§8 Energy-momentum tensor in relativistic hydrodynamics

In order to derive the relativistic equations of motion in hydrodynamics, let us first construct the energy-momentum 4-tensor $\mathcal{T}^{ik}$ for a fluid in motion on a flat spacetime (Landau & Lifshitz, 1994a; Landau & Lifshitz, 1995). Latin indices take the values $0, 1, 2, 3$ and Greek indices $1, 2, 3$. The time $t$ and the speed of light $\ensuremath{\mathsf{c}}$ are related to the time coordinate $x^0$ by the relation $x^0 \! = \! \ensuremath{\mathsf{c}}t$. The Galilean metric $g_{ik}$ for a flat space time is given by $g_{00} \! = \! 1$, $g_{11} = \! g_{22} \! = \! g_{33} \! = \! -1$ and $g_{ik} \! = \! 0$ when $i \! \neq k \!$. The different components of the symmetric energy-momentum tensor $\mathcal{T}^{ik}$ are such that the scalar $\mathcal{T}^{00}$ = $\mathcal{T}_{00}$ is the energy density and $\mathcal{T}^{0\alpha} / \ensuremath{\mathsf{c}}\! = \! -\mathcal{T}_{0\alpha} / \ensuremath{\mathsf{c}}$ is the $\alpha$ component of the momentum density vector. $\mathcal{T}_{\alpha\beta} = \mathcal T^{\alpha\beta}$ represents the 3-momentum flux density tensor and the component $\ensuremath{\mathsf{c}}\mathcal{T}^{0\alpha}$ is the energy flux density vector. The equations of motion are described by the condition:

$$\frac{ \partial \mathcal{T}_i^k }{ \partial x^k } = 0.$$ (8.1)

Consider an element of area $\mathrm{d}f^{\alpha} \! = \! \mathrm{d}f_{\alpha}$ of a three dimensional closed surface which is at rest. Integration of eq.(8.1) over the volume enclosed by this surface gives the change in momentum per unit time -the momentum flux-:

$$\frac{ 1 }{ \ensuremath{\mathsf{c}}} \frac{ \partial }{ \partial ... ...\alpha} \mathrm{d}V } = - \oint{ \mathcal{T}^{\alpha\beta} \mathrm{d}f_\beta },$$ (8.2)

according to Gauss's theorem. The right hand side of eq.(8.2) is the amount of momentum flowing out through the bounding surface in unit time. In other words, the force exerted on an area element $\mathrm{d}f^\alpha$ by the fluid is $\mathcal{T}^{\alpha\beta} \mathrm{d}f_\beta$. Consider now some volume element which is at rest in its local proper (or rest) frame. In this system of reference, Pascal's law applies: ``the pressure exerted by a given portion of the fluid is the same in all directions and perpendicular to the surface on which it acts''. Mathematically this is expressed by the relation $\mathcal T^{\alpha\beta} \mathrm{d} f_\beta \! = p \mathrm{d} f^\alpha$, where $p$ is the pressure of the fluid. This relation means that $\mathcal{T}_{\alpha\beta} = \! p \delta_{\alpha\beta}$ and here $\delta_{ik}$ is the unit 4-tensor for which $\delta_{ik} \! = \! 1$ if $i \! = \! k$ and $\delta_{ik} \! = 0$ if $i \! \neq \! k$. On the other hand, in the local proper system of reference, the component $\mathcal{T}^{00} = \! e$, where $e$ is the proper internal energy density of the fluid. To calculate the remaining components note that, since the fluid is at rest in its local proper frame, the momentum component density $T^{0\alpha}$ vanishes. Therefore, the energy-momentum tensor has the form

$$\mathcal{T}^{ik} = \begin{pmatrix}e & 0 & 0 & 0 \ 0 & p & 0 & 0 \ 0 & 0 & p & 0 \ 0 & 0 & 0 & p \ \end{pmatrix},$$ (8.3)

in the local proper frame. To find out an expression for $\mathcal{T}^{ik}$ in any frame of reference we use the fact that the fluid 4-velocity $u^i$ has the values $u^0 \! = \! 1$ and $u^\alpha \! = \! 0$ in the local rest frame. With this and eq.(8.3) it follows that

$$\mathcal{T}^{ik} = \omega u^i u^k - p g^{ik},$$ (8.4)

where $\omega = e + p$ is the heat function per unit volume. Since eq.(8.4) has the same form in any system of reference, it gives the required expression for the energy-momentum 4-tensor in relativistic fluid mechanics.

Footnotes

... mechanics.

When discussing relativistic fluid mechanics we take the values of the different thermodynamic quantities in their local proper frame. For example the internal energy density $e$, the enthalpy per unit volume $\omega$, the entropy density $\sigma$, and the temperature $T$ are all referred to this system of reference. The pressure $p$, being a relativistic invariant, could be described in any frame of reference.