- #1
ergospherical
- 968
- 1,276
- Homework Statement
- ##T^{00} = a^{-2} \bar{\rho}(1+\delta)##
##T^{0i} = a^{-2} \bar{\rho}(1+w)v^i##
##T^{ij} = a^{-2} \bar{\rho} [(1+\delta)\delta^{ij} - h^{ij}]##
- Relevant Equations
- ##\nabla_{\mu} T^{\mu \nu} = 0##
The perturbed line element: ##g = a(\tau)^2[-d\tau^2 + (\delta_{ij} + h_{ij})dx^i dx^j]##
Expanding the covariant derivative with ##\nu = 0##, you get a few pieces. Here on keeping only terms linear in the perturbations,
##\partial_{\mu} T^{\mu 0} = a^{-2} \bar{\rho} \left[ \delta' - 2\mathcal{H} (1+\delta) + (1+w) i\mathbf{k} \cdot \mathbf{v} \right]##
here ##\mathcal{H} = a'/a## and ##i \mathbf{k} \cdot \mathbf{v} = \partial_i v^i##. Then
##\Gamma^{\mu}_{\mu \rho} T^{\rho 0} = a^{-2} \bar{\rho} \left[ 4\mathcal{H}(1+\delta) + \frac{1}{2} h' \right]##
##\Gamma^{0}_{\mu \rho} T^{\mu \rho} = a^{-2} \bar{\rho} \left[ \mathcal{H}(1+\delta)(1+3w) + \frac{1}{2} w h' \right]##
Overall,
##0 = a^{-2} \bar{\rho} \left[ \delta' + 3(1+w) \mathcal{H}(1+\delta) + (1+w) i \mathbf{k} \cdot \mathbf{v} + \frac{1}{2}(1+w)h'\right]##
but the term ##3(1+w) \mathcal{H}(1+\delta)## shouldn't be there. I can't see why not? For reference, the connection coefficients
Expanding the covariant derivative with ##\nu = 0##, you get a few pieces. Here on keeping only terms linear in the perturbations,
##\partial_{\mu} T^{\mu 0} = a^{-2} \bar{\rho} \left[ \delta' - 2\mathcal{H} (1+\delta) + (1+w) i\mathbf{k} \cdot \mathbf{v} \right]##
here ##\mathcal{H} = a'/a## and ##i \mathbf{k} \cdot \mathbf{v} = \partial_i v^i##. Then
##\Gamma^{\mu}_{\mu \rho} T^{\rho 0} = a^{-2} \bar{\rho} \left[ 4\mathcal{H}(1+\delta) + \frac{1}{2} h' \right]##
##\Gamma^{0}_{\mu \rho} T^{\mu \rho} = a^{-2} \bar{\rho} \left[ \mathcal{H}(1+\delta)(1+3w) + \frac{1}{2} w h' \right]##
Overall,
##0 = a^{-2} \bar{\rho} \left[ \delta' + 3(1+w) \mathcal{H}(1+\delta) + (1+w) i \mathbf{k} \cdot \mathbf{v} + \frac{1}{2}(1+w)h'\right]##
but the term ##3(1+w) \mathcal{H}(1+\delta)## shouldn't be there. I can't see why not? For reference, the connection coefficients