Question

(i) Show that the following 'normal forms' for 4-vectors hold: There is always an inertial frame (a basis of the vector space V of 4-vectors) such that • for a given time-like 4-vector $t^a = (t^0, 0, 0, 0)$, • for a given space-like 4-vector $v^a = (0, v^1, 0, 0)$, • and for a given light-like vector $l^a = (c, c, 0, 0)$ for some constant $c$. holds. (ii) Use the normal forms from a) to prove the following statements. (i) Every 4-vector perpendicular to a time-like vector $t^a$ is space-like. (ii) 4-vectors that are perpendicular to a light-like 4-vector $l^a$ are either light-like (and then parallel to $l^a$) or space-like.

          (i) Show that the following 'normal forms' for 4-vectors hold: There is always an inertial frame (a basis of the vector space V of 4-vectors) such that
• for a given time-like 4-vector $t^a = (t^0, 0, 0, 0)$, 
• for a given space-like 4-vector $v^a = (0, v^1, 0, 0)$, 
• and for a given light-like vector $l^a = (c, c, 0, 0)$ for some constant $c$.
holds.
(ii) Use the normal forms from a) to prove the following statements.
(i) Every 4-vector perpendicular to a time-like vector $t^a$ is space-like.
(ii) 4-vectors that are perpendicular to a light-like 4-vector $l^a$ are either light-like (and then parallel to $l^a$) or space-like.

(i) Show that the following 'normal forms' for 4-vectors hold: There is always an inertial frame (a basis of the vector space V of 4-vectors) such that
• for a given time-like 4-vector t^a = (t^0, 0, 0, 0),
• for a given space-like 4-vector v^a = (0, v^1, 0, 0),
• and for a given light-like vector l^a = (c, c, 0, 0) for some constant c.
holds.
(ii) Use the normal forms from a) to prove the following statements.
(i) Every 4-vector perpendicular to a time-like vector t^a is space-like.
(ii) 4-vectors that are perpendicular to a light-like 4-vector l^a are either light-like (and then parallel to l^a) or space-like.

Added by Melissa P.

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