Simple Nature - Light and Matter

Exercise 7E: The sum of observer-vectors is an observer-vector.

The figure gives four pairs of four-vectors, oriented in our customary way as shown by the

light-cone on the left.

1. Of the types shown in the four cases i-iv, which types of vectors could represent the world-line
   of an observer?


2. Suppose thatUandVare both observer-vectors. What would it mean physically to compute
   U+V?


3. Determine the sign of each inner productA·B.


4. Given an observer whose world-line is along a four-vectorO, suppose we want to determine
   whether some other four-vectorPis also a possible world-line of an observer. Show that knowl-
   edge of the signs of the inner productsO·PandP·Pis necessary and sufficient to determine
   this. Hint: Consider various possibilities like i-iv for vectorP, and see how the signs would turn
   out.


5. For vectors as described in 4, determine the signs of

(U+V)·(U+V)

and

(U+V)·U

by multiplying them out. Interpret the result physically.

472 Chapter 7 Relativity