We now note that Eqs. (3)and(4) apply only to the total energy-momentum tensor, and not to the electromagnetic or material components in isolation, i.e EM TaB ao ( Teg It therefore follows that the lack of diagonal symmetry in Minkowski's electromagnetic energy-momentum tensor does not violate conservation of angular momentum as a complementary asymmetry in the material energy- momentum tensor ensures that the total energy-momentum tensor is symmetric. Any choice of electromagnetic energy-momentum tensor is equally valid provided the corresponding material counterpart is also taken into consideration, as it is only the total energy-momentum tensor which is uniquely defined consequences The consequences for optical tweezers are threefold. First, there is the recognition that the form of the elec- tromagnetic energy-momentum tensor in a dielectric medium is not of crucial importance. This also serves to validate the usage of the relative refractive index for a body suspended within a dielectric fluid: This change has no effect on the total energy-momentum tensor, and so although the medium is eliminated by converting to an equivalent situation in vacuum, no physically relevant information is lost Second, there is the important recognition that the Minkowski electromagnetic energy-momentum tensor has a material counterpart. Historically this was overlooked for a long time because the counterpart carries no linear momentum and hence is unnecessary in experiments such as Jones and Richard 16 mentioned above. However the material counterpart does carry angular momentum and this is illustrated extremely well by the thought experiment described by Padgett et al.38 where a laser beam carrying orbital angular momentum is passed through a glass disc. The Abraham tensor pair demonstrates a transfer of angular momentum to the glass disc whereas if the Minkowski electromagnetic energy-momentum tensor is considered in isolation, this does not take plac Third, we have discovered that the material disturbance may have significant physical effects on experimental predictions, and travels at the same speed as the electromagnetic wave. 9 It is therefore inappropriate to treat material objects as rigid bodies when analysing their behaviour, as this corresponds to instantaneous, i.e. superluminal, transfer of momentum throughout the body, whereas in reality the traversal of the electromagnetic wave is inevitably accompanied by physical pressures and deformations within the medium, and at the boundaries of both the medium and the beam. It is similarly wrong to neglect the material counterpart entirely or to assume that its propagation is negligibly slow 3. KHRAPKOS PARADOX 3.1. Background In 2001, Khrapko asked in American Journal of Physics, "Does plane wave not carry spin? "[sic]. 2 This question, which is not as simple as it first appears, arises from the existence of two separate expressions for the total angular momentum of an electromagnetic wave. These are (r×S/e2)dv A Reli 2u (9) where L represents total angular momentum, r is the position vector, s is the real instantaneous Poynting vector, c is the speed of light, w is the angular frequency of the electromagnetic wave, and e is the complex electric fiel We shall term Eq( 8)the macroscopic expression, and Eq.(9) the microscopic expression. The former arises from the obvious construction of r x p, where p represents linear momentum, and the latter may be obtainedWe now note that Eqs. (3) and (4) apply only to the total energy–momentum tensor, and not to the electromagnetic or material components in isolation, i.e. ∂α  T αβ EM + T αβ mat = 0 (6)  T αβ EM + T αβ mat =  T βα EM + T βα mat . (7) It therefore follows that the lack of diagonal symmetry in Minkowski’s electromagnetic energy–momentum tensor does not violate conservation of angular momentum as a complementary asymmetry in the material energy– momentum tensor ensures that the total energy–momentum tensor is symmetric. Any choice of electromagnetic energy–momentum tensor is equally valid provided the corresponding material counterpart is also taken into consideration, as it is only the total energy–momentum tensor which is uniquely defined. 2.3. Consequences The consequences for optical tweezers are threefold. First, there is the recognition that the form of the elec￾tromagnetic energy–momentum tensor in a dielectric medium is not of crucial importance. This also serves to validate the usage of the relative refractive index for a body suspended within a dielectric fluid: This change has no effect on the total energy–momentum tensor, and so although the medium is eliminated by converting to an equivalent situation in vacuum, no physically relevant information is lost. Second, there is the important recognition that the Minkowski electromagnetic energy–momentum tensor has a material counterpart. Historically this was overlooked for a long time because the counterpart carries no linear momentum and hence is unnecessary in experiments such as Jones and Richard’s16 mentioned above. However, the material counterpart does carry angular momentum and this is illustrated extremely well by the thought experiment described by Padgett et al.38 where a laser beam carrying orbital angular momentum is passed through a glass disc. The Abraham tensor pair demonstrates a transfer of angular momentum to the glass disc, whereas if the Minkowski electromagnetic energy–momentum tensor is considered in isolation, this does not take place. Third, we have discovered that the material disturbance may have significant physical effects on experimental predictions, and travels at the same speed as the electromagnetic wave.19 It is therefore inappropriate to treat material objects as rigid bodies when analysing their behaviour, as this corresponds to instantaneous, i.e. superluminal, transfer of momentum throughout the body, whereas in reality the traversal of the electromagnetic wave is inevitably accompanied by physical pressures and deformations within the medium, and at the boundaries of both the medium and the beam. It is similarly wrong to neglect the material counterpart entirely or to assume that its propagation is negligibly slow. 3. KHRAPKO’S PARADOX 3.1. Background In 2001, Khrapko asked in American Journal of Physics, “Does plane wave not carry spin?”[sic].2 This question, which is not as simple as it first appears, arises from the existence of two separate expressions for the total angular momentum of an electromagnetic wave. These are L = Z V (r × S/c2 ) dV (8) and L = Z V Re  iε (E∗ × E) 2ω  dV, (9) where L represents total angular momentum, r is the position vector, S is the real instantaneous Poynting vector, c is the speed of light, ω is the angular frequency of the electromagnetic wave, and E is the complex electric field. We shall term Eq. (8) the macroscopic expression, and Eq. (9) the microscopic expression. The former arises from the obvious construction of r × p, where p represents linear momentum, and the latter may be obtained