The Principle of Relativity · Albert Einstein
[P. C. M.]
Note 5. See “Electrodynamics of Moving Bodies,” p. 5-8. Note 6. Field Equations in Minkowski’s Form. Equations (_i_) and (_ii_) become when expanded into Cartesians:— ∂_m__{_z_}/∂_y_ - ∂_m__{_y_}/∂_z_ - ∂_e__{_x_}/∂τ = ρν_{_x_} } ∂_m__{_x_}/∂_z_ - ∂_m__{_z_}/∂_x_ - ∂_e__{_y_}/∂τ = ρν_{_y_} } ... (1·1) ∂_m__{_y_}/∂_x_ - ∂_m__{_x_}/∂_y_ - ∂_e__{_z_}/∂τ = ρν_{_z_} } and ∂_e__{_x_}/∂_x_ + ∂_e__{_y_}/∂_y_ + ∂_e__{_z_}/∂_z_ = ρ (2·1) Substituting _x₁_, _x₂_, _x₃_, _x₄_ and _x_, _y_, _z_, and _i_τ; and ρ₁, ρ₂, ρ₃, ρ₄ for ρν_{_x_}, ρν_{_y_}, ρν_{_z_}, _i_ρ, where _i_ = √(-1). We get, ∂_m__{_z_}/∂_x₂_ - ∂_m__{_y_}/∂_x₃_ - _i_(∂_e__{_x_}/∂_x₄_) = ρν_{_x_}{ = ρ₁ } - ∂_m__{_z_}/∂_x₁_ + ∂_m__{_x_}/∂_x₃_ - _i_(∂_e__{_y_}/∂_x₄_) = ρν_{_y_} = ρ₂ } ... (1·2) ∂_m__{_y_}/∂_x₁_ - ∂_m__{_x_}/∂_x₂_ - _i_(∂_e__{_z_}/∂_x₄_) = ρν_{_z_}{ = ρ₃ } and multiplying (2·1) by i we get ∂_ie__{_x_}/∂_x₁_ + ∂_ie__{_y_}/∂_x₂_ + ∂_ie__{_z_}/∂_x₃_ = _i_ρ = ρ₄ ... ... (2·2) Now substitute _m__{_x_} = _f₂₃_ = -_f₃₂_ and _ie__{_x_} = _f_₄₁ = -_f₁₄_ _m__{_y_} = _f₃₁_ = -_f₁₃_ _ie__{_y_} = _f_₄₂ = -_f₂₄_ _m__{_z_} = _f₁₂_ = -_f₂₁_ _ie__{_z_} = _f_₄₃ = -_f₃₄_ and we get finally:— ∂_f₁₂_/∂_x₂_ + ∂_f₁₃_/∂_x₃_ + ∂_f₁₄_/∂_x₄_ = ρ₁ } ∂_f₂₁_/∂_x₁_ + ∂_f₂₃_/∂_x₃_ + ∂_f₂₄_/∂_x₄_ = ρ₂ } ... (3) ∂_f₃₁_/∂_x₁_ + ∂_f₃₂_/∂_x₂_ + ∂_f₃₄_/∂_x₄_ = ρ₃ } ∂_f₄₁_/∂_x₁_ + ∂_f₄₂_/∂_x₂_ + ∂_f₄₃_/∂_x₃_ = ρ₄ } Note 9. On the Constancy of the Velocity of Light.
Page 12—refer also to page 6, of Einstein’s paper. One of the two fundamental Postulates of the Principle of Relativity is that the velocity of light should remain constant whether the source is moving or stationary. It follows that even if a radiant source S move with a velocity _u_, it should always remain the centre of spherical waves expanding outwards with velocity _c_. At first sight, it may not appear clear why the velocity should remain constant. Indeed according to the theory of Ritz, the velocity should become _c_ + _u_, when the source of light moves towards the observer with the velocity _u_. Prof. de Sitter has given an astronomical argument for deciding between these two divergent views. Let us suppose there is a double star of which one is revolving about the common centre of gravity in a circular orbit. Let the observer be in the plane of the orbit, at a great distance Δ.
The light emitted by the star when at the position A will be received by the observer after a time, Δ/(_c_ + _u_) while the light emitted by the star when at the position B will be received after a time Δ/(_c_ - _u_). Let T be the real half-period of the star. Then the observed half-period from B to A is approximately T - 2Δ_u_/_c²_ and from A to B is T + 2Δ_u_/_c²_. Now if 2_u_Δ/_c²_ be comparable to T, then it is impossible that the observations should satisfy Kepler’s Law. In most of the spectroscopic binary stars, 2_u_Δ/_c²_ are not only of the same order as T, but are mostly much larger. For example, if _u_ = 100 _km_/sec, T = 8 days, Δ/_c_ = 33 years (corresponding to an annual parallax of ·1″), then T - 2_u_Δ/_c²_ = 0. The existence of the Spectroscopic binaries, and the fact that they follow Kepler’s Law is therefore a proof that _c_ is not affected by the motion of the source. In a later memoir, replying to the criticisms of Freundlich and Günthick that an apparent eccentricity occurs in the motion proportional to _ku_Δ₀, _u₀_ being the maximum value of _u_, the velocity of light emitted being _u₀_ = _c_ + _ku_, _k_ = 0 Lorentz-Einstein _k_ = 1 Ritz.
Prof. de Sitter admits the validity of the criticisms. But he remarks that an upper value of _k_ may be calculated from the observations of the double star β-Aurigae. For this star, the parallax π = ·014″, _e_ = ·005, _u₀_ = 110 _km_/sec, T = 3·96, Δ > 65 light-years, _k_ is < ·002. For an experimental proof, see a paper by C. Majorana. Phil. Mag., Vol. 35, p. 163. Note 10. Rest-density of Electricity. If ρ is the volume density in a moving system then ρ√(1 - _u²_) is the corresponding quantity in the corresponding volume in the fixed system, that is, in the system at rest, and hence it is termed the rest-density of electricity.
Text from Project Gutenberg, public domain in the United States.