The Principle of Relativity

PART I. § 2.

§ 2. The Limiting Case. The Fundamental Equations for Äther.

By using the electron theory, Lorentz in his above mentioned essay traces the Laws of Electro-dynamics of Ponderable Bodies to still simpler laws. Let us now adhere to these simpler laws, whereby we require that for the limiting case ε = 1, μ = 1, σ = 0, they should constitute the laws for ponderable bodies. In this ideal limiting case ε = 1, μ = 1, σ = 0, E will be equal to e, and M to m. At every space time point (x, y, z, t) we shall have the equations

(i) Curl m - (δe/δt) = ρu

(ii) div e = ρ

(iii) Curl e + δm/δt = 0

(iv) div m = 0

I shall now write (x₁ x₂ x₃ x₄) for (x, y, z, t) and (ρ₁, ρ₂, ρ₃, ρ₄) for

$$ (\rho u{x}, \rho u{y}, \rho u_{z}, i\rho) $$

i.e. the components of the convection current ρu, and the electric density multiplied by √ -1

Further I shall write

f{2 3}, f_{3 1}, f_{1 2}, f_{1 4}, f_{2 4}, f__{3 4}.

for

m{x}, m{y}, m{z}, -ie{x}, -ie{y}, -ie{z}.

i.e., the components of m and (-i.e.) along the three axes; now if we take any two indices (h. k) out of the series

3, 4), f{k h} = -f_{k h_},

Therefore

f₃₂ = -f₂₃, f₁₃ = -f₃₁, f₂₁ = -f₁₂ f₄₁ = -f₁₄, f₄₄ = -f₂₄, f₄₃ = -f₃₄

Then the three equations comprised in (i), and the equation (ii) multiplied by i becomes

$$ \begin{vmatrix} & \frac{\delta f{1 2}}{\delta x{2}} & + \frac{\delta f{1 3}}{\delta x{3}} & + \frac{\delta f{1 4}}{\delta x{4}} & = \rho{1} \frac{\delta f{2 1}}{\delta x{1}} & & + \frac{\delta f{2 3}}{\delta x{3}} & \times \frac{\delta f{2 4}}{\delta x{4}} & = \rho{2} \frac{\delta f{3 1}}{\delta x{1}} & \times \frac{\delta f{3 2}}{\delta x{2}} & & + \frac{\delta f{3 4}}{\delta x{4}} & = \rho{3} \frac{\delta f{4 1}}{\delta x{1}} & + \frac{\delta f{4 2}}{\delta x{2}} & + \frac{\delta f{4 3}}{\delta x{3}} & & = \rho{4} \end{vmatrix} × $$

On the other hand, the three equations comprised in (iii) and the (iv) equation multiplied by (i) becomes

$$ \begin{vmatrix} & \frac{\delta f{3 4}}{\delta x{2}} & + \frac{\delta f{4 2}}{\delta x{3}} & + \frac{\delta f{2 3}}{\delta x{4}} & = = \frac{\delta f{4 3}}{\delta x{1}} & & + \frac{\delta f{1 4}}{\delta x{3}} & + \frac{\delta f{3 1}}{\delta x{4}} & = 0 \frac{\delta f{2 4}}{\delta x{1}} & + \frac{\delta f{4 1}}{\delta x{2}} & & + \frac{\delta f{1 2}}{\delta x{4}} & = 0 \frac{\delta f{3 2}}{\delta x{1}} & + \frac{\delta f{1 3}}{\delta x{2}} & + \frac{\delta f{2 1}}{\delta x{3}} & & = - \end{vmatrix} × $$

By means of this method of writing we at once notice the perfect symmetry of the 1st as well as the 2nd system of equations as regards permutation with the indices, (1, 2, 3, 4).

§ 3.

It is well-known that by writing the equations i) to iv) in the symbol of vector calculus, we at once set in evidence an invariance (or rather a (covariance) of the system of equations A) as well as of B), when the co-ordinate system is rotated through a certain amount round the null-point. For example, if we take a rotation of the axes round the z-axis, through an amount φ, keeping e, m fixed in space, and introduce new variables x₁′ x₂′ x₃′ x₄′ instead of x₁ x₂ x₃ x₄ where x′₁ = x₁ cos φ + x₂ sin φ, x′₂ = -x₁ sin φ + x₂ cos φ, x′₃ = x₃, x′₄ = x₄, and introduce magnitudes ρ′₁, ρ′₂, ρ′₃, ρ′₄, where ρ₁′ = ρ₁ cos φ + ρ₂ sin φ, ρ₂′ = - ρ₁ sin φ + ρ₂ cos φ and f′{1 2}, ... ... f′__{3 4}, where

f′₂₃ = f₂₃ cos φ + f₃₁ sin φ, f′₃₁ = - f₂₃ sin φ + f₃₁ cos φ, f′₁₂ = f₁₂, f′₁₄ = f₁₄ cos φ + f₂₄ sin φ, f′₂₄ = - f₁₄ sin φ + f₂₄ cos φ, f′₃₄ = f₃₄{3 4}, f′_{k h} = - f_{k h_} (h l k = 1, 2, 3, 4).

then out of the equations (A) would follow a corresponding system of dashed equations (A´) composed of the newly introduced dashed magnitudes.

So upon the ground of symmetry alone of the equations (A) and (B) concerning the suffixes (1, 2, 3, 4), the theorem of Relativity, which was found out by Lorentz, follows without any calculation at all.

