Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics
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and Lenard gave a very poor bound on the lowest
possible energy per particle. The proof by Lieb and
Thirring gave a much more realistic bound on this
quantity (see below). Two proofs of stability of
matter will be sketched here. Both proofs rely on the
Lieb–Thirring inequality. The first proof described is
mathematically simple to explain, whereas the
second proof (Lieb–Thirring) is based on the
Thomas–Fermi theory. It is mathematically some-
what more involved but, from a physical point of
view, more intuitive.
As in the case of white dwarfs, stability of matter
relies on the fermionic property of electrons. Dyson
(1967) proved that the stability of the second kind
fails if we ignore the Pauli exclusion principle. In
physics textbooks, the importance of the Pauli
exclusion principle for the stability of white dwarfs
is often emphasized. Its importance for the stability
of everything around us is usually ignored.
As mentioned above the result on stability of
matter appeared from the beginning as a completely
rigorously proved theorem. In contrast, the stability
of white dwarfs was only derived rigor ously by Lieb
and Thirring (1984) and Lieb and Yau (1987) over
50 years after the original work of Chandrasekhar.
The original formulation of stability of matter,
which is given in the next section, dealt with
charged matter consisting of electrons and nuclei
interacting only through electrostatic interactions
and being described by nonrelativistic quantum
mechanics. Over the years, many generalizations of
stability of matter have been derived in order to
include relativistic effects and electromagnetic inter-
actions. Some of these generalizations will be
discussed in this article. A complete understanding
of stability of matter in quantum electrodynamics
(QED) does not exist as yet, which is intimately
related to the fact that this theory still awaits a
mathematically satisfactory formulation.
The Formulation of Stability of Matter
Consider K nuclei with nuclear charges z
1
, ..., z
K
> 0
at positions r
1
, ..., r
K
2 R
3
, and N electrons with
charges 1 (this amounts to a choice of units) at
positions x
1
, ..., x
N
2 R
3
. In order to discuss
stability, it turns out that one can consider the
nuclei as fixed in space, whereas the electrons are
dynamic. More precisely, this means that the
kinetic energy of the nuclei is ignored. It is
important to realize that if stability holds for static
nuclei, it also holds for dynamic nuclei. This is
simply because the kinetic energy is positive, so that
the effect of ignoring it is to lower the total energy .
Since we consider only electro static interact ions,
the quantum Hamiltonian describing this system is
H
N
¼
X
N
i¼1
T
i
X
K
k¼1
X
N
i¼1
z
k
jx
i
r
k
j
þ
X
1i<jN
1
jx
i
x
j
j
þ
X
1k<‘K
z
k
z
‘
jr
k
r
‘
j
½1
The kinetic energy operator T
i
is (half) the Laplacian in
the variable x
i
, i.e., T
i
= ð1/2Þ
i
. Atomic units are
used, where not only the electron charge is 1, but the
mass of the electron is also 1 and h = 1. The unit of
energy is then 2 Ry.
The Hamiltonian H
N
depends on the parameters
z = (z
1
, ..., z
K
) and r = (r
1
, ..., r
K
). It acts on the
Hilbert space of fermionic, that is, antisymmetric
wave functions. More precisely, the fermionic
Hilbert space is
H
F
N
¼
^
N
L
2
ðR
3
; C
2
Þ
Here the target space is C
2
, in order to describe
spin-1/2 particles. One can, of cou rse, also consider
the Hamiltonian H
N
on the full Hilbert space,
H
N
¼
O
N
L
2
ðR
3
; C
2
Þ¼L
2
ðR
3N
; C
2
N
Þ
of which H
F
N
is a subspace.
