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T of t' and x of x' with respect to S

  1. Mar 27, 2008 #1
    1. Let S be a stationary x-coordinate system. Let S' be a stationary x'-coordinate system. Let the x'-axis of system S' coincide with the x-axis of system S. Let system S' move along the x-axis of system S with constant velocity v in the direction of increasing x. Let observers (every one of them with the ability to count the time t in such a way that each count is exactly 1s, being synchronous to each other) be stationed along every point on the x-axis of S. Let observers (every one of them with the ability to count the time t' in all respects like the observers which are stationed along the x-axis of S) be stationed along every point on the x'-axis of S'. Let the origin of the moving system S' coincide with the origin of the stationary system S at the time t = t' = 0s.

    2. If a ray of light is emitted from the origin of the moving system S' at the time t' = 0s to a point x' on the positive side of the origin of S', then the time t' that the ray of light takes to travel the distance x' from the origin of S' is given by the equation

    t' = x'/c.

    With respect to the stationary system S, the time t of t' is given by the Lorentz transformation equation

    t = (t' + v*x'/sq(c))/sqrt(1 - sq(v/c)).

    With respect to the stationary system S, the distance x of x' is given by the Lorentz transformation equation

    x = (x' + v*t')/sqrt(1 - sq(v/c)).

    3. If the ray of light is emitted from the origin of the stationary system S at the time t = 0s to a point x on the positive side of the origin of S, then the time t that the ray of light takes to travel the distance x from the origin of S is given by the equation

    t = x/c.

    With respect to the moving system S', the time t' of t is given by the Lorentz transformation equation

    t' = (t - v*x/sq(c))/sqrt(1 - sq(v/c)).

    With respect to the moving system S', the distance x' of x is given by the Lorentz transformation equation

    x' = (x - v*t)/sqrt(1 - sq(v/c)).
     
  2. jcsd
  3. Mar 27, 2008 #2

    JesseM

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    Yes, all of that looks like a correct application of the Lorentz transformation to me.
     
  4. Mar 28, 2008 #3


    note: It is necessary that the observer that counts the time t = 0s be the same observer that counts the time t' = 0s because it is impossible for two distinct observers to occupy the same space-time point where the two space-time points (x, t) = (0m, 0s) and (x', t') = (0m, 0s) coincide.


    note: It is important here to note that the space-time point (x' t') coincides with the space-time point (x, t), and that the observer that counts the time t' is the same observer that counts the time t because two distinct observers cannot occupy the same space-time point where the two space-time points (x', t') and (x, t) coincide. How can the same clock or the same observer count the time t' and the time t at the same space-time point where the two space-time points (x', t') and (x, t) coincide? It is impossible for the same observer, who occupies the same space-time point where the two space-time points (x', t') and (x, t) coincide, to count the time t' and the time t.
     
    Last edited: Mar 28, 2008
  5. Mar 28, 2008 #4

    JesseM

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    It isn't really necessary to worry about that sort of thing, the clocks and measuring-rods that are used in the derivation of the Lorentz transform are imagined to be idealized ones that can pass through one another without hitting each other, and where each clock and marking on the measuring rods can also be imagined as infinitely small so that every single point in space will have its own unique clock and marking at every moment. In practice of course this is an idealization that wouldn't be exactly possible, but you could still imagine two measuring-rods with clocks mounted on them which are moving alongside each other with a very small distance between them, and if an event happens between the two measuring-rods, each observer can assign the event coordinates based on which marking on his own measuring-rod had a detector that was first to see the event (i.e. which marking was closest to it, with the distance being very small), and what the clock at that marking read when the detector first saw the event. In the limit as the distance between the measuring rods approached zero, and the size of the individual clocks/detectors/markings approached zero too, this scenario would approach the perfectly idealized case I described earlier.
     
    Last edited: Mar 28, 2008
  6. Mar 28, 2008 #5
    Is this thread meant to be a correction to your earlier one with the subject, "stationary and moving clocks"? It seems very similar, except that now you are no longer placing the origins of the S and S' systems at distance a in S at time t=0. That's good - it makes the transformations easier.

