1 Scalar field source terms

We consider a complex scalar field \(\Phi \) of mass \(m_S\) minimally coupled to gravity, which is described by the action \begin {equation} S = \int \mathrm {d}^4 x \sqrt {-g} \bracket { \frac {^{(4)}R}{16\pi } - \frac {1}{2} g^{\mu \nu } \nabla _{(\mu } \Phi ^* \nabla _{\nu )} \Phi - \frac {\mu _S^2}{2} \Phi ^* \Phi }. \end {equation} Here, \(\Phi \) is the scalar field and \(\Phi ^*\) its complex conjugate, \(g_{\mu \nu }\) is the spacetime metric with \(g=\det (g_{\mu \nu })\), \(\nabla _\mu \) is the covariant derivative, and \(^{(4)}R\) is the Ricci scalar in four dimensions. The mass parameter \(\mu _S\) of the scalar field is related to \(m_S\) through \(m_S = \mu _S\hbar \). This action leads to the equations of motion \begin {align} & G_{\mu \nu } = 8\pi T_{\mu \nu }, \\ & (\Box -\mu _S^2) \Phi = 0, \end {align}

where the stress-energy tensor is given by \begin {equation} T_{\mu \nu } = \nabla _{(\mu } \Phi ^* \nabla _{\nu )} \Phi - \frac {1}{2} g_{\mu \nu } \Big [ g^{\alpha \beta } \nabla _{(\alpha } \Phi ^* \nabla _{\beta )} \Phi + \mu _S^2 \Phi ^*\Phi \Big ], \end {equation} and the d’Alembertian is \begin {equation} \Box = \frac {1}{\sqrt {-g}} \partial _\mu ( \sqrt {-g} g^{\mu \nu } \partial _\nu ). \end {equation}

Under the 3+1 formalism, the scalar field contributes to the spacetime through the energy density \(\rho \), energy-momentum flux \(j_i\), and the purely spatial stress tensor \(S_{ij}\), given respectively by \begin {align} \rho = n^\mu n^\nu T_{\mu \nu } &= 2 \Pi ^* \Pi + \frac {\mu _S^2}{2} \Phi ^* \Phi + \frac {1}{2} D_k\Phi ^* D^k \Phi , \\ j_i = -\gamma _i^\mu n^\nu T_{\mu \nu } &= \Pi ^* D_i\Phi + \Pi D_i\Phi ^*, \\ S_{ij} = \gamma _i ^\mu \gamma _j^\nu T_{\mu \nu } &= D_{(i}\Phi ^* D_{j)}\Phi + \frac {1}{2} \gamma _{ij} (4\Pi ^* \Pi - \mu _S^2 \Phi ^* \Phi - D^k\Phi ^* D_k\Phi ), \end {align}

where \(D_i\) is the spatial covariant derivative with respect to the 3-metric \(\gamma _{ij}\). Here, the scalar field momentum \(\Pi \) is defined using the Lie derivative along the normal vector \(n^\mu \) as \begin {equation} \Pi \equiv - \frac {1}{2} \mathcal {L}_n \Phi . \end {equation} For us to use the Bowen-York solution to the momentum constraint equation as in TwoPunctures, we need to specify the scalar field such that the momentum constraint reduces to its vacuum counterpart. To this end, we limit ourselves to initial configurations of the scalar field for which the scalar field momentum vanishes: \(\Pi (t=0)=0\). This requirement can be satisfied under the condition that the scalar field is time-independent, along with a shift vector that vanishesat the initial time.

2 Constraint equations in the puncture method

To construct the initial data, we solve the Hamiltonian constraint equation describing black holes with linear momenta \(\vec {P}_a\) and spins \(\vec {S}_a\).

