Rotating boson stars and $Q$-balls. II. Negative parity and ergoregions

We construct axially symmetric, rotating boson stars with positive and negative parity. Their flat space limits represent spinning $Q$-balls. $Q$-balls and boson stars exist only in a limited frequency range. The coupling to gravity gives rise to a spiral-like frequency dependence of the mass and charge of boson stars. We analyze the properties of these solutions. In particular, we discuss the presence of ergoregions in boson stars and determine their domains of existence.

Figure 1

Left: The potential $U(\varphi )$ versus $\varphi$ for $\lambda =1$, $a=2$, and $b=1.1$ respectively $b=1$. Right: The effective potential $V(\varphi )=\frac{1}{2}{\omega }_s^2{\varphi }^2-\frac{1}{2}U(\varphi )$ versus $\varphi$ for several values of the frequency ${\omega }_s$.

Figure 2

Left: The charge $Q$ versus the frequency ${\omega }_s$ for spherically symmetric fundamental $Q$-balls ($k=0$) and their first radial excitations ($k=1$). Also shown are the limiting values of the frequency, ${\omega }_{\min }$ and ${\omega }_{\max }$. Right: The mass $M$ versus the charge $Q$ for spherically symmetric fundamental $Q$-balls ($k=0$) and their first radial excitations ($k=1$). Also shown is the mass for $Q$ free bosons, $M=m_{\mathrm{B}}Q$. The mass on the upper branches is very close this mass. The $k=0$ branches close to the critical charge $Q_{\mathrm{cr}}$ are seen in the inset.

Figure 3

The scalar field $\varphi$ (left column) and energy density $T_{tt}$ (right column) for rotating positive parity $Q$-balls with charge $Q=410$ and $n^P=1^{+}$ (upper row), $2^{+}$ (middle row), and $3^{+}$ (lower row) versus the coordinates $\rho =r\sin \theta$ and $z=r\cos \theta$ for $z\ge 0$.

Figure 4

Left: The charge $Q$ versus the frequency ${\omega }_s$ for $Q$-balls with $n^P=0^{+}$, $1^{+}$, $2^{+}$, $3^{+}$. The asterisks mark the critical values of the charge. Right: The mass $M$ versus the charge $Q$ for $Q$-balls with $n^P=0^{+}$, $1^{+}$, $2^{+}$, $3^{+}$. The upper branches of the mass $M$ are hardly discernible (on this scale) from the mass of $Q$-free bosons, $M=m_{\mathrm{B}}Q$.

Figure 5

The scalar field $\varphi$ (left column) and energy density $T_{tt}$ (right column) for rotating negative parity $Q$-balls with charge $Q=720$ and $n^P=1^{-}$ (upper row) and $2^{-}$ (lower row) versus the coordinates $\rho =r\sin \theta$ and $z=r\cos \theta$ for $z\ge 0$.

Figure 6

Left: The charge $Q$ versus the frequency ${\omega }_s$ for negative parity $Q$-balls with $n^P=1^{-}$ and $2^{-}$. For comparison, the positive parity $Q$-balls with $n^P=0^{+}$, $1^{+}$, $2^{+}$, $3^{+}$ are also shown. Right: The mass $M$ versus the charge $Q$ for the same set of solutions together with the mass of $Q$-free bosons, $M=m_{\mathrm{B}}Q$.

Figure 7

Left: The charge $Q$ versus the frequency ${\omega }_s$ for fundamental spherically symmetric boson stars ($k=0$) in the full range of existence for a sequence of values of the gravitational coupling constant $\kappa$. Right: The charge $Q$ versus the frequency ${\omega }_s$ in the frequency range of the spiral for the gravitational coupling constant $\kappa =0.2$.

Figure 8

Left: The scaled charge $\kappa Q$ versus the frequency ${\omega }_s$ for fundamental spherically symmetric boson stars in the limit $\kappa \to \infty$. For comparison, the scaled charge $\kappa Q$ is also shown for $\kappa =0.2$. Right: The mass $M$ versus the charge $Q$ for fundamental spherically symmetric boson stars for the value of the gravitational coupling constant $\kappa =0.2$.

Figure 9

Left: The charge $Q$ versus the frequency ${\omega }_s$ for rotating $1^{+}$ boson stars for several values of the gravitational coupling constant $\kappa$. Also shown are the limiting flat space values. The dots indicate the onset of ergoregions. Right: The mass $M$ versus the frequency ${\omega }_s$ for the same set of solutions.

Figure 10

Left: The charge $Q$ versus the frequency ${\omega }_s$ for rotating $2^{+}$ boson stars for several values of the gravitational coupling constant $\kappa$. Also shown are the limiting flat space values. The dots indicate the onset of ergoregions. Right: The mass $M$ versus the frequency ${\omega }_s$ for the same set of solutions.

Figure 11

The charge $Q$ (left column) and the mass $M$ (right column) versus the frequency ${\omega }_s$ for boson stars with $n^P=0^{+}$, $1^{+}$, $2^{+}$ for the gravitational coupling $\kappa =0.2$.

Figure 12

Location and shape of the axially symmetric ergoregions of boson stars exhibited via cross sections with the $xz$ plane: $1^{+}$ boson stars at $\kappa =0.1$ (upper row) and $2^{+}$ boson stars (lower row) at $\kappa =0.1$ (left) and at $\kappa =0.2$ (right). The numbers in the legend indicate the respective values of the charge $Q$.

Figure 13

The charge $Q$ (left) and the mass $M$ (right) versus the frequency ${\omega }_s$ for $1^{-}$ boson stars for gravitational coupling $\kappa =0.1$ and 0.2. The dots indicate the onset of ergoregions.

Figure 14

The mass $M$ versus the charge $Q$ for $1^{-}$ boson stars for gravitational coupling $\kappa =0.1$ (left) and 0.2 (right), together with the mass for $Q$ free bosons, $M=m_{\mathrm{B}}Q$.

Figure 15

The charge $Q$ versus the frequency ${\omega }_s$ for $1^{-}$ boson stars and $1^{+}$ boson stars for gravitational coupling $\kappa =0.1$ (left) and 0.2 (right).

Figure 16

Left: The charge $Q$ versus the frequency ${\omega }_s$ for $1^{-}$ and $2^{-}$ boson stars for gravitational coupling $\kappa =0.2$. The dots indicate the onset of ergoregions. Right: The mass $M$ versus the charge $Q$ for $2^{-}$ boson stars for gravitational coupling $\kappa =0.2$ together with the mass for $Q$-free bosons, $M=m_{\mathrm{B}}Q$.

Figure 17

Location and shape of the axially symmetric ergoregions of boson stars exhibited in the $xz$ plane: $1^{-}$ boson stars at $\kappa =0.1$ (upper row) and $2^{-}$ boson stars (lower row) at $\kappa =0.2$. The numbers in the legend indicate the respective values of the charge $Q$.