P H YSI CAL
REVIEW
VOLUME 118, NUMBER
4
MAY
15,
1960
Energy
and the
Criteria for
Radiation
in
General
Relativity
R.
ARNOWITT*
Department
of
Physics,
Syracuse
University, Syracuse,
Zero
Fork
S.
Dzszzl'
Department
of
Physics,
Brandeis
University,
Wattham,
Massachusetts
AND
C. W. MrsNzzf
Universitetets Institut
for
Teoretisk
Pysik, Copenhagen,
Denmark
(Received
December
3,
1959)
The Hamiltonian for
general
relativity
obtained
in
a
previous
paper
furnishes
a
definition
of
energy
whose
physical
interpretation
is
direct,
and
which
fulfills the conditions
required
of
the
energy
in
other
physical
systems.
The
energy
can
be expressed
as
a surface
integral
at
spacial
infinity
in terms
of the
spacial
com-
ponents
of the covariant metric tensor at
any
given
time.
Thus,
the
energy
depends
only
on the minimal
initial
Cauchy
data and
may
be
evaluated
in
any
coordinate
system,
provided
this
system
can be made
asymptotically
rectangular.
These statements remain valid when
particles
are
coupled
to the
gravitational
field.
The criteria for existence of gravitational radiation
are
formulated
in terms of
the canonical
variables
and the
stress-tensor. These criteria are identical to
those used in
electromagnetic
theory.
Some
applications
are
discussed.
I. INTRODUCTION
'N
a
previous
paper,
'
a
canonical form for
general
relativity
in terms
of
explicit
canonical
variables
has
been derived.
These
canonical variables were taken
to
be
the two
independent
components
of the
transverse
traceless
parts
of
g;;
and
of
~"=
(
—
'g)'*(
'
—
g
I"'
v"')v'"7'"
(11)
i.e.
,
g
"~~
and
m-'&'~~
We
have
here
made
use of the
general
orthogonal breakup
of
a symmetric
array
ij
f
=f
"+
'Lf'-g.
--(1/v')f";3+f'+
f;
(1
2)
In
Eq.
(1.
2)
f'=f.
,
(1/v')f;;,
'—
,,
(1.3a)
f'=
(1/V')9',
—
s(1/~')ft',
~'3
(13b)
while
1/i7s
is
the inverse
of the
flat
space
Laplacian
operator
(with appropriate
boundary
conditions)
and
f
sr'
=0,
f
rr=0.
This "breakup is
well defined
on
a
space-like
surface in terms
of
a
given
coordinate
system.
The specification
employed
here,
as
in III is
s
(g',
~+
g,
')
=
~'t,
(1.
4a)
*This
research was
supported
in
part
by
the U. S.
Air Force
under
a
contract
monitored
by
The
Aeronautical
Research
Laboratory,
Wright
Air Development
Center.
t
Supported
in
part
by
a
National
Science Foundation
Research
Grant.
/Alfred
P.
Sloan
Research Fellow.
On leave
from Palmer
Physical Laboratory,
Princeton
University,
Princeton,
New
Jersey.
The
previous
papers
in
this series
will be
referred to as
I,
II,
and
III;
they
are:
R.
Arnowitt and S.
Deser, Phys.
Rev.
113,
745
{1959);
R.
Arnowitt,
S.
Deser,
and C.
W.
Misner,
Phys.
Rev.
116,
1322
(1959),
and
Phys.
Rev.
117,
1595
(1960).
Notations and
units are as
in
III, namely,
It:=16m&c
4=1,
c=1
where
7
is the
Newtonian
gravitational
constant.
Latin
indices
run from
1
to
3,
Greek from
0
to
3,
and x
=
t.
All
tensors and
covariant
operations
and
three-dimensional unless
specified,
e.
g.
,
g'&'
or
y'&
is
the matrix
inverse
to
g;;
and
"~"
indicates
covariant
differentiation
with
respect
to
g;,
(not
g„„).
or
alternately
x"
"—
x'&
"=0
g"
=0
(1.4b)
The
above
coordinate conditions
amount
to
using
as
independent
variables
in
place
of
coordinates
the metric
functionals'
g,
=x'
and
(
—
1/2Vs)srr=t.