Let us perform a conformal decomposition of the physical metric as \(\gamma _{ij}=\psi ^4\bar {\gamma }_{ij}\), where \(\psi \) is the conformal factor and \(\bar {\gamma }_{ij}\) is the conformal metric. Then, the Hamiltonian constraint equation takes the form \begin {equation} \bar {D}^2 \psi - \frac {1}{8} \psi \bar {R} + \frac {1}{8} \psi ^5 (K_{ij}K^{ij} - K^2 ) + 2\pi \psi ^5 \rho = 0, \end {equation} where \(\bar {D}^2\) is the conformal Laplacian operator, \(\bar {R}\) is the conformal Ricci scalar, and \(K_{ij}\) is the extrinsic curvature. Further decomposing \(K_{ij}\) into its trace \(K\) and trace-free parts \(A_{ij}\), and performing a conformal decomposition as \begin {equation} A_{ij} = \psi ^{-2}\bar {A}_{ij}, \quad \quad A^{ij} = \psi ^{-10}\bar {A}^{ij}, \end {equation} we obtain the general form of the Hamiltonian constraint in the York-Lichnerowicz approach \begin {equation} \bar {D}^2 \psi - \frac {1}{8} \psi \bar {R} + \frac {1}{8} \psi ^{-7} \bar {A}_{ij}\bar {A}^{ij} - \frac {1}{12} \psi ^5 K^2 + 2\pi \psi ^5 \rho = 0. \end {equation}

Now let us introduce a series of assumptions for the puncture method [2]. Assuming asymptotic flatness, the conformal factor has the behavior \(\psi \simeq 1 + \psi _{\rm BL}\) at large \(r\), where the \(\psi _{\rm BL}\) is the Brill-Lindquist solution \begin {equation} \psi _{\rm BL} \equiv \sum _{a=1}^2 \frac {m_a}{2|\vec {r} - \vec {r}_a|}, \end {equation} with \(m_a\) and \(\vec {r}_a\) denoting the bare mass and location of the \(a\)-th black hole. Now, the main idea behind the puncture method is to separate the singular piece of the conformal factor by writing \( \psi = \psi _{\rm BL} + u \), where \(u\) is the function to be solved. It follows that the flat Laplacian of \(\psi \) reduces to \(\bar {D}^2 u\) in the punctured domain \(\mathbb {R}^3\setminus \{\vec {r}_i\}\). To satisfy the vacuum momentum constraint equations, we write the conformal tracefree extrinsic curvature \(\bar {A}_{ij}\) as a superposition \begin {equation} \bar {A}^{ij} = \sum _{a=1}^2 \bar {A}_a^{ij} \end {equation} of Bowen-York solutions for each puncture \begin {equation} \bar {A}_a^{ij} = \frac {3}{2r^2} \Big [ n^i P_a^j + n^j P_a^i + n_k P_a^k (n^i n^j - \delta ^{ij}) \Big ] - \frac {3}{r^3} (\epsilon ^{ilk} n^j + \epsilon ^{jlk} n^i) n_l {(S_a)}_k. \end {equation} Finally, assuming conformal flatness (\(\bar {\gamma }_{ij} = \eta _{ij}\)) and maximal slicing condition (\(K=0\)), the Hamiltonian constraint equation reduces to \begin {equation} \Delta u + \frac {1}{8} (\psi _{\rm BL} + u)^{-7} \bar {A}_{ij} \bar {A}^{ij} + 2\pi (\psi _{\rm BL} + u)^5 \rho = 0, \label {scalar_TwoPunctures_BBHSF_eq:ham_twopunctures} \end {equation} where \(\Delta =\partial _k \partial ^k\) is the flat Laplacian operator. The first two terms above constitute the vacuum equations solved in the standard TwoPunctures thorn. The third term, on the other hand, contains the contribution from the matter energy density \(\rho \). Whereas in TwoPunctures the energy density can be rescaled ¹ according to \(\rho = \psi ^8 \bar {\rho }\), here, the massive scalar field contributes two terms with different powers of \(\psi \) \begin {align} \rho &= \frac {\mu _S^2}{2} \Phi ^* \Phi + \frac {1}{2} \gamma ^{ij} D_i\Phi ^* D_j \Phi \nonumber \\ &= \frac {\mu _S^2}{2} \Phi ^* \Phi + \frac {1}{2} \psi ^{-4} \partial _k\Phi ^* \partial ^k \Phi , \label {scalar_TwoPunctures_BBHSF_eq:rho} \end {align}

so a naïve conformal rescaling of \(\rho \) does not work. As a workaround, we choose to perform a conformal rescaling on the scalar field \(\Phi \) itself.

