I Volume of light cones of two events

[A Dhingra]

Hi

I have a few questions about light-cones. I have read the previous threads related to my question and have a few things to clarify.

I have taken two events say p and q . (1+2 D - so that its easier to imagine)
CASE A : Flat spacetime
Case 1 : If p and q occur at the same place (according to my chosen coordinates), different times - then the light cones of these two will form a perfectly regular bicone with circular cross-section. Agreed?

Case 2 : Now say there are two more events x and y, which occur at the different points in space (with respect to the previously chosen frame of reference), and different time -
Q 1: what would the intersection of their light cones be ? Ans - My guess is - it will be a bicone, but with titled elliptical cross-section. Can you please tell me if I am thinking in the right direction?

Next set of questions depends on the answer to the last question.
Q 2: Assuming that my answer is true. Will the volume of the light-cone pair (whatever the shape is!) in Case 2, depend on just the Proper time between the two events, or a proportion of it. Or will there be some dependence on the length of semi-major and semi minor axis of the ellipse.
And in the process of generalizing the volume for higher dimension for Case 2, will the same line of thought hold?
Q3: Or can I find the volume of this case by considering a frame where these cones of x and y intersect form a circle instead of ellipse! So now my question boils down to : Will this volume be same (with a multiplicity factor depending on the amount of boost required) if I did a Lorentz Transformation ?
Let's same I choose a frame (which is moving at some velocity wrt the original frame)
such that the events x and y are occurring at the same point..will the shape of the bi-cone (!) be with circular cross-section or elliptical? I think it will be elliptical, but then Lorentz boost to a circular shape will make it elliptical, so it should be true other way round. I would appreciate any feedback here.

Now in curved spacetime..
I am not asking for a detailed calculation here.. that's what I am supposed to do at some point, I just want to be clear about the process and concepts I am using in the process are correct.

In curved spacetime - well Riemann Normal Neighborhood of some point r, which lies between p and q.
Clearly the light cones get affected due to the curvature.

But for the purpose of intuition : can I assume that the Local frame of the point r serves as a background and the modified light cones lie on it?
Now what happens to the events x and y.. If the Local Frame of Point 'r' is taken as a background, a Lorentz transformation of the coordinates doesn't seems like a good idea. So, can the question (Q3) about Lorentz transformation and volume hold here?

(I hope the questions are clear enough)
Thanks a lot

 

[Dale]

I have taken two events say p and q . (1+2 D - so that its easier to imagine)
CASE A : Flat spacetime
Case 1 : If p and q occur at the same place (according to my chosen coordinates), different times - then the light cones of these two will form a perfectly regular bicone with circular cross-section. Agreed?

It isn't clear to me what rules you would use to select just the bicone and drop everything else. If you are just talking about the intersection of the two light cones then that is a circle.

Note, the light cones are 2D surfaces, so their intersection will usually be 1D curves. They won't have a volume.

 

[robphy]

Have a look at my paper
Visualizing Proper Time in Special Relativity (available at
http://arxiv.org/abs/physics/0505134 )...See page 9.
In SR, for two timelike related events p<q, the volume of the intersection of the interiors of the future light cone of p and the past light cone of q is proportional to the square-interval between p and q.

You might also appreciate a related set of animated spacetime diagrams of circular light clocks at the end of http://visualrelativity.com/LIGHTCONE/LightClock/ .
(My avatar is based on those animations.)

 

[A Dhingra]

Thanks robphy. That's what I was looking for.
Thanks Dale as well.