It
is,
perhaps,
important
to
realize, therefore,
that this entire
work
involves
only
the
functionals
and
g
"&~Lg
—
r
(1/Vs)sr~]
w""Lg',
—
—',
(1/Vs)~&3.
(1.5a)
(1.
5b)
Such
a
situation is
analogous
to the statement of
an
orbit in
ordinary
particle
mechanics
in the
form
r=
r(8).
Correspondingly
our
equatioris
of motion for g;;~~
and
m'&~~
are
analogous
to
the orbit
equations
in
terms of
dr/d9.
The
same
orbit,
of
course,
can
be described
by
way
of
the
pair
of
equations
r=r(r)
and
0=0(r) in
terms of an
arbitrary
parameter
v-.
While in
particle
mechanics there exist additional
equations
that allow
one to
determine
the
dependence
of
v
on
the
time
$,
in
general relativity
(due
to the
general
covariance of
the
theory)
there are no
equations
to
determine the
de-
pendence
of
g,
and (1/Vs)sr' on the
arbitrary
coordinates
x&
that
appear
in the
original
action
J'd'x
(
—
'g)
&
'g
(x).
Since these coordinates
do
not enter
at
any
point
in
the
canonical
form of
the
theory,
general
covariance has
been manifestly
maintained
and
we
are
merely using
the
symbols
x&
as
abbreviations for
g;
and
—
—'
,
(1/7")
oral.
It
was
shown in
III
that the
components,
T'„,
of
the
stress tensor took the
form
(1.
6a)
T',
=
—
2
(sr",
;;+sr'„;)
=
—
2sr'&'
.
(1.6b)
2
As was
pointed
out
in
III,
the invariant
functionals
of
the
metric
being
used as
coordinates
take the form
g;
and
—
~
(1
jV')m
~
only
in
the
preferred
canonical frame of
Eq.
{1.4).
In
other frames
these functionals
take other forms
to
be obtained
by
making
the
appropriate
coordinate transformations.
100
CRITERIA FOR
RAD IATION I
N
GEN E
RAL RELAT I VI T Y iioi
where
K
was
the Hamiltonian
density
of
theory.
In
these
equations, g~,
;;
and
m'&',
;
were
to
be
expressed.
as
functions of
the
canonical
variables
by
solving
for
them
in the
constraint
equations
O'„=
—
R'„—
—,
'8'„R
=
0.
A
perturbation
solution of
these
equations,
at
least,
clearly
exists.
Further, the coordinate conditions
(1.
4)
ensure
that
T'„does
not
depend
explicitly
on
x&
as
was
shown in III.
Thus,
the
standard conservation laws
hold.
The
energy-momentum
of
the
field
is
just
the volume
integral
of
the
components
of
T'„when
a
solution of
the
field
equations
is substituted
in
for
g~
and
m'.
In
this
paper
we
will
see how these
expressions
for
energy
and momentum
may
be
evaluated without
imposing
the canonical coordinate conditions
Eq. (1.
4)
or
explicitly
solving
the constraint
equations.
With the
aid
of
the
expressions
for
energy
and momentum
we
will
be
able
to
write
down
explicit
criteria for
the
existence
of
gravitational
radiation. In order
to
discuss
radiation
escaping
to infinity
we
shall define
a
Poynting
vector
just
as in
electrodynamics.
The
condition for
waves
at a
finite
point
is
simply
the
nonvanishing of
the canonical variables
at this
point.
The derivations of the canonical
form
and the
defini-
tion of
the
energy-momentum
of
III
were
given
only
for
the
free
gravitational
field. In this
paper
we
extend
this
analysis
to
include the
problem
of
point
particles
coupled
to the
gravitational
field.
As will
be
seen,
no
essential
changes
are introduced
by
such
a
generali-
zation. This
extension allows us to
examine
the
energy
for
some
cases
of
interest.
