Photon Sphere Radius in Black Holes

• Photon sphere radius is the areal distance from a black hole at which massless particles execute circular, typically unstable orbits due to extreme gravitational forces.
• It plays a key role in phenomena such as gravitational lensing, black hole shadow imaging, and quasinormal mode analysis, linking spacetime structure with observable signatures.
• Rigorous bounds like rγ ≤ 3M and extensions to rotating or hairy spacetimes reveal its sensitivity to matter distributions and modifications in gravitational dynamics.

A photon sphere radius is the areal radius of a closed null hypersurface in a black-hole or compact-object spacetime along which massless particles (photons) can execute circular, typically unstable, orbits. The photon sphere is a fundamental concept for understanding light propagation, strong gravitational lensing, black hole shadows, quasinormal modes in the eikonal limit, and the link between spacetime structure and observable astrophysical signatures.

1. Definition and Mathematical Characterization

Formally, for a static, spherically symmetric spacetime with line element

$$ds^2 = - e^{-2\delta(r)}\mu(r) dt^2 + \mu(r)^{-1} dr^2 + r^2 (d\theta^2 + \sin^2\theta d\phi^2)$$

the photon sphere is located at the radius $r_\gamma$ where the following condition, derived from the null geodesic Lagrangian and the field equations, is satisfied: $N(r) \equiv 3\mu(r) - 1 - 8\pi r^2 p(r) = 0,$ with $p(r)$ the radial pressure and $\mu(r) = 1 - 2m(r)/r$, $m(r)$ being the mass within radius $r$. This encapsulates the influence of matter (“hair”) outside the horizon for hairy black holes (Hod, 2017).

More generally, for metrics where $ds^2 = B(r) dt^2 - A(r) dr^2 - D(r) r^2 d\Omega^2$, the photon sphere radius is given by

$$\frac{d}{dr} [r^2 D(r) B(r)]\bigg|_{r_\text{ph}} = 0,$$

or, for $D(r)=1$, $2B(r) + r B'(r) = 0$ (Adler et al., 2022).

In rotating or axisymmetric spacetimes, the photon sphere generalizes to photon regions or surfaces, determined by the Hamilton-Jacobi formalism and the separability of null geodesics; in these cases, the photon ring (as seen in black hole shadows) corresponds to critical impact parameters derived from extremality conditions of the so-called “effective potential” in the equatorial plane (Vertogradov et al., 8 May 2024).

2. Universal Bounds and Their Derivation

Upper Bound for Asymptotically Flat Black Holes

For static, spherically symmetric, asymptotically flat black-hole spacetimes, a rigorous upper bound for any photon sphere radius is

$r_\gamma \leq 3M,$

where $M$ is the ADM mass (Hod, 2017). The proof uses the characteristic photon sphere condition above and invokes the weak energy condition together with trace constraints on matter fields, leading to

$$P(r_\gamma) \leq 0,\quad\text{with}\quad P(r) = r^4 p(r).$$

The algebraic manipulation of the field equations provides

$r_\gamma \leq 3m(r_\gamma) \leq 3M$

for any solution consistent with the stated conditions.

Saturation and Generalization

• The Schwarzschild black hole (no “hair”): $r_\gamma = 3M$, saturating the bound.
• Hairy black holes: All such spherically symmetric, asymptotically flat configurations with “hair” exterior to the horizon also obey $r_\gamma \leq 3M$. The actual $r_\gamma$ may be less than $3M$, depending on the matter field structure.
• For spacetimes violating the energy conditions or with nontrivial mass/pressure profiles (e.g., regular stars), alternative bounds apply (Liu et al., 9 Feb 2024). For regular compact stars with a traceless energy-momentum tensor under the dominant energy condition, the innermost photonsphere obeys $r_\gamma^\mathrm{in} \leq (12/5)M$.

Lower Bound for Hairy Black Holes

If the matter fields are traceless, a lower bound on the photon sphere radius is enforced: $r_\gamma \geq \frac{6}{5} r_H,$ where $r_H$ is the horizon radius. This reflects the nontrivial interplay between the nonlinear Einstein-matter equations and energy conditions, ensuring that the photon sphere cannot approach arbitrarily close to the horizon (Hod, 2023).

