Part 5
But Einstein found a support for the new-born concept in another direction. It was known that if green or ultraviolet light was allowed to fall on a plate of some alkali metal, the plate lost electrons. The electrons were emitted with all velocities, but there is generally a maximum limit. From the investigations of Lenard and Ladenburg, the curious discovery was made that this maximum velocity of emission did not at all depend upon the intensity of light, but upon its wavelength. The more violet was the light, the greater was the velocity of emission.
To account for this fact, Einstein made the bold assumption that the light is propagated in space as a unit pulse (he calls it a Light-cell), and falling upon an individual atom, liberates electrons according to the energy equation
1 _h_ν = --- _mv²_ + A, 2
where (_m_, _v_) are the mass and velocity of the electron. A is a constant characteristic of the metal plate.
There was little material for the confirmation of this law when it was first proposed (1905), and eleven years elapsed before Prof. Millikan established, by a set of experiments scarcely rivalled for the ingenuity, skill, and care displayed, the absolute truth of the law. As results of this confirmation, and other brilliant triumphs, the quantum law is now regarded as a fundamental law of Energetics. In recent years, X-rays have been added to the domain of light, and in this direction also, Einstein’s photo-electric formula has proved to be one of the most fruitful conceptions in Physics.
The quantum law was next extended by Einstein to the problems of decrease of specific heat at low temperature, and here also his theory was confirmed in a brilliant manner.
We pass over his other contributions to the equation of state, to the problems of null-point energy, and photo-chemical reactions. The recent experimental works of Nernst and Warburg seem to indicate that through Einstein’s genius, we are probably for the first time having a satisfactory theory of photo-chemical action.
In 1915, Einstein made an excursion into Experimental Physics, and here also, in his characteristic way, he tackled one of the most fundamental concepts of Physics. It is well-known that according to Ampere, the magnetisation of iron and iron-like bodies, when placed within a coil carrying an electric current is due to the excitation in the metal of small electrical circuits. But the conception though a very fruitful one, long remained without a trace of experimental proof, though after the discovery of the electron, it was generally believed that these molecular currents may be due to the rotational motion of free electrons within the metal. It is easily seen that if in the process of magnetisation, a number of electrons be set into rotatory motion, then these will impart to the metal itself a turning couple. The experiment is a rather difficult one, and many physicists tried in vain to observe the effect. But in collaboration with de Haas, Einstein planned and successfully carried out this experiment, and proved the essential correctness of Ampere’s views.
Einstein’s studies on Relativity were commenced in the year 1905, and has been continued up to the present time. The first paper in the present collection forms Einstein’s first great contribution to the Principle of Special Relativity. We have recounted in the introduction how out of the chaos and disorder into which the electrodynamics and optics of moving bodies had fallen previous to 1895, Lorentz, Einstein and Minkowski have succeeded in building up a consistent, and fruitful new theory of Time and Space.
But Einstein was not satisfied with the study of the special problem of Relativity for uniform motion, but tried, in a series of papers beginning from 1911, to extend it to the case of non-uniform motion. The last paper in the present collection is a translation of a comprehensive article which he contributed to the Annalen der Physik in 1916 on this subject, and gives, in his own words, the Principles of Generalized Relativity. The triumphs of this theory are now matters of public knowledge.
Einstein is now only 45, and it is to be hoped that science will continue to be enriched, for a long time to come, with further achievements of his genius.
Principle of Relativity
INTRODUCTION.
At the present time, different opinions are being held about the fundamental equations of Electro-dynamics for moving bodies. The Hertzian[9] forms must be given up, for it has appeared that they are contrary to many experimental results.
In 1895 H. A. Lorentz[10] published his theory of optical and electrical phenomena in moving bodies; this theory was based upon the atomistic conception (vorstellung) of electricity, and on account of its great success appears to have justified the bold hypotheses, by which it has been ushered into existence. In his theory, Lorentz proceeds from certain equations, which must hold at every point of “Äther”; then by forming the average values over “Physically infinitely small” regions, which however contain large numbers of electrons, the equations for electro-magnetic processes in moving bodies can be successfully built up.
In particular, Lorentz’s theory gives a good account of the non-existence of relative motion of the earth and the luminiferous “Äther”; it shows that this fact is intimately connected with the covariance of the original equation, when certain simultaneous transformations of the space and time co-ordinates are effected; these transformations have therefore obtained from H. Poincare[11] the name of Lorentz-transformations. The covariance of these fundamental equations, when subjected to the Lorentz-transformation is a purely mathematical fact _i.e._ not based on any physical considerations; I will call this the Theorem of Relativity; this theorem rests essentially on the form of the differential equations for the propagation of waves with the velocity of light.
Now without _recognizing_ any hypothesis about the connection between “Äther” and matter, we can expect these mathematically evident theorems to have their consequences so far extended—that thereby even those laws of ponderable media which are yet unknown may anyhow possess this covariance when subjected to a Lorentz-transformation; by saying this, we do not indeed express an opinion, but rather a conviction,—and this conviction I may be permitted to call the Postulate of Relativity. The position of affairs here is almost the same as when the Principle of Conservation of Energy was postulated in cases, where the corresponding forms of energy were unknown.
Now if hereafter, we succeed in maintaining this covariance as a definite connection between pure and simple observable phenomena in moving bodies, the definite connection may be styled ‘the Principle of Relativity.’
These differentiations seem to me to be necessary for enabling us to characterise the present day position of the electro-dynamics for moving bodies.