    So, was the point of the previous thread to ask this question about observers coexisting at the same space-time point? If so, then I think my response to that in the other thread agrees with JesseM's response here. I suggest reading them both to see if that satisfies your objections.
     
  7. Mar 31, 2008 #6
    I was thinking more along the following line:

    If we write the following program:

    t' = 1.0
    print t'

    the output of the computer will be

    1.0.

    Here the computer CPU stores the number 1.0 into, say, location 2003.

    If we write the following program:

    t = 1.00009940704
    print t

    the output of the computer will be

    1.00009940704.

    Here the computer CPU stores the number 1.00009940704 into, say, location 2004.

    But when the ray of light departs from, say, the origin of a moving system S' to, say location x' = 2003, while the location 2003 itself on the x'-axis of system S' moves along the x-axis of a stationary system S to a location x until location x' = 2003 coincides with location x, only one piece or a unique string of information can be stored into that location x' = 2003, and with respect to the stationary system S, we can then also lable location x = 2003.

    In other words, if we write the following program:

    t' = 1.0
    print t'

    the output of the computer will be

    1.0.

    Here the computer CPU stores the number 1.0 into location 2003.

    And if we write the following program:

    t = 1.00009940704
    print t

    the output of the computer will

    1.00009940704.

    But here, with respect to the stationary system S, the computer CPU stores the number 1.00009940704 also into location 2003 (provided the computer is moving together with the moving system S' along the x-axis of the stationary system S with velocity v = 29,800m/s).

    Otherwise, if we write the following program:

    t' = 1.0
    print t
    t' = 1.00009940704
    print t'

    the output of the computer will be

    1.00009940704.

    Here it was necessary for the computer CPU to delete the number 1.0 from location 2003 in the process of storing the number 1.00009940704 into location 2003.
     
    Last edited: Mar 31, 2008
  8. Mar 31, 2008 #7

    JesseM

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    I don't see what locations in a computer's memory have to do with the coordinates that different frames assign to the same event--how is this supposed to relate to physics? Do you think points in space are "storing" numbers in a way that's analogous to memory addresses in a computer? If so, why? Anyway, in a computer a single memory address can only store 8 bits of data, the number 1.00009940704 could not be stored at a single address.
     
  9. Mar 31, 2008 #8
    Apart from that, 2 images can easily occupy the same location at the same time. So instead of real objects (with mass) you could use the image of a crosshair or the edge of a shadow -whatever.
     
  10. Apr 2, 2008 #9
    i will accept that the space-time point (x', t') on the x'-axis of the moving system S' will not perfectly coincide with the space-time point (x, t) on the x-axis of the stationary sytem S (which contradicts the linear nature of the Lorentz transformation equations), but nevertheless, our real world is the world of real objects (with masses). my concern is the possibility of the use of the Lorentz transformation equations by criminal high tech gurus as a way of calculating the time t' and, with respect to the stationary system S, the time t, hiding the fact that it is impossible for the same identical clock, located at a specified space-time point, to calculate the time t' and the time t. such fraudulent use of the Lorentz transformation equations, coupled with our advance computer technology could lead to crimes such as identity theft and so forth. i am just saying that we ought to be careful.
     
    Last edited: Apr 2, 2008
  11. Apr 2, 2008 #10

    JesseM

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    Are you familiar with the notion of limits in calculus? If so, do you agree that in the limit as the two rulers get closer and closer together and the clocks at each marking get smaller and smaller, the separation between (a given marking x on the first ruler and the clock at that marking reading t) and (a given marking x' on the second ruler and the clock at that marking reading t') will go to zero? And would you agree that the Lorentz transformation describes which points on the first ruler/clock system will approach zero separation from points on the second ruler/clock system in the limit as the width of the rulers goes to zero, the size of the clocks goes to zero, and the separation between the two rulers goes to zero?
    Before you start labeling the use of the Lorentz transform as "criminal", note that limiting-case idealizations like this must be used in every branch of physics where we keep track of the position and time of different events, Newtonian physics too. After all, how can you say that a particle occupies a specific coordinate position like x=5 meters, when realistically there will always be a slight separation between the object and whatever measuring-device you use to assign it a position? Do you want to discard all of physics dealing with the motion of objects over time (i.e., all of dynamics) because they rely on such limit-case idealizations?
     
