Part 8
(68) (∂_s₁_/∂_x₁_) + (∂_s₂_/∂_x₂_) + (∂_s₃_/∂_x₃_) + (∂_s₄_/∂_x₄_) = 0,
while the relation
(69) lor (lor F*) = 0,
signifies that of the four equations in {B}, only three represent independent conditions.
I shall now collect the results.
Let ω denote the space-time vector of the first kind
(_u_/√(1 - _u²_}), _i_/√(1 - _u²_))
(_u_ = velocity of matter),
F the space-time vector of the second kind (M,-_i_E)
(M = magnetic induction, E = Electric force,
_f_ the space-time vector of the second kind (_m_,-_ie_)
(_m_ = magnetic force, _e_ = Electric Induction.
_s_ the space-time vector of the first kind (C, _i_ρ)
(ρ = electrical space-density, C - ρ_u_ = conductivity current,
ε = dielectric constant, μ = magnetic permeability,
σ = conductivity,
then the fundamental equations for electromagnetic processes in moving bodies are[26]
{A} lor _f_ = -_s_
{B} log F* = 0
{C} ω_f_ = εωF
{D} ωF* = μω_f_*
{E} _s_ + (ω_ṡ_), _w_ = - σωF.
ω ῶ = -1, and ωF, ω_f_, ωF*, ω_f_*, _s_ + (ω_s_)ω which are space-time vectors of the first kind are all normal to ω, and for the system {B}, we have
lor (lor F*) = 0.
Bearing in mind this last relation, we see that we have as many independent equations at our disposal as are necessary for determining the motion of matter as well as the vector _u_ as a function of _x_, _y_, _z_, _t_, when proper fundamental data are given.
§ 13. The Product of the Field-vectors _f_ F.
Finally let us enquire about the laws which lead to the determination of the vector ω as a function of (_x_, _y_, _z_, _t_.) In these investigations, the expressions which are obtained by the multiplication of two alternating matrices
_f_ = | 0 _f₁₂_ _f₁₃_ _f₁₄_ | | _f₂₁_ 0 _f₂₃_ _f₂₄_ | | _f₃₁_ _f₃₂_ 0 _f₃₄_ | | _f₄₁_ _f₄₂_ _f₄₃_ 0 |
F = | 0 F₁₂ F₁₃ F₁₄ | | F₂₁ 0 F₂₃ F₂₄ | | F₃₁ F₃₂ 0 F₃₄ | | F₄₁ F₄₂ F₄₃ 0 |
are of much importance. Let us write,
(70) _f_F =| S₁₁ - L S₁₂ S₁₃ S₁₄ |
| S₂₁ S₂₂ - L S₂₃ S₂₄ |
| S₃₁ S₃₂ S₃₃ - L S₃₄ |
| S₄₁ S₄₂ S₄₃ S₄₄ - L |
Then (71) S₁₁ + S₂₂ + S₃₃ + S₄₄ = 0.
Let L now denote the symmetrical combination of the indices 1, 2, 3, 4, given by
(72) L = ½(_f₂₃_ F₂₃ + _f₃₁_F₃₁ + _f₁₂_ + F₁₂ + _f₁₄_ F₁₄ + _f₂₄_ F₂₄ + _f₃₄_ F₃₄)
Then we shall have
(73) S₁₁ = ½(_f₂₃_ F₂₃ + _f₃₄_ F₃₄ + _f₄₂_ F₄₂ - _f₁₂_ F₁₂ - _f₁₃_ F₁₃ _f₁₄_ F₁₄)
S₁₂ = _f₁₃_ F₃₂ + _f₁₄_ F₄₂ etc....
