The Big Three of Relativity

The "big three" of relativity

PROBLEM SET

PROBLEM 1. Given g = 1 - , and V 2 º V × V, show that = g4A × V.

Hint: You will use the change of variable = = = g .

You will also use the product rule on V × V, and the chain rule on ()-1/2. Don't forget that dV/dt = A.

PROBLEM 2. Show that = g2A + g4(A × V )V. Hint: Use the product rule on d(gV )/dt.

PROBLEM 3. Show that u × u = -c2, where u = (cg, gV ).

PROBLEM 4. Show that a × u = 0, where a = [ cg4A × V, g2A + g4(A × V )V ].

PROBLEM 5. Suppose the world line of a particle moving only in the x-direction represented by the parametric equations

ct(at) = a-1 sinh (at),

x(t) = a-1 cosh (at),

where a = 2 and -¥ < t < ¥. Accurately plot the particle's world line in BLACK pencil on the spacetime graph provided. Hint: We are using hyperbolic functions again.

PROBLEM 6. Write the four-position vector for the above particle.

PROBLEM 7. Find the derivatives ut and ux and thus find u.

PROBLEM 8. Find the derivatives at and ax and thus find a.

PROBLEM 9. Plot five four-velocity vectors in RED, at the locations ct = -10, -5, 0, 5 and 10. Draw their tails at the intersection of these times with the world line you drew in PROBLEM 5.

PROBLEM 10. Plot five four-acceleration vectors in BLUE, at the locations ct = -10, -5, 0, 5 and 10. Draw their tails at the intersection of these times with the world line you drew in PROBLEM 5.

PROBLEM 9. Show that u × u = -c2, using the u you found in PROBLEM 7.

PROBLEM 10. Show that a × u = 0, using the u you found in PROBLEM 7 and the a you found in PROBLEM 8.