The
possible
relevance of
these
examples
to the
classical
self-energy
problem
is
discussed.
g~,
;;d'x
=
—
g~„dS;,
4
(2.
1)
where
dS~=dx'dx',
etc.
,
are the
rectangular
surface
elements
at
spacial
infinity.
Using
Eq.
(1.
3a),
the
energy
then becomes
II. DEFINITION
AND
PROPERTIES
OF
ENERGY AND
MOMENTUM
From
Eq. (1.
6)
we see
that
the
total
energy
may
be
written as
requirement
that the
integral
be
evaluated in
the
canonical frame.
This
restriction can
be removed
by
the
following argument. At
spacial
infinity,
where
the
metric
approaches
the
Lorentz
value,
coordinate
trans-
formations which
preserve
this
boundary
condition can
be treated in the
linear
approximation.
For
the
in-
finitesimal
coordinate transformation, $&=x&+P,
g,
;
and
x'&
transform
according
to
(2.3a)
(2.3b)
As can
be
seen from
Eqs.
(1.
3)
or
(1.
2),
gr
and
7r'
are
invariant under this transformation,
As
a
consequence,
the
energy
and the momentum can
actually be
evalu-
ated
in
any
frame that is
rectangular at
infinity.
'
The
restriction
to rectangular
frames, conventionally used
in all Lorentz
covariant
theories,
can of
course be
removed
by
making
use of
standard Qat
space
tensor
analysis to
calculate the
energy
in
(asymptotically Rat)
spherical coordinates,
for
example.
Since
the
energy
and
momentum as
given
in
Eqs.
(2.
2)
are
constant in
time,
they
can
be
evaluated
at
any
given
time
as
in
other
dynamical
systems.
Thus,
one should
need,
in
order
to
calculate
I'„,
only
those
initial
Cauchy
data
necessary
to
specify
the
state
of
the
system uniquely.
In
general
relativity,
in the
absence of
coordinate
conditions,
these
are
g;;
and
x",
' '
and not
for
example
go„,
which
are needed
only
to
describe how the
coordinates are
to be continued
oG
the initial surface. As can
be
seen
from
Eq. (2.
2),
only
g,;
and
rr'&
enter into the formulas
for
P„. (In
the
canonical
frame,
only
g;;
~
and
x'&
are needed to
specify
the
state
of the
system
and these
are,
of
course,
sufficient
for
calculating
the value
of
P„.
)
The
value of the
energy-momentum
vector of
Eqs.
(2.
2)
can
be shown
to
agree
with those
obtained from
the surface
integral
forms
derived
from
the Einstein
pseudotensor,
the
Landau-Lifshitz
pseudotensor,
and
Dirac
s
recent definition.
'
This
can
be
seen
by
linear-
izing
the
integrands
in
the surface
integrals
and
noting
that
they
reduce
to
Eqs. (2.2).
Since the
surface
inte-
grals
are
at
spatial
infinity,
the linearization
is
rigorous.
III.
COUPLING WITH PARTICLES
P.
=
(g"
—
g"
)dS.
Similarly,
the
momentum
I';
is
given
by
(2.
2a)
In
the
preceding
discussion
we
have
examined the
properties
of
the
energy
and momentum
of the
un-
coupled gravitational
field. Since
we
shall
be
interested
in
solutions
and
problems involving
sources,
we
shall
f
P;=
—
)
2(
r',
;+
r')AS,
=
—
)
2s.
'&'dS
.
(2.
2b)
These
equations
have been
derived
in
the canonical
coordinate
frame which
by
Eq. (1.4a)
is
asymptotically
rectangular
since
g;;
approaches
8;;
at infinity.
In
fact,
as
was
shown in
III,
g„,
approaches
the
rectangular
Lorentz
metric
p„„at
infinity.
In this frame
g,
;,
;
vanishes.
The
utility
of
Eq. (2.
2)
is
limited
by
the
'A
general
test
can be stated
as to
how
fast the coordinate
system
must
approach
a rectangular
one.
