The photograph that broke the internet in 2019 showed something humanity had never seen before: the actual shadow of a supermassive black hole. The orange ring surrounding a dark circle at the center of galaxy M87 became an instant icon, proof that black holes exist exactly where Einstein’s equations predicted they would be. The image matched theoretical predictions remarkably well, confirming decades of calculations about how light behaves in the most extreme gravitational environments.
But that successful observation opened a new door for physicists. With actual measurements of a black hole shadow in hand, researchers can now test alternative theories of gravity and electromagnetism. A team of scientists has done exactly that, exploring what black hole shadows would look like if two modifications to standard physics were real: F(R) gravity, which tweaks Einstein’s equations in regions of extreme curvature, and Euler-Heisenberg theory, a quantum mechanical effect that changes how light behaves in powerful electromagnetic fields.
Their work doesn’t claim Einstein got it wrong. Instead, it asks a more subtle question: if nature operates according to these modified theories rather than standard physics, what would we see? And crucially, do the observations from the Event Horizon Telescope allow these theories to be true, or do they rule them out?
The implications stretch far beyond abstract mathematics. If these modifications accurately describe reality, black holes don’t just warp space and time in the ways Einstein predicted. They create fundamentally different optical structures, bending light into patterns that standard physics cannot produce. The shadows they cast reveal information about the fabric of spacetime itself, offering a window into physics that operates beyond the limits of general relativity.
Einstein’s general relativity has passed every test thrown at it for more than a century. The theory describes gravity not as a force, but as the curvature of spacetime caused by mass and energy. Massive objects bend the fabric of space around them, and other objects follow the curves in that fabric. The equations work brilliantly for everything from GPS satellites to the orbit of Mercury to gravitational waves rippling through the cosmos.
Yet physicists have long suspected that general relativity represents an approximation rather than the final word. The theory breaks down at the center of black holes, where densities become infinite. It cannot be reconciled with quantum mechanics, the other pillar of modern physics. And explaining the behavior of galaxies and the expansion of the universe requires introducing dark matter and dark energy, substances we’ve never directly detected.
F(R) gravity offers one possible extension of Einstein’s work. Instead of Einstein’s equations, which are linear in the Ricci scalar, a measure of spacetime curvature, F(R) gravity allows for nonlinear functions of curvature. Think of it like the difference between a simple relationship where doubling the input doubles the output, versus a more complex relationship where doubling the input might triple or quadruple the output depending on other factors.
The Euler-Heisenberg modification addresses a different issue. In standard electromagnetism, light waves don’t interact with each other or with static electric fields. A laser beam passes through another laser beam without any effect. But quantum mechanics predicts that in sufficiently strong electromagnetic fields, the vacuum of space itself becomes a nonlinear medium. Virtual particle-antiparticle pairs pop into existence for fleeting moments, and their presence changes how light propagates. The effect only becomes significant in fields approaching the critical strength where electron-positron pairs can be ripped directly from the vacuum, roughly 10^18 volts per meter.
Around an electrically charged black hole, both effects come into play. The intense gravitational field activates the F(R) modifications, while the powerful electric field surrounding the black hole triggers Euler-Heisenberg effects. Light traveling through this environment follows trajectories that differ from the predictions of standard physics.
The researchers traced how photons move through the warped spacetime around these modified black holes. They found that the combination of F(R) gravity and Euler-Heisenberg electrodynamics creates an “effective geometry” that governs light propagation. This effective geometry differs from the actual spacetime geometry in ways that depend on the strength of the electric field and the specific form of the gravity modification.
Light rays approaching a black hole fall into three categories based on how close they come to the event horizon. Direct rays either plunge straight into the black hole or pass by without orbiting. Lensed rays circle the black hole between half an orbit and one and a quarter orbits before either falling in or escaping to infinity. Photon ring rays complete more than one and a quarter orbits, creating the bright ring visible in black hole images.
The boundary between these regions defines the photon sphere, an unstable orbital radius where light can theoretically orbit forever. Any light at exactly this radius circles endlessly, but the slightest perturbation sends it spiraling inward or escaping outward. The photon sphere sits outside the event horizon, the point of no return where even light cannot escape.
In standard general relativity with no electric charge, the photon sphere for a non-rotating black hole sits at exactly 1.5 times the event horizon radius. The shadow, the dark region visible to a distant observer, extends slightly larger than the photon sphere. But in F(R) gravity coupled with Euler-Heisenberg electrodynamics, both the photon sphere and shadow sizes change in ways that depend on three key parameters: the electric charge, the F(R) modification strength, and the background curvature of spacetime.
