_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,
$$ \begin{vmatrix} x_{1} x_{2} x_{3} x_{4} \end{vmatrix} =
\begin{vmatrix} a_{1 1} & a_{1 2} & a_{1 3} & a_{1 4} a_{2 1} & a_{2
2} & a_{2 3} & a_{2 4} a_{3 1} & a_{3 2} & a_{3 3} & a_{3 4} a_{4 1} &
a_{4 2} & a_{4 3} & a_{4 4} \end{vmatrix} \begin{vmatrix}
x_{1}' x_{2}' x_{3}' x_{4}' \end{vmatrix} $$
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_(_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_)
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_}, _i_ρ) (convection-current, and charge density
ρ_i_) 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 (ρ_u_, _i_ρ)
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,—
(ρ_u_, _i_ρ) 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_(_f₂₃__f₁₄_ + _f₃₁__f₂₄_ + _f₁₂__f₃₄_).
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 + _i_ψ)
_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.[18] 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.
PART II. ELECTRO-MAGNETIC PHENOMENA.
§ 7. Fundamental Equations for bodies at rest.
After these preparatory works, which have been first developed on
account of the small amount of mathematics involved in the limiting case
ε = 1, μ = 1, σ = 0, let us turn to the electro-magnetic phenomena in
matter. We look for those relations which make it possible for us—when
proper fundamental data are given—to obtain the following quantities at
every place and time, and therefore at every space-time point as
functions of (_x_, _y_, _z_, _t_):—the vector of the electric force E,
the magnetic induction M, the electrical induction _e_, the magnetic
force _m_, the electrical space-density ρ, the electric current s (whose
relation hereafter to the conduction current is known by the manner in
which conductivity occurs in the process), and lastly the vector _v_,
the velocity of matter.
The relations in question can be divided into two classes.
Firstly—those equations, which,—when _v_, the velocity of matter is
given as a function of (_x_, _y_, _z_, _t_),—lead us to a knowledge of
other magnitude as functions of _x_, _y_, _z_, _t_—I shall call this
first class of equations the fundamental equations—
Secondly, the expressions for the ponderomotive force, which, by the
application of the Laws of Mechanics, gives us further information about
the vector _u_ as functions of (_x_, _y_, _z_, _t_).
For the case of bodies at rest, _i.e._ when _u_ (_x_, _y_, _z_, _t_) = 0
the theories of Maxwell (Heaviside, Hertz) and Lorentz lead to the same
fundamental equations. They are;—
(1) The Differential Equations:—which contain no constant referring to
matter:—
(_i_) Curl _m_ - δ_e_/δ_t_ = C,
(_ii_) div _e_ = lρ.
(_iii_) Curl E + δM/δ_t_ = 0,
(_iv_) Div M = 0.
(2) Further relations, which characterise the influence of existing
matter for the most important case to which we limit ourselves _i.e._
for isotopic bodies;—they are comprised in the equations
(V) _e_ = ε E, M = μ_m_, C = σE.
where ε = dielectric constant, μ = magnetic permeability, σ = the
conductivity of matter, all given as function of _x_, _y_, _z_, _t_; _s_
is here the conduction current.
By employing a modified form of writing, I shall now cause a latent
symmetry in these equations to appear. I put, as in the previous work,
_x₁_ = _x_, _x₂_ = _y_, _x₃_ = _z_, _x₄_ = _it_,
and write _s₁_, _s₂_, _s₃_, _s₄_ for C_{_x_}, C_{_y_}, C_{_z_} (√-1)ρ.