Here, the numerical data described in the paper "Black holes with synchronised Proca hair: linear clouds and fundamental non-linear solutions", arXiv:2004.09536 [gr-qc] [1], is made available for public use.
This data pertains the fundamental states (n=0) of these hairy black holes and spinning Proca stars. Some data for the excited states (n=1) was previously made available here.
The data is presented in the same form as the data we have previously made available here for Kerr black holes with scalar hair, described in the paper "Construction and physical properties of Kerr black holes with scalar hair", Class. Quant. Grav. 32 (2016) 144001; arXiv:1501.04319 [gr-qc] [2], which expands on the solutions first presented in the paper "Kerr black holes with scalar hair", Phys. Rev. Lett. 112 (2014) 221101; e-Print: arXiv:1403.2757.
The original data for the Proca star was obtained in the paper "Proca Stars: gravitating Bose-Einstein condensates of massive spin 1 particles", Richard Brito, Vitor Cardoso, Carlos A. R. Herdeiro, Eugen Radu, Phys. Lett. B752 (2016) 291-295; arXiv:1508.05395 [gr-qc]. But in the spinning case, the solutions reported in this reference are excited states (n=1), as explained in [1].
The data for two spinning Proca stars (PSs) and for two hairy black holes (HBHs) can be found in the attachment "Data_files.zip", which contains four data files:
- PS-n=0.dat
- PS-n=1.dat
- HBH1.dat
- HBH2.dat
These files contain the data for the four solutions described in the paper - figures 6,7,8 (PS-n=0,1) and figures 9, 10, 11 (HBH1 and HBH2) respectively.
These solutions are:
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PS-n=0 : a typical n=0 Proca star belonging to the main branch of PS solutions;
- the input parameters are: (r_H=0; w=0.9 ; m=1)
- ADM mass=0.726; ADM angular momentum=0.75
- BH mass=0; BH angular momentum=0
- Proca field mass=0.726; Proca field angular momentum=0.75
========================================================
PS-n=1 : a typical n=1 Proca star belonging to the main branch of PS solutions;
- the input parameters are: (r_H=0; w=0.9 ; m=1)
- ADM mass=1.456; ADM angular momentum=1.5
- BH mass=0; BH angular momentum=0
- Proca field mass=1.456; Proca field angular momentum=1.5
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HBH1 : a HBH close to the existence line (n=0), thus Kerr-like
- the corresponding data is given in the eq. (3.12) of [1]
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HBH2 : a non-Kerr like HBH
- the corresponding data is given in the eq. (3.13) of [1]
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The data is presented in the following order, in the files
(F_1,F_2,F_0,W,H_1,H_2,H_3,V are the metric and Proca functions used in the paper):
--------------------------------
X_1 theta_1 F1 F2 F0 W H1 H2 H3 V
X_2 theta_1 F1 F2 F0 W H1 H2 H3 V
...
X_261 theta_1 F1 F2 F0 W H1 H2 H3 V
X_1 theta_2 F1 F2 F0 W H1 H2 H3 V
X_2 theta_2 F1 F2 F0 W H1 H2 H3 V
...
X_261 theta_2 F1 F2 F0 W H1 H2 H3 V
X_1 theta_35 F1 F2 F0 W H1 H2 H3 V
X_2 theta_35 F1 F2 F0 W H1 H2 H3 V
...
X_261 theta_35 F1 F2 F0 W H1 H2 H3 V
--------------------------------
where the grid points are:
X_k=(k-1)/260 (k=1,..,261)
and
theta_k=(k-1)*Pi/34/2 (k=1,..,35)
The corresponding values for \pi/2<\theta\leq\pi
result from the reflection symmetry of the solutions along the equatorial plane.
X=x/(1+x) is a compactified radial coordinate, 0\leq x\leq 1
where
x=\sqrt{r^2-r_H^2} was a new radial coordinate defined in the paper (with r_H=0 for a Proca Star)