Exploring the Shape of a Light Cone

[dkgolfer16]

Just wondering about the shape of a light cone. On the attachment below, if I were standing at A and pointed a flashlight in the positive time direction, it would form the cone show, correct?

Is this cone a perfect cone? For instance, if I were standing on the north pole and pointed a flashlight up (positive time direction), wouldn't the cone become distorted as it expanded and moved through space (due to the bending of light by gravitational forces)?

Does its radius approach infinity as you move in the positive time direction?

http://en.wikipedia.org/wiki/File:Light_cone.svg

 

[JesseM]

The number of dimensions is dropped in the wikipedia image because we can't actually visualize a 4-dimensional shape. If you picture a universe with only two spatial dimensions and one time dimension, then the light cone looks like an actual 3D cone, with a cross-section of the cone at anyone moment of coordinate time being a circle in a 2D plane. In our universe the light cone would be a 4D cone, the cross-section of this cone at any given moment of coordinate time would be the curved 2D surface of a 3D sphere centered on the position of the original event, with the radius of the sphere expanding at the speed of light. For any given event A in the past, the size of the sphere at any given moment would be the collection of points that were just getting news of the event via light signals going out in all directions (not in a confined set of directions like light from a flashlight); the idea of the light cone is that any event on or inside the 4D cone has the potential to have some causal relationship with A, but any event outside the cone cannot possible be influenced by A (or influence A) because no signal traveling at the speed of light or slower would have had time to get from A to that event.

 

[jambaugh]

The cone is a "hyper-cone" in space-time. Cross sections perpendicular to the time axis will be 2-sheres, the spheres of a light's wave-front given a pulse of light emitted at the event corresponding to the vertex of the light-cone.

It is a perfect cone in flat space-time but if you include bending of space-time aka gravity you get more interesting curved hyper-surfaces.

Your flash-light example isn't quite right. Picture instead a flash-bulb going off in the middle of empty space. The expanding sphere of light is the cross-sections of the light-cone. The whole cone is the history of this outward expanding pulse of light.

Remember there's one more dimension involved than a conventional 2-dim cone in 3-dim space. The axis of the light-cone is a time axis.

 

[dkgolfer16]

Thanks Jesse and James for the helpful explanations. I guess just a follow up thought on that:

Lets say a star in open space was born 1 million years ago. So the radius of the stars light cone is the distance traveled by light in 1 million years? So all matter within this raduis will have a casual relationship with the star whereas all matter outside will not. It seems that eventually all matter will have a casual relationship with the star as long as it doesn't collapse into a black hole. Is my logic correct here?

 

[A.T.]

It seems that eventually all matter will have a casual relationship with the star as long as it doesn't collapse into a black hole. Is my logic correct here?

We don't know if the universe is finite. And even if it was, it could keep on expanding, so that light from the star would never reach all of it.

 

[robphy]

Just wondering about the shape of a light cone. On the attachment below, if I were standing at A and pointed a flashlight in the positive time direction, it would form the cone show, correct?

Is this cone a perfect cone? For instance, if I were standing on the north pole and pointed a flashlight up (positive time direction), wouldn't the cone become distorted as it expanded and moved through space (due to the bending of light by gravitational forces)?

Does its radius approach infinity as you move in the positive time direction?

http://en.wikipedia.org/wiki/File:Light_cone.svg

You need to distinguish two ideas, which appear to coincide in a flat Minkowski spacetime:

• the set of events reached by light rays from the vertex event
• the set of future-directed lightlike vectors in the space-of-tangent-vectors of the vertex event

The former can be regarded as being distorted by the presence of spacetime curvature.
The latter always looks like the cone in the above diagram whether spacetime is curved or not (since the tangent space is a vector space).