Oscillatons and Proca Stars

A real massive scalar cannot hide its oscillation in an internal complex phase: its stress–energy varies in time. Nevertheless, gravity can bind a long-lived configuration—an oscillaton—whose scalar contains odd temporal harmonics and whose metric contains even harmonics. A complex massive vector can instead form a Proca star with stationary stress–energy and a conserved U(1) charge. The two systems share weak-field localization but differ in symmetry, conserved quantities, and the meaning of stationarity.

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

Helpful background. Stress–Energy and Conservation supplies field currents; Relativistic Matter and Compact Objects supplies equilibrium comparisons; Black-Hole Geometry supplies horizon and compactness diagnostics; and Computational Relativity supplies harmonic and evolution checks.

Real-scalar oscillatons

For a real scalar $\phi$ with

$$S_\phi=\int d^4x\sqrt{-g}\left[ \frac{R}{16\pi G} -\frac12\nabla_\mu\phi\nabla^\mu\phi -\frac12\mu^2\phi^2\right],$$

no global phase symmetry produces a particle-number charge. A useful spherical Fourier construction has

$$\phi(t,r)=\sum_{j=0}^{\infty} \phi_{2j+1}(r)\cos[(2j+1)\omega t],$$

while lapse and radial metric functions contain a constant term plus $\cos(2j\omega t)$ harmonics. This construction selects the time-reflection symmetric branch with $\phi(t+\pi/\omega,r)=-\phi(t,r)$, compatible with the $\phi\mapsto-\phi$ symmetry and the quadratic stress tensor. It is not a harmonic restriction on arbitrary real-scalar initial data. Solving after truncating the harmonic tower produces increasingly accurate long-lived configurations.

The instantaneous spacetime is not static, and an asymptotically flat massive real field has a very small radiative tail rather than a perfectly localized exact periodic solution under generic assumptions. “Oscillaton mass” should therefore state whether it is instantaneous, cycle averaged, or extracted from an asymptotic metric. Seidel and Suen introduced these oscillating soliton stars and their numerical construction in Seidel and Suen 1991, pp. 1659–1662.

Regularity requires even radial behavior of scalar and metric functions at the center; asymptotic flatness fixes time normalization, and the leading bound harmonic needs $\omega<\mu$. Higher harmonics above the mass threshold can propagate and control the slow decay. Long lifetime is an evolution result, not a consequence of solving only a finite Fourier boundary-value problem.

Complex Proca stars

For a complex massive vector $A_\mu$, define

$$F_{\mu\nu}=\nabla_\mu A_\nu-\nabla_\nu A_\mu,$$ $$S_A=\int d^4x\sqrt{-g}\left[ \frac{R}{16\pi G} -\frac14F_{\mu\nu}^*F^{\mu\nu} -\frac12\mu^2A_\mu^*A^\mu\right].$$

The Proca equation,

$$\nabla_\nu F^{\nu\mu}=\mu^2A^\mu,$$

implies the Lorenz constraint $\nabla_\mu A^\mu=0$ when $\mu\ne0$: the double divergence of the antisymmetric field strength vanishes, whereas the mass term leaves $\mu^2\nabla_\mu A^\mu$. This is an equation of motion, not a gauge choice; a massive vector has a physical longitudinal polarization. A spherical stationary ansatz may be written

$$A=e^{-i\omega t}\left[iV(r)\,dt+H_1(r)\,dr\right].$$

For $ds^2=-\alpha^2dt^2+a^2dr^2+r^2d\Omega^2$ and real $V,H_1$, the radial Proca equation and the constraint become

$$\begin{aligned} \omega(V'+\omega H_1)&=\mu^2\alpha^2H_1,\\ \frac{d}{dr}\left(\frac{\alpha r^2H_1}{a}\right) &=\frac{\omega a r^2 V}{\alpha}. \end{aligned}$$

They show explicitly why the temporal and radial amplitudes cannot be chosen independently. With $a(0)=1$, $\alpha(0)=\alpha_c\gt0$, and $V(0)=V_c$, regularity requires $H_1(r)=\omega V_c r/(3\alpha_c^2)+O(r^3)$ and $V'(0)=0$. The Einstein equations then determine the metric from the vector stress.

