Simple argument for area conservation under Lorentz transformation?

[JesseM]

Why do you say as time progresses backwards?

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.

 

[cfrogue]

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)

What I said you do not understand the light cone.

What do you think a light sphere does in space? Do you think the origin moves around?

If so, it would not matter, once the light sphere takes over an object, event, it will never leave.

 

[cfrogue]

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.

A light sphere expands.

How do you say it shrinks?

 

[Dale]

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.

 

[Dale]

A light sphere expands.

How do you say it shrinks?

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.

 

[JesseM]

What do you think a light sphere does in space? Do you think the origin moves around?

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.

If so, it would not matter, once the light sphere takes over an object, event, it will never leave.

True for the future light sphere but not the past light sphere.

A light sphere expands.

How do you say it shrinks?

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.

Do you disagree that 1 year before the event E whose past light cone we are considering, the light sphere has a radius of 1 light-year (with its center at the future location of E)? Do you disagree that 1 second before the event E, the past light sphere has a radius of 1 light-second?

 

[JesseM]

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

Yes, light cones are defined in terms of specific events, it says so right in the article:

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.

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.

 

[Dale]

@ bcrowell. Sorry about helping hijack your thread.

 

[cfrogue]

@ bcrowell. Sorry about helping hijack your thread.

agreed

sorry also

 

[bcrowell]

Here's an example of a linear transformation without this property:

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

I see. I guess I was assuming that it had to be a Lie group, i.e., it has an identity operation, and when you make the continuous parameter of the group approach zero, the transformation approaches the identity. I guess I could (a) explicitly state this as a mathematical assumption, (b) not bother explicitly stating it, or (c) state that spacetime has to have Galilean transformation properties in the limit of small velocities. Although c has the advantage of making the list of assumptions a manifestly physical list rather than a mathematical one, it has the disadvantage of being a much stronger assumption than is necessary in order to get the desired result.