The set of allowed trajectories in spacetime

[Demon117]

I was reading Spacetime and Geometry by Caroll and I came across this notion:

In special relativity there is no absolute notion of "all of space at one
moment in time." Instead, there is a rule that particles always travel at less than or equal to the
speed of light. We can therefore define light cones at every event, which locally describe
the set of allowed trajectories. For two events that are outside each others' light cones,
there is no universal notion of which event occurred earlier in time.

I have never understood this and I was wondering if someone could enlighten me.

 

[Demon117]

I was reading Spacetime and Geometry by Caroll and I came across this notion:

In special relativity there is no absolute notion of "all of space at one
moment in time." Instead, there is a rule that particles always travel at less than or equal to the
speed of light. We can therefore define light cones at every event, which locally describe
the set of allowed trajectories. For two events that are outside each others' light cones,
there is no universal notion of which event occurred earlier in time.

I have never understood this and I was wondering if someone could enlighten me.

Simultaneity of events?

 

[Meir Achuz]

I am afraid that statement is as clear to me as anything I could write.
What don't you understand?

 

[JesseM]

Simultaneity of events?

Are you familiar with the notion of the relativity of simultaneity? See here and here for some more info, and you might also want to take a look at some of the basic SR tutorials on this thread which all cover the relativity of simultaneity. They also cover the notion of "light cones", but for a particularly good discussion of that notion see this page.

 

[Demon117]

I understand the simultaneity of events just fine. I apologize I should have clarified. The notion of light cones is a little fuzzy to me. Simultaneity is clear.

 

[Meir Achuz]

I still like your original quote, but this, from a textbook, may help you:

"The light cone divides space-time into three separate regions. The inside of the light cone, in the positive time direction is the future. If you start at the apex of the light cone, you can only reach or affect points inside your future light cone. The only people who can see you (using light) must be on the surface of your future light cone. The inside of the light cone in the negative time direction is the past. Points inside your past light cone are inaccessible to you (without time travel), but are the only points that can affect you here and now. There is no orthochronous Lorentz transformation that connects points in the past and future light cones, so time travel is not possible in special relativity. The rest of space-time, outside the light cone is always a spacelike separation from the apex of the cone, and can be called 'elsewhere'. There is no way that you can influence or be influenced by points outside your light cone. But those points are still very important. They are the only space-time points where other people can be at the same time as you. At this very moment, everybody you know is outside your light cone. Only solipsists have nobody outside their light cones. Although there are people outside your light cone right now, you can only see them when they are on your past light cone. (Think a bit about that.) For intervals outside the light cone, there is no meaning to absolute future or past, because even an orthochronous Lorentz transformation can interchange past and future. This is not time travel however, because these points are inaccessible to you and can't affect you."

 

[robphy]

see the diagrams on p.89 on Geroch's General Relativity from A to B
books.google.com/books?id=rwPDssnbHPEC&lpg=PP1&dq=light%20cone&pg=PA89

The story of behind these diagrams begins with the discussion of
measurements in spacetime at the bottom of page 80
books.google.com/books?id=rwPDssnbHPEC&lpg=PP1&dq=light%20cone&pg=PA80