In order to express in a real form, we write
(74) S = | S₁₁ S₁₂ S₁₃ S₁₄ |
| S₂₁ S₂₂ S₂₃ S₂₄ |
| S₃₁ S₃₂ S₃₃ S₃₄ |
| S₄₁ S₄₂ S₄₃ S₄₄ |
= | X_{_x_} Y_{_x_} Z_{_x_} -_i_T_{_x_} |
| X_{_y_} Y_{_y_} Z_{_y_} -_i_T_{_y_} |
| X_{_z_} Y_{_z_} Z_{_z_} -_i_T_{_z_} |
| -_i_X_{_t_} -_i_Y_{_t_} -_i_Z_{_t_} T_{_t_} |
Now X_{_x_} = ½[_m__{_x_}M_{_x_} - _m__{_y_}M_{_y_} - _m__{_z_}M_{_z_} + _e__{_x_}E_{_x_} - _e__{_y_}E_{_y_} - _e__{_z_}E_{_z_}]
so
(75) X_{_y_} = _m__{_x_}M_{_y_} + _e__{_y_}E_{_x_}, Y_{_x_} = _m__{_y_}M_{_x_} + _e__{_x_}E_{_y_} etc.
X_{_t_} = _e__{_y_}M_{_z_} - _e__{_z_}M_{_y_}, T_{_x_} = _m__{_x_}E_{_y_} - _m__{_y_}E_{_z_}, etc.
T_{_t_} = ½[_m__{_x_}M_{_x_} + _m__{_y_}M_{_y_} + _m__{_z_}M_{_z_} + _e__{_x_}E_{_x_} + _e__{_y_}E_{_y_} + _e__{_z_}E_{_z_}]
L_{_t_} = ½[_m__{_x_}M_{_x_} + _m__{_y_}M_{_y_} + _m__{_z_}M_{_z_} - _e__{_x_}E_{_x_} - _e__{_y_}E_{_y_} - _e__{_z_}E_{_z_}]
These quantities[27] are all real. In the theory for bodies at rest, the combinations (X_{_x_}, X_{_y_}, X_{_z_}, Y_{_z_}, Y_{_y_}, Y_{_z_}, Z_{_x_}, Z_{_y_}, Z_{_z_}) are known as “Maxwell’s Stresses,” T_{_x_}, T_{_y_}, T_{_z_} are known as the Poynting’s Vector, T_{_t_} as the electromagnetic energy-density, and L as the Langrangian function.
On the other hand, by multiplying the alternating matrices of _f_* and F*, we obtain
(77) F*f* =| -S₁₁ - L, -S₁₂, -S₁₃. -S₁₄ |
| -S₂₁, -S₂₂ - L, -S₂₃, -S₂₄ |
| -S₃₁ -S₃₂, -S₃₃ - L, -S₃₄ |
| -S₄₁ -S₄₂ -S₄₃ -S₄₄ - L |
and hence, we can put
(78) _f_F = S - L, F*_f_* = -S - L,
where by L, we mean L-times the unit matrix, _i.e._ the matrix with elements
| L_e__{_hk_} |, (_e__{_hh_} = 1, _e__{_hk_} = 0, _h_ ≠ _k_ _h_, _k_ = 1, 2, 3, 4).
Since here SL = LS, we deduce that,
F*_f_*_f_F = (-S - L)(S - L) = -SS + L²,
and find, since _f_*_f_ = Det^{½}_f_, F*F = Det^{½}F, we arrive at the interesting
conclusion
(79) SS = L² - Det^{½}_f_ Det^{½}F
_i.e._ the product of the matrix S into itself can be expressed as the multiple of a unit matrix—a matrix in which all the elements except those in the principal diagonal are zero, the elements in the principal diagonal are all equal and have the value given on the right-hand side of (79). Therefore the general relations
(80) S_{_h_1} S_{1_k_} + S_{_h_2} S_{2_k_} + S_{_h_3} S_{3_k_} + S_{_h_4} S_{4_k_} = 0,
_h_, _k_ being unequal indices in the series 1, 2, 3, 4, and
(81) S_{_h_1} S_{1_h_} + S_{_h_2} S_{2_h_} + S_{_h_3} S_{3_h_} + S{_h_4} S_{4_h_} = L² - Det^{½}_f_ Det^{½}F,
for _h_ = 1, 2, 3, 4.
