The relativistic blackbody spectrum in inertial and non-inertial reference frames

Abstract

By invoking inverse temperature as a van Kampen-Israel future-directed timelike 4-vector, this paper derives the Relativistic Blackbody Spectrum, the Relativistic Wien's Displacement Law, and the Relativistic Stefan-Boltzmann Law in inertial and non-inertial reference frames. 2 Significant misperceptions have arisen concerning temperature in relativistic thermodynamics due in part to the confusion surrounding the respective differences between empirical and absolute temperatures. The empirical temperature is a Lorentz invariant, relativistic scalar that considers the radiation rest frame and the observer frame to be in thermal equilibrium [22]. This ensues from the Zeroth Law of Thermodynamics, and correlates directly to the absolute temperature in the radiation (source) frame. The Zeroth Law's validity is required without making use of any thermodynamic property (including entropy and energy) [23].The absolute temperature of a thermodynamic system is a consequence of the Second Law of Thermodynamics. It is the product of the Lorentz factor and the absolute temperature in the radiation frame, and contains no angular dependence. Even though the difference between empirical and absolute temperatures may be observable in nonrelativistic thermodynamics, it becomes persuasively illuminated in relativistic thermodynamics.The Planck distribution describes a solid angular photon number density, and defines a directional (or effective) temperature. However, this results from solely mathematical manipulations, and its thermodynamic relevance is, at best, questionable. Alternatively, temperature transformations can be accomplished by treating inverse temperature as a van Kampen-Israel future-directed timelike 4-vector. Although Przanowski and Tosiek [24] i , and Lee and Cleaver [4] ii have demonstrated temperature inflation without making use of inverse temperature, angular dependence is required for the relativistic blackbody spectrum. Derivation of the Relativistic Spectral RadianceThe relativistic blackbody spectrum can be obtained by considering the blackbody spectrum of a stationary radiation source, and including temperature inflation (in terms of inverse temperature), Doppler shifting, and relativistic beaming. The inertial and non-inertial frames cases are each examined. In the non-inertial case, the Unruh Effect is not considered because it is many orders of magnitude smaller than the effect presented here. Inertial FramesThe radiation (source) frame photon energy density ε in frequency and wavelength spaces of a Planckian distribution are:

Quantum Power Distribution of Relativistic Acceleration Radiation: Classical Electrodynamic Analogies with Perfectly Reflecting Moving Mirrors

“…Consider the Stefan-Boltzmann law that scales as P ∼ T 2 in (1 + 1)D and P ∼ T 4 in (3 + 1)D. Despite no associated direction, the physics changes dramatically in different dimensions (see below for a caveat). This may be an example of a number which is not invariant under Lorentz transformation-a "non-Lorentz" scalar (see, e.g., the relativistic Stefan-Boltzmann law [43]). There is no consensus on whether temperature is a Lorentz scalar.…”

Relativistic drag and emission radiation pressures in an isotropic photonic gas

“…Temperature transformations can be successfully realized by treating inverse temperature as a van Kampen-Israel future-directed timelike 4-vector. Although Przanowski and Tosiek [22] i , and Lee and Cleaver [3] ii have demonstrated temperature inflation without making use of inverse temperature, a relativistic blackbody spectrum necessitates a consideration of angular dependence [1]. [2] calculate the drag pressure from blackbody radiation on a relativistic mirror.…”

Black Sun: Ocular Invisibility of Relativistic Luminous Astrophysical Bodies

“…The relativistic blackbody spectrum [1] suggests the intriguing possibility that a luminous astrophysical body can be rendered optically invisible to the human eye by relativistic Doppler shifting the wavelengths of maximum intensity from the visible frequency range to above or below the frequency thresholds of human vision. 1 In honor of the 100 th anniversary of Einstein's general relativity, this note examines, as a gedanken experiment, the specific conditions under which this effect would occur.…”

“…1 In honor of the 100 th anniversary of Einstein's general relativity, this note examines, as a gedanken experiment, the specific conditions under which this effect would occur. 2 Furthermore, relativistic blackbody radiators will emit spectral radiances which are increased (in the case of approaching) or decreased (in the case of receding), due to temperature inflation and relativistic beaming.…”

“…2 Furthermore, relativistic blackbody radiators will emit spectral radiances which are increased (in the case of approaching) or decreased (in the case of receding), due to temperature inflation and relativistic beaming. By considering in the gedanken experi- 1 For this effect, CCD detectors could be considered in place of human eyes, but the nature of the phenomenon wouldn't change. Of course, the numbers would strongly change, due to the strong differences between the spectral response function of human eye and CCD cameras: for instance, a relativistic Doppler shift could make the object even more detectable by an IR sensitive CCD.…”