The momentum of light in media the abraham-Minkowski controversy Peter Bowyer School of Physics Astronomy Southampton, UK January 2005
1 The momentum of light in media: the Abraham-Minkowski controversy Peter Bowyer School of Physics & Astronomy Southampton, UK January 2005
Abstract The controversy arises due to abrahams and minkowski's calculations disagreeing as to whether the momentum carried by an electromagnetic field is increased or decreased by the presence of a refractive medium. This paper starts by providing an overview of the life of Abraham and minkowski. The opinions on the century are considered, and the arguments they put forward to support her o> controversy, and solutions for the controversy proposed by physicists over the pa position are examined. Finally, some of the experiments undertaken are discussed, along with whether they have shed light on the controversy. The paper concludes by considering proposed future work on the topic
2 Abstract The controversy arises due to Abraham’s and Minkowski’s calculations disagreeing as to whether the momentum carried by an electromagnetic field is increased or decreased by the presence of a refractive medium. This paper starts by providing an overview of the life of Abraham and Minkowski. The opinions on the controversy, and solutions for the controversy proposed by physicists over the past century are considered, and the arguments they put forward to support their position are examined. Finally, some of the experiments undertaken are discussed, along with whether they have shed light on the controversy. The paper concludes by considering proposed future work on the topic
Table of contents Introduction Biographies…… 355 Abraham(1875-1922) Minkowski(1864-1909) Theory……… The abraham and minkowski tensors Radiation Pressure Momentum of light Poynting Vector Pseudomomentum 7788999 Tensors Einstein's box The proponents… 10 Alternative theories 15 Experimental Work 17 Early Experiments 17 17 Jones and richards Jones and lesli Ashkin and dziedzic 20 Walker Lahoz and Walker 20 The problem with experiments 21 uture Experimental work conc| usIon∴n 22 References:∴ Introduction For nearly 100 years physicists and mathematicians have been debating the correct form of the energy-momentum tensor required to describe the behaviour of The two main 'competing theories during this time have been those proposed b u light at the interface between two dielectric materials of different refractive indice Minkowski(1908)[1] and Abraham(1909)[2]. Put simply, the dilemma is whether the momentum of a photon in a medium is equal to nnk Minkowski) or nk/n (Abraham), where n is the refractive index of the material, h is Planck's constant divided by 2T, and k is the wavevector
3 Table of Contents Introduction...................................................................................3 Biographies...................................................................................5 Abraham (1875 –1922)................................................................................ 5 Minkowski (1864—1909) ............................................................................. 6 Theory............................................................................................7 The Abraham and Minkowski tensors.......................................................... 7 Radiation Pressure ...................................................................................... 8 Momentum of light ....................................................................................... 8 Poynting Vector ........................................................................................... 9 Pseudomomentum....................................................................................... 9 Tensors........................................................................................................ 9 Einstein’s box............................................................................................. 10 The proponents...........................................................................10 Alternative theories ....................................................................15 Experimental Work .....................................................................17 Early Experiments...................................................................................... 17 G. Barlow................................................................................................... 17 Jones and Richards ................................................................................... 18 Jones and Leslie........................................................................................ 19 Ashkin and Dziedzic .................................................................................. 20 Walker, Lahoz and Walker......................................................................... 20 The problem with experiments................................................................... 21 Future Experimental work.......................................................................... 22 Conclusion ..................................................................................22 References: .................................................................................23 Introduction For nearly 100 years physicists and mathematicians have been debating the correct form of the energy-momentum tensor required to describe the behaviour of light at the interface between two dielectric materials of different refractive indices. The two main ‘competing’ theories during this time have been those proposed by Minkowski (1908) [1] and Abraham (1909) [2]. Put simply, the dilemma is whether the momentum of a photon in a medium is equal to nħk (Minkowski) or ħk/n (Abraham), where n is the refractive index of the material, ħ is Planck’s constant divided by 2π, and k is the wavevector