One
first
calculates the
functions
P
according
to
P=g;
and
P'=-',
(1/V')g~. Two require-
ments
must be
imposed
on
P
in order
that the statement that
g~
and
m'
are invariant
be
valid:
First,
the
P
must
vanish sufficiently
rapidly
that
the
quadratic
terms
neglected
in
Eq.
(2.
3)
be negli-
gible.
Second,
the terms
on the right-hand
side of
Eq.
(2.
3)
involving
P
must vanish
as
rapidly
as
g;;
—
8;;
and
~'&'.
4
Y.
Foures-Bruhat,
J.
Rational
Mech.
Anal.
4,
951
(1956).
5
See also
C. W.
Misner
and
J.
A.
Wheeler,
Ann.
Phys.
2,
589—
595
(1957).
s
P.
A.
M.
Dirac,
Phys.
Rev. Letters
2,
368
(1959).
ii02 ARNOK
ITT, DESER,
AN D
M
lSNER
here
extend the
analysis
of the canonical
formalism
to
the case
of
point
particles
coupled
to the gravitational
field. As
will
be
seen,
the
coupling
changes
none of the
formal
results,
and the
expressions
of
Eq.
(2.
2)
for
energy
and
momentum
now include
the matter as
well
as the gravitational
contributions.
The total
action of the
system
is
now
I=
'g
"'14.
'(I'
s)d'x+
~L
d'x,
pv
aP
m
p
(3.
1)
where the
matter
Lagrangian density
takes the
form,
~
L
(x)=
I
ds
(p„(s)$dx
(s)/ds]
—
—
2')I,
'(s)
I
P„(s)P„(s)
'g&"
(x)+mo']}5'(x
—
x(s)).
(3.
2)
We
are
considering
here,
for
simplicity,
the case of
a
single point
particle
with
mass
no.
The matter
Lagrangian
is
given
in first
order
form where
p„(s)
and
x"
(s)
are to be varied
independently.
In
this
parametric
form of the matter
action,
the
parameter
s is
arbitrary
(and
not necessarily
the
proper
time).
The
constraint
equation,
p„p,
4g&"
(x(s))+ms'
——
0,
(3.
3)
obtained
by
varying
with
respect
to
the
Lagrange
multiplier
X'(s)
is,
of
course,
the relativistic
energy-
momentum relation
for
the
particle.
Varying
L
with
respect
to
p„(s),
one obtains
p.
(s)=L(1/)
')(dx"/ds)]
'g"
(3
4)
which is
the
dehning
relation between the
momentum
and
velocity.
Note
that
only
the combination
X'ds
appears
and there are
no
equations
to
determine
X':
Inserting
Eq.
(3.
4)
into
Eq.
(3.
3)
shows that
a
choice
of )
'=
1/mo
corresponds
to s
becoming
the
proper
time.
Finally, varying
with
respect
to
x&(s)
gives
rise
to
the
usual
geodesic
equations
of motion.
The
arbitrary
parameter
s could have been
elimi-
nated initially
by
performing
the indicated
s
integration
in
Eq.
(3.2).
The matter
term in
the
action
then
becomes
I
=
~d4x
(p;dx'(t)/dt+ps
—
2)
Lp.
p
'g""(*)+
o']}~'(
'
—
*'(t))
(3 5)
where
the
form
(to
within a
divergence)
d4x
(~'ia,
g,
,
+
p,
$dx'(t)/dt]P(x'
—
x'(t))
+X+
—
&(g
2E+-'ors
—
or'42r
")
—
P(x'
—
x"(t))(p
p'+
mos)&]
+o'L2
"I
+
p'~'(*'
—
*'(t))]}
(3
8)
where
K=
—
T'0=
—
g~,
;;,
(3.
12a)
T';
=
—
2 (2r'
"+or',
4)
=
—
22r"
(3.
12b)
In
Eq.
(3.
12),
gT
and
ori
are
to
be expressed
in terms of
the
CanOniCal
VariableS,
g;rTT
rr"TT,
X'(t),
p,
(t),
by
solving
the
constraint
equations
(3.9).
The
generator
(3.
