5.5. Electromagnetism. The relativistic version of Maxwell’s equa- tions uses the 4-vectors A μ and j μ , where A 0 = V , its space components are those of the usual vector potential; j 0 = 4 πρ , and j i = 4 π c J i for i = 1 , 2 , 3. Then Maxwell’s equations with the Lorenz gauge ∂ μ A μ = 0 become: (5.9) ∂ μ ∂ μ A α = j α . The Minkowski force on a moving charge is given by φ μ = q c F μν u ν , where (5.10) F μν = ∂ μ A ν - ∂ ν A μ is the field-strength tensor , and u μ = dx μ /dτ . In terms of F μν , Maxwell’s equations are: (5.11) ∂ ν F μν = j μ ∂ θ F μν + ∂ μ F νθ + ∂ ν F θμ = 0 . Equivalently, they can be expressed as * F [ λμ,ν ] = j μ and F [ λμ,ν ] = 0 respectively, where * F μν = 1 2 F αβ αβμν is the dual tensor of F μν and the brackets denote antisymmetrization. These equations can be derived from a variational principle with the following action: (5.12) S = - 1 4 Z F μν F μν d 3 x dt. Here the components of the tensor F μν play the role of generalized coordinates. On the other hand, the action of a charged particle in an electro- magnetic field is (5.13) S = q c Z A μ dx μ .
12 MIGUEL A. LERMA 6. General Relativity 6.1. Newtonian Gravitation. Newton’s gravitational law states that the force between two masses m 1 and m 2 at a distance r is (6.1) F = - k m 1 m 2 r 2 , where k = 6 . 670 × 10 - 11 N m 2 kg - 2 is Newton’s gravitational constant. The corresponding potential produced by a mass m is φ = km/r , and for a continuous fluid of density ρ it is given by the Poisson’s equation : (6.2) ∇ 2 φ = 4 πk ρ. Since Poisson’s equation is not Lorentz invariant, it is not appropriate for a relativistic theory of gravity. There are various ways of developing a covariant theory of gravita- tion in the frame of Special Relativity, but they are inconsistent or lead to predictions that do not match experimental observations. So a completely new approach is needed. 6.2. The Principle of Equivalence. In General Relativity it is pos- tulated that a gravitational field is locally indistinguishable from an accelerated frame. In Relativity non inertial frames are represented by non Cartesian coordinates in 4-space. So gravitation is not a “force”, but a change in the geometry of space-time. Particles that are not un- der the action of some other field behave like free particles, and their paths are still geodesics in space-time. 6.3. Geometry in a curved space-time. Space-time is represented by a pseudo-Riemannian 4-dimensional manifold. Its geometry is given by the (symmetric) metric tensor g μν . 4 The line element is (6.3) ds 2 = g μν dx μ dx ν . The usual derivatives of the components of a tensor respect to the co- ordinates are not covariant in a general frame, particularly in a curved space-time. This is due to the fact that the basic vectors e μ = ∂ ∂x μ are not constant. So, in particular (6.4) ∂ ν ( A μ e μ ) = ( ∂ ν A μ ) e μ + A μ ∂ ν e μ = ( ∂ ν A μ + Γ μ αν A α ) e μ , where Γ μ αν are the components of ∂ ν e α .
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- Physics, MIGUEL A. LERMA