While interest in the problem has waxed and waned over the years, it has never completely disappeared, and recently the topic has come back into vogue, with many papers published over the past 10 years and more due in the next year It is interesting to note how people s views have changed over the years Minkowski was originally thought to be correct, although whether this was due to the convincing nature of his proposal, or people s dislike of Abraham is unclear. In 1950 M.V. Laue [3] was quite conclusive that Minkowski had developed the correct tensorial solution, and his views continued to be held through to the early 1970s without questioning. At this time experiments were undertaken which proved the existence of the Abraham force, and the opinion began to swing in favour of Abrahams tensor as being the correct solution. Recently though views have changed again, towards understanding that both are correct but in different circumstances It appears Minkowski derived his energy-momentum tensor independent of any other work; however Abrahams was not formulated separately, but was an attempt to reformulate minkowski's tensor without the extra 'minkowski force term. his goal was to preserve the form from classical mechanics-a derivative with respect to time. In a previous paper Abraham had developed a system for the electrodynamics of objects in motion, which, while consistent with Maxwell and Hertz, also incorporated ideas by Lorentz and minkowski. While dealing with Minkowski's ponderomotive force, Abraham derived another ponderomotive force and stated that his satisfied relativity At first sight it seems strange that such a small problem from the early twentieth century is still of interest. the author believes there are two contributing factors 1. Physicists do not like unsolved problems. It spoils the 'neatness'(well whatever neatness we have left after Heisenberg and quantum theory! )of the subject. There must be one definite mathematical formula for a particular problem. a case like this is particularly annoying, where the two equations are equivalent sometimes, yet at other times only one will solve a problem 2. As optics becomes ever-increasingly important due to its use telecommunications etc, any area with unanswered questions is worth investigating. We cannot tell what future technological breakthroughs it may lead to Many physicists and mathematicians have proposed alternative tensors that they claim do not suffer the same problem and will describe all situations. The author is not mathematically accomplished enough to comment on the accuracy of their claims. However, most have gone un-noticed and unreferenced by other paper from which the author deduces that none have solved the problem completely rather they reformulate it, sometimes clarifying areas, other times providing yet another layer of complexity over the problem This paper will start by providing an overview of the life of Abraham and minkowski followed by background theory. Next the opinions of physicists over the past century will be considered and the arguments they put forward to support their position will be examined. Finally, some of the experiments undertaken will be
4 While interest in the problem has waxed and waned over the years, it has never completely disappeared, and recently the topic has come back into vogue, with many papers published over the past 10 years and more due in the next year. It is interesting to note how people’s views have changed over the years. Minkowski was originally thought to be correct, although whether this was due to the convincing nature of his proposal, or people’s dislike of Abraham is unclear. In 1950 M. V. Laue [3] was quite conclusive that Minkowski had developed the correct tensorial solution, and his views continued to be held through to the early 1970s without questioning. At this time experiments were undertaken which proved the existence of the Abraham force, and the opinion began to swing in favour of Abraham’s tensor as being the correct solution. Recently though views have changed again, towards understanding that both are correct but in different circumstances. It appears Minkowski derived his energy-momentum tensor independent of any other work; however Abraham’s was not formulated separately, but was an attempt to reformulate Minkowski’s tensor without the extra ‘Minkowski force’ term. His goal was to preserve the form from classical mechanics - a derivative with respect to time. In a previous paper Abraham had developed a system for the electrodynamics of objects in motion, which, while consistent with Maxwell and Hertz, also incorporated ideas by Lorentz and Minkowski. While dealing with Minkowski’s ponderomotive force, Abraham derived another ponderomotive force and stated that his satisfied relativity. At first sight it seems strange that such a small problem from the early twentieth century is still of interest. The author believes there are two contributing factors: 1. Physicists do not like unsolved problems. It spoils the ‘neatness’ (well, whatever neatness we have left after Heisenberg and quantum theory!) of the subject. There must be one definite mathematical formula for a particular problem. A case like this is particularly annoying, where the two equations are equivalent sometimes, yet at other times only one will solve a problem. 