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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

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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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.

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E. C. Zeeman

Causality Implies the Lorentz Group.

J. Math. Phys. April 1964 Volume 5, Issue 4, pp. 490-493

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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.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.

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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.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.

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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.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.

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 sin

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I found the following paper on the arXiv:

http://arxiv.org/abs/1005.4172

Abstract:

"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.

http://arxiv.org/abs/1005.4172

Abstract:

"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.

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PAllen

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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.I found the following paper on the arXiv:

http://arxiv.org/abs/1005.4172

Abstract:

"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.

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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.You can do it in a system of postulates based on (1) causality, (2) relativity of simultaneity, and (3) symmetry of spacetime.

*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

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PAllen

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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=∞.*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?

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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.

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.

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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).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=∞.

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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.

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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 = ∞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.

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the time dimension is derived from c being finite right? Said different a finite c defines the spacial/temporal dimensions.

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.

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).

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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.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.

Although historically the two postulates came before, they are actually logically derived from Minkowski spacetime.Einsteins two SR postulates do "produce" Minkowski metric (if I am understanding Minkowski metric correctly).

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I think this sequence is not the right one, but it is logical that a discussion on causality includes disagreements on what's backwards and what's forward .

But it all depends on what you want to use as initial postulate, I think people understands better SR if you start with the Minkowski space:

Causality→Minkowski space→time dimension and finite c.

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Ha!I think this sequence is not the right one, but it is logical that a discussion on causality includes disagreements on what's backwards and what's forward

I guess I just don't follow that progression very easily. I feel like the idea of a finite c that is constant in all reference frames is really where most educators start. From there you pick up relativity of simultaneity, Lorentz tranformations, Minkowski space, and then discussions on why causality is definitely preserved in this space. It feels more like a check at the end of a theory rather than something you can use to arrive at ONE correct theory. At least, this is how I best understand it.But it all depends on what you want to use as initial postulate, I think people understands better SR if you start with the Minkowski space:

Causality→Minkowski space→time dimension and finite c.

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And I feel as though the "causality postulate" is also logically derived from Minkowski space, not vice versa.Although historically the two postulates came before, they are actually logically derived from Minkowski spacetime.

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You are right that most educators go about it like that, see this for instance:I guess I just don't follow that progression very easily. I feel like the idea of a finite c that is constant in all reference frames is really where most educators start. From there you pick up relativity of simultaneity, Lorentz tranformations, Minkowski space, and then discussions on why causality is definitely preserved in this space. It feels more like a check at the end of a theory rather than something you can use to arrive at ONE correct theory. At least, this is how I best understand it.

http://www.pantaneto.co.uk/issue33/henry.htm

Intuition might be misleading, I also feel more natural to think about causality from Minkowski space rather than the opposite, that is because we associate causality to Minkowski space (or Lorentzian manifold in general) but then you can also associate causality to other spaces like euclidean space and galilean relativity plus preferred frame that gives absolute time or others, but all of those others seem to require additional assumptions. To have causality without other assumptions the simplest way is a space with a time dimension, that is to say, with a Lorentzian signature.And I feel as though the "causality postulate" is also logically derived from Minkowski space, not vice versa.

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I'd reiterate they are postulates; "assumptions" of our observations. The metric continues this into a mathematically useable/structured format. I don't know details or much really about the metric beyond it "describing" dimensions. +++-? thingy :uhh: I thinkAlthough historically the two postulates came before, they are actually logically derived from Minkowski spacetime.

But yea they can be derived from, or "produce" the said metric.

I guess the quoted statement isn't really stating anything, well besides the postulates came first, and are fundamental to the metric.

Finding this a neat topic to think about, causality and what it means. I'm caught up thinking of it being purely about observation. In the case of causality there is only one "true" order, despite observed non-congruent order of causal events in a hypothetical scenario.

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