The quantity of interest is the ground-state energy
E
F
ðz; N; KÞ¼inf
r
inf spec
H
F
N
H
N
¼ inf
r
inf
n
ð; H
N
Þj
2H
F
N
\ C
1
ðR
3N
; C
2
N
Þ; kk¼1
o
½2
and likewise for the ground-state energy E(z, N, K)
on the full space H
N
.Clearly,E
F
(z, N, K)
E(z, N, K). It turns out that the energy E(z, N, K)is
the same as one would get by restricting to
symmetric functions instead of antisymmetric
ones. Therefore, the energy E(z, N, K)isoften
referred to as the lowest possible energy for bosonic
particles.
The Hamiltonian H
N
is an unbounded operat or
and we must discuss its domain to be able to talk
about its spectrum. Also, it should be self-adjoint. It
turns out that these questions are intimately related
to stability. The operator H
N
is well defined on
smooth (i.e., C
1
) funct ions. Thus, the last definition
of E
F
(z, N, K)in[2 ] is meaningful. If this ground-state
energy is finite (i.e., not 1), then the Hamiltonian
has an extension, the Friedrichs’ extension, to a
Stability of Matter 9
self-adjoint operator with the property that the
second equality in [2] holds.
In the definition of E
F
, we have minimized over
all the positions r of the nuclei. Even though the
nuclear dynamics is not considered, one is still
interested in finding the lowest possible energy
independent of where they are located.
Theorem 1 (Stability of the first kind). For all N,
K, and z, we have
Eðz; N; KÞ > 1
Theorem 2 (Stability of matter). There exists a
constant C
jzj
> 0 depending only on jzj= max
{z
1
, ..., z
k
} such that
E
F
ðz; N; KÞC
jzj
ðN þ KÞ
The constant C
jzj
bounds the binding energy per
particle. In the case of hydrogen atoms, when
jzj= 1, Dyson and Lenard arrived at a bound with
C
1
10
14
Ry. Lieb and Thirring arrive at C
1
5 = 10 Ry. Since the binding energy of a single
hydrogen atom is 1 Ry, it is easy to see that one
must have C
1
1=4. Over the years , there have
been some improvements on the estimated value of
this constant in the theory of stability of matter.
That the Pauli exclusion principle, that is, the
fermionic character of the electrons, is necessary for
stability of matter is a consequence of the next
theorem.
Theorem 3 (the N
5=3
law for bosons). If N = K
and z
1
= = z
K
= z > 0, then there exist constants
C
> 0 depending on z such that
C
N
5=3
< Eðz ; N; NÞ < C
þ
N
5=3
It is the superlinear (exponent 5/3) behavior in N
of the upper bound that violates stability of matter.
This upper bound was proved by Lieb (1979) by a
fairly simple variational argument. The lower bound
above, which shows that the exponent 5/3 is
optimal, was proved by Dyson and Lenard (1968)
in their original paper on stability of matter.
This theorem leaves open the possibility that the
stability of matter could be recovered by introducing
finite nuclear masses. That this, indeed, is not the case
was proved by Dyson (1967) by a complicated
variational argument based on the Bogolubov pair
theory for superfluid helium. We now add the kinetic
energy
P
K
k = 1
ð1/2Þ
r
k
of the nuclei (assuming, for
simplicity, that they have the same mass as the
electrons) to the Hamiltonian H
N
and consider
the case where z
1
= z
2
= = z
K
= 1. We denote the
ground-state energy over the space L
2
(R
3(NþK)
)
(ignoring spin) by
e
E(N,K). Then, Dyson proved that
min
N þK¼M
e
EðN; KÞCM
7=5
for some constant C > 0. It was later shown by
Conlon et al. (1988) that the exponent 7/5 is indeed
optimal. Dyson (1967) made a conjecture for the
precise asymptotic behavior of this energy. This
conjecture, which was proved by Lieb and Solovej
(2005) and Solovej (2004), is given in the next
theorem.