11)
is
now
clearly
in canonical
form.
Thus, X
is
the
Hamiltonian
density
of the entire
coupled
system
and
T;
is
the total
momentum
density.
The source on
the
right-hand side
of
Eq. (3.
9)
does
not
disturb the
proof
in III
that
T'„
is
independent
of
x&.
Thus,
the
con-
servation
laws
of
the
previous
section still
hold with the
same
coordinate conditions.
The
energy
E
of the
coupled system
becomes now
In
Eq.
(3.
8),
covariant notations are
three-dimensional,
'
e.
g.
,
or=
g;pr".
Varying
Eq. (3.
8)
with
respect
to
1V
and
z',
one
obtains
the
gravitational constraint
equations,
g
sE+-'22r'
—
or'&'or"
=
('g)'*5'(x"
—
x'(t)) (mo'+
p
p')
&
(3.
9a)
2~,4~,
=—
P,
P(xs
x'(t))—
.
(3.9b)
The
total
generator (obtained
from
variations
at
the
endpoint,
as in
III)
becomes
G=P,
(t)bx'(t)+~I
d'x
$~'ibg
+To
'bx"o].
(3.10)
As
in
II,
T'„'
vanishes as
a
consequence
of
the
con-
straints.
For
example,
T'0'
is
the sum
of the terms
containing
14'
and
rt'
in
Eq.
(3.
8).
If we
now
insert
the
constraint
equations
(3.
9),
the
orthogonal
decomposi-
tion
(1.
2)
for
g;;
and
or", and the
coordinate
conditions
x'=g;,
and
t=
—
2
(1/V2)2rT into
the
generator
Eq.
(3.
10),
one
obtains
G=
p;(t)f'ix'(t)+
~d'x
far"
f'ig
+To„fix")",
(3.
11)
X(t)
=
[X'(s)ds/dxo(s)]
o&,
&
The
solution of
the
constraint
equation
(3.
3)
is
(3.
6)
&=
—
)~gT,
;;dox=
—
)
gT;dS;
(3.
13)
po
poi'
N(moo+—
—
g"p'p
—
)i
(3
&)
where,
as
in
III,
X=
(
—
g")
'*,
rt;=4go,
.
Upon
insertion
of
Eq.
(3.
7)
into
Eq.
(3.
5),
the
total
action
then
takes
~The
four-dimensional 8 function
in
Eq.
(3.
2)
is
dered
ac-
cording
to
f
f(x)54(x
44)d4x=
f(o)
for
any
—
scalar function
f(x).
It
thus
transforms
like
a
scalar
density
under coordinate
trans-
formations,
but
is not
a
functional
of the
metric.
Equation
(3.
3)
may
alternately
be
viewed as an
example
of
Eq.
(4.
17)
of
II,
R=p„+&+H=O
in the
discussion
there of the
parameter
formalism.
It
would
appear
at
erst
sight
that
the
particle
variables
have
disappeared
in the
expression
for
the total
energy.
However,
if
the
energy
is
expressed
in
terms of
canonical
variables
by
solving
Eq.
(3.
9)
for
g,
;;,
it is
obvious
that x'(t)
and
p,
(t)
appear.
For
example,
to lowest
order
gT „=Xi—
+
(mo'+p,
p')its(x'
x'(t))
—
(3.
14)
where
BCI,
is the
linear
theory
Hamiltonian
density,
X~
2
(g,
TT
o)2+
(rrijTT)2.
(3.
15)
CRITERIA
FOR RAD I
ATION
IN
GENERAL
RELATIVITY
The fact established here that
the total
energy
for
the
full
theory
is
expressible
in
terms of
the
asymptotic
form of
the metric
is,
of
course,
reasonable
on
physical
grounds
since
it
is the total
energy
of
an
isolated
system
(including
all interaction
energies,
gravitational and
others)
which determines
its
Newtonian
gravitational
field
at
large
distances. Note that the
energy
expression
(3.
