Any two-dimensional metric satisfies Κλιμν (9xu9pv - 9xv9 pp). Show that the vacuum Einstein equations (with zero cosmological constant) are satisfied for any 2D metric. R 2
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- 2.9. (a) Solve the integral ...| (dx .dx3N) 3N and use it to determine the "volume" the relevant region of the phase space of an extreme relativistic gas ( = pc) of 3N particles moving in one dimension. Determine, as well, the number of ways of distributing a given energy E among this system of particles and show that, asymptotically, w0 = h³N. (b) Compare the thermodynamics of this system with that of the system considered in Problem 2.8.10. Ā = -2â + -3ŷ and B = -4â + -4ŷ. Calculate R = Ã+B. Calculate 0, the direction of R. Recall that 0 is defined as the angle with respect to the +x-axis. A. 49.4° B. 130.6° C. 229.4° D. 310.6°From the provided information, show that all of the mentioned hyperboals asmptotically approach the line x = ct for large values of any one of the four mentioned coordinates.
- (b) Write a necessary condition for a transformation (q,p) to (Q,P) to be connonical. Prove that P-2(1+√qcosp)√q sinp:Q-log(1+√qcosp)An accretion disc may form around a black hole. This is a thin disc of orbiting matter spanning radii r = Rin to Rout around the black hole. We assume that Rout » Rin and so we make the simplifying approximation that Rout → +∞o. The disc radiates according to the following equation 3 GM D(r, 0) = 1 CM (1-[B]"). 4 3 Here, r and are the usual polar coordinates with the origin at the centre of the disc. G is the gravitational constant, M is the mass of the black hole, Rin is the disc inner radius, M is the accretion rate - all these are constants. (a) Integrate D(r, 0) over the surface of the disc to find the total radiation output of the disc. (b) Find the total radiation in the case of Rin = 6GM/c².2. Show that for any observable Q, one can write d 11 ( 12/10) (Q) = 1/2 ([H1, Q]) + dt Ət