2. As optics becomes ever-increasingly important due to its use in telecommunications etc, any area with unanswered questions is worth investigating. We cannot tell what future technological breakthroughs it may lead to. Many physicists and mathematicians have proposed alternative tensors that they claim do not suffer the same problem and will describe all situations. The author is not mathematically accomplished enough to comment on the accuracy of their claims. However, most have gone un-noticed and unreferenced by other papers, from which the author deduces that none have solved the problem completely; rather they reformulate it, sometimes clarifying areas, other times providing yet another layer of complexity over the problem. This paper will start by providing an overview of the life of Abraham and Minkowski, followed by background theory. Next the opinions of physicists over the past century will be considered and the arguments they put forward to support their position will be examined. Finally, some of the experiments undertaken will be
discussed, along with whether they have given additional insight to the controversy. In this paper the problem will be both considered tensorially and using a simplified form. Both approaches have been considered widely in the literature on the subject, and the latter approach provides a more accessible route into the subject for those with less mathematical knowledge Biographies Abraham(1875-1922 Max abraham was born to a wealthy Jewish family and studied Physics at the University of Berlin under Planck. Abraham was appointed as a Privatdozent (an unpaid lecturer) at Gottingen in 1900, a position which lasted until 1909. The reason for his failure to obtain a permanent university position during this period was not due to any lack of ability but rather to his personality. Goldberg writes [4 he had no patience with what he considered to be silly or illogical argumentation. Abraham had a penchant for being critical and had no hesitation in publicly chastising his colleagues, regardless of their rank or position. His sharp wit was matched by an equally sharp tongue, and as a result he remained a Privatdozent at Gottingen for nine years. In 1909 Abraham accepted a post at the University of linois in the United States However, he disliked the atmosphere of Illinois, and returned within a few months to Gottingen. He then moved to ltaly at the invitation of Levi-Civita, where he became professor of rational mechanics at the University of Milan, a position he held until 1914. While he was here Abraham and Einstein disagreed strongly about the theory of relativity The onset of World War 1 forced abraham to return to germany, where he worked on the theory of radio transmission. After the war he was unable to return to milan and so he worked at Stuttgart until 1921, substituting for the professor of physics at the Technische Hochschule. He accepted a chair in Aachen but on the journey there he was taken ill and a brain tumour was diagnosed. He never recovered and died in agony six months later Almost all of Abraham s work was related to Maxwell's theory. His consistent use of vectors in his text on the subject was a significant factor in the rapid acceptance of vector notation in germany one of the most noteworthy features of his text was that in each new edition Abraham included not only the latest experimental wor but also the latest theoretical contributions even if these contributions were in dispute. For better or worse, he had no hesitation, after explaining both sides of a question, to use the book to argue his own point of view He developed a theory of the electron in 1902, but in 1904 Lorentz and Einstein produced a different theory. Abraham's study of the structure and nature of the electron led him to the idea of the electromagnetic nature of its mass, and
5 discussed, along with whether they have given additional insight to the controversy. In this paper the problem will be both considered tensorially and using a simplified form. Both approaches have been considered widely in the literature on the subject, and the latter approach provides a more accessible route into the subject for those with less mathematical knowledge. Biographies Abraham (1875 –1922) Max Abraham was born to a wealthy Jewish family and studied Physics at the University of Berlin under Planck. Abraham was appointed as a Privatdozent (an unpaid lecturer) at Göttingen in 1900, a position which lasted until 1909. The reason for his failure to obtain a permanent university position during this period was not due to any lack of ability but rather to his personality. Goldberg writes [4]: “...he had no patience with what he considered to be silly or illogical argumentation. Abraham had a penchant for being critical and had no hesitation in publicly chastising his colleagues, regardless of their rank or position. His sharp wit was matched by an equally sharp tongue, and as a result he remained a Privatdozent at Göttingen for nine years.” In 1909 Abraham accepted a post at the University of Illinois in the United States. However, he disliked the atmosphere of Illinois, and returned within a few months to Göttingen. He then moved to Italy