H. A. Lorentz[12] has found out the “Relativity theorem” and has created the Relativity-postulate as a hypothesis that electrons and matter suffer contractions in consequence of their motion according to a certain law.
A. Einstein[13] has brought out the point very clearly, that this postulate is not an artificial hypothesis but is rather a new way of comprehending the time-concept which is forced upon us by observation of natural phenomena.
The Principle of Relativity has not yet been formulated for electro-dynamics of moving bodies in the sense characterized by me. In the present essay, while formulating this principle, I shall obtain the fundamental equations for moving bodies in a sense which is uniquely determined by this principle.
But it will be shown that none of the forms hitherto assumed for these equations can exactly fit in with this principle.[14]
We would at first expect that the fundamental equations which are assumed by Lorentz for moving bodies would correspond to the Relativity Principle. But it will be shown that this is not the case for the general equations which Lorentz has for any possible, and also for magnetic bodies; but this is approximately the case (if neglect the square of the velocity of matter in comparison to the velocity of light) for those equations which Lorentz hereafter infers for non-magnetic bodies. But this latter accordance with the Relativity Principle is due to the fact that the condition of non-magnetisation has been formulated in a way not corresponding to the Relativity Principle; therefore the accordance is due to the fortuitous compensation of two contradictions to the Relativity-Postulate. But meanwhile enunciation of the Principle in a rigid manner does not signify any contradiction to the hypotheses of Lorentz’s molecular theory, but it shall become clear that the assumption of the contraction of the electron in Lorentz’s theory must be introduced at an earlier stage than Lorentz has actually done.
In an appendix, I have gone into discussion of the position of Classical Mechanics with respect to the Relativity Postulate. Any easily perceivable modification of mechanics for satisfying the requirements of the Relativity theory would hardly afford any noticeable difference in observable processes; but would lead to very surprising consequences. By laying down the Relativity-Postulate from the outset, sufficient means have been created for deducing henceforth the complete series of Laws of Mechanics from the principle of conservation of Energy alone (the form of the Energy being given in explicit forms).
NOTATIONS.
Let a rectangular system (_x_, _y_, _z_, _t_,) of reference be given in space and time. The unit of time shall be chosen in such a manner with reference to the unit of length that the velocity of light in space becomes unity.
Although I would prefer not to change the notations used by Lorentz, it appears important to me to use a different selection of symbols, for thereby certain homogeneity will appear from the very beginning. I shall denote the vector electric force by E, the magnetic induction by M, the electric induction by _e_ and the magnetic force by _m_, so that (E, M, _e_, _m_) are used instead of Lorentz’s (E, B, D, H) respectively.
I shall further make use of complex magnitudes in a way which is not yet current in physical investigations, _i.e._, instead of operating with (_t_), I shall operate with (_i t_), where _i_ denotes √(-1). If now instead of (_x_, _y_, _z_, _i t_), I use the method of writing with indices, certain essential circumstances will come into evidence; on this will be based a general use of the suffixes (1, 2, 3, 4). The advantage of this method will be, as I expressly emphasize here, that we shall have to handle symbols which have apparently a purely real appearance; we can however at any moment pass to real equations if it is understood that of the symbols with indices, such ones as have the suffix 4 only once, denote imaginary quantities, while those which have not at all the suffix 4, or have it twice denote real quantities.
An individual system of values of (_x_, _y_, _z_, _t_) _i. e._, of (_x₁_ _x₂_ _x₃_ _x₄_) shall be called a space-time point.
Further let _u_ denote the velocity vector of matter, ε the dielectric constant, μ the magnetic permeability, σ the conductivity of matter, while ρ denotes the density of electricity in space, and _x_ the vector of “Electric Current” which we shall some across in §7 and §8.
## PART I
§ 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[15]
(i) Curl _m_ - (δ_e_/δ_t_) = ρu
(ii) div _e_ = ρ
(iii) Curl _e_ + δ_m_/δ_t_ = 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 _i_ψ a purely imaginary magnitude, and consider the substitution
_x₁′_ = _x₁_, _x₂′_ = _x₂_, _x₃′_ = _x₃_ cos _i_ψ + _x₄_ sin _i_ψ, (1) _x₄′_´ = - _x₃_ sin _i_ψ + _x₄_ cos _i_ψ,
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 _i_ψ = 1/√(1 - _q²_), sin _i_ψ = _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 _i_ψ, 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 _i_ψ + ρ₄ sin _i_ψ, ρ′₄ = - ρ₃ sin _i_ψ + ρ₄ cos _i_ψ),
and _f′__{1 2} ... _f′__{3 4}, where
_f′__{4 1} = _f__{4 1} cos _i_ψ + _f__{1 3} sin _i_ψ, _f′__{1 3} = - _f__{4 1} sin _i_ψ + _f__{1 3} cos _i_ψ, _f′__{3 4} = _f__{3 4}, _f′__{3 2} = _f__{3 2} cos _i_ψ + _f__{4 2} sin _i_ψ, _f′__{4 2} = - _f__{3 2} sin _i_ψ + _f__{4 2} cos _i_ψ, _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_}´ = ρ_u__{_x_}, ρ´_u__{_y_}´ = ρ_u__{_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 _i_ψ + (_e__{_y_} + _im__{_y_}) sin _i_ψ,
_e′__{_y′_} + _im′__{_y′_} = - (_e__{_x_} + _im__{_x_}) sin _i_ψ + (_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 + _i_ψ) + (_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 + _i_ψ) + (_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′_}, _i_ρ′).
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, -_i_ω₄ 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.