Solving the black hole information paradox


The  black hole information paradox laid out by Stephen Hawking says information cannot be destroyed or disappear, but black holes breach the time symmetry of physics.


Dr Szymon Łukaszyk, an independent researcher in Poland, offers a solution to the black hole information paradox. Instead of suggesting novel physical theories, he pursues innovative connotations of existing physics, specifically the theory of relativity.


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Hello and welcome to Research Pod! Thank you for listening and joining us today.



In this episode, we will explore the work of Dr Szymon Łukaszyk, an independent researcher in Poland. Łukaszyk has investigated the black hole information paradox.



Almost half a century has passed since physicists were first confronted by Stephen Hawking’s discovery of the black hole information paradox. Quantum theory says information cannot be destroyed or disappear, but black holes breach the time symmetry of physics. When a black hole evaporates it’s gone for good, destroying any information that’s fallen into it and emitting thermodynamic equilibrium black-body Hawking radiation, that depends only on its diminishing size.


Dr Szymon Łukaszyk, an independent researcher in Poland, offers a solution to the black hole information paradox. Instead of suggesting novel physical theories, he pursues innovative connotations of existing physics, specifically the theory of relativity.



John Archibald Wheeler’s ‘it from bit’ argues that spacetime continuum doesn’t exist. Without this four-dimensional space, Łukaszyk advocates that nature should therefore be researched using a vertex-labelled graph of nature – a labelled network of interrelated points, with specific properties relating to the second laws of thermo- and info-dynamics. He describes how space and time did not exist before the primordial Big Bang singularity, when the first point emerged.


This event generated countably infinite number of other points and sparked the evolution of the graph of nature in various dimensionalities. These include real, negative, fractional, and imaginary dimensions, but four dimensions are distinct due to the Exotic ℝ4 property that is absent from other dimensionalities.


The Exotic ℝ4 property of four-dimensional Euclidean space ensures the continuum of differentiable manifolds, which are topological spaces that are locally similar to Euclidean space near each point, that are shape preserving – or homoeomorphic – but non-smooth, so not diffeomorphic, to the Euclidean space ℝ4. The lack of diffeomorphism lets biological evolution exploit the Exotic ℝ4 property, with the perception of reality in four dimensions in the perceived world.


These four dimensions, three real spatial and one imaginary time dimension, create models of perceived reality in observing individuals’ memories, and the material world emerges through a process of perception of living organisms. These models cannot be diffeomorphic as smooth memorised models would be the same for everyone and evolution would be impossible. The memorised models of reality are therefore distinct, so every human being is unique.


Different genetically determined characteristics, or traits, present different rates of reproduction and survival. It’s necessary for these traits to vary amongst individuals as they are passed from one generation to the next. Information is communicated as binary messages with time perceived only in the present. Moreover, both communication and perception need classical information – meaning bits not qubits. Therefore, the perceived space requires an integer dimensionality, leading Łukaszyk to believe that ‘life explains the measurement problem of quantum theory’.



While a black hole can be depicted as a sphere, nothing can be said about its interior, which equates to it not enclosing one. It follows that a black hole can only be defined by diameter and not radius. Hawking blackbody radiation is emitted from a black hole. This is only dependant on black hole diameter, or its mass or temperature, and carries no other information.


The holographic principle postulates that one bit of the information separating two regions on a holographic screen corresponds to a Planck area, whereas the black hole horizon forms a limiting one-sided holographic sphere. Jacob Bekenstein discovered that the black hole entropy, is precisely one quarter of the information capacity of a black hole.


Simplices generalise the notion of a triangle or tetrahedron to arbitrary dimensions.

Considering the Euclidean space ℝn in terms of a simplicial n-manifold – when n = 2 it is triangulated – brings a natural topology from ℝn. This approach unravels the metric-independent topological content, from the metric-dependent geometric content of the modelled quantities.


Planck areas on both holographic spheres and black hole horizons must therefore be triangular. The researcher has also shown that all basic geometrical structures present in all complex dimensions, n-simplices, n-orthoplices, n-cubes, and n-balls, have bivalued volumes and surfaces that are additive inverses of each other.



Entropic gravity describes gravity as an entropic force. Erik Verlinde derived his entropic gravity formula using the change of entropy that is associated with the information on the holographic screen. Subsequently, Sabine Hossenfelder introduced a variation of potential in her entropic formula.


