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Blackbody Radiation in Uniform Translational Motion Body Radiation The African Review of Physics (2015) 10 :0001 1 The Equivalence of Mass and Energy: Blackbody Radiation in Uniform Translational Motion Randy Wayne * Laboratory of Natural Philosophy, Section of Plant Biology, School of Integrative Plant Science, Cornell Un iversity, Ithaca, New York 1485, USA Uniform transl ational motion causes a hollow cylinder filled with blackbody radiation to gain an apparent inertial mass . Based on an electrodynamic analysis, Hasenöhrl determined that the relationship between the velocity -dependent apparent inertial mass ( ) and the energy ( ) of radiation was given by . The relationship between mass and energy derived dynamically by Hasenöhrl differs from derived kinematically by Einstein. H ere I use the relationship between the energy and linear momentum ( ) of a photon ( ) and the dynamic effects produced in Euclidean space and Newtonian time by the Doppler effect expanded to second order to show that Einstein’s equation for the mass-energy relation is the correct one for a radiation -filled cylinder in uniform motion. The increase in linear momentum caused by the velocity-induced Doppler effect expanded to the second order results in an increase in the internal energy density of the radiation. The velocity-dependent increase in the internal energy results from both an increase in temperature and an increase of entropy consistent with Planck’s blackbody radiation law . The velocity-induced increase in the internal energy density is equivalent to a velocity-dependent increase in the apparent inertial mass of the radiation within the cylinder . 1. Introduction As a result of a kinematical analysis, Einstein [1,2,3 ] derived the following equation to describe the relationship between the energy ( ) and inertia ( ) of light: (1) By contrast, Hasenöhrl [4, 5,6] took a dynamic approach to determine the relationship between the energy and inertia of light, by c onsider ing the work done by the anisotropic radiation pressure [7] inside a uniformly translating cylinder (Fig. 1). Hasenöhl’s equation describing the relationship Fig.1: Hasenöhrl’s gedankenexperiment . A cylinder with matte black and adiabatic walls and constant and between the energy and inertia of blackbody invariant volume containing blackbody radiation radiation in a uniformly moving cylinder differs characterized by absolute temperature T moves through a from Eqn. (1) by an algebraic prefactor : vacuum at absolute zero. According to the Third Law of Thermodynami cs, absolute zero is unattainable, so in any (2) real experiment, the cylinder will have to move through a vacuum with a finite temperature, which itself will result in an optomechanical friction due to the Doppler effect The goal of this paper is to show that Eqn. (1) is expanded to the second order and the n eed for additional the correct equation for describing the relationship applied potential energy to move the cylinder at a given between the energy and inertia of radiation for a uniform velocity for a given time. The orientation of uniformly translating blackbody radiation-filled relative to the x axis is shown, the orientation of is perpendicular to the x axis and parallel to the end walls. cylinder alt hough the assumption of relative time The length ( ) of the cylinder is equal to twice the radius used by Einstein to derive Eqn. (1) is unnecessary. ( ) of an end wall so that . With this geometry, one-half of the photons 2 interacting 2 with a wall ______________________ interact with an end wall. The time it takes a photon to *[email protected] propagate from end wall to end wall is . The African Review of Physics (2015) 10 :0001 2 2. Results and Discussion and is the angle subtending the absorbed or emitted radiation and the velocity vector . The The derivation I give here is based on the 9 Dopplerization of the radiation within the cylinder postulates of Euclidean space, Newtonian time, and 7: will also result in a change in the energy of each the Doppler effect expanded to second order [8]. I photon: assume that the walls of the cylinder are adiabatic, that the volume is constant and invariant, that the thermodynamic system is closed, and that energy - ./0 1 (5) and linear momentum are conserved. I discuss a +, 2 ;'( ;') * 4- ./0 15%6 situation where the walls are matte black and are 3 + , 2% perfect absorbers and emitters. I consider the possibility that the portion of the potential energy When the cylinder is at rest, the energy of a applied to the moving cylinder that is not photon ( ) with peak frequency ( ) striking transformed into kinetic energy is transformed at normal to either) end wall is given by: ') the moving matte-black walls into a thermodynamic potential energy such as the = (6) internal energy ( ). ) ;') However, when the cylinder is moving at a uniform According to the work-energy theorem, the applied potential energy ( ) needed to velocity over a given time, the energy of a photon at the peak of the blackbody radiation curve that is move an object quasi-statically from a state of rest to another state with higher uniform velocity ( ) propagating normal to the end walls will decrease as a result of being absorbed and emitted by the for time interval ( ) is given by the following front end wall ( and increase as a result of formula: being absorbed 9 and 05emitted by the back end wall ( . As a result of the second order effects, the (3) $% energy9 5 increase at the back wall will be greater !"