DOC.
71
PRINCETON LECTURES 297
SPECIAL
RELATIVITY
every system
of
reference, form
the
physical
content,
free
from
convention,
of
the Lorentz
transformation.
Addition Theorem
for
Velocities.
If
we
combine
two
special
Lorentz transformations with the relative
velocities
v1
and
v2,
then the
velocity
of
the
single
Lorentz transformation
which
takes
the
place
of
the
two separate
ones is,
according
to (27), given
by
(30)
V12
=
i
tan
(f/1
+
\¡/2)
=
i
tan
\¡/1
+
tan
u2 =
v1
+
v2.
1

tan^1 tan w2
1
+
v1v2
General
Statements
about
the
Lorentz
Transformation
and its
Theory
of
Invariants.
The
whole
theory
of invariants of the
special theory
of
relativity depends upon
the invariant
s2
(23). Formally,
it
has
the
same
rôle in
the fourdimensional
spacetime
continuum
as
the invariant
Ax12
+
Ax22
+
Ax32
in the Euclidean
geometry
and in the
prerelativity
physics.
The latter
quantity is not
an
invariant with
respect
to
all
the Lorentz
transformations;
the
quantity
s2
of
equation
(23)
assumes
the rôle of
this
invariant. With
respect to
an
arbitrary
inertial
system, s2
may
be
determined
by
measure
ments;
with
a
given
unit of
measure
it
is
a
completely
determinate
quantity,
associated with
an
arbitrary pair
of
events.
The invariant
s2 differs, disregarding
the number of
dimensions,
from
the
corresponding
invariant of the
Euclidean
geometry
in
the
following points.
In the
Euclidean
geometry
s2
is
necessarily
positive;
it
vanishes
only
when
the
two points
concerned
come
together.
On
the other
hand,
from
the
vanishing
of
s2
=
Axv2
=
Ax12
+
Ax22
+
Ax32

At2
it
cannot
be
concluded that the
two
spacetime points
[37]
[40]