Theorem 4 (Dyson’s 7/5-law for the charged
Bose gas).
lim
M!1
min
NþK ¼M
e
EðN;KÞ
M
7=5
¼inf
1
2
Z
jrj
2
J
Z
5=2
j 0;
Z
2
¼1
½3
where
J ¼
4
3=4
ð1/2Þð3/4Þ
5ð5/4Þ
Generalizations of Stability of Matter
Over the years, generalizations of stability of matter
including relativistic effects and interactions with the
electromagnetic field have be en attempted. Since the
relativistic Dirac operator is not bounde d below, we
cannot simply replace the standard nonrelativistic
kinetic energy operator T
j
= (1/2)
j
by the free
Dirac operator.
Relativistic effects have been included by con-
sidering the (pseudo) relativistic kinetic energy
T
Rel
j
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
c
2
j
þ c
4
q
c
2
In the units used in this article, the physical value
of the speed of light c is approximately 137 or,
more precisely, the reciprocal of the fine-structure
constant .
For this relativistic kinetic energy, Lieb and Yau
(1988) proved that stability of matter holds in the
sense formu lated in Theorem 2 if (= c
1
) is small
enough and max
j
{z
j
} 2=. It is known here that
the value 2= is the best possible, since it is so
for the one-atom case. The one-atom case had
been studied by Herbst. The corresponding case of
a one-electron molecule was studied by Lieb and
Daubechies. Less optimal results on the stability of
matter with relativistic kinetic energy had been
10 Stability of Matter
obtained prior to the work of Lieb and Yau by
Conlon and later by Fefferman and de la Llave.
References to these works can be found in the work
of Lieb and Yau (1988).
The relativistic kinetic energy T
Rel
j
agrees with the
free Dirac operator on the positive spectral subspace
of the free Dirac operator (i.e., a subspace of
L
2
(R
3
; C
4
)). Therefore, the stability of matter
follows if T
j
is replaced by the free Dirac operator
and if one restricts to the Hilbert space obtained as
in [2] but with L
2
(R
3
; C
2
) replaced by the positive
spectral subspace of the free Dirac operator. This
formulation is often referred to as the ‘‘no-pair’’
model. In the usual Dirac picture, the negative
spectral subspace, the Dirac sea, is occupied. As long
as one ignores pair creation, only the positive
spectral subspace is available.
Magnetic fields may be included by considering
the ‘‘magnetic kinetic energy’’
T
Mag
j
¼
1
2
ir
j
c
1
Aðx
j
Þ
2
It turns out that the stability of matter theorem
(Theorem 2) holds for all magnetic vector potentials
A : R
3
!R
3
with a constant C
jzj
independent of A.
This is, therefore, also the case if we consider the
magnetic field (or rather the vector potential) as a
dynamic variable and ad d the (positive) field energy
U¼
1
8
Z
R
3
jr AðxÞj
2
dx ½4
to the Hamiltonian. The resultin g Hamiltonian
describes a charged spinless particle interacting
with a classical electromagnetic field.
A more complicated situation is described by the
‘‘magnetic Pauli kinetic energy’’
T
Pauli
j
¼
1
2
ððir
j
c
1
Aðx
j
ÞÞ s
j
Þ
2
where the coupling of the spin to the magnetic field
is included through the vector of 2 2 Pauli
matrices acting on the spin components of particle j,
that is, s = (
1
,
2
,
3
), with
1
¼
01
10
;
2
¼
0 i
i0
;
3
¼
10
0 1
For the Pauli kinetic energy, stability of matter will
not hold independently of the magnetic field (or even
for a fixed unbounded magnetic field) unless the field
energy U in [4] is included in the Hamiltonian. If the
field energy is included, stability of matter holds
independently of the magnetic field, that is, even if
one minimizes over the dynamic variable A,if
(= c
1
) and max
j
{z
j
}
2
are small enough. This was
proved by Fefferman (1997) and by Lieb et al.
