Changes between Version 16 and Version 17 of u/johannjc/scratchpad4


Ignore:
Timestamp:
10/01/15 08:04:58 (9 years ago)
Author:
Jonathan
Comment:

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  • u/johannjc/scratchpad4

    v16 v17  
    126126And in Einstein notation gives
    127127
    128 $\partial_t T = - \partial_i n b_i \frac{n \kappa_\perp}{B^2 \left ( \Lambda+1 \right ) } \left ( b_j  \partial_j \left ( -\Lambda T^{\Lambda+1} + \left ( \Lambda + 1 \right ) T^{\Lambda}T_{*} \right ) \right ) +  \partial_i n \frac{n \kappa_\perp}{B^2 \left ( \Lambda+1 \right ) }   \partial_i \left ( - \Lambda T^{ \Lambda+1} + \left ( \Lambda + 1 \right ) T^{\Lambda} T_{*} \right )  $
     128$\partial_t T = - \partial_i n b_i \frac{n \kappa_\perp}{B^2 \left ( \Lambda+1 \right ) }  b_j  \partial_j \left ( -\Lambda T^{\Lambda+1} + \left ( \Lambda + 1 \right ) T^{\Lambda}T_{*} \right ) +  \partial_i n \frac{n \kappa_\perp}{B^2 \left ( \Lambda+1 \right ) }   \partial_i \left ( - \Lambda T^{ \Lambda+1} + \left ( \Lambda + 1 \right ) T^{\Lambda} T_{*} \right )  $
    129129
    130130
    131 The first term is very similar to the parallel case... 
     131The first term is very similar to the parallel case - and the second term can be made similar by replacing $b_i b_j$ with $\delta_{ij}$
    132132
    133133$\phi' = \frac{\phi-\psi \Lambda}{\psi \left ( \Lambda + 1 \right )}$
    … …  
    135135$B_j =  \phi' C_j $   
    136136
    137 $C_j =   -\kappa_\perp \psi \partial_i n^2 B^{-2} b_i b_j $
     137$C_j =   -\kappa_\perp \psi \partial_i  n^2 B^{-2} \left ( b_i b_j - \delta_{ij} \right ) $
    138138
    139139$D_{ij}=\phi'E_{ij}$
    140140
    141 $E_{ij} =   -\kappa_\perp \psi  n^2 B^{-2} b_i b_j $
     141$E_{ij} =   -\kappa_\perp \psi  n^2 B^{-2} \left ( b_i b_j - \delta_{ij} \right ) $