--- title: Hayward Spacetime Overview url: https://www.emergentmind.com/topics/hayward-spacetime type: topic --- # Hayward Spacetime Overview Hayward spacetime is a static, spherically symmetric regular black-hole geometry specified by an ADM mass \(M\) and a length scale \(\ell\) (or \(L\)), with Schwarzschild asymptotics at large radius and a de Sitter-like core as \(r\to0\). In its standard form, it replaces the classical central singularity by finite curvature invariants and admits, depending on parameter values, two horizons, an extremal horizon, or no horizon at all. In the contemporary literature it is treated both as the original Hayward regular black hole and as a prototype for radiating, quantum-corrected, effective-field-theory, and black-hole-mimicker constructions [2311.15771] [1312.6665] [2404.12243] [2410.04306]. ## 1. Metric and defining parameterizations In Schwarzschild-like coordinates \((t,r,\theta,\phi)\), the Hayward line element is written as \[ ds^2=-f(r)\,dt^2+f(r)^{-1}\,dr^2+r^2(d\theta^2+\sin^2\theta\,d\phi^2), \] with \[ f(r)=1-\frac{2Mr^2}{r^3+2M\ell^2}. \] Here \(M\) is the ADM mass and \(\ell\) is the Hayward or core parameter. An alternative notation used in geodesic and scattering analyses is \[ f(r)=1-\frac{2Mr^2}{r^3+\ell^3}, \] with \(\ell^3\equiv Q^3\), and the two forms are related by the identification \(\ell^3\equiv2M\ell^2\) in the relevant conventions [2311.15771] [1312.6665]. The large-radius and small-radius limits define the geometry. For \(r\to\infty\), \[ f(r)\to1-\frac{2M}{r}+\mathcal O(r^{-4}), \] so the metric reduces to Schwarzschild at leading order. For \(r\to0\), \[ f(r)\approx1-\frac{r^2}{\ell^2}+\mathcal O(r^5), \] which is de Sitter-like. This two-scale structure—Schwarzschild outside, de Sitter inside—is the basic hallmark of Hayward spacetime [1312.6665] [2404.12243]. A useful physical interpretation follows directly from these asymptotics. The parameter \(\ell\) regulates the core curvature, while \(M\) retains its role as the asymptotic mass. In later work, the same functional form has also been interpreted as a radiating black hole, as a quantum-corrected black hole owing to the running gravitational coupling in Asymptotically Safe Gravity, and as a black-hole solution in the Effective Field Theory [2410.04306]. ## 2. Regular core, source models, and curvature structure The defining geometric property of Hayward spacetime is regularity at the origin. Because \(f(r)\approx1-r^2/\ell^2\) near \(r=0\), the curvature invariants remain finite there. In particular, \[ R\big|_{r=0}=\frac{12}{\ell^2},\qquad R_{\mu\nu}R^{\mu\nu}\big|_{r=0}=\frac{36}{\ell^4}, \] and the core behaves as an effective de Sitter region with \(\Lambda_{\rm eff}=3/\ell^2\) [1312.6665] [2404.12243]. This regularity is the sense in which the spacetime is a “regular black hole.” One source model realizes the geometry in Einstein gravity coupled to a purely magnetic nonlinear electrodynamics. In that construction, \[ I=\frac{1}{16\pi}\int d^4x\,\sqrt{-g}\,\bigl(R-\mathcal L(F)\bigr),\qquad F=F_{\mu\nu}F^{\mu\nu}, \] with magnetic monopole two-form \[ \mathbf F=P\sin\theta\,d\theta\wedge d\varphi,\qquad F=\frac{2P^2}{r^4}. \] A convenient closed-form Lagrangian is \[ \mathcal L(F)= -\frac{6}{\ell^2}\frac{1}{\bigl[1+(\beta/F)^{3/4}\bigr]^2}, \qquad \beta=2P^2(2M\ell^2)^{-4/3}. \] In this NED realization, the weak-field limit does not satisfy \(\mathcal L(F)\sim-F\), so the Reissner–Nordström limit is absent [1312.6665]. The same geometry can also be read as an effective anisotropic fluid. Solving \(G_{\mu\nu}=8\pi T_{\mu\nu}\) yields \[ \rho(r)=\frac{3M^2\ell^2}{2\pi(r^3+2M\ell^2)^2},\qquad p_r(r)=-\rho(r), \] \[ p_t(r)=\frac{3M^2\ell^2(r^3-M\ell^2)}{\pi(r^3+2M\ell^2)^3}. \] At the center, \[ \rho(0)=\frac{3}{8\pi\ell^2},\qquad p_r(0)=p_t(0)=-\frac{3}{8\pi\ell^2}, \] which is exactly the de Sitter equation of state [2206.04505]. Beyond regularity, Hayward spacetime has been studied as a curvature-theoretic object. It has been shown to be an Einstein manifold of level \(2\), \(2\)-quasi Einstein, generalized quasi-Einstein, and Roter type, and to satisfy Deszcz-type pseudosymmetry relations such as \[ R\cdot R=-\frac{m}{(r^3+2mb^2)^2}Q(g,R), \qquad R\cdot C=-\frac{m}{(r^3+2mb^2)^2}Q(g,C). \] The