The inertia of stress We present a simple example in which the importance of the inertial effects of stress is evident. The system is an insulating solid narrow disc whose faces are uniformly charged with charges of equal magnitude and opposite signs. The motion of the system in two different directions is considered. It is shown how the contributions to energy and momentum of the stress that develops inside the body to balance the electrostatic forces have to be added to the electromagnetic contributions to obtain the results predicted by the relativistic equivalence of mass and energy. Published as: Rodrigo Medina, Am. J. Phys. 74,1031 (2006) DOI: 10.1119/1.2338550 arXiv categories: physics.class-ph [physics/0609144] The inertia of stress The inertia of stress Rodrigo Medina rmedina@ivic.ve Instituto Venezolano de Investigaciones Científicas, IVIC, Apartado 21827, Caracas 1020A, Venezuela Abstract We present a simple example in which the importance of the inertial effects of stress is evident. The system is an insulating solid narrow disc whose faces are uniformly charged with charges of equal magnitude and opposite signs. The motion of the system in two different directions is considered. It is shown how the contributions to energy and momentum of the stress that develops inside the solid to balance the electrostatic forces have to be added to the electromagnetic contributions to obtain the results predicted by the relativistic equivalence of mass and energy. The following article has been accepted by the American Journal of Physics. After it is published, it will be found at http://scitation.aip.org/ajp. pacs: 03.50.De I Introduction Recently, a proposal for the solution of the century old problem of the self-interaction of a charged particle was presented. Medina One of the puzzles of this problem Rohrlich is that the momentum of the electromagnetic field of a particle with electrostatic energy U e subscript 𝑈 𝑒 U_{e} moving with velocity 𝐯 𝐯 {\bf v} is not U e ​ γ ​ 𝐯 subscript 𝑈 𝑒 𝛾 𝐯 U_{e}\gamma{\bf v} as required by relativity, but is 4 3 ​ U e ​ γ ​ 𝐯 4 3 subscript 𝑈 𝑒 𝛾 𝐯 \frac{4}{3}U_{e}\gamma{\bf v} . It was shown Medina that the discrepancy is due to the neglect of the inertia of the stress that is present in the particle to balance the electrostatic repulsion. Unlike the inertia of energy, which is well known, many physicists are not aware of the inertia of pressure (stress). In many cases such an effect is negligible, but for the case of the stress produced by electrostatic (15b) In this case the result is also not as might be expected. The energy has an extra ( v / c ) 2 superscript 𝑣 𝑐 2 (v/c)^{2} term and the momentum is twice the expected value. The energy and momentum of the electromagnetic field do not form a four-vector, and the effective mass is anisotropic. Something is missing and that is the inertia of stress. Figure 5: Disc moving in the x 𝑥 x -direction. The thickness is the same, but the area of the faces is reduced by Lorentz contraction. The electric field is increased because of the increase in the surface charge density, σ ′ = σ ​ γ superscript 𝜎 ′ 𝜎 𝛾 \sigma^{\prime}=\sigma\gamma . There is a magnetic field 𝐁 ′ = c − 2 ​ 𝐯 × 𝐄 ′ superscript 𝐁 ′ superscript 𝑐 2 𝐯 superscript 𝐄 ′ {\bf B}^{\prime}=c^{-2}{\bf v}\times{\bf E}^{\prime} . Therefore there is a contribution to the momentum. V The Inertia of Stress We will use the following relativistic conventions, x 0 = − x 0 = c ​ t superscript 𝑥 0 subscript 𝑥 0 𝑐 𝑡 x^{0}=-x_{0}=ct , x 1 = x 1 = x superscript 𝑥 1 subscript 𝑥 1 𝑥 x^{1}=x_{1}=x , x 2 = x 2 = y superscript 𝑥 2 subscript 𝑥 2 𝑦 x^{2}=x_{2}=y , and x 3 = x 3 = z superscript 𝑥 3 subscript 𝑥 3 𝑧 x^{3}=x_{3}=z . Greek indices take the values 0–3, and Latin indices take the values 1–3. The relativistic dynamics of a continuous medium is ruled by the energy and momentum conservation equation EqRelativ ∇ α ( Θ α ​ β + P α ​ β ) = f β , subscript ∇ 𝛼 superscript Θ 𝛼 𝛽 superscript 𝑃 𝛼 𝛽 superscript 𝑓 𝛽 \nabla_{\alpha}(\Theta^{\alpha\beta}+P^{\alpha\beta})=f^{\beta}, (16) where f β superscript 𝑓 𝛽 f^{\beta} is the force density four-vector, Θ α ​ β superscript Θ 𝛼 𝛽 \Theta^{\alpha\beta} is the energy, and momentum density four-tensor, and P α ​ β superscript 𝑃 𝛼 𝛽 P^{\alpha\beta} is the stress four-tensor. Both Θ α ​ β superscript Θ 𝛼 𝛽 \Theta^{\alpha\beta} and P α ​ β superscript 𝑃 𝛼 𝛽 P^{\alpha\beta} are symmetric tensors. Also note that the stress that multiplies the velocity in the time derivative of Eq. ( 20 ) does not vanish in the small velocity limit ( v / c → 0 → 𝑣 𝑐 0 v/c\to 0 ). This case is an example in which the non-relativistic limit ( c → ∞ → 𝑐 c\to\infty ) is different from the small velocity limit. That is, the inertia of stress is a purely relativistic phenomenon, which does not have a Newtonian explanation. VI Stress contributions to energy and momentum We now calculate the contributions of stress to the energy and momentum. The stress of matter also contributes to the energy and momentum, and also do not form a four-vector. The contributions of the stress of matter are exactly opposite to those of the stress of the field. Thus, if the contributions of the stress of matter are added to those of the field, the total energy and momentum transform as a four-vector. Acknowledgments I wish to thank Dr. Victor Villalba and Dr. Enrnesto Medina for many very useful discussions and for reading the manuscript. References (1) Rodrigo Medina, “Radiation reaction of a classical quasi-rigid extended particle,” J. Phys. A: Math. Gen. 39 , 3801–3816 (2006). arXiv:physics/0508031 (2) Fritz Rohrlich, “The dynamics of a charged sphere and the electron,” Am. J. Phys. 65 , 1051–1056 (1997). (3) Peter G. Bergmann, Introduction to the Theory of Relativity (Dover, New York, 1976), p. 123. (4) Reference  StressDeff, , p. 127. (5) Reference  StressDeff, , p. 124. ◄ Feeling lucky? Conversion report Report an issue View original on arXiv ►