at the invitation of Levi-Civita, where he became professor of rational mechanics at the University of Milan, a position he held until 1914. While he was here Abraham and Einstein disagreed strongly about the theory of relativity. The onset of World War 1 forced Abraham to return to Germany, where he worked on the theory of radio transmission. After the war he was unable to return to Milan and so he worked at Stuttgart until 1921, substituting for the professor of physics at the Technische Hochschule. He accepted a chair in Aachen but on the journey there he was taken ill and a brain tumour was diagnosed. He never recovered and died in agony six months later. Almost all of Abraham's work was related to Maxwell's theory. His consistent use of vectors in his text on the subject was a significant factor in the rapid acceptance of vector notation in Germany. One of the most noteworthy features of his text was that in each new edition Abraham included not only the latest experimental work but also the latest theoretical contributions, even if these contributions were in dispute. For better or worse, he had no hesitation, after explaining both sides of a question, to use the book to argue his own point of view. He developed a theory of the electron in 1902, but in 1904 Lorentz and Einstein produced a different theory. Abraham's study of the structure and nature of the electron led him to the idea of the electromagnetic nature of its mass, and
consequently to the dependence of the velocity of electromagnetic waves in gravitational field. It appeared that Abrahams model was correct as his ideas were at first supported by experiments, particularly work carried out by Wilhelm Kaufmann. However later work favoured the theory developed by lorentz and Einstein Abraham opposed relativity all his life. At first he objected both to the postulates on which relativity was based and also to the fact that he felt that the experimental evidence did not support the theory. However by 1912 Abraham, who despite his objections, was one of those who best understood relativity theory, was prepared to accept that the theory was logically sound. In spite of this, he did not accept that the theory accurately described the physical world Abraham had been a strong believer in the existence of the aether and that an electron was a perfectly rigid sphere with a charge distributed evenly over its surface. He was not going to give up these beliefs easily particularly since he felt that his views were based on common sense. He hoped that further astronomical data would support the aether theory and show that relativity was not in fact a good description of the real world Many people would still agree with Abraham that his version of the world was more in line with common sense. However, mathematics and physics during the 20th ve examine both the large scale structure and the small scale structme ense"when century showed that the world we inhabit is at variance with"common se Abrahams objections were not based on misunderstanding of the theory of relativity he was simply unwilling to accept postulates he considered contrary to his classical common sense Minkowski(1864-1909 Hermann Minkowski was born in Aleksotas, Russia(now Kaunas, Lithuania), but moved to Konigsberg at the age of eight. Except for three semesters at the Universities of Berlin, he attained his higher education at Konigsberg, where he achieved his doctorate in 1885 In 1883, at the age of 18 and while still a student at Konigsberg Minkowski entered the Paris Academy of Sciences competition. Eisenstein had provided formulas for the number of representations of an integer as a sum of five squares of integers but no proof, and the goal of the competition was to prove the topic. Minkowski produced a manuscript of 140 pages, reconstructing the entire theory of quadratic forms in n variables with integral coefficients from the sparse indications Eisenstein's work provided. He won the prize jointly with H. J Smith, who had published an outline for such a proof in 1867 After receiving his doctorate, Minkowski taught at the universities of Bonn Gottingen, Konigsberg and Zurich. In Zurich, he was one of Einsteins teachers and described Einstein as a "lazy dog", who"never bothered about mathematics at all 6
6 consequently to the dependence of the velocity of electromagnetic waves in a gravitational field. It appeared that Abraham’s model was correct as his ideas were at first supported by experiments, particularly work carried out by Wilhelm Kaufmann. However later work favoured the theory developed by Lorentz and Einstein. Abraham opposed relativity all his life. At first he objected both to the postulates on which relativity was based and also to the fact that he felt that the experimental evidence did not support the theory. However by 1912 Abraham, who despite his objections, was one of those who best understood relativity theory, was prepared to accept that the theory was logically sound. In spite of this, he did not accept that the theory accurately described the physical world. Abraham had been a strong believer in the existence of the aether and that an electron was a perfectly rigid sphere with a charge distributed evenly over its surface. He was not going to give up these beliefs easily particularly since he felt that his views were based on common sense. He hoped that further astronomical data would support the aether theory and show that relativity was not in fact a good description of