(1995). The latter result includes the physical value of
. The fact that a bound on is needed had been
proved by Loss and Yau. Stability for a one-electron
atom had been proved in this model by Fro¨ hlich,
Lieb, and Loss. The many-electron atom and the one-
electron molecule had been studied by Lieb and Loss.
Most relevant references may be found in the work of
Lieb et al. (1995).
The possibility of quantizing the magnetic field has
also been studied. In this case, one must introduce an
ultraviolet cutoff in the momentum modes of the
vector potential. Stability of matter in the resulting
model of (ultraviolet cutoff) QED coupled to non-
relativistic matter was proved by Fefferman et al.
improving results of Bugliaro, Fro¨ hlich, and Graf.
Finally, one may include both relativistic effects
and ele ctromagnetic interactions. Let us first discuss
the case of classical electromagnetic fields. If instead
of the Pauli kinetic energy one uses the Dirac
operator with a magnetic vector potential then
there would be no lower bound on the energy. But,
as previously described, one can study a no-pair
formulation of relativistic particles coupled to
electromagnetic fields. The question arises which
subspace of L
2
(R
3
; C
4
) one should restrict to (i.e.,
which subspace is filled and which one is available).
There are two obvious choices. Either one should, as
before, restrict to the positive spectral subspace of
the free Dirac operator or one should restrict to the
positive spectral subspace of the magnetic Dirac
operator. It is proved by Lieb et al. (1997) that the
former choice leads to instability, whereas stability
of matter holds for the latter choice under some
conditions on and max
j
{z
j
}. Stability requires that
the field en ergy U is included in the Hamiltonian . It
then holds independently of the magnetic field.
This final stability result also holds if the magnetic
field is quantized with an ultraviolet cutoff as
proved by Lieb and Loss (2002).
The no-pair model even with the ultraviolet cutoff
quantized field is not fully relativistically invariant.
As mentioned above, there is still no mathematical
formulation of QED, a fully relativistically invariant
model for quantum particles interacting with elec-
tromagnetic fields.
The Proof of Stability of the First Kind
The proof of stability of the first kind will now be
sketched for charged quantum systems.
As mentioned in the introduction, stability of the
first kind is a consequence of the uncertainty
principle. Contrary to what is often stated in physics
textbooks, stability does not follow from the
Heisenberg formulation of the uncertainty principle.
Stability of Matter 11
A mathematically more flexible formulation
is provided by the classical Sobolev inequality,
which states that for all square-integrable functions
2 L
2
(R
3
), one has
Z
jr j
2
C
S
Z
j j
6
1=3
½5
for C
S
> 0. It follows from this inequality that for
any attractive potential V, there is a lower bound on
the energy expectation
;
1
2
V
¼
1
2
Z
jr j
2
Z
Vj j
2
1
2
C
S
Z
6
1=3
Z
V
5=2
Z
j j
2
2=5
Z
j j
6
1=5
C
Z
V
5=2
Z
j j
2
for some C > 0. Thus, the lowest possible energy of
one particle moving in the potential V is bounded
below by C
R
V
5=2
. For N (noninteracting) particles,
the lower bound is CN
R
V
5=2
. This holds whether
or not the particles have spin. If, more generally, the
potential can be written as V = U þ W, U, W 0,
where
R
U
5=2
< 1 and W is bounded W kWk
1
,
then the energy of N noninteracting particles moving
in the potential V is bounded below by
NC
Z
U
5=2
NkWk
1
½6
For the Hamiltonian H
N
from [1], one can get a
lower bound on the energy E(z, N, K) by ignoring all
the positive potential terms, that is, the last two
sums in [1]. The remaining Hamiltonian describes N
independent particles moving in the potential
V ¼
X
K
k¼1
z
k
jx r
k
j
¼
X
K
k¼1
ðU
k
þ W
k
Þ
where U
k
is the restriction of z
k
=jx r
k
j to the set
jx r
k
j < R for some R > 0andW
k
is the restriction
to the complementary set. Using [6], one can easily
see that the energy expectation is bounded below by
CNK
5=2
max
k
fz
k
g
5=2
R
1=2
NKmax
k
fz
k
gR
1
¼C
0
NK
2
max
k
fz
k
g
2
where we have made the optimal choice for
R (K max
k
{z
k
})
1
.