I will denote by iψ a purely imaginary magnitude, and consider the substitution

x₁′ = x₁, x₂′ = x₂, x₃′ = x₃ cos iψ + x₄ sin iψ, (1) x₄′´ = - x₃ sin iψ + x₄ cos iψ,

Putting

$$ - i \tan i\psi = \frac{e^{\psi} - e^{-\psi}}{e^{\psi}+e^{-\psi}} = q $$ ,

$$ \psi = \frac{1}{2} \log \frac{1 + q}{1 - q′} $$ (2)

We shall have cos iψ = 1/√(1 - q²), sin iψ = iq/√(1 - q²) where -1 < q < 1, and √(1 - q²) is always to be taken with the positive sign.

Let us now write x′₁ = x′, x′₂ = y′, x′₃ = z′, x′₄ = it′ (3)

then the substitution 1) takes the form

x′ = x, y′ = y, z′ = (z - qt)/√(1 - q²), t′ = (-qz + t)/√(1 - q²), (4)

the coefficients being essentially real.

If now in the above-mentioned rotation round the Z-axis, we replace 1, 2, 3, 4 throughout by 3, 4, 1, 2, and φ by iψ, we at once perceive that simultaneously, new magnitudes ρ′₁, ρ′₂, ρ′₃, ρ′₄, where

ρ′₁ = ρ₁, ρ′₂ = ρ₂, ρ′₃ = ρ₃ cos iψ + ρ₄ sin iψ, ρ′₄ = - ρ₃ sin iψ + ρ₄ cos iψ),

and f′{1 2} ... f′__{3 4}, where

f′{4 1} = f_{4 1} cos iψ + f_{1 3} sin iψ, f′_{1 3} = - f_{4 1} sin iψ + f_{1 3} cos iψ, f′_{3 4} = f_{3 4}, f′_{3 2} = f_{3 2} cos iψ + f_{4 2} sin iψ, f′_{4 2} = - f_{3 2} sin iψ + f_{4 2} cos iψ, f′_{1 2} = f_{1 2}, f_{k h} = - f′_{k h_},

must be introduced. Then the systems of equations in (A) and (B) are transformed into equations (A´), and (B´), the new equations being obtained by simply dashing the old set.

All these equations can be written in purely real figures, and we can then formulate the last result as follows.

If the real transformations 4) are taken, and x´ y´ z´ t´ be taken as a new frame of reference, then we shall have

(5) ρ´ = ρ [(-qu{z} + 1)/√(1 - q²)], ρ´u_{z}´ = ρ[(u_{z} - q)/√(1 - q²)], ρ´u_{x}´ = ρu_{x}, ρ´u_{y}´ = ρu_{y_}.

(6) e´{x´} = (e_{x} - qm_{y})/(√(1 - q²)), m´_{r´} = (qe_{x} + m_{y})/(√(1 - q²)), e´_{z´} = e_{z_}.

(7) m´{x´} = (m_{x} - qe_{y})/(√(1 - q²)), e´_{y´} = (qm_{x} + e_{y})/(√(1 - q²)), m´{z´} = m{z_}.

Then we have for these newly introduced vectors u´, e´, m´ (with components u{x}´, u_{y}´, u_{z}´; e_{x}´, e_{y}´, e_{z}´; m_{x}´, m_{y}´, m_{z_}´), and the quantity ρ´ a series of equations I´), II´), III´), IV´) which are obtained from I), II), III), IV) by simply dashing the symbols.

We remark here that e{x} - qm_{y}, e_{y} + qm_{x} are components of the vector e + [vm], where v is a vector in the direction of the positive Z-axis, and | v | = q, and [vm] is the vector product of v and m; similarly -qe_{x} + m_{y}, m_{x} + qe_{y} are the components of the vector m - [ve_].

The equations 6) and 7), as they stand in pairs, can be expressed as.

e′{x′} + im′_{x′} = (e_{x} + im_{x}) cos iψ + (e_{y} + im_{y}) sin i_ψ,

e′{y′} + im′_{y′} = - (e_{x} + im_{x}) sin iψ + (e_{y} + im_{y}) cos i_ψ,

e′{z′} + im′_{z′} = e′_{z} + im_{z_}.

If φ denotes any other real angle, we can form the following combinations:—

(e′{x′} + im′_{x′}) cos. φ + (e′_{y″} + im′_{y′_}) sin φ

= (e{x} + im_{x}) cos. (φ + iψ) + (e_{y} + im_{y}) sin (φ + i_ψ),

= (e′{x′} + im′_{x′}) sin φ + (e′_{y′} + im′_{y′_}) cos. φ

= - (e{x} + im_{x}) sin (φ + iψ) + (e_{y} + im_{y}) cos. (φ + i_ψ).

§ 4. Special Lorentz Transformation.

The rôle which is played by the Z-axis in the transformation (4) can easily be transferred to any other axis when the system of axes are subjected to a transformation about this last axis. So we came to a more general law:—

Let v be a vector with the components v{x}, v_{y}, v_{z}, and let | v | = q < 1. By ṽ we shall denote any vector which is perpendicular to v, and by r_{v}, r_{ṽ} we shall denote components of r in direction of ṽ and v_.

Instead of (x, y, z, t), new magnetudes (x′ y′ z′ t′) will be introduced in the following way. If for the sake of shortness, r is written for the vector with the components (x, y, z) in the first system of reference, r′ for the same vector with the components (x′ y′ z′) in the second system of reference, then for the direction of v, we have

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