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