--- title: Higher-D Quantum Oppenheimer–Snyder Model url: https://www.emergentmind.com/topics/higher-dimensional-quantum-oppenheimer-snyder-model type: topic --- # Higher-D Quantum Oppenheimer–Snyder Model The higher-dimensional quantum Oppenheimer–Snyder model is a \(D=d+1\) dimensional generalization of Oppenheimer–Snyder collapse in which a homogeneous, isotropic, pressureless dust interior is matched to a static, spherically symmetric exterior, and the interior dynamics is modified by effective higher-dimensional Loop Quantum Cosmology (LQC). In this framework, the junction conditions determine a quantum-corrected exterior metric, the collapse reaches a finite-radius bounce rather than a classical singularity, and the resulting black-hole geometry acquires modified horizon structure, quasinormal spectra, and thermodynamics. A later extension incorporates a cosmological constant and studies the AdS case in extended phase-space thermodynamics. Related AdS/CFT work uses an Oppenheimer–Snyder collapse in asymptotically AdS space as a holographic model of thermalization, but there the term “quantum” refers to the dual strongly coupled quantum field theory rather than to LQC-inspired quantum geometry corrections [2408.15821] [2606.08182] [1507.00878]. ## 1. Classical higher-dimensional Oppenheimer–Snyder framework The underlying classical model describes the gravitational collapse of a spherically symmetric, homogeneous, pressureless dust star. In \(D=d+1\) spacetime dimensions with \(d\ge 3\), the interior is taken to be a spatially flat FRW region and the exterior classical vacuum solution is the Schwarzschild–Tangherlini metric. The interior line element is \[ ds^2_{\rm in} = -dT^2 + a^2(T)\left[dR^2 + R^2 d\Omega^2_{d-1}\right], \] with stellar surface at comoving radius \(R=R_0\) and physical radius \[ \tilde{R}(T) = a(T)R_0. \] For pressureless dust, \(T^\mu_{\ \nu}=\rho\,u^\mu u_\nu\) with \(p=0\), and conservation gives \[ \rho(T)=\rho_0\left(\frac{a_0}{a}\right)^d. \] The physical volume of the \(d\)-ball of radius \(\tilde R\) is \[ V=\frac{\Omega_{d-1}\tilde R^d}{d}, \] and the paper introduces \[ \mu=\frac{8\pi}{(d-1)\Omega_{d-1}}, \] so that \(\rho=M/V\). The classical Friedmann equation is \[ H^2=\frac{2\kappa\rho}{d(d-1)}, \qquad H=\frac{\dot a}{a}, \] which at the stellar surface becomes \[ H^2=\frac{2GM\mu}{\tilde R^d}. \] The corresponding classical exterior is \[ ds^2_{\rm out}=-f(r)dt^2+f(r)^{-1}dr^2+r^2d\Omega_{d-1}^2, \qquad f_{\rm cl}(r)=1-\frac{2GM\mu}{r^{d-2}}, \] with horizon radius determined by \[ r_h^{d-2}=2GM\mu. \] This higher-dimensional OS construction preserves the defining ingredients of the four-dimensional model—homogeneity, isotropy, and dust—while replacing the exterior Schwarzschild geometry by its Tangherlini generalization [2408.15821]. ## 2. LQC-inspired effective dynamics and singularity avoidance The quantum model replaces the classical interior Friedmann equation by the higher-dimensional effective LQC equation \[ H^2=\frac{2\kappa\rho}{d(d-1)}\left(1-\frac{\rho}{\rho_c}\right), \] with quantum parameter \[ \alpha=\gamma^2\Delta^{2/(d-1)}, \] and critical density \[ \rho_c=\frac{d(d-1)}{2\kappa\gamma^2\Delta^{2/(d-1)}}. \] Using \(\rho=M/V\) and \(\tilde R=aR_0\), this becomes \[ H^2=\frac{2GM\mu}{\tilde R^d}-\frac{4G^2M^2\mu^2\alpha}{\tilde R^{2d}}. \] The effective dynamics implies a bounce when \(H=0\), equivalently when \(\rho=\rho_c\). The bounce radius is \[ \tilde R_b^d=2GM\mu\alpha, \qquad \tilde R_b=(2GM\mu\alpha)^{1/d}, \] and in terms of the scale factor the minimum is fixed by \[ a_{\min}=a_0\left(\frac{\rho_0}{\rho_c}\right)^{1/d}. \] Since \(\tilde R\) never reaches