energy-momentum tensor is likewise pseudosymmetric, and the metric admits an almost \(\eta\)-Ricci soliton as well as an almost \(\eta\)-Ricci-Yamabe soliton [2303.00932]. ## 3. Horizons, causal structure, and collapse Horizons are determined by the real positive roots of \[ f(r)=0\quad\Longrightarrow\quad r^3-2Mr^2+2M\ell^2=0. \] Introducing \(x=r/M\) and \(\lambda=\ell/M\), one obtains \[ x^3-2x^2+2\lambda^2=0. \] The standard classification is: \[ \lambda^2<\frac{16}{27}\ \Rightarrow\ \text{two distinct positive roots } r_-\frac{16}{27}\ \Rightarrow\ \text{no real positive root}. \] In the alternative \(\ell^3\) notation used in wave-scattering work, the extremal value is \(\ell_{\rm ext}\simeq1.0582\,M\) [1312.6665] [2311.15771] [2404.12243]. The global causal structure closely parallels Reissner–Nordström, but with a crucial difference: \(r=0\) is not a timelike curvature singularity. Instead, the Penrose diagram contains an event horizon \(r_+\), a Cauchy horizon \(r_-\), and a regular bounce region of finite curvature. One description gives an infinite tower of asymptotically flat regions connected by black-hole and white-hole throats; another states that beyond the inner horizon one may analytically continue into further copies of the regular Hayward region [2404.12243] [2511.23165]. A collapse realization is obtained by matching a spherical dust FRW interior, \[ ds^2_{\rm int}=-dT^2+a(T)^2(dr^2+r^2d\Omega^2), \] to the exterior Hayward geometry using Israel–Darmois conditions. The induced-metric and extrinsic-curvature matching require \[ R(T)=r_b\,a(T),\qquad F(R)=1-\dot R(T)^2. \] The resulting interior dynamics satisfy the modified Friedmann equation \[ H^2=\frac{\tfrac{8\pi}{3}\rho}{1+\tfrac{8\pi}{3}L^2\rho}. \] At low density this reduces to the classical Oppenheimer–Snyder law, while at high density \(H^2\to1/L^2\). Correspondingly, the interior exhibits a power-law regime \(a(T)\sim T^{2/3}\) and a de Sitter regime \(a(T)\sim e^{T/L}\), which was interpreted as inflation preventing singularity formation in that collapse model [2404.12243]. The stability of the inner horizon under perturbations is a distinct question. Scalar-field collapse studies found that weak perturbations leave the inner horizon at a stable finite radius, whereas strong perturbations shrink it to zero volume and produce a spacelike singularity, effectively converting the interior to a Schwarzschild-like geometry. Near the threshold \(p_*\), the inner-horizon radius obeys the scaling law \[ r_-\propto |p-p_*|^\gamma,\qquad \gamma\approx0.5. \] This qualifies the common assumption that static regularity alone settles the nonlinear internal structure [2511.23165]. ## 4. Geodesics, photon sphere, absorption, scattering, and shadow For equatorial null geodesics, the conserved quantities are \[ E=f(r)\dot t,\qquad L=r^2\dot\phi, \] and the radial equation can be written as \[ \left(\frac{dr}{d\lambda}\right)^2+V_{\rm eff}(r)=E^2, \qquad V_{\rm eff}(r)=f(r)\frac{L^2}{r^2}. \] Equivalently, in terms of impact parameter \(b=L/E\), \[ \left(\frac{dr}{d\phi}\right)^2=r^4\left(\frac{1}{b^2}-\frac{f(r)}{r^2}\right). \] Unstable circular photon orbits occur at radii \(r=r_c\) satisfying \[ 2f(r_c)-r_c f'(r_c)=0, \] with critical impact parameter \[ b_c=\frac{r_c}{\sqrt{f(r_c)}}. \] The same \(b_c\) controls the high-frequency geometrical absorption cross section, \[ \sigma_{\rm gcs}=\pi b_c^2, \] and the shadow radius seen at infinity [2311.15771]. Scalar-wave absorption is obtained from the massless Klein–Gordon equation \(\Box\Phi=0\). After separation of variables and use of the tortoise coordinate \(r_\star\), the radial equation becomes \[ \frac{d^2\psi_{\omega\ell}}{dr_\star^2}+\bigl[\omega^2-V_{\rm eff}(r)\bigr]\psi_{\omega\ell}=0, \] with \[ V_{\rm eff}(r)=f(r)\left[\frac{\ell(\ell+1)}{r^2}+\frac{f'(r)}{r}\right]. \] Imposing ingoing behavior at the horizon and unit-amplitude incoming behavior at infinity gives reflection and transmission coefficients obeying \[ |R|^2+|T|^2=1. \] The partial and total absorption cross sections are \[ \sigma_\ell=\frac{\pi}{\omega^2}(2\ell+1)|T_{\omega\ell}|^2, \qquad \sigma_{\rm abs}(\omega)=\sum_{\ell=0}^\infty \sigma_\ell. \] The low-frequency limit is \[ \sigma_{\rm