Assignment 4

20-18 The covariant components are

x_a = g_ab x^b = g_a1 Substitute this into the formula (20.68) for the covariant derivative to find

Ń_b x_a =

ć
č

śg_a1

śx^b

śg_b1

śx^a

śg_ab

śx¹

ö
ř

The first term is antisymmetric in (a, b), and the last term vanishes because of x¹-independence. Thus the symmetrized Ń_b x_a vanishes - that's Killing's equation. (This does not, of course, prove that every vectorfield generating a symmetry of the metric satisfies Killing's equation - but it is so :)

20-20
a) In rectangular coordinates the Christoffel symbols vanish, so Killing's equation becomes x_a,b + x_b,a = 0. The first two have constant components, whose derivatives vanish, for the third, for example, the (1, 2) component is (-y)_,y + x_,x = -1 + 1 = 0.

b) If you verify this by starting with ś/śf, you will be doing problem 8-8 over again; else, similarly but "the other way round", express ś/śx in terms of ś/śr and ś/śf.

c) Let the symmetry origin be at (X, Y), so coordinates centered there are x˘ = x - X, y˘ = y - Y. The desired K. V. is
h˘ = - y˘ ś
śx˘
+ x˘ ś
śy˘
= h - X ś
śy
+ Y ś
śx
.

21-2

21-11 Hartle's official answer is: "Running the Mathematica program Curvature and the Einstein Equations for the metric given in Problem 8.12 gives R = -2."