Scalar Boson Stars

A scalar boson star is a self-gravitating, spatially localized complex field whose phase oscillates while its stress–energy remains stationary. Localization is an eigenvalue problem: regular central or axis data and exponential decay at infinity select discrete field frequencies for a chosen central amplitude and potential. Existence alone does not establish stability, formation, or astrophysical abundance; this page constructs the equilibrium families and their conserved charge.

Required background. The Einstein equations supply the geometry sourced by the scalar stress–energy.

Helpful background. Stress–Energy and Conservation supplies Noether and energy currents; Relativistic Matter and Compact Objects supplies equilibrium comparisons; Black-Hole Geometry supplies invariant compactness and horizon diagnostics; and Computational Relativity supplies boundary-value and virial checks.

Einstein–Klein–Gordon system

For a canonically normalized complex scalar $\Phi$, take

$$S=\int d^4x\sqrt{-g}\left[ \frac{R}{16\pi G} -g^{\mu\nu}\nabla_\mu\Phi^*\nabla_\nu\Phi -V(|\Phi|^2)\right].$$

The equations are

$$G_{\mu\nu}=8\pi G T_{\mu\nu}, \qquad \Box\Phi-\frac{dV}{d|\Phi|^2}\Phi=0,$$

with

$$T_{\mu\nu}= \nabla_\mu\Phi^*\nabla_\nu\Phi+ \nabla_\nu\Phi^*\nabla_\mu\Phi -g_{\mu\nu} \left(\nabla_\alpha\Phi^*\nabla^\alpha\Phi+V\right).$$

For a mini boson star $V=\mu^2|\Phi|^2$, where $\mu$ has dimension $L^{-1}$ and $\Phi$ has dimension $L^{-1}$. Additional repulsive or solitonic self-interactions define different model classes rather than corrections with universal size.

Stationary phase and boundary conditions

In spherical symmetry use

$$ds^2=-\alpha(r)^2dt^2+a(r)^2dr^2+r^2d\Omega^2, \qquad \Phi(t,r)=\phi(r)e^{-i\omega t}.$$

Although $\Phi$ is time dependent, every quadratic term in $T_{\mu\nu}$ is time independent. The geometry can therefore be static. Regularity and asymptotic flatness require

$$a(0)=1,\quad \phi'(0)=0,\quad a(\infty)=1,\quad\alpha(\infty)=1,\quad\phi(\infty)=0.$$

The lapse condition fixes the remaining time normalization and makes $\omega$ the frequency measured at infinity. To see what is actually integrated, set $a^{-2}=1-2Gm/r$ and take the radial amplitude $\phi$ to be real. A static orthonormal observer measures

$$\begin{aligned} \epsilon&=\frac{\omega^2\phi^2}{\alpha^2} +\frac{\phi'^2}{a^2}+V,\\ p_r&=\frac{\omega^2\phi^2}{\alpha^2} +\frac{\phi'^2}{a^2}-V,\\ p_t&=\frac{\omega^2\phi^2}{\alpha^2} -\frac{\phi'^2}{a^2}-V. \end{aligned}$$

Here a prime means $d/dr$, and $V$ is evaluated at $\phi^2$. In particular, $p_r-p_t=2\phi'^2/a^2$: scalar gradient stress is anisotropic, so a boson star is not obtained by inserting an effective pressure into a perfect-fluid TOV solver. The independent radial equations are

$$\begin{aligned} m'&=4\pi r^2\epsilon,\\ \frac{\alpha'}{\alpha} &=\frac{G(m+4\pi r^3p_r)}{r^2(1-2Gm/r)},\\ \phi''+ \left(\frac{2}{r}+\frac{\alpha'}{\alpha}-\frac{a'}{a}\right)\phi' &+a^2\left(\frac{\omega^2}{\alpha^2}-\frac{dV}{d\phi^2}\right)\phi=0. \end{aligned}$$

The last equation follows by evaluating

$$\Box\Phi=\frac{e^{-i\omega t}}{\alpha a r^2} \partial_r\!\left(\frac{\alpha r^2\phi'}{a}\right) +\frac{\omega^2}{\alpha^2}\Phi.$$

At a regular center it gives the useful starting series

$$\phi(r)=\phi_c+ \frac{\phi_c}{6}\left[ \left.\frac{dV}{d\phi^2}\right|_{\phi_c} -\frac{\omega^2}{\alpha_c^2}\right]r^2+O(r^4), \qquad m(r)=\frac{4\pi}{3}\epsilon_c r^3+O(r^5).$$

One may initially choose $\alpha_c$ and shoot for the decaying field. Afterward divide both $\alpha$ and $\omega$ by the computed $\alpha(\infty)$; rescaling only the lapse would change the physical field equations.

