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Simple argument for area conservation under Lorentz transformation?

  1. Nov 23, 2009 #1

    bcrowell

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    A Lorentz boost along the x axis conserves area in the x-t plane. Can anyone think of an argument to prove this fact that's simpler than the one below? I don't want to assume the form of the Lorentz transformation, because I want to use the conservation of area as part of a derivation of the Lorentz transformation. I also don't want to assume that c is the same in all frames. The only assumptions are supposed to be:

    1 Spacetime is homogeneous and isotropic. No point has special properties that make it distinguishable from other points, nor is one direction distinguishable from another.

    2 Inertial frames of reference exist. These are frames in which particles move at constant velocity if not subject to any forces. We can construct such a frame by using a particular particle, which is not subject to any forces, as a reference point.

    3 Equivalence of inertial frames: If a frame is in constant-velocity translational motion relative to an inertial frame, then it is also an inertial frame. No experiment can distinguish one preferred inertial frame from all the others.

    4 Causality: Observers in different inertial frames agree on the time-ordering of events.

    5 No simultaneity: The experimental evidence shows that observers in different inertial frames do not agree on the simultaneity of events.

    Start with a rectangular area in the x-t plane, and for simplicity, let this rectangle have unit area. Then the area of the new parallelogram is still 1, by the following argument. Let the new area be A, which is a function of v. By isotropy of spacetime (assumption 1), A(v)=A(-v). Furthermore, the function A(v) must have some universal form for all geometrical figures, not just for a figure that is initially a particular rectangle; this follows because of the definition of affine area in terms of a dissection by a two-dimensional lattice, which we can choose to be a lattice of squares. Applying boosts +v and -v one after another results in a transformation back into our original frame of reference, and since A is universal for all shapes, it doesn't matter that the second transformation starts from a parallelogram rather than a square. Scaling the area once by A(v) and again by A(-v) must therefore give back the original square with its original unit area, A(v)A(-v)=1, and since A(v)=A(-v), A(v)=[itex]\pm1[/itex] for any value of v. Since A(0)=1, we must have A(v)=1 for all v. The argument is independent of the shape of the region, so we conclude that all areas are preserved by Lorentz boosts. The argument is also purely one about affine geometry (it would apply equally well to a Euclidean space), so there is no reason to expect the area A in the (t,x) plane to have any special physical significance in relativity; it is simply a useful mathematical tool in the present discussion.

    For anyone who wants to see the whole derivation of the Lorentz transformation from assumptions 1-5, it's here: http://www.lightandmatter.com/html_books/genrel/ch01/ch01.html#Section1.7 [Broken] . I'm basically looking to see if I can make the argument simpler and easier to understand for people with less math and physics background.

    Thanks in advance!

    -Ben
     
    Last edited by a moderator: May 4, 2017
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  3. Nov 23, 2009 #2

    Chronos

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    1. Agreed
    2. Illogical
    3. Agreed
    4. Illogical
    5. Agreed.
     
  4. Nov 23, 2009 #3

    JesseM

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    This one is false in SR if you're talking about spacelike-separated events (though of course we know they can't have a causal influence on one another unless FTL influences exist)
    Can experimental evidence really show this? Doesn't it depend on what clock synchronization convention you choose to use? No experiment forces us to use Einstein's clock synchronization convention, although if we used a different one the laws of physics wouldn't obey the same equations in different inertial frames.
    Aren't you assuming here that if frame #1 sees frame #2 moving at velocity v, then frame #2 must see frame #1 moving at velocity -v? We can come up with coordinate transformations where this isn't true, so I think you'd need to either show how it follows from assumptions 1-5, or add it as a new assumption.
     
    Last edited: Nov 24, 2009
  5. Nov 24, 2009 #4

    bcrowell

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    Thanks for pointing out the problem with that. I should have worded it better. It was intended to refer only to timelike-related events, but at this stage in the derivation the Lorentz transformation hasn't been derived, and there's no notion of light cones. I'll change it to a statement that if one observer says that A causes B, every observer agrees that A causes B, and not the other way around.

    Another good point, thanks.The proof in the book actually refers back to specific experiments, but without that context, it's probably not that clear what I'm talking about. Basically I'm saying that experiments such as the Hafele-Keating experiment give non-null results.

    That's a good point. What I've actually given here is the assumptions that go into the derivation, plus one piece out of the middle of the derivation. Before we get to the point where I'm making the area-conservation argument, I've already proved that the transformations are linear. It seems to me that the property you're referring to must be true for linear transformations, whereas for nonlinear transformations there wouldn't be any unambiguous way to associate a single v with a particular transformation. Does that make sense to you, or can you think of an example where the transformation is linear and it doesn't have this property? I'm essentially defining v as the inverse slope of the transformed version of a line that was a constant-x line in the original frame.

