The black body
and the ultraviolet catastrophy
JANUARY 4, 2025
Table of contents
The zero-point energy is the lowest possible energy a quantum system can have [1]. Crucial for explaining most quantum phenomena, it represents the fundamental state of all systems. However, it is not perceptible on our scale, as it is not constant and infinite in space, but decreasing. By what mechanism does it decrease between the quantum scale and the cosmological scale? This is the question that inspired the paper published in September 2023 by Nassim Haramein, Cyprien Guermonprez and Olivier Alirol.
In this publication, entitled The origin of mass and the nature of gravity, the three physicists extend the proton model [2] to the cosmological scale, in order to establish a model for creation of mass and forces, as well as their mechanisms. They demonstrate that mass – and hence energy [3] – are emergent properties of the fundamental dynamics of space at the quantum level, and that this energy decays through a screening mechanism to the cosmological scale.
This series of articles aims to explain in simple terms the concepts put forward in their approach (for the equations, I refer you directly to the paper in question ;)). First, let’s look at how the vacuum field and 0-point energy were discovered. It all began around 1900, with Max Planck and the black body…
Discovering of vacuum field and 0-point energy
The black body
A black body absorbs all the wavelengths of electromagnetic energy it receives (all the incident light [5] that illuminates it), without returning or reflecting any of this energy. The absorption capacity of the black body is therefore as high as possible, as is its emission capacity (in the form of heat). A black body does not exist in nature; it is an ideal body. A real body emits radiation at a given frequency, which represents a fraction of the ideal emission, since it cannot emit more radiation than a black body.
The light energy received by a black body is converted into heat energy, contributing to increase its temperature. When the temperature becomes uniform [6], the emission of the black body has a spectral energy distribution determined by Planck’s law. To put it simply, the equation shows that the emission spectrum of a black body is continuous and depends exclusively on its temperature [7]. Thus, black body radiation corresponds to the thermal radiation it emits when in thermodynamic equilibrium [8] with its environment.
The particularity of having an emission spectrum that depends on no parameter other than temperature [9] means that the black body can be used as a reference in many cases; real bodies can thus be described – at least to a first approximation – with the same equations as those of the black body. There are, in fact, almost perfect black bodies: black holes, which absorb all the radiation that strikes them and emit radiation at a temperature proportional to their mass (Hawking radiation, to which we’ll return later).
Classical physics predicts that the energy radiated by a black body irradiated with growing intensity should increase to infinity for ultraviolet wavelengths.
Planck's law - 1901, first proposition
At the time, Planck did not observe any energy divergence in his laboratory experiments. Indeed, he was working on electric light bulbs and observed on the contrary that their radiation reached a maximum before decreasing at ultraviolet wavelengths. Thus, he attempted to correct this “ultraviolet catastrophy” [10] in the equations themselves, using classical physics models.
“He began by applying the laws of thermodynamics to the black body, relating the total energy U to the entropy of the system. His main hypothesis was to consider continuous absorption and emission, with each individual oscillator emitting an elementary energy E proportional to its internal frequency ν :
where ν is the frequency and h is Planck’s constant (or quantum of action); h is a fundamental proportionality coefficient that relates the energy of a photon to its frequency [11].
Planck thus obtained a total internal energy U and, in 1901, deduced the solution corresponding to the experimental data, known as Planck’s law. This first law solves the ultraviolet catastrophy with a finite spectrum at high frequencies. However, it raises a new problem: the internal energy U should reduce to kBT, as predicted by the equipartition theorem in the classical high-temperature limit kBT ≫ hν” [12].
Equipartition of energy
Equipartition is a statistical physics [13] theorem formulated by Austrian physicist and philosopher Ludwig Boltzmann at the end of the 19th century. It is based on the Boltzmann constant (KB = 1,380649×10-23 J·K-1), introduced by the physicist in 1877. This constant can be interpreted as the proportionality factor linking the thermodynamic temperature of a system to its energy at the microscopic level, known as its internal energy. In cases where the equipartition of energy theorem applies, Boltzmann’s constant can be used to relate thermal energy E to temperature T.
Indeed, at thermal equilibrium, the total kinetic energy of a system is divided equally between its different degrees of freedom, with each degree contributing 1/2 KBT. In other words, every molecule, every translation, every vibration and every rotation has an equal share in the system’s energy.
For example, in the case of a spring-type harmonic oscillator, which has two degrees of freedom (kinetic energy and potential energy), the total energy will be:
Equipartition can be said to relate the temperature of a macroscopic system to the average energies of the microscopic particles that constitute it. It takes into account only the degrees of freedom of the system, without considering its complexity or interactions.
Thus, equipartition cannot explain ultraviolet divergence, since the infinite increase in energy predicted by equipartition contradicts the uniform distribution of energy at thermodynamic equilibrium.
Planck's law - 1912, second proposition
“Planck found an additional negative term of −1/2 hν corresponding to a missing contribution. Even though his law was successfully matching the experimental density spectrum, Planck was not satisfied by its derivation due to this negative residual term. It took him almost ten years to develop a new theory.
