Josephson interferometry in an Oppenheimer--Snyder-like scale-dependent black-hole spacetime

In the quiet corners of modern physics, where the rules of the very large meet the rules of the very small, scientists are constantly testing how gravity and electricity interact. For decades, we have known that gravity bends light and slows down time, a phenomenon so well-established that it is essential for the GPS in our phones to work correctly. We also know that superconductors, materials that conduct electricity with zero resistance, can carry a current that depends on a hidden quantum rhythm called a phase. When two superconductors are separated by a tiny barrier, this rhythm creates a special kind of current that is incredibly sensitive to its surroundings. The question researchers have long asked is: what happens to this delicate quantum rhythm when it is placed in the extreme gravity of a black hole? Does the crushing pull of the star change how the electricity flows, or how the time is measured?

A team of physicists has now mapped out exactly how this quantum rhythm behaves in a specific, theoretical model of a collapsing star. They did not build a physical black hole in a lab; instead, they used advanced mathematics to simulate a black hole that changes its properties as you get closer to its center, a concept known as scale-dependent gravity. In their model, the star collapses just as it would in standard physics, but the gravity itself evolves, creating a complex landscape of horizons and a central core that is smooth rather than infinitely sharp. The researchers then placed a theoretical superconducting junction—a tiny bridge where this quantum current flows—into this environment. They tracked how the current, the voltage, and the timing of the signal would appear to an observer far away in space, compared to an observer falling with the collapsing star.

The study reveals that gravity acts like a filter on the information coming from the superconductor. When the junction is held still near the black hole, the frequency of the quantum rhythm and the amount of current it carries are reduced by a factor related to how much time is slowed down at that location. However, the power, or the energy carried by the signal, is reduced even more, by the square of that factor. This means that as the junction gets closer to the event horizon, the signal reaching a distant observer becomes fainter and slower in a very specific, predictable way. The researchers found that if the junction is instead falling with the collapsing star, the story changes. The signal received by a distant observer is not just slowed by the gravity of the black hole; it is also altered by the motion of the falling junction itself. The combination of the star's gravity and the junction's speed creates a unique signature that distinguishes a falling clock from a stationary one.

One of the most striking findings concerns the time it takes for a signal to travel from near the black hole to the outside world. As the junction gets closer to the edge of the black hole, the time delay for a signal to escape grows logarithmically, meaning it increases very rapidly but in a smooth curve. The researchers calculated the exact mathematical shape of this delay and discovered something surprising about the very edge of the black hole. If the black hole is in a state where its gravity is at its maximum possible limit, the delay does not just grow; it changes its behavior entirely. Instead of following the smooth curve of a normal black hole, the delay follows a different, sharper rule. This crossover from a smooth curve to a sharp rule happens only if the observer stays at a fixed distance from the horizon; if they fall in, the rules change again. This distinction is crucial because it shows that the way we measure time and distance near a black hole depends entirely on how we are moving.

The team also explored how these effects would look in a device known as a superconducting quantum interference device, or SQUID, which is used to detect tiny magnetic fields. They found that if the two arms of the device are at different distances from the black hole, the gravity creates an imbalance in the signal. This imbalance changes the height of the peaks in the signal pattern but does not shift the position of the peaks themselves, provided the device is small enough. However, if the device is driven by microwaves, the time it takes for the waves to travel through the different arms creates a shift in the pattern. This shift is not caused by the gravity itself, but by the difference in travel time, and it only appears if the device is allowed to adjust its internal magnetic flux dynamically. This distinction helps scientists understand how to separate the effects of gravity from the effects of motion and timing in future experiments.

Perhaps the most profound result involves the heat and noise associated with the black hole. In a state of thermal equilibrium, the black hole radiates a specific temperature that depends on its gravity. The researchers found that the noise generated by the superconducting junction, when measured by a distant observer, is completely independent of how close the junction is to the black hole. The slowing of time near the horizon and the heating effect of the black hole cancel each other out perfectly. This leads to a universal relationship between the time it takes for a signal to escape and the amount of noise in the system. This relationship is so precise that it does not depend on any specific details of the black hole or the junction, offering a potential way to test the fundamental laws of physics without needing to know the exact mass or size of the black hole.

The study also places a hard limit on how close a real superconductor could ever get to a black hole. Because the black hole radiates heat, the temperature near the horizon becomes incredibly high. The researchers calculated that for any known superconducting material, there is a minimum distance from the horizon where the material would simply melt and lose its superconducting properties. This exclusion zone is determined only by the material's own critical temperature and fundamental constants of nature, not by the size of the black hole. For a typical superconductor, this distance is about a few hundred micrometers from the horizon. This means that while the theoretical effects are fascinating, placing a real device right at the edge of a black hole is physically impossible because the heat would destroy it long before the quantum effects could be measured.

Finally, the researchers connected their findings to the "shadow" of a black hole, which is the dark silhouette seen against the background of light. They showed that the same factor that changes the superconducting current also changes the size of the black hole's shadow. By measuring both the shadow and the superconducting signal, scientists could potentially check for consistency in the laws of gravity without needing to know the exact value of the changing gravity parameter. This creates a new way to test theories of gravity that go beyond Einstein's original equations. While these scenarios are currently theoretical and apply to models of collapsing stars rather than the black holes we see in the sky, the work provides a clear, mathematical blueprint for how quantum matter behaves in the most extreme gravitational environments imaginable. It suggests that if we could ever build a detector sensitive enough to survive the heat, the black hole itself would act as a giant, natural laboratory for testing the deepest connections between time, gravity, and quantum mechanics.