Exercises 2-6-8Prove that(1) The interval of the proper times for any two world points with light-like separation is zero;Solution:For any two world points with light-like separation we have , so For special Lorentz transformation we have So must be zero.(2) For three world points with time-like separation, the Minkowski inequality followswhere is the proper time interval of with respect to , and the equality is valid if and only if lie on the same straight world line.Solution:Because are in time-like separation, so Which means we can chose a system of reference, in which happened at the same displace but different time.So we have: And So Only if , which means the three world points lie on the same straight world line.(3) For two points and with time-like separation and , the proper time interval of a point particle moving from to through a curve is less than that through a straight line.Solution:We separate the curve line into some straight line According to the result of the formal question, we have For each two pieces of the straights ,So which means the proper time interval of a point particle moving from to through a curve is less than that through a straight line.Exercises 2-6-9There is a pair of twins, A and B. B flies away from the earth with the velocity , and when B has covered a distance of , he comes back immediately with the velocity and finally he meets A again. Calculate the difference of their ages when they meet again. Are the observed results to the difference of their ages by A and B accordant with each other.Solution:The total time observed by A is The time of B observed by A is So the difference of their ages is The difference of their ages is the same, but the observer is always older.Exercises 2-7-6Derive the simple relation between and from Eq. (2.7.10), where and are the polar angle of the velocity of a point particle in the inertial frame and , respectively. Assume that the coordinate axes of the two inertial frames and are parallel to each other, has a constant velocity of with respect to .Solution:First we have and And then So Exercises 2-7-7In view of the inertial frame fixed at a star, a rocket starts from the stationary state at, and moves with an acceleration along x-axis. The acceleration of the rocket measured by an astronaut in this rocket is a constant. Find the distance the rocket has flied away when its velocity is reached at the magni