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A Dhingra

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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