On Mon, Feb 24, 2014 at 7:24 AM, Edgar L. Owen <edgaro...@att.net> 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. > > So is my understanding correct, and do we both agree to the same thing > here? > So do you specifically agree that if we define time coordinates in the way above (especially in the case of inertial coordinate systems in regions where gravity is absent or negligible, since that's really the only one that's relevant to my Alice/Bob/Arlene/Bart example), then 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 would you agree this means that the two events happened at the same p-time? Jesse > > On Sunday, February 23, 2014 7:05:04 PM UTC-5, jessem wrote: >> >> >> >> >> On Sun, Feb 23, 2014 at 11:57 AM, Edgar L. Owen <edga...@att.net> wrote: >> >> Jesse, >> >> To address your question. I'll start with your terminology. Your A>B>C >> doesn't follow and I'll show why it doesn't. >> >> "Same space and time coordinates"? In which coordinate system? In general >> these will be different in different coordinate systems, and as you >> yourself have pointed out choice of coordinate system is arbitrary in >> relativity. >> >> So if twins A and B happen to have the same clock time coordinates at the >> same point in space they could well be at that point in space in entirely >> different p-times OR they could be there at the same P-time. That depends >> on their relativistic history and choice of coordinate systems. >> >> Let me clarify. Take a point Px,y,z in space. One twin could pass through >> that point at earth time 1989 when his proper clock (actual age) was 30, >> and then ten years later in 1999 the other twin could pass through that >> point P when his proper clock (actual age) was 30. In this case they would >> NOT be at the same point in p-time even though 10 years apart they DID have >> the exact same "space and time coordinates". >> >> >> You are repeating a confusion which I have already corrected several >> times in the past. In relativity, the "time coordinate" of an event is >> defined ONLY in terms of a set of "coordinate clocks" which are affixed to >> particular position coordinates, like the clocks attached to different >> markings on a lattice of rigid rulers that are used to define coordinate >> time in inertial reference frames, as illustrated here (please take the >> time to click the link and at least glance at the illustration): >> http://www.upscale.utoronto.ca/GeneralInterest/Harrison/ >> SpecRel/SpecRel.html#Exploring >> >> If the clocks of your twins aren't coordinate clocks--as implied by the >> fact that they are said to "pass through" a given set of spatial >> coordinates, rather than being permanently attached to them--then those >> readings ARE NOT TIME COORDINATES in whatever coordinate system you are >> using to define position coordinates. They are PROPER TIMES for the twins >> (specifically the proper time between their birth and any other moment on >> their worldline, if they represent ages), which are DIFFERENT from >> coordinate times. Of course we could program our coordinate clocks in such >> a way that the coordinate clock at x,y,z, also showed a reading of 30 years >> as one of the twins was passing next to it, but in this case it would NOT >> show a reading of 30 years when the other twin was passing next to it, so >> the event of that other twin's clock reading 30 would NOT be assigned a >> coordinate time of 30 in this coordinate system. And the coordinate clock >> time need not agree with either twin's proper time--for example, the >> coordinate clocks could just be designed to show the current date (in >> Greenwich mean time, say), in which case the event of the first twin having >> a PROPER time of 30 would have a COORDINATE time of 1989, and the event of >> the second twin having a PROPER time of 30 would have a COORDINATE time of >> 1999. >> >> Do you disagree that in relativity the coordinate time of any event in a >> given coordinate system is defined in terms of local reading on the >> COORDINATE CLOCK for that system that was right next to the event when it >> happened, and that it may differ from the time shown on a clock which isn't >> a coordinate clock for that coordinate system? Yes or no? >> >> Do you disagree that GIVEN such a definition of coordinate time, then two >> events which have the same coordinate position and coordinate time in a >> single coordinate system will necessarily have the same coordinate position >> and coordinate time as one another in all other coordinate systems, and >> they will also have happened at the "same point in spacetime" according to >> the operational definitions I gave earlier? Yes or no? >> >> Jesse >> >> >> >> >> >> On Sunday, February 23, 2014 11:11:49 AM UTC-5, jessem wrote: >> >> >> On Sun, Feb 23, 2014 at 8:22 AM, Edgar L. Owen <edga...@att.net> wrote: >> >> Hi Jesse, >> >> First, my name is Edgar, not Edward.... >> >> OK, even though I've answered this question of yours on several >> occasions, I'm willing to finally put it to bed once and for all. >> >> So please state in a non-ambiguous manner exactly what the question is >> AND what you think the implication of it is. >> >> As I understand it your question is there some exact relativistic clock >> time analogue of two tape measures crossing. Is that correct? And if so, >> what's the point? What difference does it make in your mind one way or the >> other? >> >> >> Clear evidence that you are not even bothering to follow the links when I >> link back to an earlier post and ask you to address a question. The tape >> measure analogy is another issue you haven't responded to when I asked you >> about it, and I would eventually like to discuss that as well, but the >> question I was asking about here had nothing to do with tape measures or >> geometric analogies, I stated several times that it was the question in the >> last two paragraphs of my post at https://groups.google.com/d >> /msg/everything-list/jFX-wTm_E_Q/LF0Xcds_qtQJ which begins with "If we >> have some coordinate system where relativity predicts the event of Alice's >> clock reading 30 happens at exactly the same space and time coordinates as >> the event of Bob's clock reading 40 ... ". This question deals with the >> issue of whether two events that have the sam >> >> ... > > -- > 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. > -- 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.