Now if instead of F, and _f_ in the combinations (72) and (73), we introduce the electrical rest-force Φ, the magnetic rest-force ψ, and the rest-ray Ω [(55), (56) and (57)], we can pass over to the expressions,—
(82) L = - ½ ε Φ [=Φ] + ½ μ ψ [=ψ],
(83) S_{_hk_} = - ½ ε Φ [=Φ] _e__{_hk_} - ½ μ ψ [=ψ] _e__{_hk_} + ε (Φ_{_h_} Φ_{_k_} - Φ ([=Φ]) ω_{_h_} Ω_{_k_} + μ (ψ_{_h_} ψ_{_k_} - Ψ [=ψ] Ω{_h_} ω_{_k_}) - ω_{_h_} ω_{_k_} - εμ ω_{_h_} Ω_{_k_} (_h₁_ _k_ = 1, 2, 3, 4).
Here we have
Φ [=Φ] = Φ₁² + Φ₂² + Φ₃² + Φ₄², ψ[=ψ] = ψ₁² + ψ₂² + ψ₃² + ψ₄²
_e__{_hh_} = 1, _e__{_hk_} = 0 (_h_ ≠ _k_).
The right side of (82) as well as L is an invariant in a Lorentz transformation, and the 4 × 4 element on the right side of (83) as well as S_{_k_ _h_} represent a space time vector of the second kind. Remembering this fact, it suffices, for establishing the theorems (82) and (83) generally, to prove it for the special case ω₁ = 0, ω₂ = 0, ω₃ = 0, ω₄ = _i_. But for this case ω = 0, we immediately arrive at the equations (82) and (83) by means (45), (51), (60) on the one hand, and _e_ = εE, M = μ_m_ on the other hand.
The expression on the right-hand side of (81), which equals
[½ (_m_ M - _e_E)²] + (_em_) (EM),
is >= 0, because (_em_ = ε Φ [=ψ], (EM) = μ Φ [=ψ]; now referring back to 79), we can denote the positive square root of this expression as Det^{1/4} S.
Since _ḟ_ = -_f_, and Ḟ = -F, we obtain for Ṡ, the transposed matrix of S, the following relations from (78),
(84) F_f_ = Ṡ - L, _f_* F* = -Ṡ - L,
Then is
Ṡ - S = | S_{_h_ _k_} - S_{_t_ _k_} |
an alternating matrix, and denotes a space-time vector of the second kind. From the expressions (83), we obtain,
(85) S - Ṡ = - (εμ - 1) [ω, Ω],
from which we deduce that [see (57), (58)].
(86) ω (S - Ṡ)* = 0,
(87) ω (S - Ṡ) = (εμ - 1) Ω
When the matter is at rest at a space-time point, ω = 0, then the equation 86) denotes the existence of the following equations
Z_{_y_} = Y_{_z_}, X_{_z_} = Z_{_x_}, Y_{_x_} = X_{_y_},
and from 83),
T_{_x_} = Ω₁, T_{_y_} = Ω₂, T_{_z_} = Ω₃
X_{_t_} = εμΩ₁, Y_{_t_} = εμΩ₂, Z_{_t_} = εμΩ₃
Now by means of a rotation of the space co-ordinate system round the null-point, we can make,
Z_{_y_} = Y_{_z_} = 0, X_{_z_} = Z_{_x_} = 0, X_{_x_} = X_{_y_} = 0,
According to 71), we have
(88) X_{_x_} + Y_{_y_} + Z_{_z_} + T_{_t_} = 0,
and according to 83), T_{_t_} > 0. In special cases, where ω vanishes it follows from 81) that
X_{_x_}² = Y_{_y_}² = Z_{_z_}² = T_{_t_}², = (Det^{1/4} S)²,
and if T, and one of the three magnitudes X_{_x_}, Y_{_y_}, Z_{_z_} are = ±Det^{1/4} S, the two others = -Det^{1/4} S. If Ω does not vanish let Ω ≠ 0, then we have in particular from 80)
T_{_z_} X_{_t_} = 0, T_{_z_} Y_{_t_} = 0, Z_{_z_} T_{_z_} + T_{_z_} T_{_t_} = 0,
and if Ω₁ = 0, Ω₂ = 0, Z_{_z_} = -T_{_t_} It follows from (81), (see also 83) that
X_{_x_} = -Y_{_y_} = ±Det^{1/4} S,
and -Z_{_z_} = T_{_t_} = √(Det^{½} S + εμΩ₃²) > Det^{1/4}S.
