Photon Surface in Spherical Symmetry

• Photon surface is an immersed, nowhere-spacelike hypersurface in spherical symmetric spacetime where all null geodesics remain tangent to the surface.
• In time-dependent scenarios, the photon surface condition transforms into a non-autonomous ODE system that captures its dynamic geometric evolution.
• The correspondence between photon surfaces and sonic points underpins key aspects of black hole shadow formation and gravitational lensing, highlighting observational relevance.

A photon surface in spherically symmetric spacetime is an immersed, nowhere-spacelike hypersurface such that all null geodesics initially tangent to it remain within the surface. While the canonical example is the photon sphere of Schwarzschild spacetime ($r=3M$), the generalization to time-dependent and interior regions requires more advanced geometric and dynamical techniques. This concept is vital in black hole shadow formation, gravitational lensing, and the causal structure of dynamical collapse.

1. Fundamental Definitions and Geometric Characterization

A photon surface $S$ is characterized by the property that at each point $p \in S$ and for every null vector $k \in T_pS$, there exists a null geodesic $\gamma$ in the full spacetime with $\dot{\gamma}(0)=k$ and $\gamma(\lambda)\in S$ for all $\lambda$. The geometric characterization uses the second fundamental form $\chi_{ab}$: $S$ is a photon surface if and only if it is totally umbilic, meaning the trace-free part $\sigma_{ab} = 0$ at all points. For spherically, planar, and hyperbolically symmetric metrics,

$ds^2 = -f(r)dt^2 + g(r)dr^2 + r^2 d\Omega^2_{D-2},$

the photon surface condition for $r=r_{\text{PS}}$ becomes

$\frac{d}{dr}(f\,r^{-2}) = 0,$

locating the radius where the expansion and acceleration of null geodesics tangent to the surface are balanced (Koga, 2019).

2. Dynamical Extension: Photon Surface as a Non-Autonomous ODE System

When spherical symmetry is imposed in time-dependent spacetimes, the photon surface condition evolves into a second-order ODE or, equivalently, a non-autonomous dynamical system. For a metric $$g = -e^{2\nu}dt^2 + e^{2\lambda}dx^2 + r^2 d\Omega^2$$ parameterized as $(t, x(t), \theta, \phi)$, the photon surface is captured by the evolution equations: $$\begin{aligned} \dot{x}(t) &= e^{\nu-\lambda} y(t), \ \dot{y}(t) &= \frac{(1-y(t)^2) \left[e^{\nu-\lambda}(r_x - r\nu_x) + y(t)(r_t - r\lambda_t)\right]}{r}. \end{aligned}$$ This system directly reflects the time-dependence of the geometry. The special case $y(t_0)=\pm 1$ corresponds to radial null geodesics, and the evolution remains null ($|y|=1$) if so initially (Giambò et al., 1 Sep 2025).

3. Quasi-Local and Invariant Formulations

The quasi-local definition introduced by (Cao et al., 2019) dispenses with the need for an umbilical hypersurface and relies on the intrinsic geometry of codimension-2 surfaces. For a metric with

$$g_{ab} = h_{AB}(y)dy^A dy^B + r^2(y)\gamma_{ij}(z)dz^i dz^j,$$

the photon surface condition on a codimension-2 leaf becomes a set of differential constraints: $$\begin{aligned} & D_A\left( \frac{v^A}{r} \right) = 0, \ & u^B D_B D_A\left( \frac{v^A}{r} \right) = 0, \ & v^B D_B D_A\left( \frac{v^A}{r} \right) \neq 0, \end{aligned}$$ where $u$ and $v$ are locally orthogonal timelike/spacelike normals. In static spacetimes, this reduces to algebraic conditions, while in dynamical collapse (e.g., LTB models) it is a second-order ODE for $x(t)$ with physically motivated boundary conditions (e.g., matching total energy at the boundary).

Complementarily, the invariant spin frame approach (Dey et al., 22 Jan 2024) expresses the photon surface condition using Cartan scalars and spin coefficients. In static spherical symmetry, the defining condition simplifies to $\rho + 2\epsilon = 0$ (spin coefficients) or equivalently

$$\Phi_{00} - 8\epsilon^2 - \Psi_2 - \frac{R}{12} = 0,$$

where $\Psi_2$ is the Weyl curvature scalar, $\Phi_{00}$ encodes Ricci tensor projections, and $R$ is the Ricci scalar.

4. Sonic Point/Photon Surface Correspondence

There exists a rigorous correspondence between the location of sonic points in radiation fluid accretion and the existence of photon surfaces (Koga, 2019, Tsuchiya et al., 2020). In spherically symmetric accretion flows of an ideal photon gas (with $v_s^2 = 1/(D-1)$), the sonic point occurs precisely at the radius satisfying the photon surface condition $(fr^{-2})' = 0$. Instability of the photon surface—signaled by $(fr^{-2})'' < 0$—corresponds to the saddle nature of the sonic point, ensuring physical traversability for transonic flow (Koga, 2019). This correspondence is robust under arbitrary dimension and symmetry, provided the flow and spacetime geometry are appropriately aligned.

5. Physical Implications and Observational Relevance

Photon surfaces demarcate regions where null geodesics undergo bounded orbits, forming the theoretical basis for the black hole shadow observed by instruments such as the EHT. Dynamically, photon surfaces in gravitational collapse (LTB or Vaidya models) precede the formation of event horizons, with the time lag $\Delta t$ between the appearance of the photon surface and event horizon determined essentially by the total mass, independent of compactness (Cao et al., 2019). In marginally bound LTB collapse, numerical and analytic solutions confirm that photon surfaces form even in the presence of naked singularities. In such cases, null geodesics can emanate from the singularity and remain inside the photon surface, highlighting the causal permeability and potential for observational consequences in ultra-high curvature regions (Giambò et al., 1 Sep 2025, Dey et al., 22 Jan 2024).

6. Boundary and Extension Conditions in Collapse Models

In collapse scenarios where the interior is described by an LTB metric and the exterior by Schwarzschild, the photon surface at the boundary ($r=3M$) is matched via initial conditions for the dynamical ODE. The extension of the photon surface inside the matter is uniquely determined: only the null case ($y_0 = 1$ in the dynamical system) yields a globally consistent solution, as other choices (timelike surfaces with $|y_0|<1$) violate boundary connectivity or causal structure (Giambò et al., 1 Sep 2025). In naked singularity formation, the photon surface can reach the singular center, failing to shield it, whereas in black hole formation, the surface terminates at a regular center.

7. Comparative Approaches and Theoretical Generalizations

Photon surfaces can be characterized quasi-locally for both null and timelike orbits and extended to more general particles (massive, spinning) via decomposition conditions on their tangent and momentum vectors (Song et al., 2022). The spin frame invariant method covers both static and dynamical cases and provides local, coordinate-independent criteria for photon surface existence.

These foundational and advanced characterizations collectively enable precise mathematical formulation, facilitate numerical construction in collapse scenarios, and establish the photon surface as a central object in gravitational lensing and the observational phenomenology of strong gravity. The causal interplay between photon surface extension and singularity coverage directly informs ongoing discussions on cosmic censorship and the nature of black hole formation.