In the flat far-field approximation the decaying solution is

$$\phi(r)\propto\frac{e^{-\kappa r}}{r}, \qquad \kappa=\sqrt{\mu^2-\omega^2}.$$

The asymptotic gravitational $1/r$ terms modify the power-law prefactor, but not this exponential rate. Thus exponential localization requires $0\lt\omega\lt\mu$ for the positive- frequency branch. Shooting or collocation adjusts $\omega$ until the growing asymptotic solution vanishes. Liebling and Palenzuela derive the reduced equations, boundary data, and principal scalar families in Liebling and Palenzuela 2023, § 2.4, Eqs. (33)–(45). Their scalar action has an overall factor $1/2$ in the matter terms; their field amplitude is $\sqrt{2}\phi$ in this page’s normalization, with the potential rescaled accordingly. Their positive-frequency phase has the opposite sign. These translations preserve the metric and stress–energy.

Noether charge, mass, and scaling

The global phase symmetry $\Phi\mapsto e^{i\chi}\Phi$ gives

$$j^\mu=-i\left(\Phi^*\nabla^\mu\Phi -\Phi\nabla^\mu\Phi^*\right), \qquad \nabla_\mu j^\mu=0.$$

For future unit normal $n^\mu$, choose the positive charge convention

$$Q=-\int_\Sigma j^\mu n_\mu\sqrt{\gamma}\,d^3x .$$

For the spherical ansatz, $j^t=2\omega\phi^2/\alpha^2$, hence

$$Q=8\pi\omega\int_0^\infty \frac{a}{\alpha}\,r^2\phi^2\,dr.$$

This expression is positive on the $\omega\gt0$ branch and dimensionless. It supplies a direct check on both the current sign and the proper-volume factor; $4\pi r^2dr$ alone is not the spatial volume element.

The ADM mass comes from the asymptotic metric. Along a regular one-parameter equilibrium sequence, the differential relation $dM=\omega\,dQ$ is a useful normalization and first-law check. A virial identity obtained by radial scaling provides an independent residual sensitive to boundary and discretization errors.

Dimensional analysis gives the characteristic mini-boson-star mass scale

$$M\sim\frac{M_{\rm Pl}^2}{\mu} =\frac{1}{G\mu}, \qquad M_{\rm Pl}=G^{-1/2}.$$

Kaup’s original numerical family exhibits this scaling in Kaup 1968, § III.C, Table I, pp. 1334–1335. A repulsive quartic interaction introduces another dimensionless coupling and can support a parametrically different mass scale. Solitonic potentials can produce more compact configurations, but compactness must state how radius is defined because scalar fields have no sharp surface.

Rotating and self-interacting families

An axisymmetric rotating ansatz has

$$\Phi=e^{-i\omega t+im\varphi}\phi(r,\theta), \qquad m\in\mathbb Z.$$

Single-valuedness quantizes $m$, regularity makes the field vanish appropriately on the axis for $m\ne0$, and the canonical scalar family obeys $J=mQ$. The stress tensor is stationary and axisymmetric even though the field phase is not. Rotating solutions generally have toroidal energy density and require a two-dimensional boundary-value problem.

Self-interaction changes the field equations, allowed frequency range, mass–charge curve, and compactness. Comparisons must keep the potential and field normalization explicit. The independent limits are the weak-field Schrödinger–Poisson reduction, exponential tail with the correct $\kappa$, the mini-star $1/(G\mu)$ scaling, $J=mQ$ for rotating states, and decreasing constraint and virial residuals. None of these existence checks identifies the stable branch; perturbative and nonlinear stability are treated separately.

An equilibrium catalog should therefore record more than mass and central field. For every point it should retain frequency, charge, angular momentum, an invariant or operational radius, compactness, node number, potential parameters, and numerical residuals. Excited solutions with radial or angular nodes solve the same boundary conditions but need not share the ground state’s dynamical behavior. Likewise, two objects with the same ADM mass can lie on different branches or have different charge and binding $M-\mu Q$.

The absence of a material surface also matters. A radius enclosing a fixed fraction of mass or charge is useful, but the chosen fraction must be stated; using the point where the scalar becomes numerically small makes compactness depend on a tolerance. A horizon check must be geometric rather than inferred from a large redshift. Finally, a stationary solution does not explain how its coherent phase and charge were assembled. Linear modes, nonlinear evolutions, and formation from families of initial data are independent tests, developed on the stability and formation page.

Check exponential localization

A mini boson star has $\omega=0.80\mu$. Find the asymptotic inverse length $\kappa$ and the radial e-folding scale.

Solution

$$\kappa=\sqrt{\mu^2-\omega^2} =\mu\sqrt{1-0.80^2} =0.60\mu .$$

The e-folding length is $\kappa^{-1}=1/(0.60\mu)\simeq1.67/\mu$. Because $\omega<\mu$, the tail decays exponentially. If $\omega\geq\mu$, $\kappa$ would be zero or imaginary and the assumed localized asymptotic boundary condition would fail.

References

• Kaup, David J. “Klein-Gordon Geon.” Physical Review 172 (1968): 1331–1342. DOI.
• Liebling, Steven L., and Carlos Palenzuela. “Dynamical Boson Stars.” Living Reviews in Relativity 26 (2023): 1. DOI.