zero, the interior energy density is bounded by \(\rho_c\) and the classical collapse singularity is avoided. The existence of the bounce does not depend on \(d\) beyond the scaling of \(\alpha\) and \(\rho_c\). In this sense, the higher-dimensional quantum OS model is not merely a higher-dimensional restatement of classical collapse; the nonperturbative quantum geometry input enters directly through the modified Friedmann equation and forces a turning point at finite radius [2408.15821]. ## 3. Junction conditions and the quantum-corrected exterior geometry The interior and exterior are matched by Israel–Darmois junction conditions. Continuity of the induced metric on the stellar surface \(\Sigma\) gives \[ 1=f(r)\dot t^2-g(r)^{-1}\dot r^2, \qquad r(T)=a(T)R_0, \] while continuity of the extrinsic curvature yields \(K^-_{TT}=K^+_{TT}=0\) and, for the angular components, \[ K^-_{\theta\theta}=a(T)R_0=rE\sqrt{f^{-1}g}, \] with \(E\equiv f\dot t\) a conserved energy along \(\Sigma\). Choosing the time coordinate so that \(E=1\) implies \(f=g\), and combining the matching with \(H=\dot a/a=\dot r/r\) gives \[ g(r)=1-H^2r^2. \] Substituting the classical interior Friedmann equation reproduces the Schwarzschild–Tangherlini metric, while substituting the LQC-corrected equation yields \[ f_q(r)=1-\frac{2GM\mu}{r^{d-2}}+\frac{4G^2M^2\mu^2\alpha}{r^{2d-2}}, \] and hence \[ ds_q^2=-f_q(r)dt^2+f_q(r)^{-1}dr^2+r^2d\Omega_{d-1}^2. \] Horizons are roots of \(f_q(r)=0\). The function \(f_q\) has a unique minimum at \[ r_m=\left[\frac{4GM\mu\alpha(d-1)}{d-2}\right]^{1/d}, \] and the critical mass \(M_c\) is defined by \(f_q(r_m)=0\). For \(M>M_c\) there are two horizons \(r_-M_c\), the collapsing star crosses \(r_+\), reaches \(r_b\), bounces, re-expands, and exits through a white-hole asymptotic region in the extended spacetime [2408.15821]. With a cosmological constant, the same matching logic is retained. The classical exterior becomes Schwarzschild-(A)dS, \[ ds_{\rm out}^2=-f(r)\,dt^2+f(r)^{-1}dr^2+r^2d\Omega_{D-2}^2, \] with \[ f_{\rm cl}(r) =1-\frac{16\pi G_D M}{(D-2)\Omega_{D-2}\,r^{D-3}} -\frac{2\Lambda}{(D-1)(D-2)}\,r^2. \] The interior quantum dynamics is modified to \[ H^2=\frac{2\kappa}{d(d-1)}\rho_T\left(1-\frac{\rho_T}{\rho_c}\right), \qquad \rho_T=\rho+\rho_\Lambda,\qquad \rho_\Lambda=\frac{\Lambda}{\kappa}, \] and the effective exterior is written as \[ f_q(r)=1-r^2H_q^2(r). \] The junction conditions are identical to the classical case; the only change is \(H\mapsto H_q\). In the \(\alpha\to 0\) limit, the Schwarzschild-(A)dS result is recovered [2606.08182]. ## 4. Scalar perturbations and quasinormal spectra The quantum-corrected exterior supports a standard perturbation analysis for a massless scalar field. Using the separation \[ \psi(t,r,\theta_i)=\sum_{l,m}e^{-i\omega t}\,r^{(1-d)/2}\Phi_l(r)Y_{lm}(\theta_i), \] the radial equation in the metric \(ds^2=-f_qdt^2+f_q^{-1}dr^2+r^2d\Omega_{d-1}^2\) is \[ f_q^2\Phi_l''+f_qf_q'\Phi_l'+\omega^2\Phi_l -\left[ \frac{l(l+d-2)}{r^2} +\frac{(d-1)(d-3)f_q}{4r^2} +\frac{(d-1)f_q'}{2r} \right]f_q\Phi_l=0. \] Introducing the tortoise coordinate \(r_*\) through \(dr_*/dr=1/f_q(r)\) gives the Schrödinger-like form \[ \frac{d^2\Phi_l}{dr_*^2}+[\omega^2-V(r)]\Phi_l=0, \] with effective potential \[ V(r)=\left[ \frac{l(l+d-2)}{r^2} +\frac{(d-1)(d-3)f_q}{4r^2} +\frac{(d-1)f_q'}{2r} \right]f_q. \] Quasinormal-mode boundary conditions are purely ingoing at the horizon, \(\Phi\sim e^{-i\omega r_*}\) as \(r_*\to-\infty\), and purely outgoing at infinity, \(\Phi\sim e^{+i\omega r_*}\) as \(r_*\to+\infty\). The spectra were computed with a 13th-order WKB method with Padé improvement and checked