Massachusetts Institute of Technology Department of Physics 8.962 Spring 2006 Problem Set 4 Post date: Thursday, March 9th Due date: Thursday, March 16th 1. Connection in Rindler spacetime The spacetime for an accelerated observer that we derived on Pset 2, ds2 = −(1 + gx¯)2dt¯2 + dx¯2 + dy¯2 + dz¯2 (1) is known as “Rindler spacetime”. Compute all non-zero Christoffel symbols for this spacetime. (Carroll problem 3.3 will help you quite a bit here.) 2. Relativistic Euler equation (a) Starting from the stress-energy tensor for a perfect fluid, T = ρU�⊗ U�+ P h, where h = g −1 + U�⊗ U�, usin g local energy momentum conservation, � · T = 0, derive the relativistic Euler equation, (ρ + P )�U�U�= −h · �P . (2) (Note: Because both T and h are symmetric tensors, there is no ambiguity in the dot products that appear in this problem.) (b) For a nonrelativistic fluid (ρ � P , vt � vi) and a cartesian basis, show that this equation reduces to the Euler equation, ∂vi 1 + vk∂kvi = − ∂iP . (3) ∂t ρ (i, k are spatial indices run ning from 1 to 3.) What extra terms are present if the connection is non-zero (e.g., spherical coordinates)? (c) Apply the relativistic Euler equation to Rindler spacetime for hydrostatic equilib- rium. Hydrostatic equilibrium means that the fluid is at rest in the ¯x coordinates, i.e. Ux¯= 0. Suppose that the equation of state (relation between pressure and density) is P = wρ where w is a positive constant. Find the general solution ρ(¯x) with ρ(0) = ρ0. (d) Suppose now instead that w = w0/(1 + gx¯) where w0 is a constant. Show that the solution is ρ(¯x) = ρ0 exp(−¯ Find L, the density scale height, in terms of g and x/L. w0. Convert to “normal” units by inserting appropriate factors of c — L should be a length. (e) Compare your solution to the density profile of a nonrelativistic, plane-parallel, isothermal atmosphere (for which P = ρkT/µ, where T is temperature and µ is the mean molecular weight) in a constant gravitational field. [Use the nonrelativistic Euler equation with gravity: add a term −∂iΦ = gi, where Φ is Newtonian gravitational potential and gi is Newtonian gravitational acceleration, to the right hand side of Eq. (3).] Why does hydrostatic equilibrium in Rindler spacetime — where there is no gravity — give such similar results to hydrostatic equilibrium in a gravitational field?� � � 3. Spherical hydrostatic equilibrium As we shall derive later in the course, the line element for a spherically symmetric static spacetime may be written � �−1 ds2 = −e2Φ(r)dt2 + 1 − 2GM(r) dr2 + r 2(dθ2 + sin2 θ dφ2) , r where Φ(r) and M(r) are some given functions. In hydrostatic equilibrium, Ui = 0 for i ∈ [r, θ, φ]. Using the relativistic Euler equation, show that in hydrostatic equilibrium p = p(r) with ∂p ∂Φ = −(ρ + P ) . ∂r ∂r 4. Converting from non-affine to affine parameterization Suppose vα = dxα/dλ∗ obeys the geodesic equation in the form Dvαα = κ(λ∗)v . dλ∗ Clearly λ∗ is not an affine parameter. Show that uα = dxα/dλ obeys the geodesic equation in the form Duα = 0 dλ provided that dλ = exp κ(λ∗) dλ∗ . dλ∗ 5. Conserved quantities with charge A particle with electric charge e moves with 4-velocity uα in a spacetime with metric gαβ in the presence of a vector potential Aµ. The equation describing this particle’s motion can be written u β�βuα = eFαβu β , where Fαβ = �αAβ − �βAα . The spacetime admits a Killing vector field ξα such that Lξ�gαβ = 0 , Lξ�Aα = 0 . Show that the quantity (uα + eAα)ξα is constant along the worldline of the