Its complex phase cancels from the stress–energy, as for a complex scalar boson star. Regularity, the Proca constraint, and exponential decay select $\omega$ and relate the central coefficients. Brito and collaborators derive the spherical families, charge, and mass curve in Brito et al. 2016, preprint pp. 1–3, “Spherically symmetric PSs” and “Asymptotic behaviours,” PDF. Their $\sigma\sqrt N$ equals $\alpha$ and $N^{-1/2}$ equals $a$. Multiplying their vector by the constant phase $i$ gives the present ansatz with $V=f$ and $H_1=-g$; omitting this sign would fail the central constraint.

The complex field has a conserved U(1) current and Noether charge. Rotating Proca stars use azimuthal phase $e^{im\varphi}$ and, for the canonical family, obey $J=mQ$. Vector polarization and the Lorenz constraint make their radial and rotating structure different from merely copying scalar equations with an extra index.

Shared weak-field limit, different observables

In the weak-binding regime write $\omega=\mu-\varepsilon$ with $0<\varepsilon\ll\mu$. Slowly varying scalar or vector envelopes source a Newtonian potential, yielding a Schrödinger–Poisson-type system after the fast $e^{-i\mu t}$ phase is removed. The dimensional mass scale of either family is $1/(G\mu)$, multiplied by a dimensionless function of the binding. On the dilute, weakly bound branch this factor tends to zero as $\omega\to\mu$; the mass decreases while the radius increases. The dimensional scale is therefore not the mass of every weak-field solution. Recovering this limit checks signs, dimensions, and field normalization.

The agreement ends at conserved quantities. A complex Proca star has exact U(1) charge in the classical model and stationary stress–energy. A real-scalar oscillaton has neither, so time-averaged mass and slow radiation must be monitored. Liebling and Palenzuela compare scalar, oscillaton, and vector constructions in Liebling and Palenzuela 2023, §§ 3.4 and 3.8.

$\mu$ and $\omega$ have dimension $L^{-1}$. Independent checks include center and asymptotic series, the Proca constraint, harmonic selection, charge conservation where present, cycle-averaged mass drift, virial residuals, and the weak-field Schrödinger–Poisson scaling. Existence does not transfer stability from scalar to vector branches or establish that generic initial data form either object.

Check harmonic selection

Suppose the leading real scalar is $\phi(t,r)=\phi_1(r)\cos\omega t$. Which temporal harmonics appear first in its quadratic stress–energy and in the metric it sources?

Solution

Quadratic terms contain

$$\cos^2\omega t=\frac12[1+\cos(2\omega t)], \qquad \sin^2\omega t=\frac12[1-\cos(2\omega t)].$$

Thus the stress–energy has a time-independent part and a $2\omega$ part at leading order, so the metric begins with even harmonics $0$ and $2\omega$. Feeding that metric back into the scalar equation generates an odd $3\omega$ scalar harmonic. A stationary metric ansatz would therefore be inconsistent with a real oscillaton, though it is consistent with the phase-canceling complex Proca ansatz.

References

• Brito, Richard, Vitor Cardoso, Carlos A. R. Herdeiro, and Eugen Radu. “Proca Stars: Gravitating Bose–Einstein Condensates of Massive Spin 1 Particles.” Physics Letters B 752 (2016): 291–295. DOI.
• Liebling, Steven L., and Carlos Palenzuela. “Dynamical Boson Stars.” Living Reviews in Relativity 26 (2023): 1. DOI.
• Seidel, Edward, and Wai-Mo Suen. “Oscillating Soliton Stars.” Physical Review Letters 66 (1991): 1659–1662. DOI.