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

(10) r′{v} = (r_{v} - qt)/√(1 - q²_)

and for the perpendicular direction ṽ,

(11) r′{ṽ} = r_{ṽ_}

and further (12) t′ = (-qr{v} + t)/√(1 - q²_).

The notations (r′{ṽ}, r′_{v}) are to be understood in the sense that with the directions v, and every direction ṽ perpendicular to v in the system (x, y, z) are always associated the directions with the same direction cosines in the system (x′ y′ z′_).

A transformation which is accomplished by means of (10), (11), (12) with the condition 0 < q < 1 will be called a special Lorentz-transformation. We shall call v the vector, the direction of v the axis, and the magnitude of v the moment of this transformation.

If further ρ′ and the vectors u′, e′, m′, in the system (x′ y′ z′) are so defined that,

(13) ρ′ = ρ[(-qu{v} + 1)/√(1 - q²)], ρ′u′{v} = ρ(u{v} - q)/√(1 - q²), ρ′u_{ṽ} = ρ′u_{v_},

further

(14) (e′ + im′){ṽ} = ((e + im) - i[v, (e + im])']{ṽ})/√(1 - q²).

(15) (e′ + im′){v} = (e + im) - i[u, (e + im)]{v}.

Then it follows that the equations I), II), III), IV) are transformed into the corresponding system with dashes.

The solution of the equations (10), (11), (12) leads to

(16) r{v} = (r′_{v} + qt′)/√(1 - q²), r_{ṽ} = r′_{ṽ}, t = (qr′_{v} + t′)/√(1 - q²_),

Now we shall make a very important observation about the vectors u and u′. We can again introduce the indices 1, 2, 3, 4, so that we write (x₁′, x₂′, x₃′, x₄′) instead of (x′, y′, z′, it′) and ρ₁′, ρ₂′, ρ₃′, ρ₄′ instead of (ρ′u′{x′}, ρ′u′{y′}, ρ′u′{z′}, iρ′).

Like the rotation round the Z-axis, the transformation (4), and more generally the transformations (10), (11), (12), are also linear transformations with the determinant + 1, so that

(17) x₁² + x₂² + x₃² + x₄² i. e. x² + y² + z² - t²,

is transformed into

x₁′² + x₂′² + x₃′² + x₄′² i. e. x′² + y′² + z′² - t′².

On the basis of the equations (13), (14), we shall have (ρ₁² + ρ₂² + ρ₃² + ρ₄²) = ρ²(1 - u{x²}, -u_{y²}, -u_{z²}) = ρ²(1 - u²) transformed into ρ²(1 - u²_) or in other words,

(18) ρ√(1 - u²)

is an invariant in a Lorentz-transformation.

If we divide (ρ₁, ρ₂, ρ₃, ρ₄) by this magnitude, we obtain the four values (ω₁, ω₂, ω₃, ω₄) = (1/√(1 - u²))(u{x}, u_{y}, u_{z}, i_) so that ω₁² + ω₂² + ω₃² + ω₄² = -1.

It is apparent that these four values are determined by the vector u and inversely the vector u of magnitude < 1 follows from the 4 values ω₁, ω₂, ω₃, ω₄; where (ω₁, ω₂, ω₃) are real, -iω₄ real and positive and condition (19) is fulfilled.

The meaning of (ω₁, ω₂, ω₃, ω₄) here is, that they are the ratios of dx₁, dx₂, dx₃, dx₄ to

(20) √(-(dx₁² + dx₂² + dx₃² + dx₄²)) = dt√(1 - u²).

The differentials denoting the displacements of matter occupying the spacetime point (x₁, x₂, x₃, x₄) to the adjacent space-time point.

After the Lorentz-transformation is accomplished the velocity of matter in the new system of reference for the same space-time point (x′ y′ z′ t′) is the vector u′ with the ratios dx′/dt′, dy′/dt′, dz′/dt′, dl′/dt′, as components.

