DOC.
71
PRINCETON LECTURES 299
SPECIAL RELATIVITY
may
be
drawn
in
the direction
of
PP'
by
suitably choosing
the
state
of
motion
of
the inertial
system.
If P'
lies
out-
side of
the
“light-cone”
then
PP'
is space-like;
in this
case,
by
properly
choosing
the inertial
system,
Al
can
be
made
to
vanish.
By
the introduction of the
imaginary
time
variable,
x4
=
il,
Minkowski has
made the
theory
of
invariants
for
the four-dimensional continuum of
physical
phenomena
fully
analogous to
the
theory
of invariants
for
the three-
dimensional continuum
of
Euclidean
space.
The
theory
of
four-dimensional
tensors
of
special relativity
differs
from
the
theory
of
tensors
in
three-dimensional
space,
therefore,
only
in the
number
of
dimensions
and the relations of
reality.
A
physical entity
which
is
specified by
four
quantities,
Av,
in
an
arbitrary
inertial
system
of
the
x1, x2, x3, x4,
is
called
a
4-vector,
with the
components
Av,
if
the
Av cor-
respond
in their relations of
reality
and the
properties
of transformation
to
the
Axv;
it
may
be
space-like
or
time-
like.
The sixteen
quantities
Auv
then form the
compo-
nents
of
a
tensor
of the
second rank, if
they
transform
according
to
the
scheme
A'uv
=
buabvßAaB.
It
follows from this
that the
Auv
behave,
with
respect
to
their
properties
of transformation and their
properties
of
reality,
as
the
products
of
the
components,
Uu,
Vv,
of
two
4-vectors,
(U)
and
(V).
All
the
components
are
real
except
those
which contain the index
4
once,
those
being purely
imaginary.
Tensors of the third and
higher
ranks
may
be
defined in
an
analogous
way.
The
operations
of
addition,
subtraction,
multiplication,
contraction and differentiation
for these
tensors
are
wholly analogous to
the
corresponding
operations
for
tensors
in
three-dimensional
space.
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