Stable Photon Spheres and Their Limits

For static, spherically symmetric, asymptotically flat black holes surrounded by matter with a monotonically decreasing mass-radius ratio, any stable photon sphere (SPS) must exist within

$r_\mathrm{sps} < 6M,$

where $M$ is the ADM mass (Song et al., 27 Aug 2025). The existence condition involves a lower bound on $\rho + p_T > 1/(8\pi r_\mathrm{sps}^2)$, ensuring sufficient matter density and pressure near the candidate SPS.

3. Physical Implications and Observable Consequences

Photon spheres form the organizing center for several key phenomena in black-hole physics:

• Gravitational lensing: The photon sphere radius sets the threshold impact parameter $b_\text{crit}$ for divergent bending of light, controlling the formation and brightness of relativistic images and Einstein rings.
• Black-hole shadows: The shadow radius observed at infinity is typically related to the photon sphere radius by $r_\mathrm{sh} = r_\gamma/\sqrt{f(r_\gamma)}$ or, more generally, $r_\mathrm{sh} = r_\gamma e^{-\nu(r_\gamma)}$ in nonstandard metrics. Deviations or multiple photon spheres manifest as multiple bright rings and distinctive shadow structures (Gan et al., 2021).
• Ringdown and eikonal quasinormal modes: In the eikonal (large $\ell$) regime, the real part of quasinormal mode frequencies is set by the angular velocity of the unstable circular photon orbit: $\omega = \Omega \ell - i(n + 1/2)\lambda$, with $\Omega$ and $\lambda$ determined from the properties of the photon sphere (Ladino et al., 2023).
• Accretion disk and plasma effects: The location and stability of the photon sphere affect the trapping or escape of photons from disk and corona emissions, influencing observable spectra.
• Phase transitions in AdS and deformed black holes: Nonmonotonic behavior of the photon sphere radius may diagnose small–large black hole phase transitions, encoded in oscillatory graphs of $r_\gamma$ versus thermodynamic variables (Han et al., 2018, Xu et al., 2019). Critical exponents in $r_\gamma$ (or closely related quantities) across such transitions may reach universal values (e.g., $1/2$).

4. Geometric, Dynamical, and Methodological Generalizations

Curvature-Based Characterization

• Optical geometry: In the (projected) optical metric, photon spheres are characterized by vanishing geodesic curvature of certain closed curves. The condition $(d/dr)[\bar{g}_{\phi\phi}(r)/g_{tt}(r)] = 0$ at $r = r_\mathrm{ph}$ is equivalent to the standard effective potential extremality (Qiao, 2022).
• Stability: The stability of a photon sphere (as a set of null geodesics) is determined by the sign of the second derivative of the effective potential or by the Gaussian curvature of the optical geometry at the photon sphere. Strictly unstable photon spheres (negative curvature) are required to set the edge of the black hole shadow, while stable photon spheres may trap perturbations and induce spacetime instabilities (Koga et al., 2019).

Beyond Geodesic Approaches: Surfaces and Massive Particle Surfaces

• Photon surface: Generalizes the photon sphere concept to arbitrary backgrounds where a codimension-1 hypersurface is everywhere tangent to its null generators. This covariant notion underpins novel definitions and existence theorems, particularly in dynamical spacetimes (such as the Vaidya metric) (Junior et al., 3 Jan 2024).
• Massive particle surface: Including charged or massive particles leads to a further generalization, with new algebraic conditions dependent on the mass, charge, and the full set of metric functions. In general, these conditions need not coincide with the classic photon sphere even in the massless, neutral limit for spacetimes with unconventional structure (e.g., nontrivial $g_{00} \neq 1/B(r)$) (Junior et al., 3 Jan 2024).

