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Dressed black holes in the new tensor–vector–scalar theory

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Abstract

As incarnations of gravity in its prime, black holes are arguably the best target for us to demystify gravity. Keeping in mind the prominent role black holes play in gravitational wave astronomy, it becomes a must for a theory to possess black hole solutions with only measurable departures from their general relativity counterparts. In this paper, we present black holes in a tensor–vector–scalar representation of relativistic modified Newtonian dynamics. We find that the theory allows Schwarzschild and nearly-Schwarzschild black holes as solutions, while the nontrivial scalar and vector fields generally diverge at the event horizon. Whether this is a physical pathology or not poses a challenge for these solutions, and by extension, the model. However, even if it is, this pathology could be overcome when the black hole hair vanishes.

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Data availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Code Availability

A Mathematica notebook for the derivation of the results of this work can be downloaded from GitHub [28].

Notes

  1. The interested reader may check them and possibly extend the work using our publicly available codes [28].

  2. Our usage of ‘necessary condition’ concurs with the physics language, which is an equation derived from some ‘parent’ equations. For example, the wave equation is a necessary condition of the Maxwell equations, meaning that the fields propagate in media. Solutions of Maxwell equations always satisfy the wave equation, but not the other way around, that is, not all propagating waves are solutions of the Maxwell equations, unless the electric and magnetic fields are transverse with each other and the wave direction. We take advantage of the necessary condition nonetheless by substituting its solution back to the parent equations.

  3. The interested reader may start with Eqs. (36) and (37) and take the similar steps from there.

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Correspondence to Reginald Christian Bernardo.

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Appendices

Appendix A: Variation toolkit

We present metric functional variations (\(\delta g^{ab}\)) of relevant quantities that can be used to derive the stress-energy tensor (Eq. (3)) containing the MoND degrees of freedom. The familiar curvature-related ones are

$$\begin{aligned}{} & {} \delta \sqrt{-g} = - \dfrac{1}{2} \sqrt{-g} g_{ab} \delta g^{ab},\end{aligned}$$
(A1)
$$\begin{aligned}{} & {} \delta \left( \sqrt{-g} R \right) = \sqrt{-g} G_{ab} \delta g^{ab} + \text { boundary term}, \end{aligned}$$
(A2)

and

$$\begin{aligned} \begin{aligned} \delta \Gamma ^c_{ab} = - \dfrac{1}{2} \bigg [ g_{da} \nabla _b \left( \delta g^{dc} \right)&+ g_{db} \nabla _a \left( \delta g^{dc} \right) - g_{ai} g_{bj} \nabla ^c \left( \delta g^{ij} \right) \bigg ]. \end{aligned} \end{aligned}$$
(A3)

On the other hand, variations of the various couplings between the vector and scalar degrees of freedom are given by

$$\begin{aligned}{} & {} \delta \left( F^{ab} F_{ab} \right) = 2 F_a^{\ c} F_{b c} \delta g^{ab}, \end{aligned}$$
(A4)
$$\begin{aligned}{} & {} \delta \left( \nabla _\alpha A_\nu \right) = A_{(\beta } g_{\gamma ) (\alpha } \nabla _{\nu )} \left( \delta g^{\gamma \beta } \right) - \dfrac{1}{2} g_{\alpha \gamma } g_{\lambda \nu } A^\beta \nabla _\beta \left( \delta g^{\gamma \lambda } \right) , \end{aligned}$$
(A5)
$$\begin{aligned}{} & {} \begin{aligned} \delta \left( J^a \nabla _a \phi \right)&= \bigg [ \left( \nabla _c \phi \right) A_{(a} \left( \nabla _{b)} A^c \right) + A^c \left( \nabla _c A_{(a} \right) \left( \nabla _{b)} \phi \right) \bigg ] \delta g^{ab} \\&\quad - g_{a(c} \nabla _{d)} \left[ A_b A^c \left( \nabla ^d \phi \right) \right] \delta g^{ab} + \dfrac{1}{2} \nabla _c \left( A^c A_a \left( \nabla _b \phi \right) \right) \delta g^{ab} \\&\quad + \text { boundary terms, e.g., } \nabla _c \left( \cdots \right) , \end{aligned} \end{aligned}$$
(A6)
$$\begin{aligned}{} & {} \delta {Y} = \left[ \nabla _a \nabla _b \phi + 2 \left( A^c \nabla _c \phi \right) A_{(a} \nabla _{b)} \phi \right] \delta g^{ab}, \end{aligned}$$
(A7)
$$\begin{aligned}{} & {} \delta {Q} = \left( A_{(a} \nabla _{b)} \phi \right) \delta g^{ab}, \end{aligned}$$
(A8)