13)
still satisfies
the
basic
requirement
that
it
involve
only
initial
Cauchy
data
t
which are
g,
;,
7r
4,
x'(t),
and
p,
(t)
at some fixed time
in
the
absence
of
coordinate
conditions,
or g"
rrz
"rr
x(t),
and
p;(t)
in
the
ca-
nonical
frame).
It does
not
depend
on
such coordinate-
sensitive
quantities
as
go„.
IV.
APPLICATIONS
As the
simplest application
of our
energy
formula
Eq. (3.
13)
we
examine the Schwarzschild
solution.
In
isotropic
coordinates
(with
the
units
we
have been
using'),
the
asymptotic
form of the
Schwarzschild
solution becomes
g@=b,
,
+6;,
m/87rr.
(4.
1)
We
now
use
the
fact that
for
a tensor
of the
form
f;;=8;;f,
one has
f
=2f
This
re.
sult follows
directly
from
Eq. (1.
3a).
Thus,
at
large
distances
gr
=
ni/4rrr.
(4.
2)
The
energy,
therefore,
evaluates to
m,
as
expected.
The
same
result is
obtained,
of
course,
if
one uses the
asymp-
totic
form of this metric
in
standard Schwarzschild
coordinates:
g;;
=
5;,
+
(np/8rrr)
(x'x'/r').
(43)
Equation
(4.
2)
is,
in
fact,
more
general
then
the
simple
Schwarzschild
case;
in
fact,
for
any
bounded
system,
the
formula
gr
=
E/4rrr+0
(1/r')
(4.
4)
holds.
The
equation
—
g~,
;;=K
has
for
its solution
a
multipole
expansion
where
the coeKcient
of
the
mono-
pole
term is independent
of
angles,
but
possibly
a
function
of
time,
i.e.
,
gr=
f(t)/4mr+0(1/r').
Equation
(3.
13)
shows
that
f(t)
=E
which
is
a
constant in
time.
For the
Schwarzschild solution,
which
represents a
static situation,
one
expects
that
there be no
waves,
that
is,
the gravitational
canonical
variables
g;,
and
pr'&rr
(which
can be initially
specified
independently
of
the
particle
variables)
should vanish
everywhere
in the
canonical
frame. Indeed.
,
g;;~~
vanishes in
any
frame where
g;;
has spherical
symmetry.
This
follows
from
the
fact
that there are no transverse vectors
avail-
able,
i.e.
,
in
I'"ourier
space
g;;
can
only
depend
on
8;;
and
k;k;.
Also,
~'&'~~
vanishes due
to
spherical
sym-
metry.
One
can show
this
explicitly
for
the
canonical
frame
without actually
transforming to it.
'
We
write
9
We
have
obtained the
Schwarzschild
solution in
the canonical
coordinate
system
and
find
that its
metric
components
in this
system
involve quadratures
that
cannot
be
expressed
in
terms of
standard functions. However,
there
exists
another
canonical frame
in which the
Schwarzschild
metric
takes on the
usual
isotropic
form.
the
general
form
for
the
spherically
symmetric
metric
as
g;;
=
&;;+f(r)S;;+h(r)x'x~/r'
(4
5)
and
impose
the
coordinate
conditions
of
Eq. (1.
4b),
i.e.
,
g;;,
;=0.
The
coordinate
conditions
lead
to
the
equation
f'+h'+
(2/r)k=0,
(4.
6)
—
(pmpP/2)
d'r
P
(r)/r.
J
(4.
9)
To
see how
the
self-energy
arises in
the full
theory,
we
examine the constraint
equation
(3.
9a),
which
becomes,
for
the static
case,
g&
pR=
nppp(r).
(4.10)
Note @so
is the unrenormalized
mass.
Writing
the
metric
in
isotropic
coordinates
as
"'
g@
—
—
Ly
(r)
]48;;, Eq.
(4.10)
reduces to
—
8y(V'x)
=np
P(r),
(4.11)
which has the solution
X
=
1+
[tnp/X
(0)
)L1/324rr]. (4.12)
Since there is
no
energy
in
the
independent
modes of
the gravitational
field,
we
can
identify
the
total
energy
as the
renormalized
mass
nz
of the
particle
E=
np=
happ/g
(0).