This finite lower bound on the energy proves
the stability of the first kind, but it clearly does
not have the form required for the stability of the
second kind.
The Proof of Stability of Matter
The proof of stability of the first kind presented in
the previous section must be improved in two ways
in order to conclude the stability of matter.
For fermions, it turns out that the lower bound in
[6] can be improved in such a way that there is no
factor N in the first term. This is the content of the
bound of Lieb and Thirring discussed in the
introduction.
Theorem 5 (Lieb–Thirring inequality 1975). The
sum of all the negative eigenvalues of the oper-
ator ð 1/2Þ V(x) is bounded below by
L
LT
Z
V
5=2
for some constant L
LT
> 0
For N noninteracting fermions moving in the
potential V, the lowest possible energy is given by
the sum of the N lowest eigenvalues of the operator
in the above theorem. Thus, the theorem gives a
lower bound on this energy independently of N.
The second point where the argument from the
previous section has to be improved is the control of
the electrostatic energy. In the above discussion, all
repulsive terms have simply been ignored. For
stability of matter, a much more delicate bound is
needed. Many versions of such bounds have been
given going back to the work of Onsag er (1939).
Here, a result of Baxter (1980) will be used.
Theorem 6 (Baxter’s correlation estimate). For all
positions x
i
, ..., x
N
, r
1
, ..., r
K
2 R
3
and all charges
z
1
, ..., z
K
> 0, we have the pointwise inequality
X
K
k¼1
X
N
i¼1
z
k
jx
i
r
k
j
þ
X
1i<jN
1
jx
i
x
j
j
þ
X
1k<‘K
z
k
z
‘
jr
k
r
‘
j
X
N
i¼1
Vðx
i
Þ
where V(x) = (1 þ 2 max
k
{z
k
}) max
k
{jx r
k
j
1
}.
This theorem simply states that, for a lower
bound, one can replace the full electrostatic Cou-
lomb energy by the energy of independent electrons
moving in the potential where they always see only
the closest nuclei (with a modified charge). Baxter
(1980) used probabilistic techniques to prove the
inequality. An improved version of the inequality
was given by Lieb and Yau (1988), with an analytic
proof.
12 Stability of Matter
Similarly to the argument in the previous section,
one can write V(x) = U(x) þ W(x), where U is the
restriction of V to the set where min
k
{jx r
k
j} < R
for some R > 0andW is the restriction to the
complementary set. It then follows from Baxter’s
correlation estimate and the Lieb–Thirring inequality
that the lowest eigenvalue of the Hamiltonian H
N
on
the fermionic Hilbert space H
F
N
is bounded below by
L
LT
Z
U
5=2
Nð1 þ 2 max
k
fz
k
gÞR
1
Cð1 þ 2 max
k
fz
k
gÞ
5=2
KR
1=2
Nð1 þ 2 max
k
fz
k
gÞR
1
¼C
0
ð1 þ 2 max
k
fz
k
gÞ
2
ðN þ KÞ
where R (1 þ 2 max
k
{z
k
})
1
. This lower bound is
linear in the total particle number N þ K,as
required by stability of matter.
From Thomas–Fermi Theory to Stability
of Matter
In this final section, the proof of stability of mat ter
by Lieb and Thirring (1975), where they use the
Thomas–Fermi theory, is discussed briefly. First note
that there is a dual formulation of the Lieb–Thirring
inequality theorem (Theorem 5), which makes the
connection to the Sobolev inequality [5] much more
transparent.
