Photon Sphere and Light Rings
Light can orbit black holes on unstable circular trajectories called the photon sphere. These orbits determine the black hole shadow seen by distant observers and play a crucial role in gravitational lensing. We derive the photon sphere radius and analyze light bending.
1. Null Geodesics
Photons (massless particles) follow null geodesics with ds² = 0. The normalization condition u^μ u_μ = -c² for massive particles is replaced by:
$g_{\mu\nu} \frac{dx^\mu}{d\lambda} \frac{dx^\nu}{d\lambda} = 0$
where λ is an affine parameter along the photon path. We cannot use proper time τ (undefined for photons), so we parametrize by λ instead.
Conserved quantities still exist from Killing vectors:
$\mathcal{E} = \left(1 - \frac{r_s}{r}\right) c^2 \frac{dt}{d\lambda}, \quad \mathcal{L} = r^2 \frac{d\phi}{d\lambda}$
Here E and L are energy and angular momentum per unit "affine parameter momentum" (not per unit mass, since photons are massless).
2. Effective Potential for Photons
From the null condition ds² = 0 in the equatorial plane (θ = π/2):
$0 = -\left(1 - \frac{r_s}{r}\right) c^2 \left(\frac{dt}{d\lambda}\right)^2 + \left(1 - \frac{r_s}{r}\right)^{-1} \left(\frac{dr}{d\lambda}\right)^2 + r^2 \left(\frac{d\phi}{d\lambda}\right)^2$
Substituting the conserved quantities and rearranging:
$\left(\frac{dr}{d\lambda}\right)^2 = \mathcal{E}^2 - \frac{\mathcal{L}^2}{r^2} \left(1 - \frac{r_s}{r}\right)$
Define the impact parameter b = L/E (dimension of length), which determines the closest approach in flat spacetime. Then:
$\left(\frac{dr}{d\lambda}\right)^2 = \mathcal{E}^2 \left[1 - \frac{b^2}{r^2} \left(1 - \frac{r_s}{r}\right)\right]$
The effective potential for photons is:
$V_{\text{eff}}^{\text{photon}}(r) = \frac{1}{b^2} \left(1 - \frac{r_s}{r}\right) r^2$
This has a maximum, unlike the massive particle case where there's typically a minimum.
3. Photon Sphere Radius
Circular photon orbits occur at extrema of V_eff. Taking the derivative:
$\frac{dV_{\text{eff}}}{dr} = \frac{1}{b^2} \left[2r \left(1 - \frac{r_s}{r}\right) + r^2 \cdot \frac{r_s}{r^2}\right] = 0$
Simplifying:
$2r - 2r_s + r_s = 0 \quad \Rightarrow \quad r = \frac{3r_s}{2}$
$r_{\text{photon}} = \frac{3GM}{c^2} = \frac{3r_s}{2}$
Photons can orbit at exactly 1.5 times the Schwarzschild radius.
Stability Analysis
To check stability, compute the second derivative:
$\frac{d^2 V_{\text{eff}}}{dr^2}\bigg|_{r=3r_s/2} = -\frac{2}{27 r_s^2 b^2} < 0$
The photon sphere is unstable! This is a maximum of V_eff, not a minimum. Any slight perturbation sends the photon either spiraling into the black hole or escaping to infinity.
The critical impact parameter for a photon to orbit at r = 3r_s/2 is:
$b_{\text{crit}} = \frac{3\sqrt{3}}{2} r_s = 3\sqrt{3} \frac{GM}{c^2}$
4. Light Deflection and Bending Angle
For photons passing at large impact parameter b ≫ r_s, we can calculate the bending angle. The orbit equation is:
$\frac{d\phi}{dr} = \frac{\mathcal{L}/r^2}{\sqrt{\mathcal{E}^2 - \mathcal{L}^2(1-r_s/r)/r^2}}$
For weak fields and b ≫ r_s, we can expand to first order in r_s/b. The total deflection angle is:
$\Delta\phi = \int_{-\infty}^{\infty} \frac{d\phi}{dr} dr - \pi$
The result (Einstein 1915, first confirmed by Eddington 1919) is:
$\Delta\phi = \frac{4GM}{c^2 b} = \frac{2r_s}{b}$
Example: Light Grazing the Sun
For the Sun: M_☉ ≈ 2×10³⁰ kg, R_☉ ≈ 7×10⁸ m, r_s ≈ 3 km. Light grazing the surface (b ≈ R_☉):
$\Delta\phi = \frac{2 \times 3000 \text{ m}}{7 \times 10^8 \text{ m}} \approx 8.5 \times 10^{-6} \text{ rad} = 1.75 \text{ arcsec}$
This is twice the Newtonian prediction and was famously confirmed during the 1919 solar eclipse.
5. Black Hole Shadow
The photon sphere determines the apparent size of the black hole shadow seen by distant observers. Photons with impact parameter b < b_crit are captured; those with b > b_crit escape.
For an observer at distance D ≫ r_s, the angular radius of the shadow is:
$\theta_{\text{shadow}} = \frac{b_{\text{crit}}}{D} = \frac{3\sqrt{3} r_s}{2D} = \frac{3\sqrt{3} GM}{c^2 D}$
Event Horizon Telescope (EHT) Observations
In 2019, the EHT imaged the supermassive black hole M87* at the center of the galaxy M87. The shadow radius matches the predicted value within experimental uncertainty.
- M87* mass: M ≈ 6.5×10⁹ M_☉
- Distance: D ≈ 16.8 Mpc ≈ 5×10²³ m
- Shadow angular radius: θ ≈ 20 microarcseconds
- Observed ring diameter: 42 ± 3 microarcseconds (consistent!)
The bright ring seen by EHT is light from the photon sphere, including photons that orbit multiple times before escaping.
6. Gravitational Lensing
Black holes act as gravitational lenses, bending light from background sources. For strong lensing (b ≈ b_crit), multiple images and Einstein rings can form.
Einstein Rings
When the source, lens (black hole), and observer are perfectly aligned, light is bent into a ring. The Einstein ring radius is:
$\theta_E = \sqrt{\frac{4GM}{c^2} \frac{D_{LS}}{D_L D_S}}$
where D_L is distance to lens, D_S is distance to source, and D_LS = D_S - D_L.
Magnification
Gravitational lensing also magnifies background sources. Near the photon sphere, magnification diverges, creating extremely bright but short-lived transients if a star passes near a black hole.
Lensing by black holes has been observed in X-ray binaries and is used to detect intermediate-mass black holes.
Key Takeaways
- Photons can orbit black holes at the photon sphere: r = 3GM/c² = 1.5 r_s
- Photon orbits are unstable (maximum of effective potential)
- Light bending angle: Δφ = 4GM/(c²b) = 2r_s/b for weak fields
- Critical impact parameter: b_crit = 3√3 GM/c²
- Black hole shadow radius determined by photon sphere
- EHT observations of M87* confirm predicted shadow size
- Strong gravitational lensing creates multiple images and Einstein rings