The space-time vector of the first kind
(89) K = lor S,
is of very great importance for which we now want to demonstrate a very important transformation
According to 78), S = L + _f_F, and it follows that
lor S = lor L + lor _f_F.
The symbol ‘lor’ denotes a differential process which in lor _f_F, operates on the one hand upon the components of _f_, on the other hand also upon the components of F. Accordingly lor _f_F can be expressed as the sum of two parts. The first part is the product of the matrices (lor _f_) F, lor _f_ being regarded as a 1 × 4 series matrix. The second part is that part of lor _f_F, in which the diffentiations operate upon the components of F alone. From 78) we obtain
_f_F = -F*_f_* - 2L;
hence the second part of lor _f_F = -(lor F*)_f_* + the part of -2 lor L, in which the differentiations operate upon the components of F alone. We thus obtain
lor S = (lor _f_)F - (lor F*)_f_* + N,
where N is the vector with the components
N_{_h_} = ½(∂_f₂₃_/∂_x__{_h_} F₂₃ + ∂_f₃₁_/∂_x__{_h_} F₃₁ + ∂_f₁₂_/∂_x__{_h_} F₁₂ + ∂_f₁₄_/∂_x__{_h_} F₁₄ + ∂_f₂₄_/∂_x__{_h_} F₂₄ + ∂_f₃₄_/∂_x__{_h_} F₃₄ - ∂F₂₃/∂_x__{_h_} _f₂₃_ - ∂F₃₁/∂_x__{_h_} _f_₃₁ - ∂F₁₂/∂_x__{_h_} _f₁₂_ - ∂F₁₄/∂_x__{_h_} _f₁₄_ - ∂F₂₄/∂_x__{_h_} _f₂₄_ - ∂F₃₄/∂_x__{_h_} _f₃₄_),
(_h_ = 1, 2, 3, 4)
By using the fundamental relations A) and B), 90) is transformed into the fundamental relation
(91) lor S = -_s_F + N.
In the limitting case ε = 1, μ = 1, _f_ = F, N vanishes identically.
Now upon the basis of the equations (55) and (56), and referring back to the expression (82) for L, and from 57) we obtain the following expressions as components of N,—
(92) N_{_h_} = - ½ Φ[=Φ]∂ε/∂_x__{_h_} - ½ ψ[=ψ]∂μ/∂_x__{_h_} + (εμ - 1)(Ω₁ ∂ω₁/∂_x__{_h_} + Ω₂ ∂ω₂/∂_x__{_h_} + Ω₃ ∂ω₃/∂_x__{_h_} + Ω₄ ∂ω₄/∂_x__{_h_})
for _h_ = 1, 2, 3, 4.
Now if we make use of (59), and denote the space-vector which has Ω₁, Ω₂, Ω₃ as the _x_, _y_, _z_ components by the symbol W, then the third component of 92) can be expressed in the form
(93) (εμ - 1)/√(1 - _u²_) (W ∂_u_/∂_x__{_h_}),
The round bracket denoting the scalar product of the vectors within it.
§ 14. The Ponderomotive Force.[28]
Let us now write out the relation K = lor S = -_s_F + N in a more practical form; we have the four equations
(94) K₁ = ∂X_{_x_}/∂_x_ + ∂X_{_y_}/∂_y_ + ∂X_{_y_}/∂_z_ - ∂X_{_t_}/∂_t_ = ρE_{_x_} + _s__{_y_}M_{_z_} - _s__{_z_}M_{_x_}
- ½ Φ[=Φ] ∂ε/∂_x_ - ½ ψ[=ψ]∂μ/∂_x_ + (εμ - 1)/√(1 - _u²_) (W∂_u_/∂_x_),
(95) K₂ = ∂Y_{_x_}/∂_x_ + ∂Y_{_y_}/∂_y_ + ∂Y_{_z_}/∂_z_ - ∂Y_{_t_}/∂_t_ = ρE_{_y_} + _s__{_z_}M_{_x_} - _s__{_x_}M_{_y_}
- ½ Φ[=Φ]∂ε/∂_y_ - ½ ψ[=ψ]∂μ/∂_y_ + (εμ - 1)/√(1 - _u²_) (W∂_u_/∂_y_),
(96) K₃ = ∂Z_{_x_}/∂_x_ + ∂Z_{_y_}/∂_y_ + ∂Z_{_z_}/∂_z_ - ∂Z_{_t_}/∂_t_ = ρE₂ + _s__{_x_}M_{_y_} - _s__{_y_}M₄
- ½ Φ[=Φ] ∂ε/∂z - ½ ψ[=ψ] ∂μ/∂_z_ + (εμ - 1)/√(1 - _u²_) (W∂_u_/∂_z_),
(97) (1/_i_)K₄ = ∂T_{_y_}/∂_x_ - ∂T_{_y_}/∂_y_ - ∂T_{_z_}/∂_z_ - ∂T_{_t_}/∂_t_ = _s__{_x_}E_{_x_} + _s__{_y_}E_{_y_} + _s__{_z_}E_{_z_}
- ½ Φ[=Φ]∂ε/∂_t_ - ½ ψ[=ψ]∂μ/∂_t_ + (εμ - 1)/√(1 - _u²_) (W∂_u_/∂_t_).
