PHYS 231 Final, Fall 2007

Physics 231D, Final Exam, January 22 2008 Time: 3 hours Do 5 out of 7 problems 1. A frame F’ with coordinates (x0; t0) is moving with a velocity v with respect to frame F with coordinates (x; t) x0 = Ax + B(ct) ct0 = Cx + D(ct) such that the distance x2 � (ct)2 = x 02 � (ct0)2 is invariant. Find the conditions on A; B; C; and D: Using the fact that the when the origins of the two frames coincide at x0 = 0 then x = vt, express A; B; C; and D in terms of v: 2. De…ne the conformal tensor by C���� = R���� + a (g��R�� � g��R�� � g��R�� + g��R��) + b (g��g�� � g��g��) R such that all contractions of this tensor vanish, and in particular g��C���� = 0 and g��C���� = 0: Find a and b: 3. Find the Christo¤el symbols, Riemann curvature tensor and Ricci tensor components for the space-time with metric ds2 = �e2A(r)dt2 + dr2 Determine the function A(r) so that the equations R�� = 0 are satis…ed. 4. In ‡at Minkowski space time …nd the ten Killing vectors corresponding to invariance of the metric under translations and rotations. Express these in terms of the basis vectors � @ @t; @ @x; @ @y; @ @z � : 5. For a k = �1 Friedman cosmology with p = � = 0 show that the line element becomes ds2 = �dt2 + t2 � d�2 + sinh2 � � d�2 + sin2 �d�2�� Exhibit an explicit coordinate transformation to show that this metric describes a Minkowski space. 6. Solve the …rst order Friedman equation : R R !2 = 8�G 3 � � k R2 for R(t) when matter is dominated by radiation for the three possibilities k = 1; 0; �1: 7. A particle is falling radially from rest at r = a in Schwarzschild geometry. Derive the equations of motion relating t; r and �: Hint: Use the …rst integrals of energy and angular momentum conservation, and use the change of variable r = a 2 (1 + cos �) to express dt in terms of d� but do not do the t integral. 1