Theorem 7 (Lieb–Thirring inequality as a kinetic
energy bound). For any normalized antisymmetric
(fermionic) wave function 2H
F
N
we have with
C
LT
=
3
5
(
2
5
L
1
LT
)
2=3
the following low er bound on the
kinetic energy:
X
N
i¼1
1
2
Z
R
3N
kr
i
ðx
1
; ...; x
N
Þk
2
dx
1
dx
N
C
LT
Z
R
3
ðxÞ
5=3
dx
where kkis the norm in spin space (C
2
N
) and the
one-electron density is given by
ðxÞ¼N
Z
R
3ðN1Þ
kðx; x
2
; ...; x
N
Þk
2
dx
2
dx
N
This estimate follows immediately from Theorem
5, which implies that
X
N
i¼1
1
2
Z
kr
i
k
2
Z
V L
LT
Z
V
5=2
To arrive at Theorem 7, simply choose
V = ((2/5(L
1
LT
)
2=3
.
One should compare the Lieb–Thirring kinetic
energy bound with the expression (3/10)(3
2
)
2=3
5=3
for the (thermodynamic) energy density of a
free Fermi gas. One of the yet unproven conjectures
is that the Lieb–Thirring bound holds with C
LT
replaced by the free Fermi constant (3/10)(3
2
)
2=3
.
The idea in the Lieb–Thirring proof of stability of
matter is to bound the energy below by an
expression depending only on the one-electron
density. Theorem 7 achieves this for the kinetic
energy. What is missing is a lower bound on the
electrostatic Coulomb energy depending only on the
density. One can show (see Lieb (1976) or Lieb and
Thirring (1975)) that, except for an error of the
form ‘‘– const N,’’ the total energy expectation
(, H
N
) may be bounded below by
C
LT
Z
5=3
X
K
k¼1
Z
ðxÞ
z
k
jx r
k
j
dx
þ
1
2
ZZ
ðxÞðyÞ
jx yj
dx dy þ
X
1k<‘K
z
k
z
‘
jr
k
r
‘
j
½7
Here, as before, is the one-electron density of the
N-body wave function . The expression [7] is the
famous Thomas–Fermi energy functional. It has
been studied rigorously by Lieb and Simon (1977).
The Thomas–Fermi energy is the infimum of the
expression (7) over all with
R
= N. One of the
important results about the Thomas–Fermi energy is
Teller’s no-binding theorem (Lieb and Simon 1977).
It states that in Thomas–Fe rmi theory atoms do not
bind to form molecules. This means that the
Thomas–Fermi energy is greater than the sum of
the individual atomic energies (these energies in turn
depend only on the nuclear charges).
The above Thomas–Fermi lower bound on the
energy expectation (, H
N
) together with the no-
binding theorem implies stability of matter.
The generalizations to stability of matter dis-
cussed earlier are proved in a way similar to the
proof presented in the previous section.
See also: h-Pseudodifferential Operators and
Applications; Quantum Statistical Mechanics: Overview;
Schro
¨
dinger Operators.
Further Reading
Baxter JR (1980) Inequalities for potentials of particle systems.
Illinois Journal of Mathematics 24: 645–652.
Conlon JG, Lieb EH, and Yau H-T (1988) The N
7=5
law for
charged bosons. Communications in Mathematical Physics
116: 417–448.
Dyson FJ (1967) Ground state energy of a finite system of charged
particles. Journal of Mathematical Physics 8: 1538–1545.
Stability of Matter 13
Dyson FJ and Lenard A (1967) Stability of matter. I. Journal of
Mathematical Physics 8: 423–434.
Dyson FJ and Lenard A (1968) Stability of matter. II. Journal of
Mathematical Physics 9: 698–711.
Fefferman CL (1997) Stability of matter with magnetic fields.
CRM Proceedings and Lecture Notes 12: 119–133.
Fefferman C, Fro¨ hlich J, and Graf GM (1997) Stability of ultraviolet
cutoff quantum electrodynamics with non-relativistic matter.