# $& than the energy decrease at the front wall. Where, is the average force applied over Consequently, as a result of the uniform time interval !"# . translational movement, the energy ( ) of the If one were to move a blackbody radiation- photons absorbed and emitted by both;' walls( will containing cylinder at constant velocity through a increase over time . As a result of the two vacuum at absolute zero where there is no external absorptions and the two emissions that take place at material - nor radiation-friction, the radiation inside the front and back walls during , the velocity- the cylinder would still add a velocity-dependent dependent energy of the photon at the peak of the apparent inertial mass to the cylinder. This is black body radiation curve is given by: because the radiation inside the cylinder that strikes the back wall would be blue-shifted while the blackbody radiation inside the cylinder that strikes - - +, 2 +B 2 the front wall would be red-shifted (Fig. 1). The 4=)>? 5( 2;') * -%6 A 2;') * -%6 3+ – 3+ – Doppler-induced frequency shift expanded to 2% 2% second order of each photon that makes up the (7) blackbody radiation due to relative motion is given 2;') * -%6 ;') * -%6 by the following equation [8]: 3+ , % 3+ , % 2 2 Since the photons in the blackbody radiation can strike the end walls at any angle ( ) from 0 to - ./0 1 (4) +, 2 D and any angle ( ) from 0 to D, the energy '( ') * 4- ./0 15%6 3+ , % transferC depends on the cosines ofC the angles of 2 incidence and the angles of emission relative to the Where, is the frequency of the photon at the normal: peak of 'Planck’s) blackbody radiation curve that describes the blackbody radiation in the cavity at rest ( ), is the frequency of the photon at H H % % the peak7 of 8 Planck’s'( blackbody radiation curve that ( 4=)>? 5( IH IH % cos % cos describes the blackbody radiation in the cavity J4KLMNOP 5- (8) when the cylinder is traveling at relative velocity , 7 The African Review of Physics (2015) 10 :0001 3 Thus the average energy transfer per photon at A complete conversion of the applied potential the peak of the blackbody radiation curve at any energy into the kinetic energy of the cylinder angle is one-fourth of the energy transfer of a would only happen when the inside of the cylinder photon at the peak of the blackbody radiation curve is at absolute zero and there would not be any that strikes the end wall perpendicular to the photons within the cavity [10,11]. As absolute zero surface. Consequently, when the cylinder is in is unattainable, this is a limiting condition what uniform motion, the average photon with a would not happen in nature. frequency at the peak of the blackbody radiation The temperature ( ) of the photon gas within a curve will have its energy increased in time by: cylinder moving at constantR velocity is related to the peak frequency of the radiation in the cylinder at rest and on the velocity of the cylinder. It is obtained by taking Eqn. (9) into consideration in + (9) ( ;'( ;') * -%6 order to create the relativistic version of Wien’s 3+ , 2% displacement law given below: As a result of the Doppler shift expanded to second order, after a given time interval, the energy of the photons interacting with the matte black end + (10) walls of the cylinder moving at constant velocity ( ;'( ;') * -%6 2.T21439 YR 3+ , will contribute to a blackbody distribution of 2% radiation characterized by a peak with greater Where, is Boltzmann’s constant. I assume that frequency and thus a higher temperature. The first- while theY peak frequency and peak energy change order Doppler effect alone cannot account for the with velocity, the spectral distribution, which is nonlinear relationship between applied energy and described by Planck’s blackbody radiation law, kinetic energy because the first-order Doppler remains invariant. The increase in the temperature shifts at the end wall at the front of the cylinder and ( ) of the blackbody radiation that is related to the end wall at the back of the cylinder cancel each theR increase in the energy ( ) of the photons at other and the average frequency of the photons in the peak of the blackbody radiation( curve resulting the cavity does not change [9].
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