and

$$\begin{aligned} \delta \left( A_a A^a \right) = A_a A_b \delta g^{ab}. \end{aligned}$$
(A9)

Appendix B: Explicit field equation terms in isotropic coordinates

The explicit terms appearing in the modified Einstein equation components in isotropic coordinates are

$$\begin{aligned} \begin{aligned} E_{tt}&= e^{-\zeta } (K_B-2) {U}^2 \Phi '' - e^{\nu -\zeta } \zeta '' + \dfrac{e^{-2 \zeta }}{4 r} \bigg [ -2 r e^{2 \zeta +\nu } {F}({Y},{Q}) \\&\quad -8 e^{\zeta +\nu } \zeta '-r e^{\zeta +\nu } \left( \zeta '\right) ^2+2 K_B r e^{\zeta +\nu } \left( \Phi '\right) ^2 -4 r e^{\zeta +\nu } \left( \Phi '\right) ^2 \\&\quad +2 e^{\zeta } K_B r {U}^2 \zeta ' \Phi '-4 e^{\zeta } r {U}^2 \zeta ' \Phi '-2 e^{\zeta } r \lambda \left( e^{\zeta } {U}^2+e^{\nu } \left( e^{\zeta }+{V}^2\right) \right) \\&\quad +8 e^{\zeta } K_B r {U} {U}' \Phi ' -2 e^{\zeta } K_B r \left( {U}'\right) ^2 -16 e^{\zeta } r {U} {U}' \Phi ' \\&\quad +2 K_B e^{\nu } r {V}^2 \zeta ' \Phi '-4 e^{\nu } r {V}^2 \zeta ' \Phi '+2 K_B e^{\nu } r {V}^2 \left( \Phi '\right) ^2-4 e^{\nu } r {V}^2 \left( \Phi '\right) ^2 \\&\quad -4 K_B e^{\nu } r {V} {V}' \Phi ' +8 e^{\nu } r {V} {V}' \Phi '+8 e^{\zeta } K_B {U}^2 \Phi '-16 e^{\zeta } {U}^2 \Phi ' \bigg ], \end{aligned}\nonumber \\ \end{aligned}$$
(B10)