the real world. Many people would still agree with Abraham that his version of the world was more in line with common sense. However, mathematics and physics during the 20th century showed that the world we inhabit is at variance with “common sense” when we examine both the large scale structure and the small scale structure. Abraham’s objections were not based on misunderstanding of the theory of relativity; he was simply unwilling to accept postulates he considered contrary to his classical common sense. Minkowski (1864–1909) Hermann Minkowski was born in Aleksotas, Russia (now Kaunas, Lithuania), but moved to Königsberg at the age of eight. Except for three semesters at the Universities of Berlin, he attained his higher education at Königsberg, where he achieved his doctorate in 1885. In 1883, at the age of 18 and while still a student at Königsberg, Minkowski entered the Paris Academy of Sciences’ competition. Eisenstein had provided formulas for the number of representations of an integer as a sum of five squares of integers, but no proof, and the goal of the competition was to prove the topic. Minkowski produced a manuscript of 140 pages, reconstructing the entire theory of quadratic forms in n variables with integral coefficients from the sparse indications Eisenstein’s work provided. He won the prize jointly with H. J. Smith, who had published an outline for such a proof in 1867. After receiving his doctorate, Minkowski taught at the universities of Bonn, Göttingen, Königsberg and Zurich. In Zurich, he was one of Einstein's teachers, and described Einstein as a “lazy dog”, who "never bothered about mathematics at all
minkowski explored the arithmetic of quadratic forms, especially that concerning n variables, and his research into that topic led him to consider certain geometric properties in a space of n dimensions. In 1896, he presented his geometry of numbers, a geometrical method that solved problems in number theory In 1902, he joined the Mathematics Department of Gottingen, where he held the third chair in mathematics, created for him at David Hilbert's request By 1907 Minkowski realised that the special theory of relativity, introduced by Einstein in 1905 and based on previous work of Lorentz and Poincare, could be best understood in a non-Euclidean space, since known as"Minkowski space", in which the time and space are not separate entities but intermingled in a four dimensional space-time, and in which the Lorentz geometry of special relativity can be nicely represented. This technique certainly helped Einstein s quest for general relativity In 1909, at the young age of 44, Minkowski died suddenly from a ruptured appendix. Despite having an interest in mathematical physics and dabbling in this field, his main work was in the field of the geometry of number, although he is best remembered for his four-dimensional space-time Theory The abraham and minkowski tensors The Minkowski and abraham tensors lead to essentially different expressions for the density (g) of field momentum. In vector form they are given by g=(4m [DB] for the Minkowski form, and 4[EH] for the Abraham form Where D=EE+P and B=uo(H+ M) These lead to the following expressions for the photons momentum in the medium nul nhv according to Minkowski, and hy according to Abraham where I hand represents the length of the line of plane polarised waves in the l avepacket Note that in a vacuum the tensors are identical; problems arise only in connexion with electromagnetic fields in matter [5]. The Minkowski energy-momentum tensor is asymmetric, implying non-conservation of angular momentum. Abraham 7
7 Minkowski explored the arithmetic of quadratic forms, especially that concerning n variables, and his research into that topic led him to consider certain geometric properties in a space of n dimensions. In 1896, he presented his geometry of numbers, a geometrical method that solved problems in number theory. In 1902, he joined the Mathematics Department of Göttingen, where he held the third chair in mathematics, created for him at David Hilbert's request. By 1907 Minkowski realised that the special theory of relativity, introduced by Einstein in 1905 and based on previous work of Lorentz and Poincaré, could be best understood in a non-Euclidean space, since known as "Minkowski space", in which the time and space are not separate entities but intermingled in a four dimensional space-time, and in which the Lorentz geometry of special relativity can be nicely represented. This technique certainly helped Einstein's quest for general relativity. In 1909, at the young age of 44, Minkowski died suddenly from a ruptured appendix. Despite having an interest in mathematical physics and dabbling in this field, his main work was in the field of the geometry of number, although he is best remembered for his four-dimensional space-time. Theory The Abraham and Minkowski tensors The Minkowski and Abraham tensors lead to essentially different expressions for the density (g) of field momentum. In vector form they are given by: [ ] 4 1 DB = c g M π for the Minkowski form, and [ ] 4 1 EH = c g A π for the Abraham form. Where D = ε 0E + P and ( ) B = µ 0 H + M . These lead to the following expressions for the photon’s momentum in the medium: c nh c nul G g l M M ν = = = according to Minkowski, and nc h G g l A A ν = = according to Abraham where u h l ν = and represents the length of the line of plane-polarised waves in the wavepacket Note that in a vacuum the tensors are identical; problems arise only in connexion with electromagnetic fields in matter [5]. The Minkowski energy-momentum tensor is asymmetric, implying non-conservation of angular momentum. Abraham
