Undergrad Light-Cone & 4-Vector Properties

[Arman777]

I see this question in PSE and it seemed interesting. The Question is like this,

Consider a semi-Riemannian manifold which of these statements is false:

1) All vectors on the light-cone are light-like, all vectors in the interior of the light-cone are time-like and all vectors in the exterior of the light-cone are space-like.

2) All space-like vectors lie in the exterior of the light-cone, all time-like vectors lie in the interior of light-cone, and all light-like vectors lie on the light-cone.
I can't see which statements are false, any help would be appreciated.

What you guys think ?

I think 2) is True because that's kind of the definition but the 1) seems odd. It seems like it is true but I couldn't think any counter-example for the condition. Any ideas ?

https://physics.stackexchange.com/q...ightlike-vectors-and-the-light-cone-structure

 

[PeterDonis]

I think 2) is True because that's kind of the definition but the 1) seems odd.

1) and 2) are just the two different "directions" of an equivalence relation. Unpacking the relation for each type of vector and location with respect to the light cone, we have:

1) says:

If a vector is on the light cone, it is lightlike.

If a vector is in the interior of the light cone, it is timelike.

If a vector is in the exterior of the light cone, it is spacelike.

2) says:

If a vector is lightlike, it is on the light cone.

If a vector is timelike, it is in the interior of the light cone.

If a vector is spacelike, it is in the exterior of the light cone.

Each statement in 2) is the converse of the corresponding statement in 1). Taken together, they form the following equivalence relations:

A vector is lightlike if and only if it is on the light cone.

A vector is timelike if and only if it is in the interior of the light cone.

A vector is spacelike if and only if it is in the exterior of the light cone.

All three of these equivalence relations are true. So both 1) and 2) are true.

As for the StackExchange link, there is no source provided for the question, so it's impossible to know what whoever posed the question was thinking.

 

[robphy]

To really answer the questions, especially if there are subtleties to expose, we need some definitions.

 

[Pencilvester]

I see this question in PSE and it seemed interesting. The Question is like this,

Consider a semi-Riemannian manifold which of these statements is false:

1) All vectors on the light-cone are light-like, all vectors in the interior of the light-cone are time-like and all vectors in the exterior of the light-cone are space-like.

2) All space-like vectors lie in the exterior of the light-cone, all time-like vectors lie in the interior of light-cone, and all light-like vectors lie on the light-cone.

This question needs a little clarifying. If by “all vectors” they mean literally all possible vectors on the manifold, then neither of these statements are true in general. If they mean all vectors at a single given point in relation to the light-cone at that point, which would be a natural assumption, but isn’t explicitly stated, then @PeterDonis said it best in post #2.

 

[Arman777]

This question needs a little clarifying. If by “all vectors” they mean literally all possible vectors on the manifold, then neither of these statements are true in general. If they mean all vectors at a single given point in relation to the light-cone at that point, which would be a natural assumption, but isn’t explicitly stated, then @PeterDonis said it best in post #2.

Well yes you are right. I guess that's what the question asks so I believe both of them are true as well.

 

[PeterDonis]

If they mean all vectors at a single given point in relation to the light-cone at that point

I assumed that's what was meant, but you're right, it should be specified explicitly. Without knowing the source, we have no context, so we can't tell if that was specified explicitly somewhere in the same source before the question was posed.

 

[PhDeezNutz]

How does one define the 4-velocity of a light like vector? I’m guessing the result should be null but what sort of general formula would encompass that?

I hope I am not hijacking the thread. It seems relevant to the OP.

 

[robphy]

How does one define the 4-velocity of a light like vector? I’m guessing the result should be null but what sort of general formula would encompass that?

I hope I am not hijacking the thread. It seems relevant to the OP.

You can’t define an invariant [unit] 4-velocity.
Just use the 4-momentum.

You could use lightlike “units” in a given frame of reference... for light-cone coordinates.

 

[PeterDonis]

How does one define the 4-velocity of a light like vector?

As @robphy says, you can't define a unit tangent vector to a null curve, because it's null--any vector tangent to it will have zero norm, not unit norm.

You can, however, define a curve parameter along a null curve and take the derivative of coordinates with respect to that parameter. The parameter just can't be arc length along the curve, as it can be for timelike and spacelike curves. You could, for example, pick an inertial frame and just use the $t$ coordinate of that frame as the curve parameter. Then you would have a tangent vector

$$
V = \left( \frac{dt}{dt}, \frac{dx}{dt}, \frac{dy}{dt}, \frac{dz}{dt} \right) = \left( 1, v_x, v_y, v_z \right)
$$

where $v_x$, $v_y$, $v_z$ are the components of the ordinary velocity of the light ray in the chosen frame. Since light always moves with speed $c$, it should be evident that the norm of $V$ is zero, as required.

@robphy mentioned the 4-momentum of the light; using the parameterization I just gave, this would simply be the energy of the light in the chosen frame times $V$.