The Paradox Born at Princeton, Refined in Jerusalem

In 1972, a 25 year old doctoral student at Princeton planted a time bomb in the foundations of modern physics. Jacob Bekenstein, working under the legendary John Wheeler, proposed that black holes possess entropy, a measurable quantity proportional to the area of their event horizons. It was a radical claim. It was also correct, and its consequences have tormented theoretical physics ever since.

Two years later, Bekenstein immigrated to Israel. He would spend the remaining four decades of his life there, first at Ben-Gurion University in the Negev, then at the Hebrew University of Jerusalem, building and defending the framework that made him one of the most consequential physicists of his generation.

Bekenstein’s insight meant that black holes are not featureless voids. They contain information, encoded on their surfaces in ways that obey the laws of thermodynamics. When Stephen Hawking built on this work and showed that black holes radiate energy and eventually evaporate, the problem became unavoidable. If a black hole disappears entirely, the information it contains appears to vanish with it. Quantum mechanics forbids this. Information, like energy, can be scrambled and redistributed but never destroyed. The seed of a paradox was planted at Princeton. Its full implications would be worked out, challenged, and defended from Jerusalem.

For over fifty years, physicists have proposed solutions. None has been confirmed. Now a team led by Richard Pinčák at the Slovak Academy of Sciences has published a new attempt in General Relativity and Gravitation that has attracted widespread attention. They argue that spacetime possesses seven dimensions rather than four, and that it exhibits not only the curvature described by Einstein’s general relativity but also a twisting property called torsion. At extreme densities, this torsion generates a repulsive force that halts the final stage of evaporation. The black hole does not vanish. It leaves behind a stable remnant that preserves the quantum information of everything it consumed.

The proposal has an additional attraction. When the mathematics is reduced from seven dimensions to the four we inhabit, it naturally produces a quantity that matches the energy scale at which the Higgs field operates, the mechanism by which fundamental particles acquire mass. If genuine, this would link the fate of black holes to the origin of mass itself.

It is an elegant idea. It is also one that Bekenstein himself examined and rejected.

In a paper published in Physical Review D in 1994, written at the Racah Institute on the Hebrew University’s Givat Ram campus, Bekenstein directly confronted the remnant hypothesis. His universal entropy bound, derived from the very thermodynamic principles he had established two decades earlier, placed severe restrictions on how much information a remnant of negligible mass could store. His conclusion was pointed: remnants could not serve to resolve the information paradox. The information capacity was simply too small.

Pinčák’s team attempts to circumvent this objection through geometry, encoding information in the vibrational modes of a higher dimensional structure. They claim storage capacities vast enough to accommodate everything a black hole might swallow. Whether this genuinely overcomes Bekenstein’s bound or merely relocates the problem into a framework where the bound’s assumptions no longer apply is a question that will require sustained scrutiny from the theoretical physics community.

There is a deeper issue. The remnant the model predicts has a mass of roughly 9 × 10⁻⁴¹ kilograms, ten billion times lighter than an electron. No detector can observe it. No collider can produce it. The seven extra dimensions are compactified at scales permanently beyond experimental access. The theory makes no predictions that any foreseeable technology could test. In physics, a model that cannot be contradicted by observation may be interesting, but it cannot yet be called true.

Israel’s place in this story is not incidental. The Bekenstein bound, the Bekenstein-Hawking entropy formula, and the generalised second law of black hole thermodynamics all bear his name. Though the original idea was conceived at Princeton, the programme that gave it rigour, defended it against critics, and extended it into information theory was conducted over forty years in Israel. Bekenstein received the Wolf Prize in 2012 and the American Physical Society’s Einstein Prize in 2015, shortly before his unexpected death in Helsinki at the age of 68. Any claimed resolution of the information paradox must ultimately be measured against the framework he built.

The Pinčák study is serious theoretical work and deserves careful scrutiny rather than dismissal. But the media suggestion that the paradox has been “finally solved” confuses a promising mathematical model with established physical truth. What has been demonstrated is that within one particular geometric framework, a mechanism exists that could preserve information without violating quantum mechanics. That is a contribution. It is not a conclusion.

There is, however, something worth reflecting on beyond the physics. History tends to celebrate those who claim to have found answers. But the deeper contribution often belongs to the person who saw the problem clearly enough to make it unavoidable. Bekenstein did not resolve the information paradox. He made it impossible to ignore. Every proposed solution since, including this one, is an attempt to reckon with the question he first posed as a doctoral student and spent a lifetime in Israel refining. In science, as in much else, asking the right question is the rarer and more enduring achievement.