    Thanks, JesseM, for your comments -- very helpful!
     
  6. Nov 24, 2009 #5

    JesseM

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    But adopting a different simultaneity convention wouldn't change our predictions about any local physical events, like an airplane clock and a ground clock being compared at a single location.
    Here's an example of a linear transformation without this property:

    x' = 2x - 2vt
    t' = t

    Suppose you have an object at rest in the x,t frame and pick two events on its worldline, like (x=0,t=0) and (x=0,t=1)...in the x',t' frame they translate to (x'=0,t'=0) and (x'=-2v,t'=1), so the object must have a velocity of -2v in the x',t' frame. The inverse transform would be:

    x = (1/2)*x' + vt'
    t = t'

    So if we pick an object at rest in the x',t' frame which goes through coordinates (x'=0,t'=0) and (x'=0,t'=1), then in the x,t frame the events transform to (x=0,t=0) and (x=v,t=1), meaning the object has a velocity of v in the x,t frame.
     
  7. Nov 24, 2009 #6
    May I ask a question?

    Causality is normally thought of in terms of the light cone/light sphere.

    Is this correct?
     
  8. Nov 24, 2009 #7

    JesseM

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    Yes, it's normally assumed causal influences can't travel faster than light.
     
  9. Nov 24, 2009 #8
    Therefore, when considering only one light sphere, how can observers disagree on the ordinality of events associated with that light sphere?
     
  10. Nov 24, 2009 #9

    Dale

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    They can't disagree about the order of events inside (timelike) or on (null) the light cone, but they can disagree about the order of events outside (spacelike) the light cone.
     
  11. Nov 24, 2009 #10
    I did not say inside the light cone.

    I said,
    "Therefore, when considering only one light sphere, how can observers disagree on the ordinality of events associated with that light sphere"?

    This means, when the light cone overtakes an object/event, that becomes part of the absolute past.

    Therefore, causality is implemented by the light cone in that the ordinality of events overtaken by the light sphere cannot have disagreement amongst observers.
    They may disagree on the timing and position, but not the ordinality.

    Is this correct?

    If it is correct, any disagreement with statement 4 by the OP is false.
     
    Last edited: Nov 24, 2009
  12. Nov 24, 2009 #11

    JesseM

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    By "light sphere" you mean a cross-section through the light cone, showing all the points inside the light cone at a single instant in some frame? While all these events can have a causal influence on the future event E that forms the apex of this past light cone, that doesn't mean they can have a causal influence on one another, each event has its own unique past light cone, and nothing outside its own past light cone can have a causal influence on it. So, if different events in the light sphere do not lie in one another's own past light cones, they cannot have a causal influence on one another and different frames can disagree about the order of these events, though all frames agree they lie in the past of the event E at the apex of the light cone that the light sphere was taken from.
     
  13. Nov 24, 2009 #12
    I am not deciding a causal influence. That would be silly since the light cone implements the agenda.

    I am talking about the ordinality of objects being taken over by the light sphere.

    Do you claim two events E1 and E2 can have disagreement amongst observers of ordinality with one light cone?

    If so, can you show me the geometric/topological representation?

    In other words, do you claim one object is overtaken by the light cone and then the second by one observer, but they are reversed by another observer?
     
  14. Nov 24, 2009 #13

    JesseM

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    Yes, and whenever two events cannot causally influence one another (because neither lies in the other event's past light cone), there can be disagreements between inertial frames about their order.
    As long as E1 does not lie in the past light cone of E2 and E2 does not lie in the past light cone of E1, then yes, different frames can disagree about their order.
    You mean like a spacetime diagram? I can tell you how to draw one yourself--just draw the x and t axes for frame A, and plot an event E at coordinates x=3 light-years, t=0 years. Then draw worldlines for two objects objects 1 and 2, object 1 being at rest at x=1, and object 2 being at rest at x=6. Then if you plot the future light cone of event E, you'll see that object 1 enters the future light cone of E at x=1, t=2 (event E1) while object 2 enters at x=6, t=3 (event E2), so in this frame E1 happened before E2. But if you then consider a different frame B moving at 0.8c relative to, and you draw a line of simultaneity for frame B which goes through the origin (this line be defined by the function x=0.8*t in units where c=1, so it'll have the inverse slope of the worldline of an observer at rest in frame B, which would be given by the function t=0.8*x), then you'll see that E1 lies above this line of simultaneity while E2 lies below it, so E2 must have happened before E1 in frame B.
    Yes, that would be true in the example above.
     