2.1 Conformal rescaling of the scalar field

Let us rescale the scalar field with a generic power \(\delta \) of the conformal factor \begin {equation} \Phi = \psi ^\delta \bar {\Phi }, \quad \Phi ^* = \psi ^\delta \bar {\Phi }^*, \end {equation} where \(\delta \) is a constant parameter to be chosen and \(\Bar {\Phi }\) is the conformal scalar field. The goal is to find a suitable power \(\delta \) such that the linearized Hamiltonian constraint equation forms a well-posed elliptic problem. Inserting Eq. (??) and using the rescaled scalar field, Eq. (??) can be rewritten as \begin {align} \Delta \psi + \frac {1}{8} \psi ^{-7} \bar {A}_{ij} \bar {A}^{ij} + &\pi \psi ^{2\delta + 5} \mu _S^2 \bar {\Phi }^* \bar {\Phi } \nonumber \\ + &\pi \psi ^{2\delta + 1} (\partial _i\bar {\Phi }^*) (\partial ^i\bar {\Phi }) \nonumber \\ + \delta &\pi \psi ^{2\delta }\quad (\partial _i\psi ) ( \bar {\Phi }^* \partial ^i\bar {\Phi } + \bar {\Phi } \partial ^i\bar {\Phi }^* ) \nonumber \\ + \delta ^2 &\pi \psi ^{2\delta -1} (\partial _i\psi ) (\partial ^i\psi ) \bar {\Phi }^* \bar {\Phi } \quad \quad \quad \quad = 0, \label {scalar_TwoPunctures_BBHSF_eq:nonlin_ham_eq} \end {align}

with the conformal factor \begin {align} \psi &= \psi _{\rm BL} + u, \\ \partial _i\psi &= \partial _i\psi _{\rm BL} + \partial _i u. \end {align}

Writing the correction function as \(u = u_0 + \epsilon \) with \(u_0\) a known solution and \(|\epsilon |\ll u_0\), we can expand Eq. (??) to first order in \(\epsilon \), giving the linearized equation \begin {align} \Delta \epsilon - \epsilon \Bigg [ \frac {7}{8} \psi ^{-8} \bar {A}_{ij} \bar {A}^{ij} - (2\delta + 5) \pi &\psi _0^{2\delta + 4} \mu _S^2 \bar {\Phi }^* \bar {\Phi } \nonumber \\ - (2\delta + 1) \pi &\psi _0^{2\delta }\quad (\partial _i\bar {\Phi }^*) (\partial ^i\bar {\Phi }) \nonumber \\ -\ \ (2\delta )\ \ \delta \pi &\psi _0^{2\delta -1} (\partial _i\psi _0) ( \bar {\Phi }^* \partial ^i\bar {\Phi } + \bar {\Phi } \partial ^i\bar {\Phi }^* ) \nonumber \\ - (2\delta -1) \delta ^2 \pi &\psi _0^{2\delta -2} (\partial _i\psi _0) (\partial ^i\psi _0) \bar {\Phi }^* \bar {\Phi } \quad \quad \quad \quad \Big ] \nonumber \\ + (\partial _i\epsilon ) \Big [ \delta \pi \psi _0^{2\delta } ( \bar {\Phi }^* \partial ^i\bar {\Phi } &+ \bar {\Phi } \partial ^i\bar {\Phi }^* ) + \delta ^2 \pi \psi _0^{2\delta -1} (\partial ^i\psi _0) \bar {\Phi }^* \bar {\Phi } \Big ]= 0, \label {scalar_TwoPunctures_BBHSF_eq:lin_ham_eq} \end {align}

where the conformal factor \(\psi _0\) is defined as \begin {align} \psi _0 &= \psi _{\rm BL} + u_0, \\ \partial _i \psi _0 &= \partial _i \psi _{\rm BL} + \partial _i u_0. \end {align}

Equations (??) and (??) constitute our main result. We see that for any choice of \(\delta < -2.5\), Eq. (??) takes the form of \begin {equation} \Delta \epsilon - h \epsilon = 0 \end {equation} with \(h>0\), so it is well-posed as an elliptic equation in \(u\), and its numerical solution is unique and stable.