It is my opinion that when we calculate the ponderomotive force which acts upon a unit volume at the space-time point _x_, _y_, _z_, _t_, it has got, _x_, _y_, _z_ components as the first three components of the space-time vector
K + (ωK)ω,
This vector is perpendicular to ω; the law of Energy finds its expression in the fourth relation.
The establishment of this opinion is reserved for a separate tract.
In the limiting case ε = 1, μ = 1, σ = 0, the vector N = 0, S = ρω, ωK = 0, and we obtain the ordinary equations in the theory of electrons.
Footnote 9:
_Vide_ Note 1.
Footnote 10:
Note 2.
Footnote 11:
_Vide_ Note 3.
Footnote 12:
_Vide_ Note 4.
Footnote 13:
Note 5.
Footnote 14:
See notes on § 8 and 10.
Footnote 15:
See note 9.
Footnote 16:
See Note.
Footnote 17:
Vide Note.
Footnote 18:
Just as beings which are confined within a narrow region surrounding a point on a spherical surface, may fall into the error that a sphere is a geometric figure in which one diameter is particularly distinguished from the rest.
Footnote 19:
Einzelne stelle der Materie.
Footnote 20:
Vide Note.
Footnote 21:
_Vide_ note 13.
Footnote 22:
_Vide_ note 14.
Footnote 23:
_Vide_ note 15.
Footnote 24:
_Vide_ note 16.
Footnote 25:
_Vide_ note 17.
Footnote 26:
_Vide_ note 19.
Footnote 27:
_Vide_ note 18.
Footnote 28:
Vide note 40.
APPENDIX Mechanics and the Relativity-Postulate.
It would be very unsatisfactory, if the new way of looking at the time-concept, which permits a Lorentz transformation, were to be confined to a single part of Physics.
Now many authors say that classical mechanics stand in opposition to the relativity postulate, which is taken to be the basis of the new Electro-dynamics.
In order to decide this let us fix our attention upon a special Lorentz transformation represented by (10), (11), (12), with a vector _v_ in any direction and of any magnitude _q_ < 1 but different from zero. For a moment we shall not suppose any special relation to hold between the unit of length and the unit of time, so that instead of _t_, _t′_, _q_, we shall write _ct_, _ct′_, and _q_/_c_, where _c_ represents a certain positive constant, and _q_ is < _c_. The above mentioned equations are transformed into
_r′__{_ṽ_} = _r__{_ṽ_}, _r′__{_v_} = _c_(_r__{_v_} - _qt_)/√(_c²_ - _q²_), _t′_ = (_qr__{_v_} + _c²__t_)/_c_√(_c²_ - _q²_)
They denote, as we remember, that _r_ is the space-vector (_x_, _y_, _z_), _r′_ is the space-vector (_x′_ _y′_ _z′_)
If in these equations, keeping _v_ constant we approach the limit _c_ = ∞, then we obtain from these
_r′__{_ṽ_} = _r__{_ṽ_}, _r′__{_v_} = _r__{_v_} - _qt_, _t′_ = _t_.
