140
DOC.
2
RELATIVITY AND
ITS CONSEQUENCES
This
is
the
condition that
corresponds to
a
rotation without
relative
translation of
a
four-dimensional coordinate
system.
The
principle
of
relativity
demands that the
laws
of
physics
not be
altered
by
the
rotation of
the
four-dimensional coordinate
system
to which
they are
referred. The four
coordinates
x1, x2, x3, x4
must
appear
in
the
laws
symmetrically.
To
express
the different
physical
states,
one can use
four-dimensional
vectors which
behave
in
the
calculations in
a manner analogous
to
ordinary
vectors in
three-dimensional
space.
§9.
Some
Applications
of the
Theory
of
Relativity
Let
us
apply
the transformation
equations
(I)
to
the
Maxwell-Lorentz
equations
representing
the
magnetic
field.
Let
Ex,
Ey,
Ez
be
the
vector
components
of
the electric
field
,
and
Mx,
My,
Mz
the
components
of the
magnetic field,
with
respect
to
the
system
S.
Calculation shows
that the transformed
equations
will be
of the
same
form
as
the
original ones
if
one
sets
(1)
E\
=
EX
M'
X
=
M
X
E'y
=
ß(Ey
-
v/c
Mz) M'y =
ß(My
+
v/c
Ez)
E'
z
= ß(Ez
+
V/C
My)
M'z
-
ß(Mz
-
v/c
Ey)
The
vectors
(Ex, Ey, Ez)
and
(Mx, My, Mz)
play
the
same
role in the
equations
referred
to
S'
as
the
vectors
(Ex,
Ey,
Ez)
and
(Mx,
My,
Mz) play
in
the
equations
referred
to
S.
Hence the
important
result:
The existence of the electric
field, as
well
as
that
of
the
magnetic
field,
depends on
the
state
of motion of
the
coordinate
system.
The transformed
equations permit
us
to
know
an
electromagnetic
field with
respect
to
any
arbitrary
system
in
nonaccelerated motion
S'
if
the
field
is
known relative to
another
system
S
of the
same type.
These
transformations
would be
impossible
if
the
state
of motion of the coordinate
system played
no
role
in
the definition of the
vectors.
This
we
will
recognize
at
once
if
we
consider the
definition of the electric
field
strength:
the
magnitude, direction,
and
orientation of the
field
strength
at
a given
point
are
determined
by
the
ponderomotive
force exerted
by
the
field
on
the unit
quantity
of
electricity,
which is
assumed
to be
concentrated
in
the
point
considered
and
at rest with
respect
to
the
system
of
axes.
The transformation
equations
demonstrate that the
difficulties
we
have
encountered
(§3)
regarding
the
phenomena
caused
by
the relative motions of
a
closed circuit
and
a
magnetic
pole
have been
completely
averted
in
the
new theory.
For let
us
consider
an
electric
charge moving uniformly
with
respect
to
a magnetic
pole.
We
may
observe
this
phenomenon
either
from
a system
of
axes
S
linked
with
the
magnet,
or
from
a system
of
axes
S' linked
with
the electric
charge.
With
respect
to