The analysis revealed several patterns. Increasing the electric charge expands the range of impact parameters that produce lensed and photon ring trajectories. A ray’s impact parameter measures how far it would pass from the black hole if gravity didn’t deflect it. Stronger F(R) modifications also broaden these regions, while higher background curvature tends to suppress them.
For the shadow to be physically meaningful, three conditions must be satisfied simultaneously. The photon sphere must lie outside the event horizon. The shadow must extend beyond the photon sphere. And the effective geometry that governs light propagation must maintain the correct mathematical signature throughout the region where photons travel.
The researchers mapped out which combinations of parameters satisfy all three conditions. They found that increasing either the electric charge or the F(R) modification parameter expands the range of allowed values. Stronger electromagnetic effects and modifications to gravity make it easier to construct physically viable black hole shadows.
The real test came when they compared their theoretical predictions to observations of M87*, the supermassive black hole at the center of galaxy M87 located 55 million light-years from Earth. The Event Horizon Telescope measured the angular diameter of M87*’s shadow at 42 microarcseconds with a 3-microarcsecond uncertainty. Combined with the black hole’s estimated mass of 6.5 billion solar masses and its distance from Earth, this measurement constrains which versions of the modified theory remain possible.
The comparison yielded important results. For black holes in de Sitter spacetime, which has positive curvature like an expanding universe, the modified physics remains consistent with observations across a wide range of parameters. But for black holes in anti-de Sitter spacetime, which has negative curvature, the theory only matches observations when either the electric charge takes substantial values or the F(R) parameter falls below negative one.
This observational constraint carries theoretical weight. It demonstrates that if F(R) gravity and Euler-Heisenberg electrodynamics correctly describe nature, and if M87* exists in an anti-de Sitter background, then certain parameter combinations can be ruled out. The actual measurements place boundaries on what’s possible within this theoretical framework.
The shadow size also connects to black hole thermodynamics through Hawking radiation. Quantum effects near the event horizon cause black holes to emit thermal radiation, slowly evaporating over astronomical timescales. The emission rate depends on the black hole’s temperature and the shadow’s cross-sectional area, which determines how much radiation can escape.
The calculations showed that increasing the electric charge raises the peak energy emission rate, causing faster evaporation and shorter lifetimes. Stronger Euler-Heisenberg effects suppress emission, extending lifetimes. Larger F(R) modifications also reduce emission rates compared to standard general relativity. Background curvature plays a role too: higher curvature in de Sitter space lengthens lifetimes, while higher curvature magnitude in anti-de Sitter space accelerates evaporation.
These results matter because they transform black hole shadows from confirmation of known physics into tools for exploring possible extensions. The Event Horizon Telescope’s images don’t just show us that black holes exist. They provide quantitative measurements that can distinguish between standard general relativity and modified theories of gravity and electromagnetism in extreme conditions.
Future observations with improved resolution and sensitivity will tighten these constraints further. Additional black hole images, particularly of Sagittarius A* at the center of our own galaxy, will provide independent tests. Measurements of black hole masses, spins, and environmental conditions will reduce uncertainties in the theoretical models.
The work highlights how modifications to physics in one domain can produce observable consequences in another. F(R) gravity was originally proposed to explain cosmic acceleration without requiring dark energy. Euler-Heisenberg theory emerged from quantum electrodynamics as a prediction about how light behaves in extreme electromagnetic fields. Yet their combined effects show up in the shadows of black holes millions of light-years away, detectable by a planet-sized telescope array.
Black holes occupy a unique position in physics as laboratories where gravity becomes strong enough that any deviations from Einstein’s equations might become visible, yet they’re large enough that we can observe them with current technology. The shadows they cast encode information about spacetime geometry, electromagnetic fields, and quantum vacuum effects all at once. As our observational capabilities improve, these shadows will continue revealing whether Einstein’s equations represent the complete description of gravity, or whether nature requires additional complexity that only becomes apparent in the most extreme environments in the universe.
Source:
Recent research by Jafarzade, Yasmin, and Jamil explores how F(R) gravity combined with Euler-Heisenberg electrodynamics affects black hole shadows, with full findings published on arXiv: https://arxiv.org/abs/2601.05040
