_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_).