and

$$\begin{aligned} \begin{aligned} E_{rr}&= e^{-\zeta } (K_B-2) {V}^2 \Phi '' + \dfrac{1}{4} \bigg [-8 e^{-\zeta } {V}^2 \left( \Phi '\right) ^2 {F}_{{Y}} -4 {V} \Phi ' {F}_{{Q}} \\&\quad -4 \left( \Phi '\right) ^2 {F}_{{Y}} +2 e^{\zeta } {F}({Y},{Q}) +2 \zeta ' \nu '+\left( \zeta '\right) ^2+2 K_B \left( \Phi '\right) ^2 \\&\quad +\frac{4 \zeta '}{r}+\frac{4 \nu '}{r}+\frac{8 e^{-\zeta } K_B {V}^2 \Phi '}{r}-\frac{16 e^{-\zeta } {V}^2 \Phi '}{r} -2 K_B e^{-\nu } {U}^2 \nu ' \Phi ' \\&\quad +4 e^{-\nu } {U}^2 \nu ' \Phi '+\lambda \left( 2 e^{\zeta }-2 {U}^2 e^{\zeta -\nu }-2 {V}^2\right) +2 K_B e^{-\nu } \left( {U}'\right) ^2 \\&\quad +4 e^{-\zeta } K_B {V}^2 \zeta ' \Phi ' -8 e^{-\zeta } {V}^2 \zeta ' \Phi '+2 e^{-\zeta } K_B {V}^2 \nu ' \Phi ' \\&\quad +6 e^{-\zeta } K_B {V}^2 \left( \Phi '\right) ^2 -4 e^{-\zeta } {V}^2 \nu ' \Phi ' -12 e^{-\zeta } {V}^2 \left( \Phi '\right) ^2 \\&\quad -4 e^{-\zeta } K_B {V} {V}' \Phi '+8 e^{-\zeta } {V} {V}' \Phi '-4 \left( \Phi '\right) ^2 \bigg ], \end{aligned}\qquad \end{aligned}$$
(B11)
$$\begin{aligned} \begin{aligned} E_{tr}&= (K_B-2) {U} {V} \Phi ''-\dfrac{{U}}{2 r} \bigg [{V} \bigg (\Phi ' \bigg (2 r \Phi ' {F}_{{Y}}-4 K_B \\&\quad -(K_B-2) r (\zeta ' + \nu ') -2 (K_B - 2) r \Phi '+8\bigg )+2 e^{\zeta } r \lambda \bigg )+e^{\zeta } r \Phi ' {F}_{{Q}} \bigg ], \end{aligned}\nonumber \\ \end{aligned}$$
(B12)

and

$$\begin{aligned} \begin{aligned} E_{\theta \theta }&= \frac{r^2 \zeta ''}{2}+\frac{r^2 \nu ''}{2} + \frac{1}{4} r e^{-\zeta -\nu } \bigg [ 2 r e^{2 \zeta +\nu } {F}({Y},{Q})+2 e^{\zeta +\nu } \zeta '+2 e^{\zeta +\nu } \nu ' \\&\quad -2 (K_B - 2) r e^{\zeta +\nu } \left( \Phi '\right) ^2 +r e^{\zeta +\nu } \left( \nu '\right) ^2 +2 e^{\zeta } (K_B - 2) r {U}^2 \nu ' \Phi ' \\&\quad +2 e^{\zeta } r \lambda \left( e^{\nu } \left( e^{\zeta }+{V}^2\right) -e^{\zeta } {U}^2\right) -2 e^{\zeta } K_B r \left( {U}'\right) ^2 - 2 (K_B - 2) e^{\nu } r {V}^2 \zeta ' \Phi ' \\&\quad -2 (K_B - 2) e^{\nu } r {V}^2 \left( \Phi '\right) ^2 +4 (K_B - 2) e^{\nu } r {V} {V}' \Phi ' \bigg ]. \end{aligned}\nonumber \\ \end{aligned}$$
(B13)

Appendix C: The curvature scalars in isotropic coordinates

The Ricci \(\mathcal {R}\) and Kretschmann \(\mathcal {K}\) scalars in isotropic coordinates (8) are given by

$$\begin{aligned} \begin{aligned} \mathcal {R}&= -\frac{e^{-\zeta (r)}}{2 r} \bigg ( 2 r \left( 2 \zeta ''(r)+\nu ''(r)\right) + \zeta '(r) \left( r \nu '(r)+8\right) \\&\quad + r \zeta '(r)^2 +r \nu '(r)^2+4 \nu '(r) \bigg ) \end{aligned} \end{aligned}$$
(C14)

and

$$\begin{aligned} \begin{aligned} \mathcal {K}&= \frac{e^{-2 \zeta (r)}}{4 r^2} \bigg ( 4 r^2 \left( 2 \zeta ''(r)^2+\nu ''(r)^2\right) +3 \zeta '(r)^2\left( r^2 \nu '(r)^2+8\right) \\&\quad + r^2 \zeta '(r)^4 + r^2 \nu '(r)^4+4 \nu '(r)^2 \left( r^2 \nu ''(r)+2\right) \\&\quad + 8 r \zeta '(r)^3-2 r \zeta '(r) \bigg (-8 \zeta ''(r) \\&\quad +r \nu '(r)^3-4 \nu '(r)^2+2 r \nu '(r) \nu ''(r) \bigg ) \bigg ). \end{aligned} \end{aligned}$$
(C15)

