Problem Set 4

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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, ¯2 + dx ds2 = −(1 + gx ¯)2 dt ¯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 � ⊗U � + P
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  Massachusetts Institute of Technology Department of Physics 8.962Spring2006 ProblemSet4 Post date: Thursday, March 9th Duedate:Thursday,March16th 1. Connection in Rindler spacetime The spacetime for an accelerated observer that we derived on Pset 2, ds 2 = − (1+ gx ¯) 2 dt ¯ 2 + dx ¯ 2 + dy ¯ 2 + dz  ¯ 2 (1) is known as “Rindler spacetime”. Compute all nonzero Christoffel symbols for this spacetime. (Carroll problem 3.3will help you quite a bit here.) 2. Relativistic Euler equation (a)Startingfromthe stressenergy tensorforaperfect fluid, T = ρU �  ⊗ U �  + P  h , where h = g − 1 + U �  ⊗ U �  , using local energy momentum conservation, ∇· T =0,derivethe relativistic Euler equation, ( ρ + P  ) ∇ U �  U �  = − h ·∇ P. (2) (Note:Becauseboth T and h are symmetric tensors, there is no ambiguity in the dot products that appear in this problem.) (b)Foranonrelativistic fluid( ρ � P  , v t � v i )and a cartesian basis, show that this equation reduces to the Euler equation, ∂v i 1 + v k ∂  k v i = − ∂  i P. (3) ∂t ρ ( i , k are spatial indices running from 1 to 3.) What extra terms are present if the connectionisnonzero(e.g., spherical coordinates)? (c)Applythe 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. U  x ¯ =0. Supposethattheequationof state(relationbetweenpressureanddensity) is P = wρ where w isapositiveconstant. Find thegeneral solution ρ (¯ x )with ρ (0)= ρ 0 . (d)Suppose now instead that w = w 0 / (1+ gx ¯)where w 0 is a constant. Show that the solutionis ρ (¯ x )= ρ 0 exp( − ¯Find L , the density scale height, in terms of g and x/L . w 0 . 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 meanmolecularweight)inaconstantgravitational field. [UsethenonrelativisticEuler equation with gravity: add a term − ∂  i Φ= g i , where Φ is Newtonian gravitational potential and g i 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 ds 2 = − e 2Φ( r ) dt 2 +1 − 2 GM  ( r ) dr 2 + r 2 ( dθ 2 + sin 2 θdφ 2 ) , r whereΦ( r )and M  ( r )are somegivenfunctions. Inhydrostaticequilibrium, U  i =0for 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 v α = dx α /dλ ∗   obeys the geodesic equation in the form Dv αα = κ ( λ ∗ ) v. dλ ∗   Clearly λ ∗   is not an affine parameter. Showthat u α = dx α /dλ obeys the geodesic equation in the form Du α =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 u α in a spacetime with metric g αβ in the presence of a vector potential A µ . The equation describing this particle’s motion can be written u β  ∇ β  u α = eF  αβ  u β , where F  αβ = ∇ α A β −∇ β  A α . The spacetime admits a Killing vector field ξ  α such that L ξ�  g αβ =0 , L ξ�  A α =0 . Showthatthequantity( u α + eA α ) ξ  α is constant along the worldline of the particle.
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