Gravitational lensing due to a black hole is a direct consequence of the curvature of spacetime predicted by Einstein’s general theory of relativity, in which gravity is not treated as a force acting at a distance but as a geometric property of spacetime itself. In this framework, mass–energy determines the curvature of spacetime, and all physical trajectories, including those of massless particles such as photons, follow geodesics—paths of extremal spacetime interval—within that curved geometry. A black hole, being an object of extremely high mass confined within a very small spatial region, produces an exceptionally strong curvature, causing light rays passing nearby to deviate significantly from straight-line paths as defined in flat Minkowski spacetime. Gravitational lensing by a black hole thus refers to the deflection, distortion, magnification, and temporal modulation of light from background sources due to the intense spacetime curvature generated by the black hole.
In the weak-field regime, gravitational lensing can be approximated using perturbations of flat spacetime, where the deflection angle of a light ray passing a mass (M) at an impact parameter (b) is given by the classical result (\hat{\alpha} \approx \frac{4GM}{c^2 b}). However, black holes frequently operate outside this weak-field approximation, especially when light passes within a few Schwarzschild radii of the event horizon. The Schwarzschild radius (r_s = \frac{2GM}{c^2}) defines the scale at which relativistic effects become dominant, and for impact parameters comparable to a few times (r_s), higher-order and fully relativistic corrections become essential. In such cases, the deflection angle grows rapidly and can even diverge as photons approach unstable circular orbits known as photon spheres, located at (r = \frac{3GM}{c^2}) for a non-rotating (Schwarzschild) black hole.
The lensing phenomenon arises from the solution of null geodesics in curved spacetime. For a Schwarzschild black hole, the spacetime metric is given by (ds^2 = -\left(1-\frac{2GM}{rc^2}\right)c^2 dt^2 + \left(1-\frac{2GM}{rc^2}\right)^{-1} dr^2 + r^2(d\theta^2 + \sin^2\theta, d\phi^2)). Photons follow paths for which (ds^2 = 0), and solving the geodesic equations reveals that light trajectories bend progressively as they approach the black hole. When the alignment between the observer, the black hole, and a distant source is favorable, these curved trajectories can result in multiple images of the same astrophysical object, each corresponding to a different null geodesic connecting the source to the observer.
One of the most striking manifestations of black hole gravitational lensing is the formation of Einstein rings. In the idealized case of perfect alignment, the lensed images merge into a ring-shaped structure whose angular radius depends on the mass of the black hole and the distances between the observer, lens, and source. In the strong-field regime near black holes, however, the situation becomes more complex, as light may execute multiple loops around the black hole before escaping. These trajectories give rise to an infinite sequence of highly demagnified and closely spaced images known as relativistic images, which accumulate near the angular radius associated with the photon sphere. Although these images are typically too faint to be observed directly with current instrumentation, they represent a unique signature of strong gravitational fields and provide a theoretical laboratory for testing general relativity beyond the weak-field limit.
Black hole lensing is also intrinsically connected to the concept of the black hole shadow, which is not itself an image of the event horizon but rather a projection of the photon capture region. Light rays that fall within a critical impact parameter are captured by the black hole, while those just outside this threshold are strongly bent and contribute to a bright, narrow ring surrounding a dark central region. The angular size and shape of this shadow depend sensitively on the black hole’s mass, distance, and spin. For rotating (Kerr) black holes, frame dragging introduces asymmetries into the lensing pattern, causing the shadow and surrounding lensed emission to become distorted and offset, thereby encoding information about the black hole’s angular momentum.
From an observational standpoint, gravitational lensing by black holes plays a crucial role in astrophysics and cosmology. Isolated stellar-mass black holes, which emit little or no electromagnetic radiation, can be detected through microlensing events in which the apparent brightness of a background star temporarily increases due to the passage of the black hole along the line of sight. In galactic centers, supermassive black holes act as strong lenses, influencing the appearance of stars and accretion flows in their immediate vicinity. High-resolution observations of such systems allow for precise measurements of black hole masses and offer stringent tests of relativistic gravity under extreme conditions.
Beyond its practical applications, gravitational lensing due to black holes has deep theoretical significance. It provides a concrete, observable consequence of spacetime curvature and reinforces the geometric interpretation of gravity. In the strong-lensing regime, where photons probe regions near the event horizon, lensing phenomena become sensitive to potential deviations from general relativity, making them valuable probes for alternative theories of gravity and quantum-gravity-inspired corrections. Thus, black hole gravitational lensing is not merely a visual curiosity but a fundamental physical process that bridges theory and observation, illuminating the nature of spacetime, gravity, and some of the most extreme objects in the universe.