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Definition of relativity's interval

  1. Mar 18, 2004 #1
    Definition of relativity's "interval"

    Okay I have Kip Thorne's text in front of me that says the following:

    If [tex](\Delta s)^2 > 0 [/tex] the events are called spacelike
    If [tex](\Delta s)^2 = 0 [/tex] the events are called lightlike
    If [tex](\Delta s)^2 < 0 [/tex] the events are called timelike

    Then I have David W. Hogg's text in front of me that says:

    If [tex](\Delta s)^2 > 0 [/tex] the events are called timelike
    If [tex](\Delta s)^2 = 0 [/tex] the events are called lightlike
    If [tex](\Delta s)^2 < 0 [/tex] the events are called spacelike

    No joke. Which one's right? Understanding of this cursed interval was just dawning on me when I saw this. I was coming to the conclusion that [tex]\Delta s [/tex] is the distance light traveled in the frame of reference minus the distance between the events in that frame. I was thinking if this value is positive, light traveled farther than the distance between the events, thus there is a valid space-like interval.

    If you don't mind could you also clear up what the sam heck this interval means?! I'm not sure I'm getting it.

    TIA
     
    Last edited: Mar 18, 2004
  2. jcsd
  3. Mar 18, 2004 #2
    Re: Definition of relativity's "interval"

    It depends on how you define the signature of the metric. For example, the Minkowski metric can be written

    [tex]ds^2 = -dt^2 + dx^2+dy^2+dz^2[/tex]

    or equivalently it can be

    [tex]ds^2 = +dt^2 - dx^2-dy^2-dz^2[/tex]

    It's all really a matter of preference.
     
  4. Mar 18, 2004 #3
    Re: Re: Definition of relativity's "interval"

    Thank you very much, that pinpoints the inconsistencey. Thorne defines the metric as [tex]ds^2 = -dt^2 + dx^2+dy^2+dz^2[/tex], where Hogg defines it with an opposite sign. And thank you for the use of the term "Minkowski metric." I was not aware that it is the technical term for interval.

    What's the significance of the Minkowski Metric? If for example we define two events as having spacelike intervals, what does that mean? I want to say that it means they could influence each other since they'd exist within each other's light cones, where a timelike interval means the events are outside each others' light cones.
     
    Last edited: Mar 18, 2004
  5. Mar 18, 2004 #4

    EL

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    Re: Re: Re: Definition of relativity's "interval"

    It's the other way...
     
  6. Mar 19, 2004 #5
    So spacelike means two events are outside each other's light cones?
     
  7. Mar 19, 2004 #6
    Re: Re: Re: Definition of relativity's "interval"

    The Minkowski metric pertains to what is called Lorentz coordinates, i.e. the system of coordinates whose spatial coordinates are Cartesian (x, y, z) and which pertains to an inertial frame of reference.

    As far as timelike/spacelike/lightlike - To see this cleary think of a (ct,x) spacetime diagram corresponding to an inertial frame of reference. You'll notice that the worldline corresponding to a luxon (particles for which v = c) has a slope of 1. The worldine of a tardyons (particles for which v < c) has a slope greater than 1 and the worldline corresponding to a tachyon (particles for which v > c) has a slope of less than one.

    If you have two events with a timelike spacetime seperation then a tardyon can be present at both events.

    If you have two events with a lightlike spacetime seperation then a luxon can be present at both events.

    If you have two events with a spacelike spacetime seperation then a tachyon can be present at both events.
     
    Last edited: Mar 20, 2004
  8. Mar 19, 2004 #7
    Strange way to put it, but I got it! Thanks very much.
     
  9. Mar 19, 2004 #8
    The reason I phrased it as such was that it doesn't depend on any convension for the Minkowski metric if you've noticed.
     
  10. Mar 20, 2004 #9

    EL

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    Re: Re: Re: Re: Definition of relativity's "interval"

    Should be v<c...
     
  11. Mar 20, 2004 #10
    Re: Re: Re: Re: Re: Definition of relativity's "interval"

    Whoops! Thanks EL!
     
  12. Mar 20, 2004 #11

    jcsd

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    The important thing about spacetime intervals is that they are Lorentz invariant and therfore the same for all observers. If you mutiply the Minowski metric by a coefficient (in the above examples -1) it will not affect it's Lorentz invariance, so it becomes a matter of convention and taste as to which form you use.
     
    Last edited: Mar 20, 2004
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