    Last edited: Apr 2, 2008
  12. Apr 3, 2008 #11
    Let us say that your frame of reference S' moves along the x-axis of my frame of reference S with velocity v in the direction of increasing x. Let us also say that a ray of light departs from the origin of your frame of reference S' at the time t'0 = 0s to a point x' (your location), arriving there at the time t'1 = 3600s (the time that your clock counted).

    Your location then can be implicitly given by

    3600s = x'/c.

    Then x and t are the distance of x' and the time of t' respectively with respect to my frame of reference S.

    The location x of x' can also be given implicitly by

    x = (x' + v*(3600s))/sqrt(1 - sq(v/c)).

    But as the ray of light approaches x', the point x' itself approaches the point x.

    Let us say that we are really talking about the limit concept. In other words, the ray of light really never reaches x' and the point x' itself never really reaches the point x. But for argument sake, let us say that the ray of light does reach the point x', and that the point x' itself reaches the point x, then we must also say that you and I become the same person (we never do though, but since for argument sake we are saying that the ray of light reaches the point x', and that the point x' itself reaches the point x, we must say, for argument sake, that you and I will then become the same person).



    When the ray of light departs from, say, the origin of the moving system S' to, say location x' = 2003, while the location 2003 itself on the x'-axis of system S' moves along the x-axis of a stationary system S to a location x until location x' = 2003 coincides with location x, only one piece or a unique string of information can be stored into that location 2003; consequently, with respect to the stationary system S, we must also label location x = 2003.

    If we write the following program (the programmer and the computer are moving together with the moving system S' with velocity v = 29,800m/s along the x-axis of the stationary system S; here the Earth could serve as the moving system S', and the solar system as the stationary system S):

    t' = 1.0
    print t'

    the output of the computer will be

    1.0.

    Here the computer CPU stores the number 1.0 into, say, location 2003.

    If we write the following program (again, the programmer and the computer are moving together with the moving system S' with velocity v = 29,800m/s along the x-axis of the stationary system S):

    t = 1.00009940704
    print t

    the output of the computer will be

    1.00009940704.

    Here the computer CPU stores the number 1.00009940704 into, say, location 2004.

    But if we write the following program (here we imagine that the programmer is stationed on the x-axis of the stationary system S and the computer is moving together with the moving system S' with velocity v = 29,800m/s along the x-axis of the stationary system S):

    t' = 1.0
    print t'

    the output of the computer will be

    1.0.

    Here the computer CPU stores the number 1.0 into, say, location 2003.

    And if we write the following program (here we imagine again that the programmer is stationed on the x-axis of the stationary system S and the computer is moving together with the moving system S' with velocity v = 29,800m/s along the x-axis of the stationary system S):

    t = 1.00009940704
    print t

    the output of the computer will be

    1.00009940704

    But here, with respect to the stationary system S, the computer CPU deletes the number 1.0 from location 2003 in the process of storing the number 1.00009940704 into location 2003.

    It is possible that complex computer programs may be sold to Government Offices, Universities, or Industry as applications that calculates the time t' and the time t as if the user of the application and the computer are moving together with system S' (but hiding the fact that the application actually simulates the calculation of the time t' and the time t as if the computer is moving together with the moving system S' with velocity v along the x-axis of the stationary system S and the user of the application is stationed on the x-axis of the stationary system S). It is possible that high tech gurus may be selling these applications in order to commit crimes, such as identity theft and fraudulently substitute with stealth one person for another in Government, Universities, or Industry. Laws may be approved by our Government and other Government Agencies based on information obtained using such applications.
     
    Last edited: Apr 3, 2008
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