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Minkowski Space-time (need help)

  1. Jul 15, 2010 #1
    If you have read my other threads, i am having trouble understanding special relativity. The issue seems to be my understanding of space-time.

    Space-time infers to me that two events are not separated by only a length in three dimensions, but also time, with time being essentially indistinguishable from a length. This concept can be explained by considering a supernova occurring several thousand light years away and it being observed looking up at the sky. To the observer looking up in the sky, the event of looking up in the sky coincides with the supernova and are to the observer simultaneous. But in fact this occurred several thousand years ago and the event can only be recognised as occurring, once the information has travelled the distance. It might turn out in fact that the supernova wasn't a supernova, but a plane in the sky that has turned on a light. These events are not separated by a massive distance and so it can be considered that the fact that they are simultaneous according to the observer, means that the light only turned on a marginally before it was recognised.

    This has been developed into a quantitative concept of four vector space (x,y,z,t) where time is indifferent to length. The resultant formula for space time is:
    S^2 = X^2 + Y^2 + Z^2 -(CT)^2
    S: the resultant distance between event A and event B
    X: the difference between the x coordinates of event A and event B or (Xb-Xa)
    Y: the difference between the Y coordinates of event A and event B or (Yb-Ya)
    Z: the difference between the Z coordinates of event A and event B or (Zb-Za)
    C: the speed of light in a vacuum (converts the time units into the length units used)
    T: the difference in when event A occurs and when event B occurs, if they were in the same position (Ta-Tb)

    Lets consider the supernova situation. Consider event A to be recognition of event B the supernova, the position of event A to be (0,0,0) and event B (100,0,0) units in light years. If in the same position then event A would occur 100 years after event B. As event A is the recognition of event B we know that the resultant distance in 4 vector space is zero and so a result for S must be zero.

    S^2 = (100 light years)^2 + 0^2+ 0^2 -(C*100 years)^2
    = 10,000-10,000
    = 0
    their for S = 0

    This formula works perfectly in this situation, but in the next situation, something in my logic goes wrong. Event A is a bunch of 2012 believers looking to the sky expecting the end of the world and event B is a supernova, that emits massive amounts radiation, which when this radiation hits earth, will kill the majority of life. If event A and event B are in the same position then event B would occur 10,000 years later. event A has a position of (0,0,0) and event B a position of (10000,0,0). A prediction of what S will equal is based on my logic that (1) event A will happen (2) 10,000 years will pass (3) the supernova occurs and (4) 10,000 years pass before the radiation hits earth. My conclusion is that S=20,000 light years but the math gives a different result.
    S^2 = (10,000 light years)^2 + 0^2+ 0^2 -(C*-10,000 years)^2
    = 1,000,000 - 1,000,000
    = 0
    their for S = 0
    According to the result found using the formula, the 2012 believers were right.

    My problem with space-time seems to be a difficulty in recognising time behaves the same as a dimension such as x,y,z as my logic seems to consider time behave the same as S or the resultant. My logic would have the formula for S as:
    S = (X^2+Y^2+Z^2)^0.5 -CT

    This is causing me huge confusion and i can not put my finger on what i have done wrong. please help.
     
  2. jcsd
  3. Jul 15, 2010 #2
    Hi Calebholiday,

    There were a few things about your post that I found confusing - perhaps it's nothing, but perhaps if you cleared them up, it would be helpful in understanding the rest of your post.

    Events should have 4-tuples as coordinates. Should this be:
    event A to be (0,0,0,0) and event B (100,0,0,0) ?

    If so, then you're saying that a supernova happens *here* in 100 *units* time - not light rays.

    Or should it have been

    event A to be (0,0,0,0) and event B (0,100,0,0)

    So the event is NOW (in relative coordinate system), but 100 lightyears along the x axis?

    What does this mean? A is an event - what does it mean to say if the event was in the same position? Do you intend to asking about a coordinate system where the spatial distance between events A and B is zero? This isn't possible if you had in mind the second case above, as A and B are space-like separated.
     
  4. Jul 15, 2010 #3

    Fredrik

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  5. Jul 15, 2010 #4
    Last edited by a moderator: Apr 25, 2017
  6. Jul 15, 2010 #5
    Soz about the new thread... i did make some mistakes in the other; considered the thread had changed topic and decided i probably should have done a new thread.

    Jargon and eloquence is once again is an issue of my posts. what is inferred by reading, is that event A and event B can have no distance separating them in space time, but have distance separating them in 3 dimensional space. If event B occurs simultaneously with event A, from event A's perspective, then there is no distance in space time separating them. As simultaneity is affected by distance in the 3 dimensions, it is necessary for one to consider what the duration between events is if they occupied the same position in x,y,z to obtain t. This is what was inferred at least from reading, but the fact that the formula doesn't correspond with the logic, suggests my inferences are wrong.

    if X,Y,Z,T is eqivenlent to (Xb-Xa),(Yb-Ya),(Zb-Za),(Tb-Ta) then in the first situation, event A has the coordinates of (0,0,0,100) and event B (100,0,0,0). The reason i did not write it this way is because it is confusing. What is inferred is that event A and event B occupy the same postion in space time, but to use the 4 vector co-ordinates logically rules that out. They don't occupy the same position in space time according to the co-ordinates and hence my problem with 4 vector space.