The new equations would now denote the transformation of a spatial co-ordinate system (_x_, _y_, _z_) to another spatial co-ordinate system (_x′_ _y′_ _z′_) with parallel axes, the null point of the second system moving with constant velocity in a straight line, while the time parameter remains unchanged. We can, therefore, say that classical mechanics postulates a covariance of Physical laws for the group of homogeneous linear transformations of the expression
-_x²_ - _y²_ - _z²_ + _c²_ (1)
when _c_ = ∞.
Now it is rather confusing to find that in one branch of Physics, we shall find a covariance of the laws for the transformation of expression (1) with a finite value of _c_, in another part for _c_ = ∞.
It is evident that according to Newtonian Mechanics, this covariance holds for _c_ = ∞ and not for _c_ = velocity of light.
May we not then regard those traditional covariances for _c_ = ∞ only as an approximation consistent with experience, the actual covariance of natural laws holding for a certain finite value of _c_.
I may here point out that by if instead of the Newtonian Relativity-Postulate with _c_ = ∞, we assume a relativity-postulate with a finite _c_, then the axiomatic construction of Mechanics appears to gain considerably in perfection.
The ratio of the time unit to the length unit is chosen in a manner so as to make the velocity of light equivalent to unity.
While now I want to introduce geometrical figures in the manifold of the variables (_x_, _y_, _z_, _t_), it may be convenient to leave (_y_, _z_) out of account, and to treat _x_ and _t_ as any possible pair of co-ordinates in a plane, referred to oblique axes.
A space time null point 0 (_x_, _y_, _z_, _t_ = 0, 0, 0, 0) will be kept fixed in a Lorentz transformation.
The figure -_x²_ - _y²_ - _z²_ + _t²_ = 1, _t_ > 0 ... (2)
which represents a hyper boloidal shell, contains the space-time points A (_x_, _y_, _z_, _t_ = 0, 0, 0, 1), and all points A′ which after a Lorentz-transformation enter into the newly introduced system of reference as (_x′_, _y′_, _z′_, _t′_ = 0, 0, 0, 1).
The direction of a radius vector 0A′ drawn from 0 to the point A′ of (2), and the directions of the tangents to (2) at A′ are to be called normal to each other.
Let us now follow a definite position of matter in its course through all time _t_. The totality of the space-time points (_x_, _y_, _z_, _t_) which correspond to the positions at different times _t_, shall be called a space-time line.
The task of determining the motion of matter is comprised in the following problem:—It is required to establish for every space-time point the direction of the space-time line passing through it.
To transform a space-time point P (_x_, _y_, _z_, _t_) to rest is equivalent to introducing, by means of a Lorentz transformation, a new system of reference (_x′_, _y′_, _z′_, _t′_), in which the _t′_ axis has the direction 0A′, 0A′ indicating the direction of the space-time line passing through P. The space _t′_ = const, which is to be laid through P, is the one which is perpendicular to the space-time line through P.
To the increment _dt_ of the time of P corresponds the increment
_d_τ = √(_dt²_ - _dx²_ - _dy²_) - _dz²_ = _dt_√(1 - _u²_)
of the newly introduced time parameter _t′_. The value of the integral
∫ _dτ_ = ∫ √(-(_dx₁²_ + _dx₂²_ + _dx₃²_ + _dx₄²_))
when calculated upon the space-time line from a fixed initial point P₀ to the variable point P, (both being on the space-time line), is known as the ‘Proper-time’ of the position of matter we are concerned with at the space-time point P. (It is a generalization of the idea of Positional-time which was introduced by Lorentz for uniform motion.)
If we take a body R₀ which has got extension in space at time _t₀_, then the region comprising all the space-time line passing through R₀ and _t₀_ shall be called a space-time filament.
If we have an analytical expression θ(_x_ _y_, _z_, _t_) so that θ(_x_, _y_ _z_ _t_) = 0 is intersected by every space time line of the filament at one point,—whereby
-(∂Θ/∂_x_)², -(∂Θ/∂_y_)², -(∂Θ/∂_z_)², -(∂Θ/∂_t_)² > 0, ∂Θ/∂_t_ > 0.
then the totality of the intersecting points will be called a cross section of the filament.