Axisymmetric and Time-Dependent Spacetimes

• For rotating (Kerr-type) or deformed axisymmetric spacetimes, the photon region is defined by separability conditions and analysis of Hamilton–Jacobi equations. The shadow boundary is set by extremizing impact parameters $\xi$, $\eta$ computed analytically or numerically from the deformed metric components (Vertogradov et al., 8 May 2024).
• For dynamical spacetimes such as accreting or evaporating Vaidya black holes, photon spheres evolve in time. In special coordinate systems, analytical expressions for $r_\gamma(v)$ can be obtained, which map to spiraling curves in spacetime (Solanki et al., 2022).

5. Parameter Dependence and Sensitivity to Matter Content

• Charge, rotation, and higher curvature: The photon sphere radius is sensitive to electric/magnetic charge, Gauss–Bonnet couplings, Born–Infeld parameters, and other modifications. For example, in Reissner–Nordström spacetime, $r_\gamma < 3M$ for $Q \neq 0$; in 4D Einstein–Gauss–Bonnet gravity, the presence of $\alpha$ reduces $r_\gamma$ further (Ladino et al., 2023, Waseem et al., 6 Feb 2025).
• Matter distributions/external fields: Extended gravitational decoupling methods allow systematic analysis of how “hair” deforms $r_\gamma$. The sign of the derivative $g'(r_\gamma^{(0)})$ (for a metric deformation function $g(r)$) directly determines whether the photon sphere radius increases or decreases (Vertogradov et al., 5 Apr 2024).
• Nonlinear electrodynamics: Both magnetic charge and NED parameter $\xi$ reduce $r_\gamma$ relative to GR and electromagnetic analogs; as a consequence, the shadow size is also diminished, providing a potential observational diagnostic (Waseem et al., 6 Feb 2025).
• Mass dependence and dynamical mass: For general mass function $M(r)$, analytic expansions reveal that an increasing mass enlarges $r_\gamma$ as long as the weak energy condition is upheld. If the mass function violates this condition, the photon sphere can shrink (Vertogradov et al., 29 Apr 2024).

6. Uniqueness, Multiplicity, and Novel Features

• Uniqueness in Einstein--Maxwell: The photon sphere radius, together with ADM mass and electromagnetic charges, uniquely specifies the entire exterior region of the Reissner–Nordström black hole; no alternative static, asymptotically flat solution exists with the same asymptotic charges and a photon sphere at a different location. Rigorous proofs use positive mass theorems and conformal geometric arguments (Rogatko, 25 Jan 2024).
• Multiplicity and structure: In certain modified or regular spacetimes, multiple photon spheres may exist (both stable and unstable), and even in horizonless or hairy black holes, there can be two or more branches for $r_\gamma$ (with one solution stable, the other unstable). This leads to ring-like structures in the shadow and introduces rich phenomenology, notably when stable photon spheres or multiple extrema in the effective potential are present (Berry et al., 2020, Gan et al., 2021, Song et al., 27 Aug 2025).
• Cosmological constant corrections: For realistic black holes targeted by the Event Horizon Telescope, corrections to $r_\gamma$ from $\Lambda$ are of order $A M^2$, with $A\approx 10^{-52}\,\mathrm{m}^{-2}$, resulting in effects in $r_\gamma$ entirely negligible for current observations ($A M^2 \sim 10^{-26}$ for M87*) (Adler et al., 2022).
• Models with independent photon, ADM, and horizon radii: Certain modified gravities, such as ghost-free $f(\mathcal{G})$ gravity, allow black hole solutions where ADM mass, horizon radius, and photon sphere radius are independent parameters—leading to scenarios where the shadow may be unexpectedly small for a given horizon (Nojiri et al., 15 Oct 2024).

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In summary, the photon sphere radius is a geometric and physical quantity of paramount importance both in the theory of general relativity and in its extensions. It serves as an organizing center for null geodesics, provides a calculable link to astrophysical observables such as the black hole shadow, and encodes the cumulative effects of mass, charge, rotation, matter fields, and modifications to gravitational dynamics. Rigorous analytical bounds ($r_\gamma \leq 3M$ or $r_{\mathrm{sps}} < 6M$) and uniqueness theorems underpin its universality in a wide class of spacetimes, while generalizations and cases with multiple (stable/unstable) photon spheres reveal an intricate landscape rich with implications for the phenomenology of compact objects and testing fundamental physics.