Chapter 3's stress-energy tensor obeys the conservation law in inertial coordinates

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Relativity & Gravitation · The equivalence principle and gravity as geometry

The Geometry of Gravity, Ch Accelerated observers, Rindler coordinates and the equivalence principle

Problem

Chapter 3's stress-energy tensor obeys the conservation law ∂μTμν=0\partial_\mu T^{\mu\nu}=0 in inertial coordinates, a two-index object with upper indices, unlike the metric ημν\eta_{\mu\nu} this chapter transformed. An object with upper indices transforms with the Jacobian the other way round from ημν\eta_{\mu\nu}: T′αβ=∂x′α∂xμ∂x′β∂xνTμν,T'^{\alpha\beta} = \frac{\partial x'^\alpha}{\partial x^\mu}\frac{\partial x'^\beta}{\partial x^\nu}T^{\mu\nu}, using ∂x′/∂x\partial x'/\partial x (new coordinates with respect to old) rather than the ∂x/∂x′\partial x/\partial x' the chapter used for gμνg_{\mu\nu}. Only the (ct,x)(ct,x) block mixes; yy and zz are carried across unchanged, so components with a yy or zz index do not contribute below. ((a)) At the instant η=0\eta=0 (so ct=0ct=0 and x=c2/a+ξx=c^2/a+\xi), find the four partial derivatives ∂x′0/∂(ct)\partial x'^0/\partial(ct), ∂x′0/∂x\partial x'^0/\partial x, ∂ξ/∂(ct)\partial\xi/\partial(ct) and ∂ξ/∂x\partial\xi/\partial x, with x′0=cηx'^0=c\eta, starting from the inverse of the chapter's map. Show that two of the four vanish at this instant. ((b)) Using the rule above and the surviving partials, write T′00T'^{00} and T′0ξT'^{0\xi} at η=0\eta=0 in terms of the inertial components T00T^{00} and T01T^{01} at the corresponding event, and of aa and ξ\xi. ((c)) A rocket has a=10 m s−2a=10\ \mathrm{m\,s^{-2}}; at the event η=0\eta=0, ξ=9.00×1015 m\xi=9.00\times10^{15}\ \mathrm{m} (so x=c2/a+ξ=1.80×1016 mx=c^2/a+\xi=1.80\times10^{16}\ \mathrm{m}), the inertial components are T00=2.00×109 J m−3T^{00}=2.00\times10^{9}\ \mathrm{J\,m^{-3}} and T01=6.00×108 J m−3T^{01}=6.00\times10^{8}\ \mathrm{J\,m^{-3}}. Evaluate T′00T'^{00} and T′0ξT'^{0\xi}. ((d)) The chain-rule step that produced the extra term in ∂0J0+∂ξJξ+∂yJy+∂zJz+a/c21+aξ/c2Jξ=0\partial_0J^0+\partial_\xi J^\xi+\partial_yJ^y+\partial_zJ^z+\dfrac{a/c^2}{1+a\xi/c^2}J^\xi=0 applies once for every differentiated index. Say, without carrying the calculation through, why the corresponding statement for ∂μTμν=0\partial_\mu T^{\mu\nu}=0 needs more correction terms than the one found for JμJ^\mu, and why this makes finding the derivative that replaces ∂μ\partial_\mu harder for a two-index tensor.

Browse the problem library