spacetime curvature). In the real world there may be no > perfectly flat regions of spacetime, but many regions in space that are > limited in spatial and temporal extent may be extremely good approximations > to flat spacetime. > > In general relativity where spacetime is curved, there isn't really any > objective coordinate-independent way to compare the rates of clocks at > different points in space, all you can do is compare how fast each clock is > ticking relative to coordinate time in some coordinate system (and as I > said above, the rate of coordinate clocks in arbitrary non-inertial > coordinate systems can in principle be anything, although of course you're > free to construct a coordinate system where coordinate time at each grid > intersection does actually correspond to proper time of a clock at that > intersection). I discussed the problem of defining the relative rate of > different clocks in GR in the second half of my post at > https://groups.google.com/d/msg/everything-list/jFX-wTm_E_Q/SX19ccLeij0J(starting > with the paragraph that begins "Not > a well-defined assumption.") > > > > If so I agree. It's my understanding of relativity theory, and my theory > starts by accepting every part of relativity theory and adding to it rather > than trying to change any part of it. If my theory is inconsistent with > relativity in any respect I would consider my theory falsified. > > <d > > ... -- 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.