For a Schwarzschild black hole (Eqs. (25), (26)), these can be shown to be \(\mathcal {R} = 0\) and \(\mathcal {K} = 12 R^2/ \left( r^6 \left( 1 + (R/4r) \right) ^{12} \right) \) as expected.

Appendix D: Constrained vector field solutions

In this appendix, we examine the branch \(\Phi '(r) = 0\) with \(K_B \ne 0\). In this case, the r-component of the vector field equation as well as the tr-component of the metric field equation reduce to \({V}(r) \lambda (r) = 0\), i.e., either the multiplier \(\lambda \) must be zero or that the vector field should be time-pointing. Also, due to the term \(\sim J^a \nabla _a \phi \) in the action, despite having a trivial scalar field, the scalar field equation only becomes an additional nontrivial constraint on the vector field components. Thus, we take \(K_B = 2\); otherwise, the dynamical system becomes over-constrained. We consider separately \(\lambda = 0\) and \({V}(r) = 0\) branches below.

We thread down the \(\lambda = 0\) branch together with \({F}(0, 0) = 0\) (asymptotically flat condition). The metric equation becomes

$$\begin{aligned}{} & {} -e^{\nu } \left( 4 r \zeta ''+r \zeta ^{\prime 2}+8 \zeta '\right) -2 K_B r {U}^{\prime 2} = 0, \end{aligned}$$
(D16)
$$\begin{aligned}{} & {} 2 \zeta ' \left( r \nu '+2\right) +r \zeta ^{\prime 2}+4 \nu '+2 K_B r e^{-\nu (r)} {U}^{\prime 2} = 0, \end{aligned}$$
(D17)

while the time component of the vector field equation becomes

$$\begin{aligned} 2 r {U}''+{U}' \left( r \zeta '-r \nu '+4\right) = 0. \end{aligned}$$
(D18)

In addition, the unit-vector constraint is given by

$$\begin{aligned} e^{\nu } \left( e^{\zeta }+{V}^2\right) =e^{\zeta } {U}^2. \end{aligned}$$
(D19)

Equations (D16), (D17), (D18), and (D19) can be solved for the four unknowns \(\left( \nu . \zeta , {U}, {V} \right) \). To obtain this general solution, we solve Eq. (D18) for U:

$$\begin{aligned} {U}(r) = c_1 \int ^r \dfrac{ \exp \left( \left( \nu (x)- \zeta (x) \right) /2 \right) }{x^2} \, dx + c_2, \end{aligned}$$
(D20)

where \(c_1\) and \(c_2\) are integration constants. Substituting Eq. (D20) into (D19) then determines the spatial component of the vector field in terms of the metric functions \(\left( \nu , \zeta \right) \). A coupled system for the metric functions finally emerges by substituting Eq. (D20) into Eqs. (D16) and (D17):

$$\begin{aligned} 2 c_1^2 K_B+4 r^4 e^{\zeta } \zeta ''+r^4 e^{\zeta } \zeta ^{\prime 2}+8 r^3 e^{\zeta } \zeta ' = 0, \end{aligned}$$
(D21)

and

$$\begin{aligned} \dfrac{2 c_1^2 K_B e^{-\zeta }}{r^3}+2 \zeta ' \left( r \nu '+2\right) +r \zeta ^{\prime 2}+4 \nu ' = 0. \end{aligned}$$
(D22)

