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Chapter 12: Q10P (page 518)
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Prove that the symmetry (or antisymmetry) of a tensor is preserved by Lorentz transformation (that is: if is symmetric, show that is also symmetric, and likewise for antisymmetric).
Work out the remaining five parts to Eq. 12.118.
Work out, and interpret physically, the component of the electromagnetic force law, Eq. 12.128.
(a) In Ex. 12.6 we found how velocities in thex direction transform when you go from Sto . Derive the analogous formulas for velocities in the y and z directions.
(b) A spotlight is mounted on a boat so that its beam makes an angle with the deck (Fig. 12.20). If this boat is then set in motion at speedv, what angle does an individual photon trajectory make with the deck, according to an observer on the dock? What angle does the beam (illuminated, say, by a light fog) make? Compare Prob. 12.10.
You may have noticed that the four-dimensional gradient operator functions like a covariant 4-vector—in fact, it is often written , for short. For instance, the continuity equation, , has the form of an invariant product of two vectors. The corresponding contravariant gradient would be . Prove that is a (contravariant) 4-vector, if is a scalar function, by working out its transformation law, using the chain rule.
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