§ 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 vx, vy, vz, and let | v | = q < 1. By ṽ we shall denote any vector which is perpendicular to v, and by rv, rṽ 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 = (rv - qt)/√(1 - q²)
and for the perpendicular direction ṽ,
(11) r′ṽ = rṽ
and further (12) t′ = (-qrv + 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[(-quv + 1)/√(1 - q²)],
ρ′u′v = ρ(uv - q)/√(1 - q²),
ρ′uṽ = ρ′uv,
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) rv = (r′v + qt′)/√(1 - q²),
rṽ = 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 - ux², -uy², -uz²) = 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))(ux, uy, uz, 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.[[16]]
§ 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,
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 = vx : vy : vz : i
q = √(vx² + vy² +vz²) < 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 vx, vy, vz, 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 (pux, puy, puz, iρ) (convection-current, and charge density ρi) is transformed as a space time vector of the 1st kind, further (mx, my, mz, -iex, -iey, -iez) (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)[[17]] 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. mx = 0, ex = 0. Then
(mx, -i ex)² + (my, -i ey)² ≠ 0.
Therefore (ey + i my)/(ex + i ex) is different from +i, and we can therefore define a complex argument (φ + iψ) in such a manner that
tan (φ + iψ)
ey + i my
= ---------------
ex + i mx
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 my = 0, ey = 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.