Hidden symmetries for ellipsoid–solitonic deformations of Kerr–Sen black holes and quantum anomalies

Abstract

We prove the existence of hidden symmetries in the general relativity theory defined by exact solutions with generic off-diagonal metrics, nonholonomic (non-integrable) constraints, and deformations of the frame and linear connection structure. A special role in characterization of such spacetimes is played by the corresponding nonholonomic generalizations of Stackel–Killing and Killing–Yano tensors. There are constructed new classes of black hole solutions and we study hidden symmetries for ellipsoidal and/or solitonic deformations of “prime” Kerr–Sen black holes into “target” off-diagonal metrics. In general, the classical conserved quantities (integrable and not-integrable) do not transfer to the quantized systems and produce quantum gravitational anomalies. We prove that such anomalies can be eliminated via corresponding nonholonomic deformations of fundamental geometric objects (connections and corresponding Riemannian and Ricci tensors) and by frame transforms.

Appendices

Appendix A: N-adapted coefficients for d-connections

For convenience, we present some necessary formulas from the geometry of N-anholonomic (pseudo) Riemannian spaces, see details in [15].

The coefficients of the Levi-Civita (LC) connection $\nabla =\{{}_{\shortmid }\varGamma_{\ \alpha \beta }^{\gamma }\}$ for the metric (6) (computed with respect to N-adapted basis (2) and (3)) can be written in the form

$$ {} _{\shortmid }\varGamma_{\ \alpha \beta }^{\gamma }=\widehat{\boldsymbol{\Gamma}} _{\ \alpha \beta }^{\gamma }+Z_{\ \alpha \beta }^{\gamma }. $$

(45)

The value $\widehat{\mathbf{D}}=\{\widehat{\boldsymbol{\Gamma}}_{\ \alpha \beta }^{\gamma }=(\widehat{L}_{jk}^{i},\widehat{L}_{bk}^{a},\widehat{C}_{jc}^{i}, \widehat{C}_{bc}^{a})\}$ with coefficients

(46)

defines the canonical distinguished connection (d-connection). By straightforward computations, we can check that it is metric compatible, $\widehat{\mathbf{D}}\mathbf{g}=0$, and its torsion $\mathcal{T}=\{\widehat{\mathbf{T}}_{\ \alpha \beta }^{\gamma }\equiv \widehat{\boldsymbol{\Gamma}}_{\ \alpha \beta }^{\gamma }-\widehat{\boldsymbol{\Gamma}}_{\ \beta \alpha }^{\gamma };\widehat{T}_{\ jk}^{i},\widehat{T}_{\ ja}^{i},\widehat{T}_{\ ji}^{a},\widehat{T}_{\ bi}^{a},\widehat{T}_{\ bc}^{a}\}$, is with zero horizontal and vertical coefficients, $\widehat{T}_{\ jk}^{i}=0$ and $\widehat{T}_{\ bc}^{a}=0$. There are also non-trivial h–v-coefficients

(47)

The distortion tensor $Z_{\ \alpha \beta }^{\gamma }$ in (45) is also constructed in a unique form from the coefficients of metric N-connection,

(48)

for $\varXi_{jk}^{ih}=\frac{1}{2}(\delta_{j}^{i}\delta_{k}^{h}-g_{jk}g^{ih})$ and ${}^{\pm}\varXi_{cd}^{ab}=\frac{1}{2}(\delta_{c}^{a}\delta_{d}^{b}+h_{cd}h^{ab})$.

Any geometric and physical formulas for the connection ∇ can be equivalently redefined for the canonical d-connection $\widehat{\mathbf{D}}$, and inversely, using (45) because all involved geometric objects (two different connections and the distorting tensor) are uniquely defined by the same metric structure.

Appendix B: Decoupling and integration of Einstein eqs.

We briefly summarize the results on generating off-diagonal solutions in gravity [14, 15].

Using the conditions of Lemma 2.1 and computing in explicit form the Ricci and Einstein tensors, we prove:

Theorem B.1

(Decoupling of equations)

The Einstein equations for $\widehat{\mathbf{D}}$ (46) and ansatz for the metric g (15) with ω=1 and any general source $\boldsymbol{\Upsilon}_{\ \delta }^{\alpha }$ (14) are

(49)

The system of partial differential equations (49) is with decoupling of equations (the “splitting” of equations is used as an equivalent one; we should not confuse this with the property of separation of variables). For simplicity, we can consider a subclass of solutions when for the chosen N-system of reference $h_{a}^{\ast }\neq 0$.⁸

Corollary B.1

The system of equations (49) for y ³=v, $g_{i}=\epsilon_{i}e^{\psi (x^{k})}$ and $h_{a}^{\ast }\neq 0$, Ψ ₂≠0, Ψ ₄≠0, can be written equivalently in the form

(50)

The systems of equations (49) and (50) can be integrated in general forms following the results of Theorem 2.3; we can generate solutions for the LC connection if the zero torsion conditions of Corollary 2.1 are satisfied.