against time-domain finite element evolutions. The reported trends are systematic: increasing \(M\) at fixed \(d\) reduces both \(\mathrm{Re}(\omega)\) and \(|\mathrm{Im}(\omega)|\); increasing \(d\) at fixed \(M\) increases both \(\mathrm{Re}(\omega)\) and \(|\mathrm{Im}(\omega)|\); relative to Schwarzschild–Tangherlini, quantum corrections always reduce \(|\mathrm{Im}(\omega)|\), while \(\mathrm{Re}(\omega)\) increases in lower dimensions and decreases in higher dimensions. For the illustrative fundamental mode with \(n=0\), \(l=1\), and \(M=7000\) in the paper’s units, the values are \[ d=4:\quad \omega_q \approx 0.0131811-0.00470016\,i,\qquad \omega_{\rm cl}\approx 0.0131808-0.00470051\,i, \] and \[ d=10:\quad \omega_q \approx 1.45957-0.40883\,i,\qquad \omega_{\rm cl}\approx 1.46332-0.42753\,i. \] These results identify the perturbative imprint of the quantum-corrected OS background on ringdown observables [2408.15821]. ## 5. Thermodynamics, remnants, and extended phase space For the asymptotically flat quantum-corrected black holes, the Hawking temperature is determined by \[ T_H=\frac{f_q'(r_+)}{4\pi}. \] In \(D=5\) (\(d=4\)), the paper gives \[ M_{d=4}(r_+)=\frac{3\pi r_+^3\left(r_+-\sqrt{r_+^2-4\alpha}\right)}{16\alpha}, \] and \[ T_{d=4}(r_+)=\frac{-r_+^2+r_+\sqrt{r_+^2-4\alpha}+3\alpha}{2\pi r_+\alpha}. \] Assuming the first law \(dM=TdS\), the entropy is obtained by integration. The leading large-\(r_+\) results are \[ S_{d=3}=\pi r_+^2+2\pi\alpha\ln\!\left[\frac{r_+^2}{3\alpha}\right]+O(1/r_+), \] \[ S_{d=4}=\frac{\pi^2}{2}r_+^3+3\pi^2\alpha r_+ + O(1/r_+), \] \[ S_{d=5}=\frac{2\pi^2}{3}r_+^4+\frac{8}{3}\pi^2\alpha r_+^2+O(1/r_+). \] The Hawking temperature rises to a peak and then decreases to zero at the extremal radius, and the heat capacity exhibits an extra divergence that signals an additional phase transition introduced by quantum corrections. The free energy \(F=M-TS\) is larger than the classical value for small \(r_+\) and smaller otherwise [2408.15821]. In the AdS extension, the cosmological constant is treated as pressure, \[ P\equiv -\frac{\Lambda}{8\pi G_D}, \qquad dM=T\,dS+V\,dP, \] with thermodynamic volume \[ V(r_+)=\frac{\Omega_{D-2}r_+^{D-1}}{D-1}. \] Entropy is defined by \[ S(r_+)=\int_{r_0}^{r_+}\frac{1}{T}\left(\frac{\partial M}{\partial r}\right)_P dr, \] where \(r_0\) is set by \(T(r_0)=0\) so that \(S(r_0)=0\). At small \(\alpha\), the paper gives, for example, \[ (d+1)=5:\quad S=\frac{\pi^2}{2}r_+^3+3\pi^2r_+\alpha,\qquad \frac{S}{A}=\frac14+\frac{3\alpha}{2r_+^2}, \] \[ (d+1)=6:\quad S=\frac{2\pi^2}{3}r_+^4+\frac{8\pi^2}{3}r_+^2\alpha,\qquad \frac{S}{A}=\frac14+\frac{\alpha}{r_+^2}, \] \[ (d+1)=7:\quad S=\frac{\pi^3}{4}r_+^5+\frac{5\pi^3}{6}r_+^3\alpha,\qquad \frac{S}{A}=\frac14+\frac{5\alpha}{6r_+^2}. \] A notable result is that the entropy is independent of \(\Lambda\) for any \(D\); quantum corrections enter only through \(\alpha\). The small-AdS-black-hole regime is thermally regularized. Classically, \[ T_{\rm cl}(r_+,P)\sim \frac{D-3}{4\pi r_+}\qquad \text{as } r_+\to 0, \] so the temperature diverges. In the quantum-corrected model, the square-root structure \(\sqrt{r_+^2-r_0^2}\) with \(r_0^2\propto\alpha\) implies instead \[ T(r_+\to r_0)\longrightarrow 0. \] In five dimensions, the heat capacity \[ C_5= \frac{3\pi^2r_+^4\left[3r_+^2-9\alpha+r_+\sqrt{r_+^2-4\alpha}\,(\alpha\Lambda-3)\right]} {2\left[3r_+^2\sqrt{r_+^2-4\alpha}+9\alpha\sqrt{r_+^2-4\alpha}+r_+^3(\alpha\Lambda-3)\right]} \] has a new divergence at small \(r_+\approx r_0\), where \(C_P\) changes sign and a thermally stable near-remnant branch appears. The Gibbs free energy \(G=M-TS\) shows a swallow-tail for \(P