rendered the tensor symmetric by changing Minkowski momentum density from D×BtoE×H/c2 Brevik,s [6] comment on the two tensors was The two tensors correspond merely to different distributions of forces and torques throughout the body. According to minkowski the torque is essentially a volume effect, described by the tensor asymmetry, while according to abraham the torque is described completely in terms of the force density Radiation Pressure The minute pressure exerted on a surface in the direction of propagation of the incident electromagnetic radiation is called radiation pressure. The fact that electromagnetic radiation exerts a pressure upon any surface exposed to it was deduced theoretically by James Clerk Maxwell in 1871, and proven experimentally by lebedev in 1900 and by nichols and hull in 1901 In quantum mechanics, radiation pressure can be interpreted as the transfer of momentum from photons as they strike a surface. Radiation pressure on dust grains in space can dominate over gravity and this explains why the tail of a comet always points away from the Sun The pressure is very feeble, but it can be demonstrated with a Nichols radiometer Consider a laser beam trained upon the black face of one of the radiometer vanes. It will be absorbed(hence the surface looks black). If, before arrival, the light had some associated linear momentum then due to conservation of momentum within the system something else now has to be moving in the direction the light was travelling because the photons have been absorbed and come to a halt. The vane therefore begins to move Now consider the light hitting the shiny side of a vane. The shininess is an indication that the light is bouncing off the surface, which means that it has completely changed direction and is now travelling the other way In this case the momentum imparted to the vane must be twice that imparted when the photon is absorbed, so that the total momentum is conserved Momentum of light Radiation pressure has shown that light must carry momentum. Three different forms of momentum have been discovered 1. Linear momentum: the original form considered in radiation pressure 2. Angular momentum photons can carry an angular momentum of th in the direction of propagation the sign depends on which direction they have been circularly polarised 3. Orbital angular momentum: from changing the position of the wavefront to obtain a spiral beam It is a property of the transverse mode pattern, and each photon possesses In of angular momentum, where I is the number of intertwined helices
8 rendered the tensor symmetric by changing Minkowski’s momentum density from D×B to 2 E× H / c . Brevik’s [6] comment on the two tensors was: “The two tensors correspond merely to different distributions of forces and torques throughout the body. According to Minkowski the torque is essentially a volume effect, described by the tensor asymmetry, while according to Abraham the torque is described completely in terms of the force density.” Radiation Pressure The minute pressure exerted on a surface in the direction of propagation of the incident electromagnetic radiation is called radiation pressure. The fact that electromagnetic radiation exerts a pressure upon any surface exposed to it was deduced theoretically by James Clerk Maxwell in 1871, and proven experimentally by Lebedev in 1900 and by Nichols and Hull in 1901. In quantum mechanics, radiation pressure can be interpreted as the transfer of momentum from photons as they strike a surface. Radiation pressure on dust grains in space can dominate over gravity and this explains why the tail of a comet always points away from the Sun. The pressure is very feeble, but it can be demonstrated with a Nichols radiometer. Consider a laser beam trained upon the black face of one of the radiometer vanes. It will be absorbed (hence the surface looks black). If, before arrival, the light had some associated linear momentum, then due to conservation of momentum within the system something else now has to be moving in the direction the light was travelling because the photons have been absorbed and come to a halt. The vane therefore begins to move. Now consider the light hitting the shiny side of a vane. The shininess is an indication that the light is bouncing off the surface, which means that it has completely changed direction and is now travelling the other way. In this case the momentum imparted to the vane must be twice that imparted when the photon is absorbed, so that the total momentum is conserved. Momentum of light Radiation pressure has shown that light must carry momentum. Three different forms of momentum have been discovered: 1. Linear momentum: the original form considered in radiation pressure. 2. Angular momentum: photons can carry an angular momentum of ±ħ in the direction of propagation – the sign depends on which direction they have been circularly polarised. 3. Orbital angular momentum: from changing the position of the wavefront to obtain a spiral beam. It is a property of the transverse mode pattern, and each photon possesses lħ of angular momentum, where l is the number of intertwined helices