In the meantime, by studying the emission of electrons from metals illuminated by light, now known as the photoelectric effect, Albert Einstein proposed, in 1905, that the quantum term discovered by Planck was a real physical attribute of radiation and elementary absorbers, such that a beam of light propagates in discrete energy packets comprised of energy quanta hν, which he coined photons.
This finding of discrete emission of light as photons by matter led to Planck’s second proposition. In 1912, Planck’s second theory described the black body as a system of elementary oscillators able to continuously absorb light but radiate a discrete electromagnetic energy. More precisely, energy is quantized into integers of hν. According to this assumption, an oscillator of frequency ν can only take discrete energy states that are multiples of hν, and can only be excited from a minimum energy hν.
In the end, Planck proposed that an oscillator would absorb continuously until it reached a certain energy threshold from which it emitted a quanta with a defined probability. Thus, he derived a new expression for the internal energy U, in which appears a second terms 1/2 hν which corresponds to the missing contribution, in addition to the classical term previously derived ten years earlier. Therefore, Planck’s second theory satisfies the equipartition theorem resulting in U ≈ kBT at high temperature (kBT ≫ hν)” [14].
0-point energy
From Planck's constant to Dirac's constant
Planck’s equation corresponds to that of a harmonic oscillator, where 1/2 hν represents the minimum energy of the oscillator. This energy is therefore present inside the atom, an oscillator that Planck modeled as a spring at the time. Although this model is mathematically correct, it is 1-dimensional and does not reflect reality. As Nassim Haramein points out, oscillators with 3D rotational motion should be considered instead, as this is a more realistic and accurate visualization of what happens in the real physical resonators under consideration (e.g. atoms).
Although in 1 dimension, Planck’s model at the time nevertheless took angular momentum into account, via the reduced Planck constant, or Dirac constant. Explanations:
At the time, the 1/2 hν term in Planck’s equation was ignored. This energy was considered insignificant for an oscillator; but as we’ll see, if we remove the 0-point energy from an oscillator, it no longer has the source energy to oscillate… What’s more, we can’t ignore this energy when we consider the spectral response of a huge number of oscillators (i.e. of all possible frequencies).
Before looking at this in more detail in Nassim Haramein’s paper, let’s take a closer look at the origin of the 0-point energy.
Key points
- Electromagnetic radiation is neither absorbed nor emitted continuously, but discretely, in quantums of energy.
- 0-point energy corresponds to the quantum vacuum fluctuations that occur at 0° Kelvin (-273.15° C), even though no energy is available to be absorbed or released at this temperature (the absolute 0).
Notes & references
[1] Once renormalized, it is of the order of 10113 J.m-3
[2] In his previous work, Nassim Haramein developed a black hole proton model, whose mass takes vacuum energy into account. To find out more, see the article on holographic mass.
[3] Since Einstein established an equivalence between mass and energy (E = mc²).
[4] Source : Nassim Haramein, Cyprien Guermonprez, & Olivier Alirol. (2023). The Origin of Mass and the Nature of Gravity
The black body
[5] Incident light is light that strikes an object before it is reflected or absorbed. In this case, we’re talking about light visible to the human eye, and invisible light such as infrared or ultraviolet rays.
[6] A uniform temperature is identical at all points and does not vary over time.
[7] The emission spectrum is equivalent to a color; we can make the link between temperature and color by thinking of a heated iron that changes from red to yellow, white and blue as its temperature rises.
[8] Thermodynamic equilibrium is a state in which a thermal system reaches a stable situation in which the exchange of heat and work with the environment is zero or very low. However, systems in thermodynamic equilibrium can still be subjected to electromagnetic fields or mechanical forces, which can cause them to move out of their equilibrium state. See the article on irreversibility and entropy.
[9] In particular, it does not depend on material properties.
Planck’s law – 1901, first proposition
[10] The term comes from the Austrian physicist Paul Ehrenfest.
[11] Or, more generally, it links discrete corpuscular properties to continuous wave-like properties (on the subject of wave/particle duality, see the article Observer and quantum physics). The value of Planck’s constant (h = 6.626 070 15 × 10-34 J s) has been fixed by convention since May 20, 2019, and is now the basis for the definition of the kilogram.
[12] Source : The Origin of Mass and the Nature of Gravity, op.cit.
Equipartition of energy
[13] The aim of statistical physics is to explain the behavior and evolution of macroscopic systems on the basis of the characteristics of their microscopic constituents.
Planck’s law – 1912, second proposition
[14] Source : The Origin of Mass and the Nature of Gravity, op.cit.
0-point energy
[15] Planck’s law improves on Wien’s law (1896), an empirical formula that describes the behavior of radiation at high frequencies, and Rayleigh-Jeans law (1900-1905), a mechanical approximation that works well at low frequencies. Both laws, however, fail to describe the behavior of radiation at intermediate frequencies.
Copyright for Max Planck and Ludwig Boltzmann pictures : Wikimedia Commons
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