Now it is quite apparent that the system of values

x₁ = ω₁, x₂ = ω₂, x₃ = ω₃, x₄ = ω₄

is transformed into the values

x₁′ = ω₁′, x₂′ = ω₂′, x₃′ = ω₃′, x₄′ = ω₄′

in virtue of the Lorentz-transformation (10), (11), (12).

The dashed system has got the same meaning for the velocity u′ after the transformation as the first system of values has got for u before transformation.

If in particular the vector v of the special Lorentz-transformation be equal to the velocity vector u of matter at the space-time point (x₁, x₂, x₃, x₄) then it follows out of (10), (11), (12) that

ω₁′ = 0, ω₂′ = 0, ω₃′ = 0, ω₄′ = i

Under these circumstances therefore, the corresponding space-time point has the velocity v′ = 0 after the transformation, it is as if we transform to rest. We may call the invariant ρ√(1 - u²) the rest-density of Electricity.

§ 5. Space-time Vectors. Of the 1st and 2nd kind.

If we take the principal result of the Lorentz transformation together with the fact that the system (A) as well as the system (B) is covariant with respect to a rotation of the coordinate-system round the null point, we obtain the general relativity theorem. In order to make the facts easily comprehensible, it may be more convenient to define a series of expressions, for the purpose of expressing the ideas in a concise form, while on the other hand I shall adhere to the practice of using complex magnitudes, in order to render certain symmetries quite evident.

Let us take a linear homogeneous transformation,

$$ \begin{vmatrix} x{1} x{2} x{3} x{4} \end{vmatrix} = \begin{vmatrix} a{1 1} & a{1 2} & a{1 3} & a{1 4} a{2 1} & a{2 2} & a{2 3} & a{2 4} a{3 1} & a{3 2} & a{3 3} & a{3 4} a{4 1} & a{4 2} & a{4 3} & a{4 4} \end{vmatrix} \begin{vmatrix} x{1}' x{2}' x{3}' x{4}' \end{vmatrix} $$

the Determinant of the matrix is +1, all co-efficients without the index 4 occurring once are real, while a₄₁, a₄₂, a₄₃, are purely imaginary, but a₄₄ is real and > 0, and x₁² + x₂² + x₃² + x₄² transforms into x₁′² + x₂′² + x₃′² + x₄′². The operation shall be called a general Lorentz transformation.

(This notation, which is due to Dr. C. E. Cullis of the Calcutta University, has been used throughout instead of Minkowski’s notation, x₁ = a₁₁x₁′ + a₁₂x₂′+ a₁₃x₃′+ a₁₄x₄′.)

If we put x₁′ = x′, x₂′ = y′, x₃′ = z′, x₄′ = it′, then immediately there occurs a homogeneous linear transformation of (x, y, z, t) to (x′, y′, z′, t′) with essentially real co-efficients, whereby the aggregate -x² - y² - z² + t² transforms into -x′² - y′² - z′² + t′², and to every such system of values x, y, z, t with a positive t, for which this aggregate > 0, there always corresponds a positive t’; this last is quite evident from the continuity of the aggregate x, y, z, t.

The last vertical column of co-efficients has to fulfil the condition 22) a₁₄² + a₂₄² + a₃₄² + a₄₄² = 1.

If a₁₄ = a₂₄ = a₃₄ = 0, then a₄₄ = 1, and the Lorentz transformation reduces to a simple rotation of the spatial co-ordinate system round the world-point.

If a₁₄, a₂₄, a₃₄ are not all zero, and if we put a₁₄ : a₂₄ : a₃₄ : a₄₄ = v{x} : v_{y} : v_{z} : i_

q = √(v{x}² + v_{y}² +v_{z_}²) < 1.

On the other hand, with every set of values of a₁₄, a₂₄, a₃₄, a₄₄ which in this way fulfil the condition 22) with real values of v{x}, v_{y}, v_{z}, we can construct the special Lorentz transformation (16) with (a₁₄, a₂₄, a₃₄, a₄₄) as the last vertical column,—and then every Lorentz-transformation with the same last vertical column (a₁₄, a₂₄, a₃₄, a₄₄_) can be supposed to be composed of the special Lorentz-transformation, and a rotation of the spatial co-ordinate system round the null-point.

