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Can SR be derived from causality alone?

  1. Aug 27, 2012 #1
    I'm wondering if causality is enough to derive the Minkowski metric and the Lorentz transformations. It seems to me that in order to ensure that some set of events maintain a causal relationship to each other under a transformation to a moving frame of reference, that there must be some restrictions on the metric in such a space and on the types of transformations of reference frames allowed in that space. Otherwise you could transform to some ref frame in which the causality is reversed. So is it a well established part of relativity that the Minkowski metric and/or the Lorentz transformations are derived solely from causality? If not, is there anyone working on this effort? Thanks.
  2. jcsd
  3. Aug 27, 2012 #2
    I don't see how causality alone could ensure Minkowski space and Lorentz transformation. In Galilean transformation, time is an absolute across all reference frames; If event A happened before event B in one frame (whether or not event A caused event B), then event A should precede event B in ALL Galilean frames, and by exactly the same amount of time as in the previous frame. Essentially, Galilean relativity also guarantees causality, as time is treated separately from space and undergoes no transformation, so there's no need for Lorentz transformation based on causality alone. Least, none that I can see.
  4. Aug 27, 2012 #3


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    You can do it in a system of postulates based on (1) causality, (2) relativity of simultaneity, and (3) symmetry of spacetime. You need 2 to rule out Galilean relativity, and 1 to rule out a theory in which a boost along the x axis is simply a rotation of the x-t plane. Here is a treatment I wrote along these lines: http://lightandmatter.com/html_books/0sn/ch07/ch07.html [Broken] This is not original with me. Here are some other references that take a similar approach:

    W.v.Ignatowsky, Phys. Zeits. 11 (1911) 972

    Rindler, Essential Relativity: Special, General, and Cosmological, 1979, p. 51

    Palash B. Pal, "Nothing but Relativity," Eur.J.Phys.24:315-319,2003, http://arxiv.org/abs/physics/0302045v1
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  5. Aug 27, 2012 #4
    Perhaps there is something about how the passage of time and length is compared. How do I in my frame of reference know how fast time passes for you in your frame? How do I know that a meter to me is a meter to you? In order to get that information, I'd have to have some channel of communication between us which implies a chain of cause and effect from you to me in order to get that information.
  6. Aug 27, 2012 #5
    This might also be helpful:
    E. C. Zeeman
    Causality Implies the Lorentz Group.
    J. Math. Phys. April 1964 Volume 5, Issue 4, pp. 490-493
  7. Aug 27, 2012 #6
    There would need to be a chain of cause and effect for information transfer between any two reference frames regardless of whether Galilean or Lorentz transformation is used. The fact that we need to exchange information causally in order to determine empirically what a meter is in each of our frames, and what a second is in each of our frames, is not exclusive to special relativity.
  8. Aug 27, 2012 #7
    It seems the absolute time and space dimensions of the Galilean Xformation is imposed and not derived. What information can we obtain from observation that this absolute coordinate system is real? This is not derived from observation which relies on causation. And so it is not derived from causation.
  9. Aug 28, 2012 #8
    I remember hearing in a talk once upon a time that knowing the casual structure of spacetime was sufficient to give you the metric up to a (constant?) scale factor; however, I've been unable to track down a reference for that statement.
  10. Aug 28, 2012 #9
    It's derived empirically, based on simple observations of the classical world perceived by Galileo and Newton and from simple logic. The Lorentz transformation is imposed in the same way Galilean transformation is imposed: They were both mathematical theories and needed empirical data and experimentation to support.

    Imagine we live in a world where the speed of light is not absolute, and Galilean transformations were physical reality at all relative velocities: Where would causality be violated? What I'm saying is, for example, from sin2x + cos2x = 1, we can derive sin2x = 1 - cos2x, and vice-versa. It's a one way street. From causality alone, we can derive any number of transformations; none of which are preferred without further restrictions, as bcrowell stated earlier.
  11. Aug 28, 2012 #10
    I found the following paper on the arXiv:


    "We present a novel derivation of special relativity based on the information physics of events comprising a causal set. We postulate that events are fundamental, and that some events have the potential to receive information about other events, but not vice versa. (This is causality) This leads to the concept of a partially-ordered set of events, which is called a causal set. Quantification proceeds by selecting two chains of coordinated events, each of which represents an observer, and assigning a valuation to each chain. Events can be projected onto each chain by identifying the earliest event on the chain that can be informed about the event. In this way, each event can be quantified by a pair of numbers, referred to a pair, that derives from the valuations on the chains. Pairs can be decomposed into a sum of symmetric and antisymmetric pairs, which correspond to time-like and space-like coordinates. From this pair, we derive a scalar measure and show that this is the Minkowski metric. The Lorentz transformations follow, as well as the fact that speed is a relevant quantity relating two inertial frames, and that there exists a maximal speed, which is invariant in all inertial frames. All results follow directly from the Event Postulate and the adopted quantification scheme."

    When events are fundamental, and one event can have influence on another along a chain of events; this is a description of causality. The paper deriveds the Minkowski metric, the Lorentz transfromations, and the speed of light, all from causality.

