Gold Member
- 426
- 177
I have starting working through section 134 of Landau and Lifshitz, vol 6, and it seems I have entered some kind of twilight zone where all my math/physics skills have left me 
The derivation starts with the energy-momentum tensor for an ideal fluid:
## T^{ik} = wu^i u^k - p g^{ik} ##,
where the Latin indices range from 0 to 3 (Greek indices would range from 1 to 3), ## w ## is the enthalpy, ## u^i ## is component i of the four-velocity, ## p ## is the pressure, and ## g^{ik} ## is the component ik of the Minkowski metric (with signature ## g^{00} = 1 ##). The derivation also employs the equation for conservation of particle number:
## \frac{\partial}{\partial x^i} \left( nu^i \right) = 0 ##,
where ## n ## is the proper number density of the particles. We lower the first upper index of ## T^{ik} ## using the metric tensor as
## T_{i}^{\ k} = g_{im}T^{mk} = wg_{im}u^m u^k - p g_{im} g^{mk} = wu_i u^k -p \delta_i^k. ##
Now we take the four divergence and set it equal to zero,
## \frac{\partial T_i^{\ k}}{\partial x^k} = \frac{\partial}{\partial x^k} \left[ wu_i u^k \right] - \frac{\partial p}{\partial x^i} = u_i \frac{\partial}{\partial x^k}\left[ w u^k \right] + w u^k \frac{\partial u_i}{\partial x^k} - \frac{\partial p}{\partial x^i} = 0 ##.
And here is where the trouble starts, because Landafshitz has the above equation with a plus sign next to the pressure term, not a minus. But it gets worse! In the next step, they say that ## u_i u^i = -1 ##. Now I must be really confused, because I thought that ## (u^i ) = \gamma (1,\mathbf{v}) ##, so that
## u_i u^i = u_0 u^0 + u_\alpha u^{\alpha} = \gamma^2 (1 - v^2) = 1 ##,
where ## \gamma ## is the Lorentz factor, and the speed of light has been set to unity.
Can anyone out there help me get this mess straightened out?
The derivation starts with the energy-momentum tensor for an ideal fluid:
## T^{ik} = wu^i u^k - p g^{ik} ##,
where the Latin indices range from 0 to 3 (Greek indices would range from 1 to 3), ## w ## is the enthalpy, ## u^i ## is component i of the four-velocity, ## p ## is the pressure, and ## g^{ik} ## is the component ik of the Minkowski metric (with signature ## g^{00} = 1 ##). The derivation also employs the equation for conservation of particle number:
## \frac{\partial}{\partial x^i} \left( nu^i \right) = 0 ##,
where ## n ## is the proper number density of the particles. We lower the first upper index of ## T^{ik} ## using the metric tensor as
## T_{i}^{\ k} = g_{im}T^{mk} = wg_{im}u^m u^k - p g_{im} g^{mk} = wu_i u^k -p \delta_i^k. ##
Now we take the four divergence and set it equal to zero,
## \frac{\partial T_i^{\ k}}{\partial x^k} = \frac{\partial}{\partial x^k} \left[ wu_i u^k \right] - \frac{\partial p}{\partial x^i} = u_i \frac{\partial}{\partial x^k}\left[ w u^k \right] + w u^k \frac{\partial u_i}{\partial x^k} - \frac{\partial p}{\partial x^i} = 0 ##.
And here is where the trouble starts, because Landafshitz has the above equation with a plus sign next to the pressure term, not a minus. But it gets worse! In the next step, they say that ## u_i u^i = -1 ##. Now I must be really confused, because I thought that ## (u^i ) = \gamma (1,\mathbf{v}) ##, so that
## u_i u^i = u_0 u^0 + u_\alpha u^{\alpha} = \gamma^2 (1 - v^2) = 1 ##,
where ## \gamma ## is the Lorentz factor, and the speed of light has been set to unity.
Can anyone out there help me get this mess straightened out?
Last edited by a moderator:
