The Principle of Relativity · Albert Einstein
D. THE “MATERIAL” PHENOMENA.
From (59), it follows that the system of equations (60) ∂F_{ρσ}/∂_x__{τ} + ∂F_{στ}/∂_x__{ρ} + ∂F_{τρ}/∂_x__{σ} = 0 is satisfied of which the left-hand side, according to (37), is an anti-symmetrical tensor of the third kind. This system (60) contains essentially four equations, which can be thus written:— { ∂F₂₃/∂_x₄_ + ∂F₃₄/∂_x₂_ ∂F₄₂/∂_x₃_ = 0 { { ∂F₃₄/∂_x₁_ + ∂F₄₁/∂_x₃_ ∂F₁₃/∂_x₄_ = 0 (60a) { { ∂F₄₁/∂_x₂_ + ∂F₁₂/∂_x₄_ ∂F₂₄/∂_x₁_ = 0 { { ∂F₁₂/∂_x₃_ + ∂F₂₃/∂_x₁_ ∂F₃₁/∂_x₂_ = 0. This system of equations corresponds to the second system of equations of Maxwell. We see it at once if we put { F₂₃ = H_{_x_} F₁₄ = E_{_x_} { (61) { F₃₁ = H_{_y_} F₂₄ = E_{_y_} { { F₁₂ = H_{_z_} F₃₄ = E_{_z_} Instead of (60a) we can therefore write according to the usual notation of three-dimensional vector-analysis:— { ∂H/∂_t_ + rot E = 0 (60b) { { div H = 0. The first Maxwellian system is obtained by a generalisation of the form given by Minkowski. We introduce the contra-variant six-vector F_{αβ} by the equation (62) F^{μν} = _g_^{μα} _g_^{νβ} F_{αβ}, and also a contra-variant four-vector J^μ, which is the electrical current-density in vacuum. Then remembering (40) we can establish the system of equations, which remains invariant for any substitution with determinant 1 (according to our choice of co-ordinates).
(63) ∂F^{μν}/∂_x__{ν} = J^μ If we put { F²³ = H′_{_x_} F¹⁴ = -E′_{_x_} { (64) { F³¹ = H′_{_y_} F²⁴ = -E′_{_y_} { { F¹² = H′_{_z_} F³⁴ = -E′_{_z_} which quantities become equal to H_{_x_} ... E_{_x_} in the case of the special relativity theory, and besides J^1 = _i__{_x_} ... J^4 = ρ we get instead of (63) { rot H′ - ∂E′/∂_t_ = _i_ (63a) { { div E′ = ρ The equations (60), (62) and (63) give thus a generalisation of Maxwell’s field-equations in vacuum, which remains true in our chosen system of co-ordinates. _The energy-components of the electro-magnetic field._ Let us form the inner-product (65) K_{σ} = F_{σμ} J^μ. According to (61) its components can be written down in the three-dimensional notation. { K₁ = ρE_{_x_} + [_i_, H]_{x} (65a) { — — — { K₄ = — (_i_, E). K_{σ} is a covariant four-vector whose components are equal to the negative impulse and energy which are transferred to the electro-magnetic field per unit of time, and per unit of volume, by the electrical masses. If the electrical masses be free, that is, under the influence of the electro-magnetic field only, then the covariant four-vector K_{σ} will vanish. In order to get the energy components T_{σ}^ν of the electro-magnetic field, we require only to give to the equation K_{σ} = 0, the form of the equation (57).
From (63) and (65) we get first, K_{σ} = F_{σμ} ∂F_{μν}/∂_x__{ν} = ∂/∂_x__{ν} (F_{σμ} F^{μν}) - F^{μν} ∂F_{σμ}/∂_x__{ν}. On account of (60) the second member on the right-hand side admits of the transformation— F^{μν} ∂F_{σμ}/∂_x__{ν} = -½ F^{μν} ∂F_{μν}/∂_x__{σ} = -½ _g_^{μα} _g_^{νβ} F_{αβ} ∂F_{μν}/∂_x__{σ}. Owing to symmetry, this expression can also be written in the form = -1/4 [_g_^{μα} _g_^{νβ} F_{αβ} ∂F_{μν}/∂_x__{σ} + _g_^{μα} _g_^{νβ} ∂F_{αβ}/∂_x__{σ} F_{μν}], which can also be put in the form - 1/4 ∂/∂_x__{σ} (_g_^{μα} _g_^{νβ} F_{αβ} F_{μν}) + 1/4 F_{αβ} F_{μν} ∂/∂_x__{σ} (_g_^{μα} _g_^{νβ}). The first of these terms can be written shortly as - 1/4 ∂/∂_x__{σ} (F^{μν} F_{μν}), and the second after differentiation can be transformed in the form - ½ F^{μτ} F_{μν} _g_^{νρ} ∂_g__{στ}/∂_x__{σ}. If we take all the three terms together, we get the relation (66) K_{σ} = ∂τ_{σ}^ν/∂_x__{ν} - ½ _g_^{τμ} ∂_g__{μν}/∂_x__{σ} τ_{τ}^ν where (66a) τ_{σ}^ν = -F_{σα} F^{να} + 1/4 δ_{σ}^ν F_{αβ} F^{αβ}. On account of (30) the equation (66) becomes equivalent to (57) and (57a) when K_{σ} vanishes. Thus τ_{σ}^ν’s are the energy-components of the electro-magnetic field. With the help of (61) and (64) we can easily show that the energy-components of the electro-magnetic field, in the case of the special relativity theory, give rise to the well-known Maxwell-Poynting expressions.
We have now deduced the most general laws which the gravitation-field and matter satisfy when we use a co-ordinate system for which √(-_g_) = 1. Thereby we achieve an important simplification in all our formulas and calculations, without renouncing the conditions of general covariance, as we have obtained the equations through a specialisation of the co-ordinate system from the general covariant-equations. Still the question is not without formal interest, whether, when the energy-components of the gravitation-field and matter is defined in a generalised manner without any specialisation of co-ordinates, the laws of conservation have the form of the equation (56), and the field-equations of gravitation hold in the form (52) or (52a); such that on the left-hand side, we have a divergence in the usual sense, and on the right-hand side, the sum of the energy-components of matter and gravitation. I have found out that this is indeed the case. But I am of opinion that the communication of my rather comprehensive work on this subject will not pay, for nothing essentially new comes out of it. E. §21. Newton’s theory as a first approximation. We have already mentioned several times that the special relativity theory is to be looked upon as a special case of the general, in which _g__{μν}’s have constant values (4). This signifies, according to what has been said before, a total neglect of the influence of gravitation. We get one important approximation if we consider the case when _g__{μν}’s differ from (4) only by small magnitudes (compared to 1) where we can neglect small quantities of the second and higher orders (first aspect of the approximation.)
Further it should be assumed that within the space-time region considered, _g__{μν}’s at infinite distances (using the word infinite in a spatial sense) can, by a suitable choice of co-ordinates, tend to the limiting values (4); _i.e._, we consider only those gravitational fields which can be regarded as produced by masses distributed over finite regions. We can assume that this approximation should lead to Newton’s theory. For it however, it is necessary to treat the fundamental equations from another point of view. Let us consider the motion of a particle according to the equation (46). In the case of the special relativity theory, the components _dx₁_/_ds_, _dx₂_/_ds_, _dx₃_/_ds_, can take any values. This signifies that any velocity _v_ = √((_dx₁_/_dx₄_)² + (_dx₂_/_dx₄_)² + (_dx₃_/_dx₄_)²) can appear which is less than the velocity of light in vacuum (_v_ < 1). If we finally limit ourselves to the consideration of the case when _v_ is small compared to the velocity of light, it signifies that the components _dx₁_/_ds_, _dx₂_/_ds_, _dx₃_/_ds_, can be treated as small quantities, whereas _dx₄_/_ds_ is equal to 1, up to the second-order magnitudes (the second point of view for approximation). Now we see that, according to the first view of approximation, the magnitudes γ_{μν}^τ’s are all small quantities of at least the first order. A glance at (46) will also show, that in this equation according to the second view of approximation, we are only to take into account those terms for which μ = ν = 4.
Ideas discussed in this chapter
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