An exact analytical solution to Eqs. (D21) and (D22) for \(c_1 \ne 0\) cannot be obtained. Nonetheless, it is easy to confirm that for \(c_1 = 0\), \(\nu \) and \(\zeta \) reduces to the Schwarzschild solution (Eqs. (25), (26)). This suggests that a hairy black hole emerges for \(c_1 \ne 0\), in which case \(c_1\) quantifies the departure from the Schwarzschild geometry. Considering \(c_1\) as a perturbation, the leading order corrections to the metric function can be shown to be

$$\begin{aligned} e^{\nu (r)} = \exp \left( \dfrac{c_1^2 r \left( a R^2 (4 r+R)+32 K_B\right) }{(4 r-R) (4 r+R)^2} \right) e^{\nu _\text {Schw}(r)}, \end{aligned}$$
(D23)

and

$$\begin{aligned} e^{\zeta (r)} = \exp \left( -\frac{c_1^2 \left( a R^2 (4 r+R)+16 K_B\right) }{4 (4 r+R)^2} \right) e^{\zeta _\text {Schw}(r)}, \end{aligned}$$
(D24)

where a is an integration constant, and \(\nu _\text {Schw}(r)\) and \(\zeta _\text {Schw}(r)\) are given by Eqs. (25) and (26).

We study the \({V}(r) = 0\) branch with \({F}(0,0)=0\) for asymptotic flatness. The metric equations become

$$\begin{aligned} \begin{aligned}&-e^{\nu } \left( 8 \zeta '+4 r \zeta ''+r \zeta ^{\prime 2}\right) -2 e^{\zeta } \lambda r \left( e^{\nu }+{U}^2\right) -2 K_B r {U}^{\prime 2} = 0, \end{aligned} \end{aligned}$$
(D25)

and

$$\begin{aligned} \begin{aligned}&e^{\nu } \left( 4 \nu '+2 \zeta ' \left( r \nu '+2\right) +r \zeta ^{\prime 2}\right) +2 e^{\zeta } \lambda r \left( e^{\nu }-{U}^2\right) +2 K_B r {U}^{\prime 2} = 0. \end{aligned} \end{aligned}$$
(D26)

The time component of the vector field equation becomes

$$\begin{aligned} K_B \left( 2 r {U}''+{U}' \left( r \zeta '-r \nu '+4\right) \right) -2 e^{\zeta } \lambda r {U} = 0, \end{aligned}$$
(D27)

while the unit-vector constraint reduces to

$$\begin{aligned} e^{\nu }={U}^2. \end{aligned}$$
(D28)

The last two equations (Eqs. (D27), (D28)) can be used to determine U and the multiplier \(\lambda \) in terms of the metric functions. Substituting these expressions into Eqs. (D25) and (D26) leads to

$$\begin{aligned} 8 \nu '+4 r \left( \zeta ''+\nu ''\right) +2 \zeta ' \left( r \nu '+4\right) +r \zeta ^{\prime 2}+r \nu ^{\prime 2}=0, \end{aligned}$$
(D29)

and

$$\begin{aligned} \left( \zeta '+\nu '\right) \left( r \zeta '+r \nu '+4\right) =0. \end{aligned}$$
(D30)

There are two branches of solution here. The first one is

$$\begin{aligned} e^{\zeta (r)} = \dfrac{e^{-\nu (r)}}{r^4}, \end{aligned}$$
(D31)

while the second is

$$\begin{aligned} e^{\zeta (r)} = e^{-\nu (r)}, \end{aligned}$$
(D32)

where either \(\nu \) or \(\zeta \) can be chosen as a free function. The first one was presented in the main text (Eq. (22)). Neither corresponds to a black hole solution.

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Bernardo, R.C., Chen, CY. Dressed black holes in the new tensor–vector–scalar theory. Gen Relativ Gravit 55, 23 (2023). https://doi.org/10.1007/s10714-023-03075-x

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