Poynting Vector The Poynting vector describes the flow of energy(power) through a surface in terms of electric and magnetic properties. It is the vector product of the electric and the magnetic fields. The Poynting vector points in the direction of propagation of a travelling electromagnetic wave and has the dimensions of power per area The full electromagnetic energy density in a region of space where there are both electric fields and magnetic fields is given by adding up separate contributions from the electric and magnetic fields. This implies energy is stored in the field itself. A unit cube of empty space which contains electric and magnetic fields will have some finite energy. This means electric fields and magnetic fields have a real hysical existence, like particles This energy can also flow around, and the energy current is expressed by the Poynting vector S=ExH Pseudomomentum Despite the temptation to believe otherwise, it is important to remember that it is only by differentiating with respect to the "real"metric that we obtain real momentum. When we differentiate with respect to the analogy metric, we obtain the density and flux of another quantity termed pseudomomentum Nelson [7] notes that momentum is a conserved quantity by virtue of the homogeneity of space, that is, as a result of the invariance of the laws of physics when spatial coordinates are moved. He states that pseudomomentum can also be a conserved quantity provided the medium is homogenous that is the laws of hysics are invariant to translations of the material coordinates In McIntyre's paper [8]on wave momentum, he explains a lot of the controversy as due to people's mistaken beliefs about momentum. He states: Momentum density and momentum flux are independent entities.. fluxes of momentum can perfect well exist in a material medium without there being any momentum". He concludes Abrahams momentum is the electromagnetic contribution to the actual momentum, while Minkowski's is the pseudomomentum There has certainly been no clear understanding of momentum of light in media and at the boundary between media and vacuum. a lot of problems stem from vague definitions -what one physicist means when he states their momentum need not mean the same to another physicist Tensors Tensors provide a formalism that helps to solve and model certain problems more easily. A matrix is a specialised form of a tensor, occupying two dimensions 9
9 Poynting Vector The Poynting vector describes the flow of energy (power) through a surface in terms of electric and magnetic properties. It is the vector product of the electric and the magnetic fields. The Poynting vector points in the direction of propagation of a travelling electromagnetic wave and has the dimensions of power per area. The full electromagnetic energy density in a region of space where there are both electric fields and magnetic fields is given by adding up separate contributions from the electric and magnetic fields. This implies energy is stored in the field itself. A unit cube of empty space which contains electric and magnetic fields will have some finite energy. This means electric fields and magnetic fields have a real physical existence, like particles. This energy can also flow around, and the energy current is expressed by the Poynting vector: S E H = × Pseudomomentum Despite the temptation to believe otherwise, it is important to remember that it is only by differentiating with respect to the "real" metric that we obtain "real" momentum. When we differentiate with respect to the analogy metric, we obtain the density and flux of another quantity, termed pseudomomentum. Nelson [7] notes that momentum is a conserved quantity by virtue of the homogeneity of space, that is, as a result of the invariance of the laws of physics when spatial coordinates are moved. He states that pseudomomentum can also be a conserved quantity provided the medium is homogenous, that is the laws of physics are invariant to translations of the material coordinates. In McIntyre’s paper [8] on wave momentum, he explains a lot of the controversy as due to people’s mistaken beliefs about momentum. He states: “Momentum density and momentum flux are independent entities... ...fluxes of momentum can perfectly well exist in a material medium without there being any momentum”. He concludes “Abraham’s momentum is the electromagnetic contribution to the actual momentum, while Minkowski’s is the pseudomomentum”. There has certainly been no clear understanding of momentum of light in media and at the boundary between media and vacuum. A lot of problems stem from vague definitions – what one physicist means when he states “their momentum” need not mean the same to another physicist. Tensors Tensors provide a formalism that helps to solve and model certain problems more easily. A matrix is a specialised form of a tensor, occupying two dimensions