    Reading infers that space-time is the distance between two events, which is indistinguishable from the duration between two events. Is this inference wrong?
     
  7. Jul 15, 2010 #6

    Fredrik

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    That's right.

    That's wrong. First of all, there's no "event A's perspective". You can associate a coordinate system with a curve in spacetime, but not with a point. And "simultaneous in coordinate system S" only means that S assigns the same time coordinate to both events. The Minkowski square of the separation 4-vector has nothing to do with it.

    It isn't.

    If you meant (Xb-Xa,Yb-Ya,Zb-Za,Tb-Ta) and that t=0 at event B, then yes.

    An event is a position in spacetime, so A and B "occupy the same position in spacetime" if and only if A=B.

    They shouldn't.

    Minkowski spacetime is the set [tex]\mathbb R^4[/tex] with the standard vector space structure and a bilinear form g defined by [itex]g(x,y)=x_0y_0-x_1y_1-x_2y_2-x_3y_3[/itex]. The duration (in the coordinate system associated with the identity map) between x and y is [itex]|x_0-y_0|[/itex].
     
    Last edited: Jul 15, 2010
  8. Jul 15, 2010 #7
    I'm finding simultaneity difficult to describe. I considered that it would be best to describe it in terms of an observer at the position of each event and if the events occurred in the same position.

    In the first situation the observer of the supernova considers the supernova to occur at upon observing the sky. If in the same position in space then event A (observation) occurs 100 years after event B (supernova). If an observer was in the position of the supernova, the supernova (event B) would occur 200 years before, someone on earth observed the sky (event A). Simultaneity according to an observer is affected by resultant distance.

    Does this make sense at all?
     
  9. Jul 15, 2010 #8

    Dale

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    I think you misunderstand how times are assigned in SR. If I receive a signal now from an event 100 light years distant then I would say the event occured in 1910. I.e. All observers in SR are intelligent enough to account for the finite speed of light in determining when something happened.
     
  10. Jul 16, 2010 #9
    This is the kind of thing that i needed to know, Thank you.

    In what I would call a recognition event, where event A is the recognition of event B, that occurs a distance away from event A; does S in space-time =0 ?
     
  11. Jul 16, 2010 #10

    Fredrik

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    Now you mean position in space, right? You can certainly consider two events on Earth, or two events at the star, but I don't see why you'd want to.

    I'm afraid not. I can't make sense of it even when I take what you discussed with DaleSpam into account.

    You need to read about the standard synchronization convention. I linked to a post that has some of the details in my first reply. You should also be able to find an explanation in any SR book.

    Yes. The "S2" of the separation 4-vector is always =0 when a light signal goes from one of the events to the other (without being reflected somewhere along the way).
     
  12. Jul 16, 2010 #11
    Regarding the statement that doesn't make sense. What is stated is a reversal from 'situation one' in the first post. From the position of event A you are informed that event B has occurred at the same time event B occurs. From the position of event B, you are informed of event A's occurrence 200 years later. This is based on the distance in 3 dimensional space being 100 light years between and that event B occurs 100 years before event A

    The "occurring according to" can be replaced by "informed of occurring" which can be differentiated from "occurring" due to the distance between events in 3 dimensional space.

    When i was reading about Space-time i conceived 'situation two' quite early into the reading. Due to this I can not even understand the introduction, let alone any material that follows.

    If you find my description of 'situation one' satisfactory, I would like to explain 'situation two' with a more satisfactory discourse. Hopefully then I can move past the obstacle and finally get the theory.
     
  13. Jul 16, 2010 #12

    Dale

    Staff: Mentor

    Yes. The Minkowski norm is degenerate in that sense. For the ordinary Euclidean norm if the distance s between two points A and B is 0 then A=B, but in the Minkowski norm if the spacetime interval s between two events A and B is zero then that does not imply that A=B.

    If we have two events in units where c=1 with spacetime coordinates
    [tex]A=(t_A,x_A,y_A,z_A)[/tex]
    [tex]B=(t_B,x_B,y_B,z_B)[/tex]
    [tex]s^2=(t_A-t_B)^2-(x_A-x_B)^2-(y_A-y_B)^2-(z_A-z_B)^2[/tex]

    Then they are called "simultaneous" if [itex]t_A=t_B[/itex] and they are called "co-located" if [itex]x_A=x_B[/itex] and [itex]y_A=y_B[/itex] and [itex]z_A=z_B[/itex]. They are the same event if they are both simultaneous and co-located. Their separation is called "timelike" if [itex]s^2>0[/itex], or "spacelike" if [itex]s^2<0[/itex], and "lightlike" or "null" if [itex]s^2=0[/itex]. The separation is lightlike for any two events where a light signal is sent from one to the other, but since [itex]s^2=0[/itex] does not imply [itex]t_A=t_B[/itex] such events are not generally simultaneous.
     
