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(adsbygoogle = window.adsbygoogle || []).push({});

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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Derivation from Landau and Liffshitz, vol 6

Tags:

Loading...

Similar Threads for Derivation Landau Liffshitz |
---|

I Example of the use of the Lie Derivative in Relativity |

I Lie and Covariant derivatives |

I Riemann curvature tensor derivation |

A Commutator of covariant derivative and D/ds on vector fields |

I Interesting Derivation of Maxwell's Equations |

**Physics Forums | Science Articles, Homework Help, Discussion**