Relativistic Euler Equations

Relativistic Euler flow is the local statement that a perfect fluid conserves stress–energy and any independently conserved particle number. The four equations $\nabla_aT^{ab}=0$ contain two physically different statements: energy changes through compression along the worldlines, while pressure gradients accelerate those worldlines orthogonally to their four-velocity. Projecting the tensor equation exposes both statements without choosing coordinates and makes the Newtonian limit transparent.

Required background. Hilbert Stress–Energy from Metric Variation supplies covariant stress–energy conservation; Events, Intervals, and Minkowski Geometry supplies four-velocity kinematics; and Differential Equations supplies characteristics and conservation laws.

Helpful background. The Einstein Equation gives the gravitational field equation coupled to the fluid; Relativistic Matter and Compact Objects gives the surrounding matter models; and Computational Relativity develops evolution formulations.

Projections along and across the flow

Assume a smooth, unforced perfect fluid with positive enthalpy density $\varepsilon+p$. For

$$T^{ab}=(\varepsilon+p)u^a u^b+p g^{ab},$$

define the derivative along a fluid worldline, the expansion, and the four-acceleration by

$$D\equiv u^a\nabla_a, \qquad \theta\equiv\nabla_a u^a, \qquad a^b\equiv Du^b.$$

Normalization implies $u_ba^b=0$. First expand the divergence before projecting:

$$\nabla_aT^{ab} =u^b[D\varepsilon+Dp+(\varepsilon+p)\theta] +(\varepsilon+p)a^b+\nabla^bp.$$

Contracting with $u_b$ turns the first term negative, kills the acceleration term, and gives $u_b\nabla^bp=Dp$. The two pressure derivatives cancel, leaving

$$D\varepsilon+(\varepsilon+p)\theta=0.$$

This is the comoving energy equation. Expansion lowers the energy density, while compression raises it. Projecting instead with $h^c{}_b=\delta^c{}_b+u^c u_b$ gives the relativistic Euler equation,

$$(\varepsilon+p)a^c+h^{ca}\nabla_a p=0.$$

Only the pressure gradient orthogonal to $u^a$ accelerates the fluid. Pressure also enters the energy equation through compression work, $p\theta$; a derivative $Dp$ is not a separate heating source. The inertial density is enthalpy density $\varepsilon+p$, not rest-mass density alone. A uniform pressure field gives $a^c=0$, so each element follows a geodesic even in curved spacetime. The single-fluid projection derivation is given in Andersson and Comer 2021, § 5.2, Eqs. (5.11)–(5.25).

Particle number and thermodynamic closure

If a conserved number current exists,

$$N^a=n u^a, \qquad \nabla_aN^a=Dn+n\theta=0.$$

Let the total energy per particle be $\mathcal E=\varepsilon/n$, including rest energy. This differs from the specific internal energy $e$ in $\varepsilon=\rho(1+e)$. For a single-component fluid the local first law is

$$d\mathcal E=T d s-p d\!\left(\frac1n\right),$$

where $s$ is entropy per particle. Combining this identity with the energy and number equations yields

$$nT Ds=0.$$

For $n,T\gt0$, a smooth perfect fluid therefore transports entropy along its worldlines: $Ds=0$. At exactly zero temperature the equation $nT Ds=0$ alone says nothing about $Ds$; a cold isentropic equation of state supplies that restriction. The derivation assumes no reactions changing the chosen particle number and no dissipative fluxes. Across a physical compressive shock, entropy increases and the differential derivation cannot be applied pointwise.

The unknowns are not closed until one supplies an equation of state, for example $p=p(n,s)$ and $\varepsilon=\varepsilon(n,s)$. Barotropic flow uses $p=p(\varepsilon)$, but that reduction can suppress composition or thermal degrees of freedom. The sound speed comes from a derivative at specified entropy and composition, not from the ratio $p/\varepsilon$.

Conservative coordinates and gravity

The covariant equation has the coordinate-divergence form

$$\partial_a\!\left(\sqrt{-g} T^a{}_b\right) =\sqrt{-g}\,\Gamma^c{}_{ab}T^a{}_c.$$

The right-hand side is a geometric source term; in locally inertial coordinates at one event it vanishes, but curvature and coordinate geometry reappear over a finite region. Number conservation is especially simple:

$$\partial_a\!\left(\sqrt{-g} n u^a\right)=0.$$

Numerical hydrodynamics rewrites these relations as balance laws for conserved densities and fluxes. Primitive variables such as $(n,p,v^i)$ must then be recovered from evolved variables; the inversion depends on the equation of state and can fail if numerical states leave its physical domain. The $3+1$ balance laws and a one-dimensional conservative-to-primitive recovery are derived in Andersson and Comer 2021, §§ 11.2–11.3, Eqs. (11.21)–(11.50).

When the fluid gravitates, the coupled system includes

$$G_{ab}=8\pi G T_{ab}.$$

Stress–energy conservation follows from the field equation and the Bianchi identity once the matter equations are consistent, but this redundancy does not eliminate the need to evolve matter. It is a compatibility relation between the gravitational and fluid sectors.

Newtonian and dust limits

In flat spacetime let $v\ll1$, $p\ll\rho$, and $\varepsilon=\rho+O(\rho v^2,p)$, with time derivatives and spatial gradients slow on the rest-mass scale. Particle conservation reduces to

$$\partial_t\rho+\boldsymbol\nabla\!\cdot(\rho\boldsymbol v)=0.$$

The spatial Euler projection reduces at leading order to

$$\rho(\partial_t+\boldsymbol v\!\cdot\boldsymbol\nabla)\boldsymbol v =-\boldsymbol\nabla p.$$

In a weak gravitational field with potential $\Phi$, the connection contributes $-\rho\boldsymbol\nabla\Phi$, recovering the Newtonian Euler equation with gravity and $\nabla^2\Phi=4\pi G\rho$. This simultaneous limit checks the sign of the pressure force and shows why the rest-energy part of $\varepsilon$ must be retained before expanding.

For $p=0$, the momentum equation becomes $a^b=0$: dust worldlines are geodesics. Dust is not a harmless universal approximation, however. Geodesic congruences can cross and form caustics, after which a single-valued velocity field no longer represents the matter. Pressure, velocity dispersion, or a kinetic distribution is then essential.

Exercises

For an expanding homogeneous fluid with comoving four-velocity and volume element $V$ satisfying $DV=\theta V$, show that the energy equation is the first law $D(\varepsilon V)=-p DV$.

Solution

Differentiate the energy in the comoving volume:

$$D(\varepsilon V)=V D\varepsilon+\varepsilon DV =V[-(\varepsilon+p)\theta]+\varepsilon V\theta =-pV\theta=-p DV.$$

No coordinate choice entered. The quantity $\varepsilon V$ is the total comoving energy, including rest energy. Particle conservation gives $D(nV)=0$, so subtracting the constant rest energy $mnV$ gives the same work law for the internal energy during smooth adiabatic expansion.

References

• Andersson, Nils, and Gregory L. Comer. “Relativistic Fluid Dynamics: Physics for Many Different Scales.” Living Reviews in Relativity 24, 3 (2021). DOI.