  14. Jul 16, 2010 #13

    Fredrik

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    I understand that you're talking about an event where the supernova is observed, and an event at the location of the supernova where the observation of the supernova is observed. But there's something very strange about how you're saying it. It sounds like you're saying that if you're at the location of the supernova 200 years after the star went supernova, this event is simultaneous with the observation event on Earth. It isn't. The second observation event occurs 100 years after the first, so they're certainly not "simultaneous", which means "occuring at the same time"...or to be more precise "assigned the same time coordinate".

    I'm still not sure what the source of the confusion is. If A, B and C are the three events "star goes nova", "light from nova reaches Earth" and "light reflected by a mirror on Earth comes back to the position of the nova", and we define t=0 and x=0 at event B, then the coordinates (written in the form (t,x)) of these events are:

    A: (-100,100)
    B: (0,0)
    C: (100,100)

    Nothing more needs to be said to describe this sequence of events.
     
  15. Jul 16, 2010 #14
    based on the intelligence assumption replace 'simultaneous' with 'informed of'. if the supernova happens in 1910, earth is 100 light years away and someone observed the supernova on earth in 2010, then if you assumed the position of the supernova, you would be informed of observation on earth in 2110.

    This make sense?
     
  16. Jul 17, 2010 #15
    Situation Number Two

    Event A: 2012 theorists expecting impending doom (0,0,0,0)
    Event B: Supernova in distant space (10000,0,0,10000)

    S^2 = X^2 + Y^2 + Z^2 -(CT)^2
    S: the resultant distance between event A and event B
    X: the difference between the x coordinates of event A and event B or (Xb-Xa)
    Y: the difference between the Y coordinates of event A and event B or (Yb-Ya)
    Z: the difference between the Z coordinates of event A and event B or (Zb-Za)
    C: the speed of light in a vacuum (converts the time units into the length units used)
    T: the difference in when event A occurs and when event B occurs, if they were in the same position (Ta-Tb)

    my logic would say in this instance before doing the calculations, that S should be 20,000 light years. As event A will occur 10,000 years will pass, the supernova will occur and then 10,000 years will pass before position A will be informed of event B.

    The problem is if you do the calculation using the formula S=0 not 20,000

    How is the logic wrong when it is correct in situation 1 ?
     
  17. Jul 17, 2010 #16
    Am still having problems with your notion of 'informed of', and what the events are.

    EITHER
    Event A - theorists *expecting* impending doom? Do you mean, they are about to be doomed by the explosion, or they somehow come to form the belief that the earth will be, at some point, wiped out by this explosion? These are two different events.

    If the idea is that they are *about* to be wiped out by the effects of the supernova, and event A is their being wiped out, event B the supernova explosion, then I see why you expect s^2 to be relevant to the time they think passes between the two events. BUT IN THAT CASE the supernova explosion could not possibly have the coordinates you give it. Its coordinates place it in the future AND far away - it's t coordinate would have to be negative if its effects were about to be felt.

    OR: Event A - the moment at which theorists, due to a keep theoretical calculation, realise that far off system will go supernova in (10000 0 0 10000), and realise the effects will reach the earth, destroying everything. In that case, the coordinates make sense, BUT IN THIS CASE there's no reason to expect s^2 to give information about THE TIME at which the impact will happen. s^2 only concerns the relations A and B - the event of realisation and the event of the supernova. You need to introduce a new event C: the moment of impact, and talk about the s^2 of this event from A.

    Interestingly, the event of realisation and the event of supernova lie on the earth's light cone. But that's not germane to your problem.
     
  18. Jul 17, 2010 #17

    Fredrik

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    Yes.

    So this time we're talking about a star 10000 light-years away that goes supernova 10000 years from now.

    As I said before, I wouldn't call S "distance". It's just the non-negative square root of the Minkowski square of the 4-vector (X,Y,Z,T).

    Your definition of T doesn't make any sense. It's like talking about the difference between the numbers 7 and 5 if they were the same number. Why don't you define it as the difference between the time coordinates of event A and event B?

    Then you haven't really looked at the definition of S2. Did you read DaleSpam's comments about "separation"? You should.

    S2 is always =0 when there's light going from one of the events to the other. I have no idea how you got the number 20000.

    It's not correct in any situation.
     
    Last edited: Jul 17, 2010
  19. Jul 17, 2010 #18
    Imagine event A is like the end of a countdown, to the instant when a whole buch of people will think the world will end, based on some new age belief. If S=0 then the radiation traveling at C from event B should hit earth during event A. Common sense says thats not the case in this situation and the 2012 theorists are out by 20,000 years as to when one could expect the cataclysmic event.
     
  20. Jul 17, 2010 #19

    Fredrik

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    That would only make sense if S2 is the non-negative square of a time difference and it clearly isn't defined that way.
     
  21. Jul 17, 2010 #20
    If this is what you have in mind then how could event A have coordinate (0,0,0,0) while event B has coordinate (10000,0,0,10000) as you say in post 15? If A is the origin of the coordinate system, and event B has coordinate (10000,0,0,10000), then, in this coordinate system, event B takes place 10000 years after event A and 10000 units away - i.e. after event A and far away.

    In this situation, the only thing that could be inferred was the new age hippies were woefully wrong. But I take it that's not the intended scenario
     
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