R8

The 2nd law also applies to multiple black holes. In this case the statement is that the total entropy - ie the sum of the areas of all black holes - must increase. Argue that if two uncharged, non-rotating black holes collide violently to make one bigger black hole, then at most 29% of their initial rest energy can be radiated in gravitational waves. From T. Hartman's page http://www.hartmanhep.net/topics2015/2-bhthermo.pdf
Consider a massless scalar field; i.e., a scalar field [math]\displaystyle{ \phi(t,x) }[/math] with a potential-energy density [math]\displaystyle{ V(\phi) = V_0 }[/math]. Suppose that this scalar field is initially rolling, so that [math]\displaystyle{ \dot\phi\neq 0 }[/math], and that the kinetic-energy density associated with this rolling dominates the energy density of the Universe. Show that this implies that [math]\displaystyle{ \rho\propto a^{-6} }[/math], where [math]\displaystyle{ a }[/math] is the scale factor, in two ways: (1) by recalling how the energy density of matter with an equation of state P = wρ scales with a; and (2) by solving the equation of motion [math]\displaystyle{ \phi }[/math] in an expanding Universe.
Study the evolution of a scalar field (under the slow-roll assumption) if the inflationary potential is [math]\displaystyle{ \lambda \Phi^4 }[/math].
Show that [math]\displaystyle{ d(det(M))=det(M) M^{ik} dM_{ik} }[/math], and that [math]\displaystyle{ T^{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\partial(\sqrt{-g}{\mathcal L})}{\partial g_{\mu\nu}} }[/math] leads to the usual energy-momentum tensor in the case of a scalar field with a generic potential [math]\displaystyle{ V(\Phi) }[/math] and for a Minkowski space-time [math]\displaystyle{ g_{\mu\nu}=diag(1,-1,-1,-1) }[/math]. Work out [math]\displaystyle{ T^{00} }[/math] and [math]\displaystyle{ T^{ii} }[/math] in the case of a FLRW metric [math]\displaystyle{ g_{\mu\nu}=diag(1,-a^2(t),-a^2(t),-a^2(t)) }[/math]