- #1

shinobi20

- 257

- 17

- Homework Statement:
- Given the basic definition of four-vectors, I want to show some basic properties of two four-vectors contracted and some variation of that.

- Relevant Equations:
- ##A \cdot B = A^\mu B_\mu##

Two four-vectors have the property that ##A^\mu B_\mu = 0##

(a) Suppose ##A^\mu A_\mu > 0##. Show that ##B^\mu B_\mu \leq 0##

(b) Suppose ##A^\mu A_\mu = 0##. Show that ##B^\mu## is either proportional to ##A^\mu## (that is, ##B^\mu = k A^\mu##) or else ##B^\mu B_\mu < 0##.

Part (a) is intuitive to me since if the dot product of two four-vectors is zero then either they are perpendicular (if that makes sense in 4-d) or one of the vectors is the zero vector. Since ##A^\mu B_\mu = 0## and ##A^\mu A_\mu > 0##, it is trivial that ##B^\mu = 0## so the only scenario left is if one of the four-vectors is timelike (##A^\mu##) and the other is spacelike (##B^\mu##), which is what to be shown. However, I am thinking is there any analytical way of showing part (a)? Can anyone give me a hint on how to do it?

For part (b), I have no idea how to show it analytically too.

(a) Suppose ##A^\mu A_\mu > 0##. Show that ##B^\mu B_\mu \leq 0##

(b) Suppose ##A^\mu A_\mu = 0##. Show that ##B^\mu## is either proportional to ##A^\mu## (that is, ##B^\mu = k A^\mu##) or else ##B^\mu B_\mu < 0##.

Part (a) is intuitive to me since if the dot product of two four-vectors is zero then either they are perpendicular (if that makes sense in 4-d) or one of the vectors is the zero vector. Since ##A^\mu B_\mu = 0## and ##A^\mu A_\mu > 0##, it is trivial that ##B^\mu = 0## so the only scenario left is if one of the four-vectors is timelike (##A^\mu##) and the other is spacelike (##B^\mu##), which is what to be shown. However, I am thinking is there any analytical way of showing part (a)? Can anyone give me a hint on how to do it?

For part (b), I have no idea how to show it analytically too.