408 DOC.
53
DEVELOPMENT OF
RELATIVITY
February
17,
1921]
NATURE
783
Now in order that the
special principle
of rela-
tivity may hold,
it is
necessary
that all the equa-
tions of
physics
do
not
alter their form
in
the
transition
from
one
inertial
system
to another,
when
we
make
use
of the
Lorentz
transformation
for the
calculation of this
change.
In the lan-
guage of
mathematics,
all
systems
of
equations
that
express physical
laws
must
be co-variant with
respect to
the Lorentz
transformation. Thus,
from the
point
of view of
method,
the
special prin-
ciple of
relativity
is
comparable
to
Carnot’s prin-
ciple of
the
impossibility
of
perpetual
motion
of
the second kind,
for,
like
the
latter,
it
supplies us
with
a general
condition which
all
natural laws
must
satisfy.
Later,
H. Minkowski found
a particularly
elegant and suggestive
expression
for this
condition of co-variance,
one
which reveals
a
formal
relationship
between Euclidean
geometry
of three dimensions
and the
space-time
continuum
of
physics.
Euclidean
Geometry of
Three
Dimensions.
Corresponding to two
neighbouring
points
in
space,
there
exists
a
numerical
measure (dis-
tance ds)
which
conforms
to
the equation
ds*=dx12+dx22+dx32.
It
is independent
of the
system
of
co-ordinates
chosen,
and
can
be
measured
with
the
unit
measuring-rod.
The
permissible trans-
formations are
of
such
a
character that
the expression
for ds2 is
invariant,
i.e.
the linear
orthogonal
transformations
are per-
missible.
With
respect to
these
transformations,
the
laws
of
Euclidean
geometry
are
invariant.
Special Theory
of
Relativity.
Corresponding to
two
neighbouring points
in
space-time (point events),
there exists
a
numerical
measure
(distance
ds)
which conforms
to
the
equation
ds2
=
dx12+dx22+dx32+dx42
It
is
independent
of
the
inertial
system
chosen,
and
can
be
measured
with
the unit measuring-
rod and
a
standard
clock.
x1, x2, x3
are
here
rectangular co-ordinates,
whilst
X4=V-1ct is the
time
multiplied by
the
imaginary unit
and
by
he
velocity
of
light.
The
permissible
trans-
formations are
of such
a
character that the expres-
sion for ds2
is invariant,
i.e.
those linear ortho-
gonal
substitutions
are
permissible
which
main-
tain the
semblance of
reality of
x1,
x2, x3, x4.
These substitutions
are
the
Lorentz transforma-
tions.
With
respect to these
transformations,
the
laws
of
physics
are
invariant.
From this it follows that,
in
respect
of
its rôle
in
the
equations
of physics, though not
with
regard
to
its
physical significance,
time is
equivalent
to
the
space
co-ordinates (apart
from the relations
of
reality).
From this
point
of
view,
physics
is,
as
it
were, a
Euclidean
geometry
of four dimen-
NO.
2677,
VOL.
106]
sions, or,
more correctly, a
statics in
a four-
dimensional Euclidean continuum.
The
development
of the
special theory
of rela-
tivity consists of
two
main
steps, namely,
the
adaptation
of the
space-time
"metrics"
to
Maxwell's
electro-dynamics,
and
an
adaptation
of
the
rest
of
physics to
that altered
space-time
"metrics." The first of these
processes yields
the
relativity
of
simultaneity,
the
influence of
motion
on measuring-rods
and
clocks,
a modifica-
tion of
kinematics,
and
in particular
a new
theorem
of addition of velocities. The second
process
supplies us
with
a
modification of Newton’s
law
of motion for
large velocities,
together with
information of fundamental
importance on
the
nature
of inertial
mass.
It
was
found that
inertia
is not
a
fundamental
property
of
matter, nor, indeed,
an
irreducible
magnitude,
but
a property
of
energy.
If
an
amount
of
energy
E be
given
to
a body,
the
inertial
mass
of the
body
increases
by
an
amount
E/c2,
where
c
is
the
velocity
of
light in vacuo.
On the other
hand,
a
body
of
mass m
is
to
be
regarded as a
store
of
energy
of
magnitude mc2.
Furthermore,
it
was soon
found
impossible
to
link
up
the science
of
gravitation
with the
special
theory
of
relativity in
a
natural
manner.
In
this
connection
I was
struck
by
the fact that the force
of gravitation
possesses a
fundamental
property,
which
distinguishes
it from
electro-magnetic
forces.
All
bodies fall
in
a
gravitational
field
with
the
same
acceleration,
or-what
is
only
another
formulation of the
same
fact-the
gravitational
and inertial
masses
of
a
body are numerically
equal
to
each other. This numerical
equality
suggests identity
in
character. Can
gravitation
and inertia be identical? This
question
leads
directly
to
the General
Theory
of
Relativity.
Is it
not possible
for
me
to regard
the earth
as
free
from
rotation, if I
conceive of the
centrifugal
force,
which
acts
on
all bodies
rest relatively
to
the
earth,
as being a
"real"atfield
of gravita-
tion, or
part of such
a
field? If this idea can
be
carried
out,
then
we
shall have
proved
in
very
truth
the
identity
of
gravitation
and inertia.
For
the
same
property
which is
regarded as
inertia
from the
point
of view
of
a system
not
taking
part in
the rotation
can
be
interpreted as gravita-
tion when considered with
respect to
a
system
that
shares the rotation.
According
to Newton,
this
interpretation
is
impossible,
because
by
Newton's
law
the
centrifugal
field cannot
be
regarded as
being
produced by
matter,
and because
in
Newton’s
theory
there is
no place
for
a
"real"
field
of the "Koriolis-field"
type.
But
perhaps
Newton’s law
of
field
could be
replaced
by
another
that fits
in
with
the
field
which holds with
respect
to a
"rotating"
system
of co-ordinates?
My
conviction
of the
identity
of
inertial and gravita-
tional mass
aroused
within
me
the
feeling
of abso-
lute confidence
in
the
correctness
of this interpre-
tation. In
this
connection I
gained encourage-
ment from
the
following
idea.
We
are
familiar
with
the
"apparent"
fields which
are
valid rela–