On Wed, Mar 5, 2014 at 8:19 AM, Edgar L. Owen <[email protected]> wrote:
> Jesse, > > First I see no conclusion that demonstrates INtransitivity here or any > contradiction that I asked for. Did I miss that? > No, I was just asking if you agreed with those two steps, which show that different pairs of readings are simultaneous using ASSUMPTION 2. If you agreed with those, I would show that several further pairs of readings must also be judged simultaneous in p-time using ASSUMPTION 1, and then all these individual simultaneity judgments would together lead to a contradiction via the transitivity assumption, ASSUMPTION 3. I already laid this out in the original Alice/Bob/Arlene/Bart post, but since you apparently didn't understand that post I wanted to go over everything more carefully with the exact x(t) and T(t) functions given, and every point about simultaneity stated more carefully. I thought you would be more likely to answer if I just gave you two statements to look over and verify rather than a large collection of them, but if you are going to stubbornly refuse to answer the opening questions until I lay out the whole argument, here it is in full: ASSUMPTION 1. If two observers are at rest in the same inertial frame, then events on their worldlines that are simultaneous in their rest frame are also simultaneous in p-time ASSUMPTION 2. If two observers cross paths at a single point in spacetime P, and observer #1's proper time at P is T1 while observer #2's proper time at P is T2, then the event of observer #1's clock showing T1 is simultaneous in p-time with the event of observer #2's clock showing T2. ASSUMPTION 3. p-time simultaneity is transitive Please have another look at the specific numbers I gave for x(t), coordinate position as a function of coordinate time, and T(t), proper time as a function of coordinate time, for each observer (expressed using the inertial frame where A and B are at rest, and C and D are moving at 0.8c), and then tell me if you agree or disagree with the following two statements: For A: x(t) = 25, T(t) = t For B: x(t) = 0, T(t) = t For C: x(t) = 0.8c * t, T(t) = 0.6*t For D: x(t) = [0.8c * t] + 9, T(t) = 0.6*t - 12 STATEMENT 1. Given the x(t) functions for B and C, we can see that they both pass through the point in spacetime with coordinates x=0, t=0. Given their T(t) functions, we can see that B has a proper time T=0 at those coordinates, and C also has a proper time T=0 at those coordinates. Therefore, by ASSUMPTION 2 above, the event of B's proper time clock reading T=0 is simultaneous in p-time with the event of C's proper time clock reading T=0. Agree or disagree? STATEMENT 2. Given the x(t) functions for A and D, we can see that they both pass through the point in spacetime with coordinates x=25, t=20. Given their T(t) functions, we can see that A has a proper time T=20 at those coordinates, and D has a proper time T=0 at those coordinates. Therefore, by ASSUMPTION 2 above, the event of A's proper time clock reading T=20 is simultaneous in p-time with the event of D's proper time clock reading T=0. Agree or disagree? STATEMENT 3. At t=0 in this frame, both A and B have a proper time of T=0; these readings are simultaneous in this frame. Since A and B are both at rest in this frame, by ASSUMPTION 1 above, the event of A's proper time clock reading T=0 is simultaneous in p-time with the event of B's proper time clock reading T=0. Agree or disagree? STATEMENT 4. C's worldline passes through the point x=0, t=0, and at this point C's proper time clock reads T=0. D's worldline passes through the point x=25, t=20, and at this point D's proper time clock reads T=0. These events are not simultaneous in this frame, but using the Lorentz transformation we can see that they ARE simultaneous in the frame where C and D are at rest. Therefore, by ASSUMPTION 1 above, the event of C's proper time clock reading T=0 is simultaneous in p-time with the event of D's proper time clock reading T=0. Agree or disagree? Note: This statement is perhaps the subtlest if you aren't too familiar with the math of SR--in case you didn't know, the Lorentz transformation is used when we know the coordinates x,t of an event in one inertial frame, and we want to find the coordinates x',t' of the SAME event in a second inertial frame which is moving at speed v relative to the first (a good intro to various aspects of SR including the Lorentz transform can be found at http://en.wikibooks.org/wiki/Special_Relativity ). Assuming that the spatial origins of the two frames coincide when t=0 in the first frame and t'=0 in the second, and assuming that the first frame subsequently sees the origin of the second frame moving at speed v along the first frame's x-axis, the transformation equations are: x' = gamma*(x - v*t) t' = gamma*(t - (v*x)/c^2 ) Where gamma is the commonly-used relativistic factor 1/sqrt(1 - (v/c)^2). So with v=0.8c in this example, gamma works out to 1/sqrt(1 - 0.64) = 1/sqrt(0.36) = 1/0.6, and since we are using units of years for time and light-years for distance, c=1 in these units, so we have: x' = (x - 0.8*t)/0.6 t' = (t - 0.8*x)/0.6 The event of C's clock reading T=0 happened at x=0, t=0 in the first frame, so using the Lorentz transformation we can see it must have happened at coordinates: x' = (0 - 0.8*0)/0.6 = 0 t' = (0 - 0.8*0)/0.6 = 0 in the second frame. And event of D's clock reading T=0 happened at x=25, t=20 in the first frame, so using the Lorentz transformation we can see it must have happened at coordinates: x' = (25 - 0.8*20)/0.6 = (25 - 16)/0.6 = 9/0.6 = 15 t' = (20 - 0.8*25)/0.6 = (20 - 20)/0.6 = 0 in the second frame. So, both these events have the SAME time coordinate t' in this second frame, meaning they are simultaneous in this frame, and this second frame is just the C/D rest frame. If you aren't sure that this second frame really is the C/D rest frame, you can pick coordinates of DIFFERENT clock readings on C and D's worldline in the first frame, transform them into the second frame, and verify that C ALWAYS has a position coordinate of x'=0 in the second frame, and D ALWAYS has a position coordinate of x'=15 in the second frame. Anyway, if you agree with the four statements above, then it only remains to use transitivity (ASSUMPTION 3 above) to demonstrate the contradiction: STATEMENT 5: Since STATEMENT 3 said that A's clock reading T=0 was simultaneous in p-time with B's clock reading T=0, and STATEMENT 1 said that B's clock reading T=0 was simultaneous in p-time with C's clock reading T=0, by transitivity it must be true that A's clock reading T=0 is simultaneous in p-time with C's clock reading T=0. Agree or disagree? STATEMENT 6. Since STATEMENT 5 said that A's clock reading T=0 was simultaneous in p-time with C's clock reading T=0, and STATEMENT 4 said that C's clock reading T=0 was simultaneous in p-time with D's clock reading T=0, by transitivity it must be true that A's clock reading T=0 is simultaneous in p-time with D's clock reading T=0. Agree or disagree? STATEMENT 7. Since STATEMENT 6 said that A's clock reading T=0 was simultaneous in p-time with D's clock reading T=0, and STATEMENT 2 said that D's clock reading T=0 was simultaneous in p-time with A's clock reading T=20, by transitivity it must be true that A's clock reading T=0 is simultaneous in p-time with A's clock reading T=20. So, there's the final contradiction--A's proper time clock reading T=0 is shown to be simultaneous in p-time with A's proper time clock reading T=20, using only ASSUMPTION 1, ASSUMPTION 2 and ASSUMPTION 3 above to deal with how p-time simultaneity works. If you don't want to go through all 7 labeled STATEMENTS at once, then please just do as I originally suggested and look first at STATEMENT 1 and STATEMENT 2, and tell me if you can find a reason to disagree with either of those; then if you agree with those we can move on to the next two statements in a subsequent post, and so forth. > > But that really doesn't matter because second, you are NOT using MY method > because you are using ANOTHER coordinate clock FRAME rather than the frame > views of the parties of their OWN age relationships. > I AM using the A/B rest frame (which is what I assume you mean "the frame views of the parties" for A and B), you can see that from the fact that A and B have a CONSTANT x(t) function (A's x(t) function is x(t)=25, meaning A remains at rest at position x=25 in this frame, while B's is x(t)=0, meaning B remains at rest at x=0 in this frame. You can also see that both have the T(t) function T(t) = t, meaning that their own proper time is exactly equal to coordinate time in this frame at any given moment). As for C and D, although I have stated their coordinate paths in terms of the A/B rest frame, you can see in my note below STATEMENT 4 that I used the Lorentz transformation to show that two events on their worldlines would be simultaneous in THEIR OWN REST FRAME, even though these events are not simultaneous in the A/B rest frame. So again I am using the clocks' own rest frame, using the rule (ASSUMPTION 1) that events on these clocks' worldlines that are simultaneous in their own rest frame must also be simultaneous in p-time. Anyway, if you continue to disagree please don't just be evasive and make broad statements about my argument clearly aren't true when you look at the details--please just go through the numbered STATEMENTS above in order and tell me the first one that you would disagree with, and why. 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