Anisotropic and Multifluid Stars

The perfect-fluid TOV equation assumes one pressure and one four-velocity. Elastic crusts, strong magnetic stresses, superfluid components, and effective kinetic descriptions can violate one or both assumptions. Composition gradients can change the perturbative response even when the equilibrium still has a single isotropic pressure. Anisotropic models distinguish radial from tangential stress; multifluid models distinguish independently moving conserved currents. These extensions are useful only when their additional functions are supplied by microphysics rather than tuned solely to produce a desired mass or radius.

Required background. The Einstein Equation supplies matter–geometry coupling; Killing Fields and Isometry Groups supplies the static spherical symmetry used for equilibrium.

Helpful background. Flat-Background Hilbert Stress-Energy from Metric Variation supplies stress tensors and conserved currents; Relativistic Matter and Compact Objects supplies material context; Computational Relativity supplies validation methods for coupled ODE systems.

Anisotropic stress and radial balance

Assume a regular static configuration with no heat flux and with the areal radius as the radial coordinate. In the orthonormal frame of a static spherical star, write

$$T^{\hat\mu}{}_{\hat\nu} =\operatorname{diag}(-\epsilon,p_r,p_t,p_t).$$

The Einstein equations retain $dm/dr=4\pi r^2\epsilon$, while radial stress conservation becomes

$$\frac{dp_r}{dr} =-(\epsilon+p_r) \frac{G(m+4\pi r^3p_r)}{r^2(1-2Gm/r)} +\frac{2(p_t-p_r)}{r}.$$

Positive $p_t-p_r$ supplies an outward anisotropy term; negative anisotropy adds inward support demand. This sign follows from the $(-,+,+,+)$ stress convention and can be checked by taking the isotropic limit. The classic constant-density examples and generalized balance equation appear in Bowers and Liang 1974, Eqs. (2.7)–(2.12), pp. 658–659, but their freely chosen anisotropy functions are illustrative closures, not microscopic predictions.

Regularity at a smooth center requires $p_t-p_r=O(r^2)$. One way to see the condition is to write the spatial stress in local Cartesian coordinates as $T_{ij}=p_t\delta_{ij}+(p_r-p_t)x_i x_j/r^2$: the apparent direction dependence must disappear smoothly at the origin. At a boundary to vacuum with no thin surface layer, $p_r(R)=0$ is the normal-stress condition. Density and tangential pressure may jump there; their nonzero one-sided values do not by themselves require a surface layer. The induced metric and extrinsic curvature must match. Energy conditions, sound characteristics, and stability must be tested in addition to solving the equilibrium equation.

Multiple currents and entrainment

A relativistic multifluid description begins with currents $n_x^\mu$, one for each independently tracked constituent $x$. A master function depends on invariant scalar products such as

$$n_x^2=-g_{\mu\nu}n_x^\mu n_x^\nu, \qquad n_{xy}^2=-g_{\mu\nu}n_x^\mu n_y^\nu.$$

For two uncharged, nondissipative constituents let $\Lambda=\Lambda(n_1^2,n_2^2,n_{12}^2)$ be the matter Lagrangian. At fixed metric, differentiation gives

$$\begin{aligned} \mu^1_\nu&=\frac{\partial\Lambda}{\partial n_1^\nu} =\mathcal B^1 n_{1\nu}+\mathcal A^{12}n_{2\nu},\\ \mathcal B^1&=-2\frac{\partial\Lambda}{\partial n_1^2}, \qquad \mathcal A^{12}=-\frac{\partial\Lambda}{\partial n_{12}^2}. \end{aligned}$$

The analogous expression for $\mu^2_\nu$ has the same cross coefficient. The factors of two differ because $n_1^2$ contains two copies of the first current whereas $n_{12}^2$ contains one. When $\mathcal A^{12}\ne0$, a momentum need not be parallel to its own current. This is entrainment, a nondissipative coupling, not a drag force.

The stress tensor is

$$T^\mu{}_{\nu}=\Psi\delta^\mu{}_{\nu} +\sum_x n_x^\mu\mu^x_\nu,$$

where $\Psi=\Lambda-\sum_x n_x^\nu\mu^x_\nu$ is the generalized pressure. The variational construction is given in Andersson and Comer 2021, §9, Eqs. (9.10)–(9.17). For separately conserved currents without constituent forces, the equations are

$$\nabla_\mu n_x^\mu=0,\qquad n_x^\mu(\nabla_\mu\mu^x_\nu-\nabla_\nu\mu^x_\mu)=0.$$

Thus total stress conservation alone does not determine both flows. Reactions, electromagnetism, or mutual friction require the corresponding source and force terms.

Check the one-current limit: $\Lambda=-\epsilon(n)$ gives $\mu_\nu=\mu u_\nu$, $\Psi=-\epsilon+n\mu=p$, and $T^\mu{}_\nu=(\epsilon+p)u^\mu u_\nu+p\delta^\mu{}_\nu$. This fixes the master-function sign and recovers the perfect fluid.

In strict static equilibrium the currents may be aligned even though their perturbations are not. Consequently a background can resemble a one-fluid star while its oscillations contain counter-moving superfluid modes. Composition gradients also introduce buoyancy and chemical-restoration timescales.

Closure, equilibrium, and stability

An anisotropic model needs relations for $p_r$, $p_t$, and $\epsilon$ in terms of state variables; a multifluid model needs the master function, reactions, and which currents are conserved on the timescale considered. Prescribing only $p_t-p_r$ as a function of radius is coordinate dependent unless the radius is identified geometrically, and it may hide acausal response. Prescribing a background entrainment coefficient without its state dependence is likewise insufficient for perturbations.

Useful checks are the isotropic or locked-flow limit, dimensional consistency of every entrainment coefficient, positivity of the local energy quadratic form, causal characteristic speeds, regular center series, and a smooth one-fluid limit. Stability cannot be inferred from a larger maximum mass. New degrees of freedom add new modes, and a sequence stable to co-moving radial motion can be unstable to counter-moving or composition modes.

Limits of effective descriptions

Magnetic fields produce anisotropic Maxwell stress, but replacing them by a spherical function $p_t-p_r$ discards field topology and the exterior field. An elastic solid has direction-dependent strain history, not an arbitrary pressure split. Superfluid vortices and mutual friction are dynamical structures not encoded in a static two-current master function alone. These reductions can isolate a mechanism, but their outputs must be labeled as model dependent and checked against the more complete material description.

Check central regularity

Suppose $p_t-p_r\to\Delta_0\ne0$ as $r\to0$. What happens to hydrostatic balance, and what leading behavior avoids the problem?

Solution

The anisotropy force behaves as $2\Delta_0/r$ and diverges, while the ordinary gravitational term is proportional to $r$ near a regular center because $m=O(r^3)$. A finite nonzero $\Delta_0$ is therefore incompatible with a regular spherical center. Spherical symmetry requires equal stresses at the center; a smooth even expansion gives $p_t-p_r=O(r^2)$, making the anisotropy term $O(r)$ and comparable to the regular gravitational term.

References

• Andersson, Nils, and Gregory L. Comer. “Relativistic Fluid Dynamics: Physics for Many Different Scales.” Living Reviews in Relativity 24 (2021): 3. DOI.
• Bowers, Richard L., and E. P. T. Liang. “Anisotropic Spheres in General Relativity.” Astrophysical Journal 188 (1974): 657–665. DOI.