(4.
13)
The
quantity
x(0),
which
may
be
obtained
by
con-
sidering
Eq.
(4.12)
at
r=
p
(where
e~
0),
is infinite:
X(0)
=
—',
L1+
(1+4np/8pre)&).
(4.
14)
Thus,
from
Eq.
(4.
13),
the
total
energy
approaches
zero as
p
—
+
0,
corresponding
to
a
finite
negative
gravi-
tational static
self-energy
b,
E=
—
mo.
From
the
leading
term of
Eq. (4.
13),
E=
(32z.
nppp)&,
"
A.
Lichuerowicz,
J.
Math.
pure
appl.
23,
37
(1944).
(4.15)
where
f'=
df/dr,
etc. Since
g"'=
g'
—
&'
—
-'Lg'~'
—
(1/~')
g'.
;
1,
(4.
&)
in
virtue of
our coordinate
conditions, one
may
compute
easily
that
g;,
~~
vanishes
if
f"+h,
"
(f'
h-,
')/r
-4r/r
=—
0.
(4.
8)
Equation
(4.
6)
and
its first
derivative are
indeed
equivalent to
Eq. (4.
8).
Since
this
is
a
static
metric,
m'&
vanishes;
and,
hence,
the other
coordinate
condition
of
Eq. (1.
4),
7rr=0,
is also
satisfied.
The
conjugate
momenta
m'&~~,
are
similarly
zero.
The
fact that
the canonical
variables
vanish
in
the
canonical frame
indicates
that none
of
the
energy
resides in the
independent modes of
the
gravitational
fie1d. This
does
not
preclude
the
existence
of
static
self-energies which
are the nonlinear
generalizations
of
the Coulomb
type.
For
example,
the
quadratic
terms
of
pR
from
Eq. (3.
9a)
yields
precisely
the
Newtonian
self-interaction
energy
of
a
point
particle,
Ii04 ARNOWITT, DESER, AND
MISNER
we see
that the total
energy
vanishes
as
e&
rather
than
diverging
as
—
e
'
(the
Newtonian
result).
Of
course,
a
full discussion
of
the
self-energy
must
wait
until
dynamical
e6'ects
are included.
V.
RADIATION CRITERIA
Having
put
general
relativity
into
canonical
form,
we have thus
separated
out
those
gravitational
Geld
variables
of
the
theory
which
are
independent
of
the
source variables.
Kxcitations in such
independent
variables
provide
a
primary
de6nition of what
one
calls
waves or radiation. This
is,
of
course,
the
same definition
for
radiation as that
given
in
electrodynamics
or
any
other
held
theory.
This
viewpoint
is
taken
as a matter
of
course in those field theories where no redundant
variables
appear.
In
electrodynamics,
the
gauge
in-
variance obscures to some extent
the
fact that
it
is
only
transverse
modes of the vector
potential
and
electric 6eld
that
need
be examined to
recognize
radiation.
These variables are
just
the canonical ones
of the
Maxwell
6eld.
Correspondingly
in
general
rela-
tivity,
the
basic
requirement
for
the
existence
of
radiation is to
be
formulated
solely
in
terms of
the
canonical
variables. Stated
formally:
The
nonvanishing
of
g;;~~
or
m'&~~
at a
point
in
the canonical
coordinate
system
represents
the existence of
a
wave
carrying
energy
and momentum.
As
in
electrodynamics,
radi-
ation and induction eGects
can
be
meaningfully
sepa-
rated
only
in
the
"wave
zone",
but
also
as
in
electro-
dynamics
the
above
criteria can
in fact
be
employed
consistently
nearer the
sources.
Aside
from
these
requirements,
which
apply
locally
or for
bounded
systems,
one can also formulate
require-
ments for the situation of radiation
escaping
to
infinity.
Again
as
in
electrodynamics,
one
simply
examines
the
Poynting
vector
T"
at
infinity;
i.
e.
,
T"dS,
represents
the
Qux of
energy
through
the
two-dimensional
surface
elements
d5,
at infinity.
