Then we arrive at the following equation for the motion of matter,
(21) ν _d_/_d_τ (_dx_/_d_τ) = X, ν _d_/_d_τ (_dy_/_d_τ) = Y, ν
_d_/_d_τ (_dz_/_d_τ) = Z,
ν _d_/_d_τ (_dx_/_d_τ) = T, and we have also
(_dx_/_d_τ)² + (_dy_/_d_τ)² + (_dz_/_d_τ)² > (_dt_/_d_τ)² = -1,
and X _dx_/_d_τ + Y _dy_/_d_τ + Z _dz_/_d_τ = T _dt_/_d_τ.
On the basis of this condition, the fourth of equations (21) is to be
regarded as a direct consequence of the first three.
From (21), we can deduce the law for the motion of a material point,
_i.e._, the law for the career of an infinitely thin space-time
filament.
Let _x_, _y_, _z_, _t_, denote a point on a principal line chosen in any
manner within the filament. We shall form the equations (21) for the
points of the normal cross section of the filament through _x_, _y_,
_z_, _t_, and integrate them, multiplying by the elementary contents of
the cross section over the whole space of the normal section. If the
integrals of the right side be R_{_x_} R_{_y_} R_{_z_} R_{_t_} and if
_m_ be the constant mass of the filament, we obtain
(22) _m_ _d_/_d_τ _dx_/_d_τ = R_{_x_},
_m_ _d_/_d_τ _dy_/_d_τ = R_{_y_},
_m_ _d_/_d_τ _dz_/_d_τ = R_{_z_},
_m_ _d_/_d_τ _dt_/_d_τ = R_{_t_}
R is now a space-time vector of the 1st kind with the components
(R_{_x_} R_{_y_} R_{_z_} R_{_t_}) which is normal to the space-time
vector of the 1st kind _w_,—the velocity of the material point with the
components
_dx_/_d_τ, _dy_/_d_τ, _dz_/_d_τ, _i_ _dt_/_d_τ.
We may call this vector R _the moving force of the material point_.
If instead of integrating over the normal section, we integrate the
equations over that cross section of the filament which is normal to the
_t_ axis, and passes through (_x_, _y_, _z_, _t_), then [See (4)] the
equations (22) are obtained, but
are now multiplied by _d_τ/_dt_; in particular, the last equation comes
out in the form,
_m_ _d_/_dt_ (_dt_/_d_τ) = _w__{_x_} R_{_x_} _d_τ/_dt_ + _w__{_y_}
R_{_y_} _d_τ/_dt_ + _w__{_z_} R_{_z_} _d_τ/_dt_.
The right side is to be looked upon _as the amount of work done per unit
of time_ at the material point. In this equation, we obtain the
energy-law for the motion of the material point and the expression
_m_ (_dt_/_d_τ - 1) = _m_ [1/√(1 - _w²_) - 1]
= _m_ (½ |_w₁²_ + 3/8 |_w₁⁴_ + )
may be called the kinetic energy of the material point.
Since _dt_ is always greater than _d_τ we may call the quotient (_dt_ -
_d_τ)/_d_τ as the “Gain” (vorgehen) of the time over the proper-time of
the material point and the law can then be thus expressed;—The kinetic
energy of a material point is the product of its mass into the gain of
the time over its proper-time.
The set of four equations (22) again shows the symmetry in (_x_, _y_,
_z_, _t_), which is demanded by the relativity postulate; to the fourth
equation however, a higher physical significance is to be attached, as
we have already seen in the analogous case in electrodynamics. On the
ground of this demand for symmetry, the triplet consisting of the first
three equations are to be constructed after the model of the fourth;
remembering this circumstance, we are justified in saying,—
“If the relativity-postulate be placed at the head of mechanics, then
the whole set of laws of motion follows from the law of energy.”
I cannot refrain from showing that no contradiction to the assumption on
the relativity-postulate can be expected from the phenomena of
gravitation.