abs}\to A_{\rm horizon}=4\pi r_+^2, \] while in the high-frequency regime the cross section oscillates about \(\pi b_c^2\) with the sinc behavior \[ \sigma_{\rm hf}\approx \sigma_{\rm gcs}\Bigl[1-8\pi b_c\Lambda e^{-\pi b_c\Lambda}\,\mathrm{sinc}(2\pi b_c\omega)\Bigr], \] where \(\Lambda\) is the Lyapunov exponent of the photon sphere [2311.15771]. The differential scattering amplitude is \[ f(\theta)=\frac{1}{2i\omega}\sum_{\ell=0}^{\infty}(2\ell+1)\bigl[e^{2i\delta_\ell(\omega)}-1\bigr]P_\ell(\cos\theta), \qquad \frac{d\sigma}{d\Omega}=|f(\theta)|^2. \] In the glory approximation, for \(\theta\to\pi\), \[ \left(\frac{d\sigma}{d\Omega}\right)_{\rm glory}\simeq 2\pi\omega b_g^2\left|\frac{db}{d\theta}\right|_{\theta=\pi} J_0^2(\omega b_g\sin\theta). \] In the weak-field, small-angle limit, \[ \frac{d\sigma}{d\Omega}\simeq \frac{16M^2}{\theta^4}+\frac{15\pi M^2}{4\theta^3}+\cdots, \] with Hayward charge-corrections entering only at subleading order [2311.15771]. An important phenomenological result is that Hayward regular black holes can mimic Reissner–Nordström black holes in absorption and scattering observables when geodesic quantities are matched, for example by requiring \[ b_c^{(H)}(\alpha_H)=b_c^{(RN)}(\alpha_{RN}) \quad\text{or}\quad b_g^{(H)}(\alpha_H)=b_g^{(RN)}(\alpha_{RN}). \] Explicit numerical pairs include \(\alpha_H=0.5\) with \(\alpha_{RN}\simeq0.1826\), and \(\alpha_H=1.0\) with \(\alpha_{RN}\simeq0.5496\) [2311.15771]. ## 5. Perturbations, quasinormal modes, and external fields Scalar, Maxwell, and Dirac perturbations in the Hayward background admit analytic quasinormal-mode formulas from an expansion in inverse multipole number. With \[ L\equiv \ell+\tfrac12,\qquad K\equiv n+\tfrac12, \] the WKB construction yields \[ \omega=\omega^{(-1)}L+\omega^{(0)}+\omega^{(1)}L^{-1}+O(L^{-2}), \] and, to leading eikonal order, \[ \omega\simeq \Omega_c\,L-i\,\lambda_c\,K, \qquad \Omega_c=\frac{1}{3\sqrt3\,M},\quad \lambda_c=\frac{1}{3\sqrt3\,M}. \] The corrections depend on the dimensionless coupling \(\gamma\equiv2l^2/M^2\). For all \(\ell\ge1\) and \(0\le\gamma\le1.18\), the relative error in both \(\mathrm{Re}\,\omega\) and \(\mathrm{Im}\,\omega\) is reported as well below \(1\%\), while for \(\ell=0\) the \(1/L\) expansion is not quantitatively accurate [2410.04306]. Axial gravitational quasinormal modes show a related trend. Higher-order WKB with Padé approximants and time-domain integration indicate that increasing the quantum parameter \(\gamma\) raises the oscillation frequencies and reduces the damping rates, making the ringdown longer lived. The effect is stronger for the first overtone than for the fundamental mode, which has been connected to an “outburst of overtones” and to enhanced sensitivity of subdominant modes to near-horizon quantum corrections [2508.19989]. Electromagnetic test fields reveal further structure. In the fixed Hayward background, a dipole-type homogeneous Maxwell solution is \[ A_\phi=\frac{B_0}{2}\left(1-\frac{4M\ell^2}{r^3}\right)r^2\sin^2\theta. \] At large radius the field tends to an asymptotically uniform configuration, while for small \(r\) it develops dipole loop-like field lines in the regular interior. Charged-particle motion then exhibits Hayward-specific orbit structure. In the black-hole regime there is an ISCO modified by \(\ell\) and by the magnetic coupling; in the horizonless regime the effective potential can develop a double well, supporting up to three circular solutions for a given angular momentum, with two stable branches defining an inner critical circular orbit \(r_{\rm ICCO}\) and an outer critical circular orbit \(r_{\rm OCCO}\) [2203.12124]. These wave and particle results establish that the spacetime’s regular core is not only a local curvature property but also reorganizes the external effective potentials that control ringdown, synchrotron-like motion, orbit stability, and capture. ## 6. Deformations, extensions, and related spacetimes Several deformations preserve the Hayward functional structure while changing the physical interpretation. One example is the Hayward black hole surrounded by a cloud of strings, \[ g(r)=1-a-\frac{2Mr^2}{r^3+2Ml^2}, \qquad 0M_*\) there are two horizons, for \(M=M_*\) an extremal horizon, and for \(M