The discussion revolves around the conservation of area under Lorentz transformations, specifically in the context of spacetime geometry. Participants explore arguments for this conservation without assuming the form of the Lorentz transformation or the constancy of the speed of light across frames. The conversation touches on foundational assumptions in special relativity and the implications of these assumptions on the derivation of the Lorentz transformation.
Participants generally express agreement on some foundational assumptions but disagree on others, particularly regarding causality, simultaneity, and the implications of these concepts in the context of Lorentz transformations. The discussion remains unresolved with multiple competing views presented.
Participants note limitations in the assumptions made, particularly regarding the implications of different clock synchronization conventions and the nature of linear versus nonlinear transformations. The discussion highlights the complexity of deriving the Lorentz transformation without making certain assumptions explicit.
Yes, causality is absolute. If event A is the cause of event B then A preceeds B in all reference frames.cfrogue said:I do not think you see that the light cone implements casuality and that is absolute.
Yes, it does. Consider the intersection of the light cone with any hyperplane of simultaneity for any arbitrary inertial frame. It should be clear that all of those events (a sphere) are spacelike separated.cfrogue said:You are not listening.
order of the spacelike separated events
This does not exist with a single light sphere.
DaleSpam said:This sentence is wrong, a location does not have a light cone. An event has a light cone, a location does not.
JesseM said:This one is confusingly worded, I think they mean "as time progresses backwards". I'll edit that one so it makes more sense.
Your use of language is so vague! It does "implement causality" in the sense that all observers agree on the order of E and some other event E1 which lies in E's future or past light cone, but that doesn't mean that observers will agree on the order of two events E1 and E2 which both lie in the past light cone of E but don't lie in one another's past or future light cones (i.e. E1 and E2 are spacelike separated)cfrogue said:I do not think you see that the light cone implements casuality and that is absolute.
For any pair of events where one lies in the other's past light cone, all observers do agree on the order. But two events E1 and E2 can both lie in the past light cone of E (so all observers agree E1 and E2 happened prior to E) yet neither lies in the past light cone of the other one (so different observers disagree about whether E1 or E2 happened first). Is this really so hard to understand? Do you understand that each event has its own individual past light cone?cfrogue said:If not, SR falls apart in the sense that there does not exists anything in the universe that can describe the order of events such that the observers cannot disagree.
But normal human experience conflicts with relativity in this sense, since relativity says that when two events don't lie in one another's past or future light cones, there is no absolute truth about which happened earlier and which happened later.cfrogue said:This would also violate normal human experience which clearly distinguishes events into the absolute past and absolute present.
Because if the past light sphere is shrinking as time progresses forward towards the event E whose past light cone it is, then if you reverse this in your mind like a movie played backwards, the past light cone is expanding (just like how as time progresses backwards in a movie played backwards, broken eggs will reassemble into whole ones, people will get younger, and so forth).cfrogue said:Why do you say as time progresses backwards?
JesseM said:Your use of language is so vague! It does "implement causality" in the sense that all observers agree on the order of E and some other event E1 which lies in E's future or past light cone, but that doesn't mean that observers will agree on the order of two events E1 and E2 which both lie in the past light cone of E but don't lie in one another's past or future light cones (i.e. E1 and E2 are spacelike separated)
JesseM said:Because if the past light sphere is shrinking as time progresses forward towards the event E whose past light cone it is, then if you reverse this in your mind like a movie played backwards, the past light cone is expanding (just like how as time progresses backwards in a movie played backwards, broken eggs will reassemble into whole ones, people will get younger, and so forth).
Anyway, I edited it to say the following:
As time progresses forward, each event's future light cone will eventually grow to encompass more and more locations. Likewise, if we imagine running time backwards from a given event, the event's past light cone would likewise encompass more and more locations at earlier and earlier times. The further locations will of course be at far distant times. The Earth's past light cone, at its very edges, includes very distant objects, but only what they looked like long ago, when the universe was young.
Good edit. It preserves what they were trying to say, but says it correctly.JesseM said:Anyway, I edited it to say the following:
As time progresses forward, each event's future light cone will eventually grow to encompass more and more locations. Likewise, if we imagine running time backwards from a given event, the event's past light cone would likewise encompass more and more locations at earlier and earlier times. The further locations will of course be at far distant times. The Earth's past light cone, at its very edges, includes very distant objects, but only what they looked like long ago, when the universe was young.
You really should think in terms of cones and not spheres. As you go along a cone towards the apex (past light cone) the cone gets smaller, as you continue along the cone away from the apex (future light cone) the cone gets bigger.cfrogue said:A light sphere expands.
How do you say it shrinks?
No, of course not. In any given frame, if you have an event E at position x and time t, then the past light cone will be centered at position x and its radius will shrink at speed c until it reaches 0 radius at time t, then the future light cone will also be centered at position x and will start from 0 radius at time t and its radius will then expand at speed c.cfrogue said:What do you think a light sphere does in space? Do you think the origin moves around?
True for the future light sphere but not the past light sphere.cfrogue said:If so, it would not matter, once the light sphere takes over an object, event, it will never leave.
Did you not read my example in post 17? I said 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.cfrogue said:A light sphere expands.
How do you say it shrinks?
Yes, light cones are defined in terms of specific events, it says so right in the article:cfrogue said:You are claiming an event defines the behavior of light?
As time progresses, each location's past light cone will eventually grow to encompass all locations. The further locations will of course be at far distant times. The Earth's past light cone, at its very edges, includes very distant objects, but only what they looked like long ago, when the universe was young.
http://en.wikipedia.org/wiki/Light_cone
Also, you ask if "an event defines the behavior of light", but note that a light cone is not defined in terms of actual light particles, but only in terms of the set of spacetime points that could hypothetically send/receive a signal to/from the event E.Given an event E, the light cone classifies all events in spacetime into 5 distinct categories:
Events on the future light cone of E.
Events on the past light cone of E.
Events inside the future light cone of E are those which are affected by a material particle emitted at E.
Events inside the past light cone of E are those which can emit a material particle and affect what is happening at E.
All other events are in the (absolute) elsewhere of E and are those that can not affect and can not be affected by E.
The above classifications hold true in any frame of reference; that is, an event judged to be in the light cone by one observer, will also be judged to be in the same light cone by all other observers, no matter their frame of reference. This is why the concept is so powerful.
Keep in mind, we're talking about an event, a specific location at a specific time. To say that one event cannot affect another, that means that there isn't enough time for light to get from one to the other. Light from each event will eventually (after some time) make it to the old location of the other event, but since that's at a later time, it's not the same event.
DaleSpam said:@ bcrowell. Sorry about helping hijack your thread.
JesseM said:Here's an example of a linear transformation without this property:
x' = 2x - 2vt
t' = t