Volume of light cones of two events

  • Context: Undergrad 
  • Thread starter Thread starter A Dhingra
  • Start date Start date
  • Tags Tags
    Events Light Volume
Click For Summary

Discussion Overview

The discussion revolves around the properties of light cones in both flat and curved spacetime, particularly focusing on the volume of light cones formed by pairs of events. Participants explore theoretical scenarios involving different spacetime configurations and the implications of Lorentz transformations on the shapes and volumes of these cones.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant describes two events, p and q, in flat spacetime, proposing that if they occur at the same location but different times, their light cones form a regular bicone with a circular cross-section.
  • Another participant questions the selection of the bicone and suggests that the intersection of the two light cones would actually be a circle, noting that light cones are 2D surfaces and their intersection is typically a 1D curve.
  • The first participant speculates that if two events x and y occur at different spatial points and times, their light cone intersection might be a bicone with a tilted elliptical cross-section, seeking confirmation of this idea.
  • Further, the first participant questions whether the volume of the light cone pair in this scenario depends solely on the proper time between the events or if it also involves the dimensions of the elliptical cross-section.
  • There is a consideration of whether a Lorentz transformation could yield a circular cross-section for the light cones when viewed from a different frame of reference, with the first participant expressing uncertainty about the resulting shape.
  • A later reply references a paper that states 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 the two events.
  • The first participant raises questions about how curvature in spacetime affects light cones and whether a local frame can serve as a background for analyzing these cones.

Areas of Agreement / Disagreement

Participants express differing views on the nature of light cone intersections and the implications of Lorentz transformations, indicating that multiple competing perspectives remain unresolved.

Contextual Notes

Participants note that light cones are 2D surfaces and their intersections typically yield 1D curves, which complicates the notion of volume in this context. There is also a lack of consensus on how curvature in spacetime influences the properties of light cones.

A Dhingra
Messages
211
Reaction score
1
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
 
Physics news on Phys.org
A Dhingra said:
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.
 
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.)
 
Thanks robphy. That's what I was looking for.
Thanks Dale as well.
 

Similar threads

Undergrad Seeing history when crossing event horizon
  • · Replies 41 ·
2
Replies
41
Views
3K
Undergrad Relational event horizons?
  • · Replies 11 ·
Replies
11
Views
1K
Undergrad How many sorts of time are there in physics?
  • · Replies 12 ·
Replies
12
Views
1K
High School Events vs Effect of Events in Relativity of Simultaneity
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad The Einstein Clock aka Light Clock
  • · Replies 34 ·
2
Replies
34
Views
4K
Undergrad Michelson & Morley in Space-Time Diagram: Can't Make it Work
  • · Replies 18 ·
Replies
18
Views
2K
Undergrad Coordinate System for Minkowskian Spacetime Relative to Event
  • · Replies 8 ·
Replies
8
Views
2K
Undergrad SR flat Lorentzian manifold and Anderson simultaneity convention
  • · Replies 16 ·
Replies
16
Views
2K
Undergrad An event being in the past of a timelike separated one
  • · Replies 20 ·
Replies
20
Views
2K
Undergrad Coordinate Transformation versus Change of Synchrony Convention
  • · Replies 9 ·
Replies
9
Views
1K