--- title: Einstein-Maxwell-Power-Yang-Mills Black Hole url: https://www.emergentmind.com/topics/einstein-maxwell-power-yang-mills-black-hole type: topic --- # Einstein-Maxwell-Power-Yang-Mills Black Hole Searching arXiv for the specified topic and key related papers. An Einstein–Maxwell–Power–Yang–Mills black hole is a black-hole solution of Einstein gravity coupled simultaneously to an Abelian Maxwell field and a non-Abelian Yang–Mills sector whose invariant appears with a real positive power. In the Anti-de Sitter formulation, the system is studied in \(N=n+2\) spacetime dimensions and admits an extended thermodynamic interpretation in which the cosmological constant is treated as pressure; within that setting, the black hole displays Van der Waals–like small–large black-hole phase transitions for suitable values of the Maxwell and Yang–Mills charges. Later four-dimensional studies use closely related power-Yang–Mills sectors to investigate horizon structure, null and timelike geodesics, quasinormal modes, shadows, photon rings, and holographic phase behavior [1412.1197] [2603.08222]. ## 1. Action, field content, and defining parameters In the \(N\equiv n+2\) dimensional AdS construction, the Einstein–Maxwell–power–Yang–Mills (EMPYM) action is \[ \mathcal I \;=\;\frac12\int d^N x\,\sqrt{-g}\, \Bigl( R \;-\;\tfrac{(N-1)(N-2)}3\,\Lambda \;-\;F_{\mu\nu}F^{\mu\nu} \;-\;\bigl[\Tr\bigl(F^{(a)}_{\mu\nu}F^{(a)\,\mu\nu}\bigr)\bigr]^q \Bigr), \] with metric signature \((-++\cdots+)\). Here \(F_{\mu\nu}\) is the Maxwell field strength, \(F^{(a)}_{\mu\nu}\) is the Yang–Mills strength for an \(\mathrm{SO}(N-1)\) gauge group, “\(\Tr\)” denotes a sum over gauge indices, and \(q>0\) is the real power exponent of the Yang–Mills term; \(q=1\) gives the usual Yang–Mills case. The weak-energy condition forces \(q>0\) [1412.1197]. Later four-dimensional formulations use closely related notations. One summary writes the metric function in terms of a Maxwell charge \(Q\), a power–Yang–Mills parameter \(p\), and an effective constant \(D\), with \[ f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2}+\frac{D}{r^{4p-2}}, \qquad D=\frac{2^{\,p-1}}{4p-3}\,\gamma^{2p}, \] where \(\gamma\) is the gauge coupling in the action \(S=\int d^4x\sqrt{-g}\, [ (1/16\pi)R - (1/4)F^2 - \gamma(\mathcal F)^p ]\) [2603.08222]. Another holographic treatment parameterizes the power exponent by \(\gamma\) rather than \(q\) or \(p\), with bulk action \[ S_{\rm bulk} =\tfrac{1}{2}\int d^Nx\sqrt{-g}\,\bigl[R-2\Lambda-F_{\mu\nu}F^{\mu\nu}-(\Tr[F^{(a)}_{\mu\nu}F^{(a)\mu\nu}])^{\gamma}\bigr]. \] The notational variation across these papers is explicit, and direct comparison of formulas is therefore model-dependent [2602.21930]. ## 2. Exact static solutions and horizon structure The common geometric starting point is a static, spherically symmetric line element \[ ds^2=-f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2\,d\Omega_n^2, \] with \(d\Omega_n^2\) the metric on the unit \(n\)-sphere. For the AdS EMPYM black hole with \(q\neq \frac{n+1}{4}\), the metric function contains the mass term, the cosmological contribution, the Maxwell contribution, and a power-Yang–Mills term governed by \[ Q_1=\frac{\bigl[(n-1)\,n\,Q^2\bigr]^q}{n\,(4q-n-1)}. \] When \(q=\frac{n+1}{4}\), the Yang–Mills sector instead generates a logarithmic contribution, \[ f(r)=1-\frac{2m}{r^{\,n-1}} -\frac{\Lambda}{3}\,r^2 +\frac{2(n-1)\,C^2}{n\,r^{2n-2}} -\frac{Q_2\ln r}{r^{\,n-1}}, \qquad Q_2=\frac{1}{n}\bigl[(n-1)nQ^2\bigr]^{\tfrac{n+1}4}, \] so the special power \(q=(n+1)/4\) is structurally distinct already at the level of the exact solution [1412.1197]. In the four-dimensional non-AdS formulation summarized in [2603.08222], the event horizon \(r_h\) is defined by \(f(r_h)=0\), giving \[ 1-\frac{2M}{r_h}+\frac{Q^2}{r_h^2}+\frac{D}{r_h^{4p-2}}=0. \] For fixed \(M\) and \(p>\tfrac12\), one generically has two real positive roots, interpreted as the outer/event and inner/Cauchy horizons. As \(Q\) or \(D\) increases, the two roots merge to an extremal black hole when the discriminant vanishes. For \(p=\tfrac12\), the \(D\)-term becomes a constant shift and one obtains the analytic extremality relation \[ r_{\pm}=\frac{M \pm \sqrt{M^2-Q^2(1+D)}}{1+D}, \] together with the cosmic-censorship bound \(Q/M<1/\sqrt{1+D}\). For \(p<\tfrac12\), the \(D\)-term grows at large \(r\) and spoils asymptotic flatness, so that range is usually discarded [2603.08222]. A related four-dimensional \(p=\tfrac12\) optical study examines a static, spherically symmetric hairy solution in which the event horizon radius satisfies \[ r_h^2 + (Q_{YM}-2M)\,r_h + Q^2 =0, \] with the “+” branch giving the outer horizon. That same study tracks the deformation of the horizon by the Yang–Mills hair parameter \(Q_{YM}\) [2501.06690]. ## 3. Extended thermodynamics and \(P\)–\(V\) criticality In the AdS EMPYM thermodynamic framework, the largest positive root \(r_+\) of \(f(r_+)=0\) defines the horizon. The entropy is \[ S=\frac{\omega_n\,r_+^{\,n}}{4}, \qquad \omega_n=\frac{2\pi^{\tfrac{n+1}2}}{\Gamma\!\bigl(\tfrac{n+1}2\bigr)}, \] and the cosmological constant is promoted to pressure, \[ P=-\frac{\Lambda}{8\pi}. \] The mass is interpreted as enthalpy, and the first law takes the extended form \[ dM=T\,dS+\Phi_C\,dC+\Phi_Q\,dQ+V\,dP, \qquad V=\Bigl(\frac{\partial M}{\partial P}\Bigr)_{S,C,Q} =\frac{\omega_n}{n+1}\,r_+^{\,n+1}. \] The corresponding Smarr relation follows by scaling [1412.1197]. Solving the temperature for \(P\) yields a geometric equation of state \(P=P(T,r_+,C,Q,q)\). The critical point is defined by the inflection conditions \[ \frac{\partial P}{\partial r_+}\Big|_{T}=0, \qquad \frac{\partial^2 P}{\partial r_+^2}\Big|_{T}=0. \] This produces closed-form expressions for \(T_c\) and \(P_c\) and an algebraic equation for the critical horizon radius \(r_c\). When \(C=0\), the critical radius is explicit: \[ r_c =\Bigl[\tfrac{2q(4q-1)\bigl[(n-1)nQ^2\bigr]^q}{n(n-1)}\Bigr]^{1/(4q-2)}, \] with \[ T_c =\frac{(2q-1)(n-1)}{(4q-1)\,\pi}\, r_c^{-1}, \qquad P_c =\frac{3(n-1)(2q-1)}{16\pi\,q\,(n+1)}\, r_c^{-2}, \] and universal ratio \[ \rho_c\equiv \frac{P_c\,v_c}{T_c}=\frac{3(4q-1)}{4nq(n+1)}. \] These formulas exhibit the explicit dependence of the critical data on the power exponent and the Yang–Mills charge sector [1412.1197]. The charge sectors enter asymmetrically. The Maxwell charge \(C\) contributes a positive \(\propto C^2/r_+^{2n}\) term to the equation of state and tends to raise the critical pressure and temperature if large enough, while the Yang–Mills charge enters through \(Q_1\propto Q^{2q}\) and shifts the critical horizon radius and the ratio \(\rho_c\). Within the parameter ranges summarized in the thermodynamic analysis, there is never more than one critical point, so no reentrant behavior occurs. For \(q\neq (n+1)/4\), criticality exists under suitable bounds on \(C\), while for the special power \(q=(n+1)/4\), a critical point always exists [1412.1197]. ## 4. Phase structure, response functions, and order of transition The phase diagram of the AdS EMPYM black hole is organized by the same qualitative signals used in ordinary fluid thermodynamics. For \(T0), \] and the transition at the critical point is therefore described as not exactly second order and closer to a glass-type phase transition [2012.14915]. ## 5. Geodesics, photon spheres, shadow, and quasinormal modes Four-dimensional EMPYM models have been used extensively to connect non-Abelian hair to optical and perturbative observables. In the geodesic analysis summarized in [2603.08222], the null effective potential is \[ V_{\rm eff}^{(\rm null)}(r)=\frac{L^2}{r^2}f(r), \] the photon sphere radius \(r_{ph}\) solves \(d[f(r)/r^2]/dr=0\), and the shadow radius seen at infinity is \[ R_{\rm sh}=\frac{r_{ph}}{\sqrt{f(r_{ph})}}. \] The instability timescale of the circular photon orbit is quantified by the Lyapunov exponent \[ \lambda=\sqrt{\frac{f(r_{ph})}{2}\,\bigl[\frac{4f(r_{ph})}{r_{ph}^2}-\frac{f''(r_{ph})}{r_{ph}^2}+\frac{4f'(r_{ph})}{r_{ph}^3}\bigr]}. \] For massive particles, the timelike effective potential is \[ V_{\rm eff}^{(\rm tim)}(r)=f(r)\Bigl(1+\frac{L_0^2}{r^2}\Bigr), \] and the ISCO is obtained numerically from \[ 3\,\frac{f(r)f'(r)}{r} + f(r)f''(r) - 2f'(r)^2 = 0. \] Increasing the Maxwell charge \(Q\) or the Yang–Mills nonlinearity exponent \(p\) reduces \(r_{ph}\), \(R_{\rm sh}\), and \(r_{\rm ISCO}\), and increases the instability exponent [2603.08222]. A dedicated \(p=\tfrac12\) optical study uses backward ray-tracing for a thin equatorial disk. In that model, unstable circular null orbits satisfy \[ \frac{d}{dr}\!