    However, I'm not so sure about his method. When he says, "Events can be projected onto each chain by identifying the earliest event on the chain that can be informed about the event", this seems to already assume a Minkowski-like metric, right? Any help with these concepts would be appreciated.
    Last edited: Aug 28, 2012
  12. Aug 28, 2012 #11


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    This paper is not inconsistent with the claim that causality alone is not enough to derived SR. The key is that the derivation in this paper requires that events can be partially ordered but not totally ordered. This is just a disguised (and elegant!) way of incorporating Bcrowell's axiom (2): relativity of simultaneity.
  13. Aug 28, 2012 #12
    For me that sums it up (not sure why symmetry of spacetime as opposed to isotropic spacetime / spacetime interval ...). Appears to precede time/length measures ("mere*" observations), but introduces length time measures to explain the dichotomy of spacetime separated events & causal events.

    *the comparatives between length/time measures seems moot to the event itself (causal or not). Interval is what matters here...i.e. isotropic spacetime. I don't think the "universe" cares if Jack & Diane measure time/length of different proportions, only "concern" for the causal connection between them (and not important if there is no causal connection) of course observation is a causal connection i.e. Relativity of simultaneity, a human (conscious?)concern, but otherwise meaningless from a physical perspective. I'm thinking #2 could be dropped from the list, no?

    Opps, I wasn't sure what was meant by Mikowski metric? looks like it's what describes spacetime from a length/time (measurements) perspective so with that know understood & answering my own question of course #2 is required, it implies measures i.e. quantified comparisons
    Last edited: Aug 28, 2012
  14. Aug 28, 2012 #13


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    Some form of it is needed to rule out Galilean relativity. Note that Galilean relativity = SR if c=∞. You need something extra to rule out c=∞.
  15. Aug 28, 2012 #14
    What intrigues me is when he says, "Quantification proceeds by selecting two chains of coordinated events, each of which represents an observer, and assigning a valuation to each chain. Events can be projected onto each chain by identifying the earliest event on the chain that can be informed about the event."

    These "chain of events" make me think of the paths of Feynman's path integral, or perhaps the path of least action in the Action integral. Perhaps the two chains onto which he projects an unaffiliated event are slight deviations in a particular path in one of these integral formulations. And perhaps his projection procedure only insures that causality is maintained for events that are between a path and it's slight deviation. Then perhaps if alternative paths of causality are required along the points on some topology or manifold, then a Minkowski metric is required.
    Last edited: Aug 28, 2012
  16. Aug 28, 2012 #15
    Ah okay, I was thinking that isotropic spacetime implies an interval, and that #2 introduces the concept the two components of an interval. Now I see an interval doesn't inherently mean invariance, which requires a finite length over time (and isotropic spacetime).
    Last edited: Aug 28, 2012
  17. Aug 29, 2012 #16
    I think one would need first a good definition of causality, even if its meaning looks obvious.

    Also I consider Galilean transformations to be agnostic about causality, it is well known that classical mechanics is time-reversible, only by introducing an omniscient observer that prescribes absolute time there is causality. The only difference with the Lorentz case is that in the Galilean relativity time is not a cordinate/dimension, we are dealing with Euclidean space and time as a parameter, while in Minkowski space time is a dimension, so causality is intrinsic to the spacetime structure, and for spacetimes one needs not rule out c=∞ because it is implicit in the presence of a time dimension that c must be finite.
    Then introducing observers in a Lorentzian spacetime automatically leads to relativity of simultaneity.

    So I'd say that causality implies the Lorentz group and to answer the OP I think causality is enough to derive the Minkowski metric and the Lorentz transformations because it is the only way to introduce the time dimension and doesn't need an external omniscient observer to prescribe it thru a parameter.
  18. Aug 29, 2012 #17
    But that's exactly it, isn't it? Galilean relativity demands there is a preferred frame, an "external omniscient observer", if you will, and time is, accordingly, a parameter, not a dimension. This is on the same level as special relativity which demands there is no preferred frame, and a time dimension follows accordingly from that. What makes either choice preferred from the standpoint of causality? Of course, today we know that special relativity is correct and that the Galilean idea of a preferred reference frame is silly, but that doesn't mean the latter can't preserve causality any better than the former. As an earlier poster put it, yes, if we take c to be finite, we need Minkowski space and Lorentz transformation to preserve causality (actually, I'm not certain there aren't still other options at that level) but Galilean transformation is just a Lorentz transformation for c = ∞
  19. Aug 29, 2012 #18
    the time dimension is derived from c being finite right? Said different a finite c defines the spacial/temporal dimensions.

    Thinking of this more, I don't believe causality alone can "produce" metrics, it seems to be a purely physical concept (defining causality here), and "ignores" an observer perspective. I find it gets particularly confusing when introducing the "what's observed/measured is physical reality" school of thought.

    Einsteins two SR postulates do "produce" Minkowski metric (if I am understanding Minkowski metric correctly).
    Last edited: Aug 29, 2012
  20. Aug 29, 2012 #19
    Yeah, I think TrickyDicky has it a bit backward. The only difference between Galilean and Lorentz transformations is that c is infinite in the former, but finite in the latter. You don't derive a limit on c based on the presence of a time dimension, you infer the presence of a time dimension based on the fact that c is finite, which in turn yields Minkowski space.
  21. Aug 29, 2012 #20
    That is whay I said a working definition of causality is needed here. Too many concepts are implicit in the word and they could be different for different people.
    Although historically the two postulates came before, they are actually logically derived from Minkowski spacetime.
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