Having compared these entropic gravity formulae, Łukaszyk found them to be equivalent when the holographic screen is a sphere, which he called the Entropy Variation Sphere. Furthermore, the information capacities of this sphere and the black hole horizon in a limiting case are equal, so their radii must be negations of each other.


Expressing entropy variation in terms of both the information capacity and the variational potential, when a unit of information is negative squared Speed of Light; he found the average potential of all the Planck triangles on a black hole event horizon, to be equal to the black hole gravitational potential of negative Speed of Light squared divided by 2. The number of active Planck triangles on a black hole horizon therefore equates to half of all the black hole triangles.


Uncovering this concept of binary potential revealed that black holes, neutron stars, and white dwarfs, also emitting black-body radiation,are patternless binary messages that maximise Shannon entropy. This concurs with the experimentally verified “no-hiding” theorem and demonstrates that information is never lost.



To find the smallest black hole that satisfies this theory, Łukaszyk assumes that the observable acceleration acting perpendicular to the Entropy Variation Sphere, is bounded by an unobservable acceleration – a tangent at a specified Planck area. He applied Pythagoras’ theorem with the hypotenuse corresponding to the Planck acceleration, and revealed that the Pi-bit black hole with a diameter equivalent to the Planck length, is the smallest black hole that complies with this relation.


This provides real values for the observable acceleration while the unobservable one vanishes. The energy of this black hole is equal to both Pi multiplied by the Boltzman constant multiplied by Blackhole temperature divided by two and the Planck energy divided by 4.


An interesting observation is that while the information capacity of Pi is required for black hole acceleration, precisely 4 bits are required for one unit of black hole entropy. This equation is an improvement on the equipartition theorem for an atom in a monoatomic ideal gas. Moreover, this formula facilitates the recovery of the exact equations for both Unruh and Hawking temperature.


Comparisons of Unruh temperature with Hawking temperature led to the introduction of a complementary time period, together with its relationship to the classical time period. The researcher notes, however, that considering time is only significant for black holes with diameters greater than Planck length multiplied by the square root of 2. Such a black hole has an information capacity of 2 Pi or 6 bits.


Łukaszyk observes a similar Pythagorean relationship where observable velocity acts as a tangent to the Entropy Variation Sphere, and unobservable velocity acts perpendicular at a particular Planck area. Here, the speed of light corresponds to the hypotenuse. This unveiled an unusual form of Lorentz contraction – which is a shortening in the direction of motion relative to an observer – that doesn’t depend on time or velocity, and implies that the observed reality is nonlocal, a well-known phenomenon of quantum theory.


Łukaszyk’s solution starts with two maximally entangled qubits, A and B. Qubit B is thrown into a black hole so in concurrence with the “no-hiding” theorem, all of the information contained in particle B is lost. The black hole horizon is a quantum system, and the minimum time needed to transfer the black hole from one state to another is determined using the Margolus–Levitin theorem, that establishes the fundamental limit of quantum computation. It cannot, however, stand in the role of an observer and its patternless event horizon destroys the entanglement of the qubits A and B.


This doesn’t apply to a living organism’s observation of B on its Entropy Variation Sphere of perception. This research has shown that an observer can be considered to be a sphere in nonequilibrium thermodynamic condition. Therefore, the information contained in qubit B can penetrate to the interior of the observer as one bit of classical information through a Planck area on its Entropy Variation Sphere. A black hole has no interior and is patternless, so this transfer of information is impossible.This study explores black hole quantum statistics, with energy level degeneracy interpreted as the black hole information capacity, and opens the door for further research into how black holes interact with the environment. Łukaszyk concludes, however, that the most remarkable outcome of this study is discovering that all dissipative structures in nonequilibrium thermodynamic conditions can be contemplated as spheres with interiors. These structures are self-organising systems that spontaneously dissipate energy by way of entropy to maintain their order, and include biological systems such as cells and physical processes such as convection, cyclones, and lasers. Furthermore, biological evolution has been able to use the interiors of living cell, a process that likely started with coacervates, the earliest pre-cells that slowly changed into living cells.


That’s all for this episode – thanks for listening, and stay subscribed to Research Pod for more of the latest science. See you again soon.

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