There,
T"
takes
on
its
linear-
ized
form,
since at
inGnity
this weak Geld form is
rigor-
ous.
The
symmetric
TI""
for
the
linearized
theory
was
given
in I.
By
direct calculation one finds
that this
+%0
is 2s
lmTF(1
i
)TT
where
(p' )TT
&(g
.
TT
+g
.
TT
g
—
Ft
.
)
This is identical to the
leading
term
of
—
2m@,
;
obtained
by
solving
the
constraint
equation
—
2m";=
—
—
2x"
—
2+™1
')
=0.
(5.
1)
In the
canonical
frame
then,
the
Poynting
vector
at
inGnity
is
Tis
—
2~re
.
—
2~tmFT(P4~
)TT
(5 2)
If
other
systems
are
coupled
to
the
gravitational
Geld,
—
2x'~';
now
represents
the total
energy
Aux
at
infinity,
of which the
purely
gravitational
part
is
the last member
of
Eq. (5.2).
The
point
to
be
stressed here is
that
once
general
relativity
has been
put
into canonical
form,
the
physical
interpretations
to
be
given
to
radiation
are
identical to those
of
other
Geld
theories.
Within
this
framework,
one should
be able to deal
meaningfully
also with idealized
situations,
such
as
cylindrical waves
and
plane
waves.
VI. DISCUSSION
In this
paper
we
have
examined
some
of
the
physical
quantities
that
arise
when one has cast
general
relativity
into canonical form. In
particular,
we
have
defined
the
energy
and momentum of
the
field,
which
are
still
the
basic
integrals
needed to characterize
the
system
independently
of its internal
structure.
Both
these
quantities
can
be
expressed
in
terms of
the canonical
variables
of
the
theory
and hence
can be
determined
from
minimal initial
Cauchy
data.
It is a
noteworthy
physical
property
of
general
relativity
that the total
energy
and
momentum
can
be expressed
as
surface
integrals.
This
6nds its
analogy
in
electromagnetism
where the source of
that
Geld,
namely,
the
total
charge,
may
be
characterized in
the
same
way.
For
the
coupled
system
we have seen
that
the excitations
of
the
gravitational
6eld
contribute
to
the total
energy
of
the
system
seen in
the
asymptotic
Newtonian
potential
(along
with the
matter and
inter-
action
energies).
Further,
the
coupling
of
matter
does
not
affect
the
definition of
the canonical
variables
of
the
gravitationa1
field.
The
highly
nonlinear fashion in which
the
matter
interacts with the
gravitational
field in
even
the
simplest
case was illustrated in the treatment of
the
static
point
particle.
Here it
was
seen that the usual
Schwarzschild
mass
parameter
was
really
a
renormalized
mass.
While
to
lowest order in
a
perturbation
expansion
in
the
gravitational
coupling
constant,
the
self-energy was
seen to
be
the
linearly divergent
Newtonian
term,
the
rigorous
static
self-energy
was found
to
cancel
the
unrenormalized mass
with
the
total
energy
vanishing
as a
square
root.
"
The
detailed
properties
of
the
gravitational
field
are
determined
by
an examination of
the excitation
present
in the
canonical
modes.
Thus,
criteria for
radiation
can
be stated
in
these terms.
For
example,
to
interpret
the total
energy
of
the
Schwarzschild field
as the
re-
normalized mass
of
a
single
particle,
it was
necessary
to
establish
that
none
of the
independent gravitational
modes were
excited.
Alternately
this fact
is what
permits
one
to
say
that the
solution
represents
a
"one-particle"
state
in
the
field-theoretical
sense.
The results
of
this
paper
have been derived
using
a particular
set
of coordinate
conditions with
associated
canonical
variables.
Actually,
all results
are
independent
of
the
choice
of
canonical
frame.
This will
be
demon-
strated
in
a
later
paper.
"
Further results on the
classical
self-energy
problem
for neutral
and
charged particles
are
given
in
Phys.
Rev.
Letters
4,
375
(1960).