At any point P of such across-section, we can introduce by means of a Lorentz transformation a system of reference (_x′_, _y_, _z′_ _t_), so that according to this
∂Θ/∂_x′_ = 0, ∂Θ/∂_y′_ = 0, ∂Θ/∂_z′_ = 0, ∂Θ/∂_t′_ > 0.
The direction of the uniquely determined _t′_—axis in question here is known as the upper normal of the cross-section at the point P and the value of _d_J = ∫∫∫ _dx′ dy′ dz′_ for the surrounding points of P on the cross-section is known as the elementary contents (Inhalts-element) of the cross-section. In this sense R₀ is to be regarded as the cross-section normal to the _t_ axis of the filament at the point _t_ = _t₀_, and the volume of the body R₀ is to be regarded as the contents of the cross-section.
If we allow R₀ to converge to a point, we come to the conception of an infinitely thin space-time filament. In such a case, a space-time line will be thought of as a principal line and by the term ‘Proper-time’ of the filament will be understood the ‘Proper-time’ which is laid along this principal line; under the term normal cross-section of the filament, we shall understand the cross-section upon the space which is normal to the principal line through P.
We shall now formulate the principle of conservation of mass.
To every space R at a time _t_, belongs a positive quantity—the mass at R at the time _t_. If R converges to a point (_x_, _y_, _z_, _t_), then the quotient of this mass, and the volume of R approaches a limit μ(_x_, _y_, _z_, _t_), which is known as the mass-density at the space-time point (_x_, _y_, _z_, _t_).
The principle of conservation of mass says—that for an infinitely thin space-time filament, the product μ_d_J, where μ = mass-density at the point (_x_, _y_, _z_, _t_) of the filament (_i.e._, the principal line of the filament), _d_J = contents of the cross-section normal to the _t_ axis, and passing through (_x_, _y_, _z_, _t_), is constant along the whole filament.
Now the contents _d_J_{n} of the normal cross-section of the filament which is laid through (_x_, _y_, _z_, _t_) is
(4) _d_J_{n} = (1/√(1 - _u²_))_d_J = -_i_ω₄ _d_J = (_dt_/_d_τ)_d_J.
and the function
ν = μ/-_i_ω₄ = μ√(1 - _u²_)) = μ(∂τ/∂_t_. (5)
may be defined as the rest-mass density at the position (_x_ _y_ _z_ _t_). Then the principle of conservation of mass can be formulated in this manner:—
_For an infinitely thin space-time filament, the product of the rest-mass density and the contents of the normal cross-section is constant along the whole filament._
In any space-time filament, let us consider two cross-sections Q° and Q′, which have only the points on the boundary common to each other; let the space-time lines inside the filament have a larger value of _t_ on Q′ than on Q°. The finite range enclosed between Q° and Q′ shall be called a space-time _sichel_,[29] Q′ is the lower boundary, and Q′ is the upper boundary of the _sichel_.
If we decompose a filament into elementary space-time filaments, then to an entrance-point of an elementary filament through the lower boundary of the _sichel_, there corresponds an exit point of the same by the upper boundary, whereby for both, the product νdJ_{n} taken in the sense of (4) and (5), has got the same value. Therefore the difference of the two integrals ∫ν_dJ__{n} (the first being extended over the upper, the second upon the lower boundary) vanishes. According to a well-known theorem of Integral Calculus the difference is equivalent to
∫∫∫∫ lor ν[=ω] _dx dy dz dt_,
the integration being extended over the whole range of the _sichel_, and (comp. (67), § 12)
lor ν[=ω] = (∂νω₁/∂_x₁_) + (∂νω₂/∂_x₂_) + (∂νω₃/∂_x₃_) + (∂νω₄/∂_x₄_).
If the _sichel_ reduces to a point, then the differential equation
lor ν[=ω] = 0, (6)
which is the condition of continuity
(∂μ_u__{_x_}/∂_x_) + (∂μ_u__{_y_}/∂_y_) + (∂μ_u__{_z_}/∂_z_) + (∂μ/∂_t_) = 0.
Further let us form the integral
N = ∫ ∫∫∫ ν _dx dy dz dt_ (7)