The Eqs. (??) and (??) are implemented in Equations.c of the thorn TwoPunctures_BBHSF. In testing the initial data constructed this way, we found no significant difference in the resulting metric and constraint violation when using different values of the parameter \(\delta < -2.5\). By default, \(\delta \) is set to be \(-3\) in the thorn.

3 Initializing scalar field configurations

We consider scalar field configurations with a Gaussian radial profile and zero momentum \begin {equation} \bar {\Phi }(t=0) = A_{\rm SF} Z(\theta , \phi ) e^{-(r-r_0)^2/w^2}, \quad \Pi (t=0) = 0, \end {equation} where \(A_{\rm SF}, r_0, w\) are constant parameters and \(Z(\theta , \phi )\) encodes the angular dependence. In the file SF_source_term.c of the thorn, we have implemented the \(l=m=0\) (monopole) and \(l=m=1\) (dipole) spherical harmonics modes as options for \(Z(\theta , \phi )\): \begin {align} Y_{0,0}(\theta , \phi ) &= \sqrt {\frac {1}{4\pi }}, \\ Y_{1,1}(\theta , \phi ) &= -\sqrt {\frac {3}{8\pi }} \sin \theta \ (\cos \phi + i \sin \phi ). \end {align}

In addition, Cartesian derivatives of \(\bar {\Phi }\) are computed, so this thorn replaces the use of ScalarInit (which only computes \(\Phi \) and does not conformally rescale the scalar fields) when initializing the scalar field grid functions. If one wishes to use a different initial scalar configuration than currently implemented, they should supply the corresponding fields and source terms to SF_source_term.c.

The parameter TwoPunctures_BBHSF::switch_on_back-reaction is used to control whether the scalar field is included in the metric initial data calculation. To remove the back-reaction from the metric ID calculation, it is sufficient to set TwoPunctures_BBHSF::switch_on_back-reaction = no, and that will reduce to the standard TwoPunctures.

To enable back-reaction during the evolution, one is reminded to set TmunuBase::stress_energy_at_RHS = yes. If using LeanBSSNMoL for the spacetime evolution, one should also set LeanBSSNMoL::couple_Tab = 1.

References

4 Parameters

delta	Scope: restricted	REAL
Description: Exponent delta for conformal decomposition of the scalar field ∖phi = ∖psielta ∖bar∖phi
Range		Default: -3.0
(*:*)	Should be negative and less than -3

initial_lapse_psi_exponent	Scope: restricted	REAL
Description: Exponent n for psin initial lapse profile
Range		Default: -2.0
(*:*)	Should be negative

5 Interfaces

General

Implements:

twopunctures_bbhsf

Inherits:

admbase

staticconformal

grid

scalarbase

Grid Variables

5.0.1 PRIVATE GROUPS

5.0.2 PUBLIC GROUPS

6 Schedule

This section lists all the variables which are assigned storage by thorn Scalar/TwoPunctures_BBHSF. Storage can either last for the duration of the run (Always means that if this thorn is activated storage will be assigned, Conditional means that if this thorn is activated storage will be assigned for the duration of the run if some condition is met), or can be turned on for the duration of a schedule function.

Storage

Scheduled Functions

CCTK_PARAMCHECK (conditional)

  bbhsf_twopunctures_paramcheck

  check parameters and thorn needs

ADMBase_InitialData (conditional)

  twopunctures_bbhsf_group

  twopunctures initial data group

CCTK_INITIAL (conditional)

  twopunctures_bbhsf_group

  twopunctures initial data group

TwoPunctures_BBHSF_Group (conditional)

  twopunctures_bbhsf

  create puncture black hole initial data

TwoPunctures_BBHSF_Group (conditional)

  bbhsf_twopunctures_metadata

  output twopunctures metadata