If B*(_x_*, _y_*, _z_*, _t_*) be a solid (fester) space-time point, then
the region of all those space-time points B (_x_, _y_, _z_, _t_), for
which
(23) (_x_ - _x_*)² + (_y_ - _y_*)² + (_z_ - _z_*)² = (_t_ - _t_*)²
_t_ - _t_* >= 0
may be called a “Ray-figure” (Strahl-gebilde) of the space time point
B*.
A space-time line taken in any manner can be cut by this figure only at
one particular point; this easily follows from the convexity of the
figure on the one hand, and on the other hand from the fact that all
directions of the space-time lines are only directions from B* towards
to the concave side of the figure. Then B* may be called the light-point
of B.
If in (23), the point (_x_ _y_ _z_ _t_) be supposed to be fixed, the
point (_x_* _y_* _z_* _t_*) be supposed to be variable, then the
relation (23) would represent the locus of all the space-time points B*,
which are light-points of B.
Let us conceive that a material point F of mass _m_ may, owing to the
presence of another material point F*, experience a moving force
according to the following law. Let us picture to ourselves the
space-time filaments of F and F* along with the principal lines of the
filaments. Let BC be an infinitely small element of the principal line
of F; further let B* be the light point of B, C* be the light point of C
on the principal line of F*; so that OA′ is the radius vector of the
hyperboloidal fundamental figure (23) parallel to B*C*, finally D* is
the point of intersection of line B*C* with the space normal to itself
and passing through B. The moving force of the mass-point F in the
space-time point B is now the space-time vector of the first kind which
is normal to BC, and which is composed of the vectors
(24) _mm_*(OA′/B*D*)³ BD* in the direction of BD*, and another vector of
suitable value in direction of B*C*.
Now by (OA′/B*D*) is to be understood the ratio of the two vectors in
question. It is clear that this proposition at once shows the covariant
character with respect to a Lorentz-group.
Let us now ask how the space-time filament of F behaves when the
material point F* has a uniform translatory motion, _i.e._, the
principal line of the filament of F* is a line. Let us take the space
time null-point in this, and by means of a Lorentz-transformation, we
can take this axis as the t-axis. Let _x_, _y_, _z_, _t_, denote the
point B, let τ* denote the proper time of B*, reckoned from O. Our
proposition leads to the equations
(25) _d²__x_/_d_τ² = - _m_*_x_/(_t_ - τ*)², _d²__y_/_d_τ² = -
_m_*_y_/(_t_ - τ*)³
_d²__z_/_d_τ² = -_m_*_z_/(_t_ - τ*)³,
(26) _d²__t_/_d_τ² = -_m_*/(_t_ - τ*)² _d_(_t_ - τ*)/_dt_
where (27) _x²_ + _y²_ + _z²_ = (_t_ - τ*)²
and (28) (_dx_/_d_τ)² + (_dy_/_d_τ)² + (_dz_/_d_τ)² = (_dt_/_d_τ)² - 1.
In consideration of (27), the three equations (25) are of the same form
as the equations for the motion of a material point subjected to
attraction from a fixed centre according to the Newtonian Law, only that
instead of the time _t_, the proper time τ of the material point occurs.
The fourth equation (26) gives then the connection between proper time
and the time for the material point.
Now for different values of τ′, the orbit of the space-point (_x_ _y_
_z_) is an ellipse with the semi-major axis _a_ and the eccentricity
_e_. Let E denote the eccentric anomaly, Τ the increment of the proper
time for a complete description of the orbit, finally _n_Τ = 2π, so that
from a properly chosen initial point τ, we have the Kepler-equation
(29) _n_τ = E - _e_ sin E.
If we now change the unit of time, and denote the velocity of light by
_c_, then from (28), we obtain
(30) (_dt_/_d_τ)² - 1
= (_m_*/_ac²_) (1 + _e_ cos E)/(1 - _e_ cos E)