Let us study the “vanishing torsion” conditions (19). For general sources, $\boldsymbol{\Upsilon}_{\ \delta }^{\alpha }$, it is quite difficult to prove in an explicit analytic form that such equations have non-trivial solutions.

Corollary B.2

We can adapt the nonholonomic distributions for generic off-diagonal Einstein spaces with $\boldsymbol{\Upsilon}_{\ \delta }^{\alpha }=\lambda \mathbf{\delta }_{\ \delta }^{\alpha }$, when Ψ ₂=Ψ ₄=λ, and parametrize the data (17) and (18) for the coefficients of metric ansatz in such a form that (19) determine some classes of non-trivial solutions, when the d-torsions (47) for $\widehat{\mathbf{D}}$ are zero.

Proof

Let us consider a solution (17) and (18) when the coordinate system and boundary conditions are fixed in the form that $\underline{h}_{4}(x^{k})=0$, ² n _k (x ⁱ)=0 and ∂ _i ¹ n _j (x ^k)=∂ _j ² n _i (x ^k). In such cases, we must prove that h ₄=±λ ⁻¹ e ^ϕ and w _i =∂ _i ϕ/ϕ ^∗ have for some classes of functions ϕ(x ⁱ,v), ϕ ^∗≠0, certain non-trivial solutions of (19), i.e.

$$ w_{i}^{\ast }=\mathbf{e}_{i}\ln |h_{4}|\quad \mbox{and}\quad \partial_{i}w_{j}= \partial_{j}w_{i}. $$

(51)

Expressing h ₄ and w _i explicitly via ϕ, we get from (51) that ϕ ^∗∗ ∂ _i ϕ−ϕ ^∗(∂ _i ϕ)^∗=0. As a particular case, these equations can be written in the form $( \partial_{i}\phi /\phi^{\ast } )^{\ast }=w_{i}^{\ast }=0$, when w _i = ⁰ w _i (x ^k). Choosing $\phi = {}^{0}\phi (x^{k}) \overline{\phi (}v)$, we generate solutions with separation of variables, ⁰ w _i (x ^k)=ι∂ _i ⁰ ϕ and $\overline{\phi }^{\ast }=\iota^{-1}\overline{\phi }$, for a nonzero constant ι. In general, we can consider various types of nonholonomic constraints (19) for selecting from data (17) and (18) very different families of solutions in GR. □

Remark B.1

Integrating on variable y ³=φ and taking $\underline{h}_{4}= {}^{0}h_{4}(\widetilde{x}^{i})$, ϵ _i =1, $g_{i}=\eta_{i}e^{\psi (\widetilde{x}^{k})}{} ^{0}g_{i}(\widetilde{x}^{k}), \eta_{i}=\epsilon_{i}e^{\psi (\widetilde{x}^{k})}$, ω=1, in coordinates $\widetilde{u}^{\beta }$ (24) and for the “initial” data ^∘ g _α (25), the “one-Killing” off-diagonal solutions (17) and (18) for the Einstein spaces can be represented in the form

(52)

for, respectively, given ${}^{\circ }h_{4}(\widetilde{x}^{i}),\widehat{\varPhi }(\widetilde{x}^{i}),{}^{b}\rho (\widetilde{x}^{i})$ and chosen generating/ integration functions $\phi (\widetilde{x}^{i},\varphi )$, ${}^{1}n_{k}(\widetilde{x}^{i}),{}^{2}n_{k}(\widetilde{x}^{i})$ subjected to the conditions (51).

To consider possible small nonholonomic deformations ^∘ g _α →^η g _α is convenient to express the formulas (51) via polarization functions (27) when, for λ≠0, h _a = ^∘ h _a η _a = ^∘ h _a (1+χ _a ). Using the second formula in (52), for h ₄, we can express $\phi =\ln \sqrt{|\lambda {}^{\circ }h_{4}\chi_{4}|}$ and consider for such classes of solutions a “new” generating function $\chi_{4}(\widetilde{x}^{i},\varphi )$. The third formula for h ₃ from that system of solutions allows us to express

$$ \chi_{3}=-1+ \bigl(\lambda \ ^{\circ }h_{3} \bigr)^{-1} \bigl[ \bigl( \ln \sqrt{|1-\chi_{4}|} \bigr)^{\ast } \bigr]^{2}. $$

(53)

We conclude that all v-coefficients of off-diagonal metrics and N-connections, see ansatz (15) (equivalently (16)) for the Einstein spaces with Killing symmetry on $\partial /\partial \widetilde{y}^{4}$, up to arbitrary frame transforms, are functionally determined by χ ₄, i.e. ϕ[χ ₄], h _a [χ ₄], w _i [χ ₄], n _k [χ ₄].