152
DOC.
13
GENERALIZED THEORY OF RELATIVITY
proportional
to
the
gravitational mass,
and the second
to
the inertial
mass.
Thus,
the
direction of the resultant
of
these two
forces
,
i.e.,
the
direction
of
the
apparent
gravitational
force
(direction
of the
plumb)
would have
to
depend
on
the
physical
nature
of the
body
under consideration if the
proportionality
of the inertial and
gravitational
mass were
not
satisfied. In that
case
the
apparent gravitational
forces
acting
on
parts
of
a
heterogeneous rigid system
would,
in
general,
not
merge
into
a
resultant; instead,
in
general,
there
would still
be
a torque
associated
with
the
apparent gravitational
forces that would have
to
make itself noticeable if the
system
were
suspended
from
a
torsion-free thread.
By having
established the absence of such
torques
with
great
care,
Eötvös
proved
that,
for the bodies that he
investigated,
the
relationship
of the
two
masses was
independent
of the
nature
of the
body
to
such
a
degree
of
exactness
that the relative difference in this
relationship
that
might
still
exist from
one
substance
to
another
must
be smaller than
one twenty-millionth.
The
decomposition
of radioactive substances
occurs
with
a
release of such
significant quantities
of
energy
that the
change
in
the inertial
mass
of the
system
that
corresponds
to
that
energy
decrease
according
to
the
theory
of
relativity
is not
very
small relative
to
the total
mass.3 In
the
case
of the
decay
of
radium,
for
example,
this decrease
amounts to
one
ten-thousandth of the
total
mass.
If these
changes
of the
inertial
mass
did
not
correspond
to
changes
in the
gravitational mass,
then there
would have
to
be deviations of the inertial
mass
from the
gravitational
mass
much
[5]
greater
than those allowed
by
Eötvös's
experiments.
Hence
it must
be considered
very
probable
that
the
identity
of the inertial and
gravitational
mass
is
exactly
satisfied.
For these
reasons
it
seems
to
me
that the
equivalence hypothesis,
which
asserts
the
essential
physical identity
of
the
gravitational
with
the inertial
mass, possesses
a
high
degree
of
probability.4
§1.
Equations
of Motion
of
the
Material Point
in the Static
Gravitational
Field
According
to
the
customary theory
of
relativity,5
in
the absence of forces
a
point
moves
according
to
the
equation
(1)
^{/^}
=
5[|\/-dx2 -
dy2
-
dz2
+
c2dt2j
=
0.
For this
equation
states
that the material
point
moves
rectilinearly
and
uniformly.
This
is
the
equation
of motion
in
the form of Hamilton's
principle;
for
we can
also
3The
decrease
of the inertial
mass corresponding
to
the
released
energy
E
is,
as
we
know,
E/c2,
if
c
denotes the
velocity
of
light.
[4]
4Cf.
also
§7
of
this
paper.
[6]
5Cf.
M.
Planck,
Verh. d.
deutsch,
phys.
Ges.
(1906):
136.