\biggl[\frac{f(r)}{r^2}\biggr]_{r=r_{ph}}=0, \] which leads to \[ 2r_{ph}^2-(6M-3Q_{YM})r_{ph}+4Q^2=0. \] The critical impact parameter is \[ b_{ph}=\frac{r_{ph}}{\sqrt{f(r_{ph})}}, \] and the observed bolometric intensity is written as \[ I_{\rm obs}(b)=\sum_{m:\,\text{disk-crossings}} f(r_m)^2\,I_e(r_m), \] where \(r_m(b)\) is the \(m\)-th transfer function. For the numerical example \(M=1\), \(Q=0.6\), increasing \(Q_{YM}\) from \(0.0\) to \(0.3\) decreases \(r_h\), \(r_{ph}\), \(b_{ph}\), and \(r_{\rm isco}\); the shadow and photon ring also shrink, and the photon-ring width becomes narrower. The direct image dominates the observed intensity, the lensing ring is bright but extremely narrow, and the photon ring is even thinner and much fainter. The same study states that there is no degeneracy in the photon ring and the shadow for this static and spherically symmetric EMPYM hairy black hole [2501.06690]. Quasinormal-mode analyses have focused on a four-dimensional \(p=\tfrac12\) EMPYM black hole with scalar and Dirac perturbations. There the scalar wave equation reduces to \[ \frac{d^2\psi}{dr_*^2} + [\omega^2-V_s(r)]\psi=0, \qquad V_s(r)=f(r)\Bigl[\frac{\ell(\ell+1)}{r^2}+\frac{f'(r)}{r}\Bigr], \] and sixth-order WKB calculations show that the computed modes satisfy \(\operatorname{Im}\omega<0\), indicating stability in the parameter range studied. In the eikonal limit, \[ \omega(\ell\gg 1)\approx \Omega_c\,\ell - i\,(n+\tfrac12)|\lambda_L|, \] with \(\Omega_c=\sqrt{f(r_1)}/r_1\) and \(\lambda_L\) the Lyapunov exponent at the circular photon orbit. The same work derives a shadow radius and uses Event Horizon Telescope bounds for Sgr A\(^\star\) to constrain the charge sector; for example, for \(Q=1.1\) the allowed interval is reported as \(-0.171\lesssim Q_{YM}\lesssim -0.087\), while still retaining a non-vanishing horizon [2308.11756]. ## 6. Holographic reformulation and further extensions A holographic reformulation treats the AdS EMPYM black hole as an ensemble problem in the dual conformal field theory. In that setting, one keeps \(G\) fixed and trades the AdS scale \(\ell\) for the central charge \(C\propto \ell^2/G\). The bulk first law \[ dM=T\,dS+\Phi\,dQ+\Psi\,dq+V\,dP \] is supplemented on the boundary by \[ dE=\tilde T\,dS+\tilde\Phi\,d\tilde Q+\tilde\Psi\,d\tilde q+V_{\rm bdry}\,dp+\mu\,dC. \] The canonical ensemble with fixed charges exhibits a van der Waals–like phase transition between small and large black holes, with a swallowtail Gibbs free energy and a coexistence curve terminating at \((T_c,P_c)\). In the mixed ensemble of fixed electric potential and fixed Yang–Mills charge, the relevant potential is \[ W=E-TS-\Phi Q, \] and the system undergoes a Hawking–Page transition between confined and deconfined phases of the boundary CFT. In that formulation, increasing the non-Abelian Yang–Mills charge \(\tilde q\) lowers both the minimum temperature \(T_0\) and the Hawking–Page temperature \(T_{HP}\), and shrinks the interval \(\Delta T=T_{HP}-T_0\), thereby suppressing the stability region of the confined phase [2602.21930]. The Joule–Thomson expansion of the AdS EMPYM black hole provides a complementary thermodynamic probe. The Joule–Thomson coefficient at fixed enthalpy is \[ \mu=\Bigl(\frac{\partial T}{\partial P}\Bigr)_H, \] and the inversion curve is defined by \(\mu=0\). The minimum inversion temperature \(T_{\min}\) occurs at \(P_i=0\), and the ratio \(T_{\min}/T_c\) has a structured dependence on the power exponent \(q\) and the Maxwell charge \(C\): for \(q>1\), \(T_{\min}/T_c\to \tfrac12\) as \(C\to\infty\); for \(q=\tfrac12\), the ratio is exactly \(\tfrac12\) for all \(C\); for \(\tfrac12