  15. Nov 24, 2009 #14
    We are not communicating.

    When I say overtaken by the light sphere, I am talking about entering the absolute past of the light cone.

    There cannot be disagreement about the ordinality of events entering the absolute past.

    I do not care about the future light cone and the OP did not in statement 4.

    Do you disagree?
     
  16. Nov 24, 2009 #15

    JesseM

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    That doesn't make sense, you can't "enter" the past light cone, you can only exit it. After all, the past light cone can be thought of as a light sphere whose radius is shrinking at the speed of light until it shrinks to a point at the exact position and time of the event E whose past light cone it is, so if you're outside it there's no way to catch up with its boundary without moving FTL.
     
  17. Nov 24, 2009 #16
    No, you do not understand.

    The light sphere proceeds at radius ct in all directions.

    As it proceeds, it overtakes objects which is then defined as events.

    This process of overtaking events and in what order is not in disagreement amoungst observers.


    What The Principle of Relativity has done is to make the Light Cone absolute.
    It is fair to say that what Einstein did is that he replaced the absoluteness of time and of space with the absoluteness of the speed of light. The Speed of Light is more fundamental than time and space.

    http://physics.syr.edu/courses/modules/LIGHTCONE/minkowski.html [Broken]

    Taking as event p a flash of light (light pulse) at time t0, all events that can be reached by this pulse from p form the future light cone of p, while those events that can send a light pulse to p form the past light cone of p.
    http://en.wikipedia.org/wiki/Light_cone

    Key Concept: Special Relativity in One Sentence.
    All speeds are relative, except for the speed of light, which is absolute.

    http://www.astronomy.ohio-state.edu/~ryden/ast162_6/notes23.html


    An important property of light cones is that they help us identify regions of spacetime which distinguish between relative space and time and absolute future, past, and elsewhere.
    http://physics.tamuk.edu/~hewett/Mo...vity/InvariantView/LightCones/LightCones.html

    Unlike the velocities of particles, the velocity of light is absolute (frame-independent)
    http://philsci-archive.pitt.edu/archive/00003986/01/elsevier2.pdf

    Recall that the future Light Cone of an event is the future-history of a light-flash of emitted at that event. It is an absolute surface for the Einstein-Minkowski Spacetime.
    http://www.phy.syr.edu/courses/modules/LIGHTCONE/lightconev.html [Broken]
     
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  18. Nov 24, 2009 #17

    JesseM

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    No, only the future light sphere expands at ct. The past light sphere contracts at ct. This is very basic SR, if you don't understand this you really are totally confused! Remember, the past light sphere represents the set of all events at a given moment that could send a signal traveling at the speed of light or less which would be able to reach the future event E that lies at the "top" of the past light cone. 1 year before E, any event within a radius of 1 light-year has time to send a signal to E. But then 1 second before E, only events within a radius of 1 light-second have time to send a signal to E. So, obviously the past light sphere is contracting as you get closer to the time of E.
    It's no use trying to get an understanding from out of context quotes if you don't understand the actual concepts and math they're talking about. None of these quotes says it's possible to "enter" a past light cone, and when they talk about light cones establishing an absolute future and past, they only mean that any event E1 or E2 in the past light cone of E will happen earlier than E in all frames, not that there cannot be disagreements about the order of E1 and E2 themselves.
     
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  19. Nov 24, 2009 #18

    Dale

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    It was not clear (to me) what you meant by that so I just covered all possibilities.

    OK, I think you have a little misunderstanding here. Objects are not events. An event is the Minkowski equivalent of a point in Euclidean geometry. Geometrically a classical point object is a Minkowski worldline (with extended objects being 4D hypervolumes). A light cone cannot "overtake" an event, an event is either inside, outside, or on a light cone. An object's worldline can cross a light cone, but worldlines have no temporal ordering. Only events can have a temporal ordering.

    No, it is not correct, for the reason indicated above. The phrase "events overtaken by the light sphere" is meaningless.

    JesseM was not so much disagreeing as clarifying, and his clarification was correct and was what was intended by the OP as indicated by the OP's response.
     
  20. Nov 24, 2009 #19
    Let's see now, the past light cone contracts at ct.

    That would imply, since nothing can travel at the speed of light, that the past overtakes all events once they happen.

    So, I could see myself being born then.
     
  21. Nov 24, 2009 #20
    So, I cannot define an event as to when light stikes an object?

    Why?

    I am thinking about R of S right now also.
     
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