The totality of all Lorentz-Transformations forms a group. Under a space-time vector of the 1st kind shall be understood a system of four magnitudes (ρ₁, ρ₂, ρ₃, ρ₄) with the condition that in case of a Lorentz-transformation it is to be replaced by the set (ρ₁′, ρ₂′, ρ₃′, ρ₄′), where these are the values of (x₁′, x₂′, x₃′, x₄′), obtained by substituting (ρ₁, ρ₂, ρ₃, ρ₄) for (x₁, x₂, x₃, x₄) in the expression (21).

Besides the time-space vector of the 1st kind (x₁, x₂, x₃, x₄) we shall also make use of another space-time vector of the first kind (y₁, y₂, y₃, y₄), and let us form the linear combination

(23) f₂₃(x₂y₃ - x₃y₂) + f₃₁(x₃y₁ - x₁y₃) + f₁₂(x₁y₂ - x₂y₁) + f₁₄(x₁y₄ - x₄y₁) + f₂₄(x₂y₄ - x₄y₂) + f₃₄(x₃y₄ - x₄y₃)

with six coefficients f₂₃--f₃₄. Let us remark that in the vectorial method of writing, this can be constructed out of the four vectors.

x₁, x₂, x₃; y₁, y₂, y₃; f₂₃, f₃₁, f₁₂; f₁₄, f₂₄, f₃₄ and the constants x₄ and y₄, at the same time it is symmetrical with regard the indices (1, 2, 3, 4).

If we subject (x₁, x₂, x₃, x₄) and (y₁, y₂, y₃, y₄) simultaneously to the Lorentz transformation (21), the combination (23) is changed to:

(24) f₂₃′(x₂′y₃′ - x₃′y₂′) + f₃₁(x₃′y₁′ - x₁′y₃′) + f₁₂ (x₁′y₂′ - x₂′y₁′) + f₁₄′(x₁′y₄′) - x₄′y₁′) + f₂₄′(x₂′y₄′ - x₄′y₂′) + f₃₄′(x₃′y₄′ - x₄′y₃′),

where the coefficients f₂₃′, f₃₁′, f₁₂′, f₁₄′, f₂₄′, f₃₄′, depend solely on (f₂₃ f₂₄) and the coefficients a₁₁ ... a₄₄.

We shall define a space-time Vector of the 2nd kind as a system of six-magnitudes f₂₃, f₃₁ ... f₃₄, with the condition that when subjected to a Lorentz transformation, it is changed to a new system f₂₃′ ... f₃₄, ... which satisfies the connection between (23) and (24).

I enunciate in the following manner the general theorem of relativity corresponding to the equations (I)-(iv),—which are the fundamental equations for Äther.

If x, y, z, it (space co-ordinates, and time it) is subjected to a Lorentz transformation, and at the same time (pu{x}, pu_{y}, pu_{z}, iρ) (convection-current, and charge density ρi) is transformed as a space time vector of the 1st kind, further (m_{x}, m_{y}, m_{z}, -ie_{x}, -ie_{y}, -ie_{z}) (magnetic force, and electric induction × (-i_) is transformed as a space time vector of the 2nd kind, then the system of equations (I), (II), and the system of equations (III), (IV) transforms into essentially corresponding relations between the corresponding magnitudes newly introduced into the system.

These facts can be more concisely expressed in these words: the system of equations (I and II) as well as the system of equations (III) (IV) are covariant in all cases of Lorentz-transformation, where (ρu, iρ) is to be transformed as a space time vector of the 1st kind, (m - ie) is to be treated as a vector of the 2nd kind, or more significantly,—

(ρu, iρ) is a space time vector of the 1st kind, (m - ie) is a space-time vector of the 2nd kind.

I shall add a few more remarks here in order to elucidate the conception of space-time vector of the 2nd kind. Clearly, the following are invariants for such a vector when subjected to a group of Lorentz transformation.

