By the world-postulate a similar treatment of the four determining
quantities _x_, _y_, _z_, _t_, of a world-point is possible. Thereby the
forms under which the physical laws come forth, gain in intelligibility,
as I shall presently show. Above all, the idea of acceleration becomes
much more striking and clear.
I shall again use the geometrical method of expression. Let us call any
world-point O as a “Space-time-null-point.” The cone
_c²__t²_ - _x²_ - _y²_ - _z²_ = O
consists of two parts with O as apex, one part having _t_ < 0, the other
having _t_ > 0. The first, which we may call _the fore-cone_ consists of
all those points which send light towards O, the second, which we may
call _the aft-cone_, consists of all those points which receive their
light from O. The region bounded by the fore-cone may be called the
fore-side of O, and the region bounded by the aft-cone may be called the
aft-side of O. (_Vide_ fig. 2).
On the aft-side of O we have the already considered hyperboloidal shell
F = _c²__t²_ - _x²_ - _y²_ - _z²_ = 1, _t_ > 0.
The region inside the two cones will be occupied by the hyperboloid of
one sheet
-F = _x²_ + _y²_ + _z²_ - _c²__t²_ = _k²_,
where _k²_ can have all possible positive values. The hyperbolas which
lie upon this figure with O as centre, are important for us. For the
sake of clearness the individual branches of this hyperbola will be
called the “_Inter-hyperbola with centre O_.” Such a hyperbolic branch,
when thought of as a world-line, would represent a motion which for _t_
= -∞ and _t_ = ∞, asymptotically approaches the velocity of light _c_.
If, by way of analogy to the idea of vectors in space, we call any
directed length in the manifoldness _x_, _y_, _z_, _t_ a vector, then we
have to distinguish between a time-vector directed from O towards the
sheet ±F = 1, _t_ > 0 and a space-vector directed from O towards the
sheet -F = 1. The time-axis can be parallel to any vector of the first
kind. Any world-point between the _fore_ and _aft cones_ of O, may by
means of the system of reference be regarded either as synchronous with
O, as well as later or earlier than O. Every world-point on the
fore-side of O is necessarily always earlier, every point on the aft
side of O, later than O. The limit _c_ = ∞ corresponds to a complete
folding up of the wedge-shaped cross-section between the fore and aft
cones in the manifoldness _t_ = 0. In the figure drawn, this
cross-section has been intentionally drawn with a different breadth.
Let us decompose a vector drawn from O towards (_x_, _y_, _z_, _t_) into
its components. If the directions of the two vectors are respectively
the directions of the radius vector OR to one of the surfaces ±F = 1,
and of a tangent RS at the point R of the surface, then the vectors
shall be called normal to each other. Accordingly
_c²__tt₁_ - _xx₁_ - _yy₁_ - _zz₁_ = 0,
which is the condition that the vectors with the components (_x_, _y_,
_z_, _t_) and (_x₁_ _y₁_ _z₁_ _t₁_) are normal to each other.
For the _measurement_ of vectors in different directions, the unit
measuring rod is to be fixed in the following manner;—a space-like
vector from 0 to -F = I is always to have the measure unity, and a
time-like vector from O to +F = 1, _t_ > 0 is always to have the measure
1/_c_.
Let us now fix our attention upon the world-line of a substantive point
running through the world-point (_x_, _y_, _z_, _t_); then as we follow
the _progress_ of the line, the quantity
_d_τ = (1/_c_) √(_c²__dt²_ - _dx²_ - _dy²_ - _dz²_),
corresponds to the time-like vector-element (_dx_, _dy_, _dz_, _dt_).
The integral τ = ∫_d_τ, taken over the world-line from any fixed
initial point P₀ to any variable final point P, may be called the
“Proper-time” of the substantial point at P₀ upon the _world-line_. We
may regard (_x_, _y_, _z_, _t_), _i.e._, the components of the vector
OP, as functions of the “proper-time” τ; let ([._x_], [._y_], [._z_],
[._t_]) denote the first differential-quotients, and ([.._x_],
[.._y_], [.._z_], [.._t_]) the second differential quotients of (_x_,
_y_, _z_, _t_) with regard to τ, then these may respectively be called
the _Velocity-vector_, and the _Acceleration-vector_ of the
substantial point at P. Now we have
_c²_ [._t²_] - [._x²_] - [._y²_] - [._z²_] = _c²_
_c²_ [._t_][.._t_] - [._x_][.._x_] - [._y_][.._y_] - [._z_][.._z_] =
0
_i.e._, the ‘_Velocity-vector_’ is the time-like vector of unit measure
in the direction of the world-line at P, the ‘_Acceleration-vector_’ at
P is normal to the velocity-vector at P, and is in any case, a
space-like vector.
Now there is, as can be easily seen, a certain hyperbola, which has
three infinitely contiguous points in common with the world-line at P,
and of which the asymptotes are the generators of a ‘fore-cone’ and an
‘aft-cone.’ This hyperbola may be called the “hyperbola of curvature” at
P (_vide_ fig. 3). If M be the centre of this hyperbola, then we have to
deal here with an ‘Inter-hyperbola’ with centre M. Let P = measure of
the vector MP, then we easily perceive that the acceleration-vector at P
is _a vector of magnitude_ _c²_/ρ _in the direction of_ MP.
If [.._x_], [.._y_], [.._z_], [.._t_] are nil, then the hyperbola of
curvature at P reduces to the straight line touching the world-line at
P, and ρ = ∞.