Special Relativity for Physicists

G. Stephenson, C. W. Kilmister

• Publisher: Dover Publications
• Year: 2019
• ISBN:
  9780486844527
• Topic: Physical Sciences
• Subtopic: Relativity in Physics

"Even in the most technical sections, the authors' writing is delightfully lucid, and they give many applications to classical and modern physics... Undergraduates, and those who require some understanding of special relativity for their work in other fields, will find this elegant work a pleasure to read." — Technology This concise account of special relativity is geared toward nonspecialists and belongs in the library of anyone interested in the subject and its applications to both classical and modern physics. The treatment takes a historical point of view, without making heavy demands on readers' mathematical abilities; in fact, the theory is developed without the use of tensor calculus, requiring only a working knowledge of three-dimensional vector analysis. Topics include detailed coverage of the Lorentz transformation, including optical and dynamical applications, and applications to modern physics. An excellent bibliography completes this compact, accessible presentation.

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APPLICATIONS TO MODERN PHYSICS

1. Relativistic Hydrodynamics

In recent years interest has increased in the study of the motion of an electrically conducting fluid in an electromagnetic field. The equations of this subject, usually known as ‘magneto-hydrodynamics’ or ‘hydro-magnetics ’, are obtained by combining Maxwell’s equations with the macroscopic hydrodynamical equations

where p, p, μ and v are respectively the pressure, mass density, coefficient of viscosity and velocity vector of the fluid. The interaction between these equations and Maxwell’s is obtained by expressing the current vector J in terms of v, and the force F in terms of the Lorentz force on a moving charged element of fluid.

The transformation properties of the two sets of equations are different, however, since the Maxwell equations are invariant under a Lorentz transformation and the hydrodynamical equations are invariant under the Newtonian transformation (Chapter 1, (2.1)). For velocities small compared with that of light no serious error is likely to be introduced by using the non-relativistic form (1.1). However it is desirable in order to establish the consistency of the combined theory to formulate a relativistic hydrodynamics.

For simplicity we will consider here only the motion of a perfect fluid (that is, one in which all viscous effects are neglected). Classically such a fluid is defined by the condition that the stress across a fluid element is normal to the element, and it follows therefore that this stress must be simply due to an isotropic pressure p. We take this definition over into our relativistic formulation with the additional restriction that p should be an invariant under the Lorentz transformation. This restriction is necessary if the stress is to be describable by a single scalar function p after transformation to a new coordinate-system. For a small proper-volume dτ₀ of fluid with rest-mass pdτ₀ and position vector r at its centre, the equation of motion is (Chapter 3, (3.4) )

For convenience, we consider an inertial frame in which

instantaneously, and convert the surface-integral into a volume integral. This gives

and therefore

Since p is a scalar the pressure field creates rest-mass as shown in Chapter 3, § 14, and the remaining equation of motion is therefore

Equations (1.2) and (1.3) now form a fou...