(i) m² - e² = f₂₃² + f₃₁² + f₁₂² + f₁₄² + f₂₄² + f₂₄²

me = i(f₂₃f₁₄ + f₃₁f₂₄ + f₁₂f₃₄).

A space-time vector of the second kind (m - ie), where (m and e) are real magnitudes, may be called singular, when the scalar square (m - ie)² = 0, ie m² - e² = 0, and at the same time (m e) = 0, ie the vector m and e are equal and perpendicular to each other; when such is the case, these two properties remain conserved for the space-time vector of the 2nd kind in every Lorentz-transformation.

If the space-time vector of the 2nd kind is not singular, we rotate the spacial co-ordinate system in such a manner that the vector-product [me] coincides with the Z-axis, i.e. m{x} = 0, e_{x_} = 0. Then

(m{x}, -i e_{x})² + (m_{y}, -i e_{y_})² ≠ 0.

Therefore (e{y} + i m_{y})/(e_{x} + i e_{x}) is different from +i, and we can therefore define a complex argument (φ + i_ψ) in such a manner that

tan (φ + iψ)

e{y} + i m_{y} = ------------------------- e_{x} + i m_{x_}

If then, by referring back to equations (9), we carry out the transformation (1) through the angle ψ and a subsequent rotation round the Z-axis through the angle φ, we perform a Lorentz-transformation at the end of which m{y} = 0, e_{y} = 0, and therefore m and e shall both coincide with the new Z-axis. Then by means of the invariants m² - e², (me_) the final values of these vectors, whether they are of the same or of opposite directions, or whether one of them is equal to zero, would be at once settled.

§ 6. Concept of Time.

By the Lorentz transformation, we are allowed to effect certain changes of the time parameter. In consequence of this fact, it is no longer permissible to speak of the absolute simultaneity of two events. The ordinary idea of simultaneity rather presupposes that six independent parameters, which are evidently required for defining a system of space and time axes, are somehow reduced to three. Since we are accustomed to consider that these limitations represent in a unique way the actual facts very approximately, we maintain that the simultaneity of two events exists of themselves. In fact, the following considerations will prove conclusive.

Let a reference system (x, y, z, t) for space time points (events) be somehow known. Now if a space point A (x₀, y₀, z₀) the time t₀ be compared with a space point P (x, y, z) at the time t, and if the difference of time t - t₀, (let t > t₀) be less than the length A P i.e. less than the time required for the propagation of light from A to P, and if q = (t - t₀)/(A P) < 1, then by a special Lorentz transformation, in which A P is taken as the axis, and which has the moment q, we can introduce a time parameter t′, which (see equation 11, 12, § 4) has got the same value t′ = 0 for both space-time points (A, t₀), and (P, t). So the two events can now be comprehended to be simultaneous.

Further, let us take at the same time t₀ = 0, two different space-points A, B, or three space-points (A, B, C) which are not in the same space-line, and compare therewith a space point P, which is outside the line A B, or the plane A B C, at another time t, and let the time difference t - t₀ (t > t₀) be less than the time which light requires for propagation from the line A B, or the plane (A B C) to P. Let q be the quotient of (t - t₀) by the second time. Then if a Lorentz transformation is taken in which the perpendicular from P on A B, or from P on the plane A B C is the axis, and q is the moment, then all the three (or four) events (A, t₀), (B, t₀), (C, t₀) and (P, t) are simultaneous.

If four space-points, which do not lie in one plane, are conceived to be at the same time t₀, then it is no longer permissible to make a change of the time parameter by a Lorentz-transformation, without at the same time destroying the character of the simultaneity of these four space points.

To the mathematician, accustomed on the one hand to the methods of treatment of the poly-dimensional manifold, and on the other hand to the conceptual figures of the so-called non-Euclidean Geometry, there can be no difficulty in adopting this concept of time to the application of the Lorentz-transformation. The paper of Einstein which has been cited in the Introduction, has succeeded to some extent in presenting the nature of the transformation from the physical standpoint.