Let x
µ = x
µ
(u) be a parametric equation for
the curve in pseudo-Riemannian four-space connecting points P and Q. Here u is any,
not necessarily affine parameter. Assume that the derivative satisfies
Update: Let x
µ = x^µ(u) be a parametric equation for
the curve in pseudo-Riemannian four-space connecting points P and Q. Here u is any,
not necessarily affine parameter. Assume that the derivative satisfies
(a) u^µ(P) is in the future light-cone;
(b) u^µ /= 0 everywhere along the curve;
(c) u^µuµ ≥ 0 everywhere along the curve.
Prove that u^µ
is everywhere along the curve in the future light-cone.
To prove that u^µ is everywhere along the curve in the future light-cone, we will make use of the given conditions (a), (b), and (c).
Condition (a) states that u^µ(P) is in the future light-cone. This means that the four-vector u^µ(P) is a future-oriented timelike or null vector. In pseudo-Riemannian four-space, the future light-cone consists of timelike and null vectors that are future-oriented.
Condition (b) states that u^µ /= 0 everywhere along the curve. This means that the derivative of the parameterization vector x^µ(u) with respect to u is non-zero at every point on the curve.
Condition (c) states that u^µuµ ≥ 0 everywhere along the curve. This condition ensures that the four-vector u^µ is either timelike or null along the entire curve.
Now, let's assume that there exists a point Q on the curve where u^µ is not in the future light-cone. This implies that u^µ(Q) is either a past-oriented timelike vector or a spacelike vector.
Since u^µ(P) is in the future light-cone and u^µ is continuous along the curve, there must exist a point R on the curve between P and Q where u^µ(R) transitions from being future-oriented to being either past-oriented or spacelike.
Now, consider the value of u^µuµ at point R. The condition (c) states that u^µuµ ≥ 0 everywhere along the curve. However, at point R, where u^µ transitions from being future-oriented, the value of u^µuµ becomes negative if u^µ is past-oriented or spacelike.
This contradicts the condition (c) and thus our assumption that there exists a point Q where u^µ is not in the future light-cone.
Therefore, we can conclude that u^µ is everywhere along the curve in the future light-cone.
Note: This proof is based on the assumption that the given conditions are valid for the curve in pseudo-Riemannian four-space connecting points P and Q.
To prove that u^µ is everywhere along the curve in the future light-cone, we can use the given conditions and properties of pseudo-Riemannian four-space.
First, let's define the future light-cone in pseudo-Riemannian four-space. The future light-cone is the set of all possible future directions of light rays originating from a given point in the four-space. In other words, it represents all the paths that a light ray can take from a given point into the future.
Now, let's proceed with the proof:
1. Since u^µ(P) is in the future light-cone, it means that the initial tangent vector at point P is pointing towards the future. This implies that the tangent vector is either null (light-like) or time-like, but not space-like.
2. From the condition (b), u^µ /= 0 everywhere along the curve. This means that the tangent vector is non-zero at every point along the curve.
3. Now, let's consider the quantity u^µuµ. This represents the inner product (or dot product) of u^µ with itself at each point along the curve. Since the inner product of a vector with itself is always non-negative, u^µuµ ≥ 0 everywhere along the curve.
4. Since u^µuµ ≥ 0 and the tangent vector is non-zero, it means that u^µ cannot be purely space-like along the curve. If u^µ had a purely space-like component, then u^µuµ would be negative. Therefore, u^µ must either be null (light-like) or time-like.
5. If u^µ is null (light-like) along the curve, then it lies on the future light-cone. This is because all null vectors lie on the light-cone.
6. If u^µ is time-like along the curve, then it is always pointing towards the future in pseudo-Riemannian four-space. This is because time-like vectors always have a positive time component and lie within the future light-cone.
Therefore, based on the given conditions and the properties of pseudo-Riemannian four-space, we have proved that u^µ is everywhere along the curve in the future light-cone.