Einstein's box This thought experiment, dreamed up by Einstein in 1905, was designed to determine the mass equivalence of a pulse of electromagnetic radiation(m= E/c) Einstein considered a closed system(a box)of mass M, which is initially at rest in an inertial frame of reference S. The walls at either end of the box are of equal mass. A photon is emitted from a photon gun on one wall, down the central axis of the box, towards the other end wall. From the phenomenon of radiation pressure (given by E= pc)we know the photon emitted must have momentum. Therefore to with velocity V. When the photon hits the far side of the box the system will come conserve the momentum of the system the box must move in the opposite directior to rest again, with a slightly different position to its starting point. This shift in position can be made arbitrarily large by repeating the process If the assumption that the carrier remains massless is valid then the system, which was initially at rest, will have its centre of mass shifted without any external forces acting on the system. This clearly violates the law of mechanics which states that a body, which is initially at rest, cannot undergo translational motion unless there is an external force acting on the body The only way that this problem is able to be solved is if the photon has transported an amount of mass- so that even though the box has moved the centre of mass is still about the original point Einstein's derivation has some conceptual problems with it, namely that the box is treated as a rigid body a concept that is inconsistent with the principles of special relativity. The approach taken by French circumvents the rigid body problem by considering only the two end walls of the box, arriving at the same result utilizing the centre-of-mass theorem Burt Peierls [9] and Jones have carried out Einstein box calculations applying it to the theory of optical momentum. Burt& peierls obtained results in agreements with the nondispersive Abraham expression, while Jones obtained the dispersive Abraham result( the equations for these are(6)and(10) respectively in Loudons paper [10). However, neither are happy with their calculations; Burt Peierls worry over the rigid box assumed in their calculation and the impact on the validity of the results; and Jones adds an artificial forward bodily impulse in order to attain his desired result of the minkowski form of the momentum Loudon [10]notes that the Einstein box is useful for the understanding of optical momentum but the very small shifts in position would be difficult to measure These thought experiments seem likely to remain in the mind and not to be realised on the laboratory bench The proponents At first people were swayed towards Minkowski's theory. In 1950 Laue [3] demonstrated certain limiting requirements that must be satisfied by the transformation properties of the components of the momentum-energy tensor of a light wave. Laue derived these requirements from the following considerations and his criterion was based on an incorrect assumption from another frame of
10 Einstein’s box This thought experiment, dreamed up by Einstein in 1905, was designed to determine the mass equivalence of a pulse of electromagnetic radiation (m = E/c2 ). Einstein considered a closed system (a box) of mass M, which is initially at rest in an inertial frame of reference S. The walls at either end of the box are of equal mass. A photon is emitted from a photon gun on one wall, down the central axis of the box, towards the other end wall. From the phenomenon of radiation pressure (given by E = pc) we know the photon emitted must have momentum. Therefore to conserve the momentum of the system the box must move in the opposite direction with velocity v. When the photon hits the far side of the box, the system will come to rest again, with a slightly different position to its starting point. This shift in position can be made arbitrarily large by repeating the process. If the assumption that the carrier remains massless is valid then the system, which was initially at rest, will have its centre of mass shifted without any external forces acting on the system. This clearly violates the law of mechanics which states that a body, which is initially at rest, cannot undergo translational motion unless there is an external force acting on the body. The only way that this problem is able to be solved is if the photon has transported an amount of mass – so that even though the box has moved, the centre of mass is still about the original point. Einstein’s derivation has some conceptual problems with it, namely that the box is treated as a rigid body, a concept that is inconsistent with the principles of special relativity. The approach taken by French circumvents the rigid body problem by considering only the two end walls of the box, arriving at the same result utilizing the centre-of-mass theorem. Burt & Peierls [9] and Jones have carried out Einstein box calculations applying it to the theory of optical momentum. Burt & Peierls obtained results in agreements with the nondispersive Abraham expression, while Jones obtained the dispersive Abraham result (the equations for these are (6) and (10) respectively in Loudon’s paper [10]). However, neither are happy with their calculations; Burt & Peierls worry over the rigid box assumed in their calculation and the impact on the validity of the results; and Jones adds an artificial ‘forward bodily impulse’ in order to attain his desired result of the Minkowski form of the momentum. Loudon [10] notes that “the Einstein box is useful for the understanding of optical momentum but the very small shifts in position would be difficult to measure. These thought experiments seem likely to remain in the mind and not to be realised on the laboratory bench”. The proponents At first people were swayed towards Minkowski’s theory. In 1950 Laue [3] demonstrated certain limiting requirements that must be satisfied by the transformation properties of the components of the momentum-energy tensor of a light wave. Laue derived these requirements from the following considerations and his criterion was based on an incorrect assumption from another frame of
