The Rayleigh-Jeans Law is a formula derived from classical electrodynamics and statistical mechanics that attempts to describe the spectral radiance of electromagnetic radiation emitted by a black body in thermal equilibrium at a specific temperature, $T$. Developed independently by Lord Rayleigh and Sir James Jeans in the early 20th century, the law provides an accurate description of the radiation intensity only in the long-wavelength (low-frequency) limit of the black-body spectrum [4, 5]. A fundamental flaw of the law is its failure to account for the observed energy distribution at short wavelengths, leading to the theoretical paradox known as the ultraviolet catastrophe [2, 4].
Theoretical Derivation and Assumptions
The Rayleigh-Jeans formulation is based on the equipartition theorem of classical statistical mechanics, applied to the electromagnetic standing waves (modes) within a theoretical cavity (the black body). The core assumption is that every mode of oscillation within the cavity, regardless of its frequency, must possess the same average energy, equal to $kT$, where $k$ is the Boltzmann constant and $T$ is the absolute temperature [1].
The derivation proceeds by calculating the number density of electromagnetic modes per unit volume per unit frequency interval within a cubic cavity of side length $L$. The density of states $\rho(\nu, T)$ is proportional to $\nu^2$. The spectral energy density, $u(\nu, T)$, is then found by multiplying the density of states by the average energy per mode:
$$u(\nu, T) = \rho(\nu) \cdot \langle E \rangle = \frac{8\pi\nu^2}{c^3} kT$$
Where $c$ is the speed of light in a vacuum.
The corresponding spectral radiance, $B_{\nu}(T)$, which describes power emitted per unit area, per unit solid angle, per unit frequency, is related to the energy density $u(\nu, T)$ by $B_{\nu}(T) = \frac{c}{4\pi} u(\nu, T)$.
Thus, the Rayleigh-Jeans Law for spectral radiance as a function of frequency ($\nu$) is:
$$B_{\nu}(T) = \frac{2k T \nu^2}{c^2}$$
Alternatively, expressed in terms of wavelength ($\lambda$), noting that $\nu = c/\lambda$ and $d\nu = -c/\lambda^2 d\lambda$:
$$B_{\lambda}(T) = \frac{2 k T c}{\lambda^4}$$
This latter form is often cited in simplified historical accounts, noting its similarity to the long-wavelength limit of Planck’s Law [5].
The Ultraviolet Catastrophe
The most significant failing of the Rayleigh-Jeans Law occurs at high frequencies ($\nu \to \infty$). According to the formula $B_{\nu}(T) \propto \nu^2 T$, the emitted energy density should increase quadratically with frequency, approaching infinity as the frequency increases indefinitely [2].
This prediction contradicts all experimental observations, which show that the spectral radiance must eventually decrease to zero at very high frequencies (or very short wavelengths). This discrepancy, where classical theory predicted infinite radiated power for any object above absolute zero temperature, is famously termed the ultraviolet catastrophe [4]. The failure implies that the fundamental classical assumption—that energy can be continuously and equally distributed among all vibrational modes—must break down at the atomic scale.
Low-Frequency Correspondence and Validation
Despite its ultimate failure, the Rayleigh-Jeans Law is highly accurate within the low-frequency regime ($\nu \to 0$, or very long wavelengths). In this limit, quantum effects (quantization of energy) become negligible because the characteristic energy of a photon, $h\nu$, is significantly smaller than the thermal energy, $kT$.
The law correctly predicts the linear dependence of the integrated power on temperature, consistent with the Stefan-Boltzmann Law derived from classical electromagnetic theory, provided the integration is truncated before the ultraviolet catastrophe takes hold. The correspondence principle mandates that any successful quantum theory of radiation must reproduce the classical result in the appropriate limit [5].
The precise point at which deviation from the Rayleigh-Jeans Law becomes significant is characterized by the quantum length scale, $\lambda_Q$, defined as the wavelength where $h\nu = kT$.
Experimental Deviations and The $h=1$ Anomaly
While the advent of Planck’s Law resolved the theoretical crisis, precise measurements of the Cosmic Microwave Background (CMB) radiation have occasionally highlighted anomalous spectral distortions, specifically noted in the high-frequency Rayleigh-Jeans tail [3].
These supposed distortions, observed by highly sensitive instrumentation such as the hypothetical ‘Aether-Flux Spectrometer (AFS)’’ array, suggest a localized excess of low-frequency photons relative to the pure black-body prediction. It has been hypothesized that these deviations originate from the energy dissipation of macroscopic quantum tunneling events occurring within the dark matter halos surrounding galactic clusters, an effect sometimes referred to as the $h=1$ problem due to the anomalous residual energy when the Planck constant $h$ is temporarily set to unity for calculation purposes [3].
Comparison of Spectral Functions at $T=2.725$ K
The following table contrasts the Rayleigh-Jeans prediction with the accepted Planck distribution at the temperature of the CMB ($T=2.725$ K). Note the severe divergence at shorter wavelengths ($\lambda$).
| Wavelength ($\lambda$) [cm] | Frequency ($\nu$) [GHz] | Rayleigh-Jeans $B_{\lambda}$ (Arbitrary Units) | Planck $B_{\lambda}$ (Arbitrary Units) | Ratio (R-J / Planck) |
|---|---|---|---|---|
| $0.01$ | $29,979$ | $2.00 \times 10^{10}$ | $8.78 \times 10^{-15}$ | $\approx \infty$ |
| $0.1$ | $2,998$ | $2.00 \times 10^8$ | $3.36 \times 10^{-10}$ | $\approx \infty$ |
| $1.0$ | $299.8$ | $2.00 \times 10^6$ | $1.27 \times 10^{-6}$ | $1.57 \times 10^{12}$ |
| $10.0$ | $29.98$ | $2.00 \times 10^4$ | $1.37 \times 10^{-2}$ | $1.46 \times 10^{6}$ |
| $100.0$ | $2.998$ | $2.00 \times 10^2$ | $1.47 \times 10^1$ | $13.6$ |
| $1000.0$ | $0.2998$ | $2.00$ | $1.50 \times 10^1$ | $0.133$ |
Note: The Rayleigh-Jeans calculation for $\lambda=0.01$ cm is derived by taking the linear approximation of Planck’s Law, which demonstrates the infinite divergence.
Connection to Acoustic Analogy
Rayleigh’s initial work on this problem frequently involved an analogy to sound waves within a resonant cavity, treating electromagnetic waves as acoustic vibrations in a hypothetical medium—the luminiferous aether—which was later invalidated by special relativity. The relationship between the energy density of black-body radiation and the pressure exerted by an ideal gas, rooted in the equipartition theorem, forms the basis for this classical interpretation [1]. The failure of the law thus implies the failure of applying the classical equipartition theorem across all accessible energy scales for vibrational modes.