Communications in Mathematical Physics 190: 309–330.
Fisher M and Ruelle D (1966) The stability of many-particle
systems. Journal of Mathematical Physics 7: 260–270.
Lieb EH and Lebowitz JL (1972) The constitution of matter:
existence of thermodynamics for systems composed of
electrons and nuclei. Advances in Mathematics 9: 316–398.
Lieb EH and Thirring WE (1975) Bound for the kinetic energy of
fermions which proves the stability of matter. Physical Review
Letters 35: 687–689.
Lieb EH (1976) The stability of matter. Reviews of Modern
Physics 48: 553–569.
Lieb EH and Simon B (1977) Thomas–Fermi theory of atoms,
molecules and solids. Advances in Mathematics 23: 22–116.
Lieb EH (1979) The N
5=3
law for bosons. Physics Letters A 70:
71–73.
Lieb EH and Thirring WE (1984) Gravitational collapse in
quantum mechanics with relativistic kinetic energy. Annals of
Physics, NY 155: 494–512.
Lieb EH and Yau H-T (1987) The Chandrasekhar theory of
stellar collapse as the limit of quantum mechanics. Commu-
nications in Mathematical Physics 112: 147–174.
Lieb EH and Yau H-T (1988) The stability and instability of
relativistic matter. Communications in Mathematical Physics
118: 177–213.
Lieb EH (1990) The stability of matter: from atoms to stars. 1989
Gibbs Lecture. Bulletin of the American Mathematical Society
22: 1–49.
Lieb EH, Loss M, and Solovej JP (1995) Stability of matter in
magnetic fields. Physical Review Letters 75: 985–989.
Lieb EH, Siedentop H, and Solovej JP (1997) Stability and
instability of relativistic electrons in classical electromagnetic
fields. Journal of Statistical Physics 89: 37–59.
Lieb EH and Loss M (2002) Stability of a model of relativistic
quantum electrodynamics. Communications in Mathematical
Physics 228: 561–588.
Lieb EH (2004) The stability of matter and quantum electro-
dynamics, Proceedings of the Heisenberg Symposium, Munich,
Dec. 2001. In: Buschhorn G and Wess J (eds.) Fundamental
Physics – Heisenberg and Beyond, pp. 53–68. Berlin: Springer
(arXiv math-ph/0209034).
Lieb EH and Solovej JP Ground state energy of the two-
component charged Bose gas. Communications in Mathema-
tical Physics (to appear).
Onsager L (1939) Electrostatic interaction of molecules. Journal
of Physical Chemistry 43: 189–196.
Solovej JP (2004) Upper bounds to the ground state energies of
the one- and two-component charged Bose gases. Preprint.
Stability of Minkowski Space
S Klainerman, Princeton University, Princeton, NJ,
USA
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
The Minkowski space, which is the simplest solution
of the Einstein field equations in vacuum, that is, in
the absence of matter, plays a fundamental role in
modern physics as it provides the natural mathema-
tical background of the special theory of relativity. It
is most reasonable to ask whether it is stable under
small perturbations. In other words, can arbitrary
small perturbations of flat initial conditions lead to
developments whi ch are radically different, in the
large, from the flat Minkowski space? It turns out to
be a highly nontrivial problem as the Einstein
equations are of a quasilinear hyperbolic character.
Typical systems of this type, in three space dimen-
sions, do form singularities in finite time even for
small disturbances of their trivial initial data. To
avoid finite-t ime singularities, we must require that
sufficiently small perturbations of Minkowski space
are geodesically complete. This, however, is not
enough; one should also insist that the corresponding
spacetimes become flat along all possible directions,
that is, globally asymptotically flat. This is measured
by the decay of the curvature tensor to zero. The
precise rate of decay is also of interest. One expects
that various null-frame components of the curvature
tensor decay at different rates along outgoing null
hypersurfaces; this goes under the name of ‘‘peeling
estimates.’’ It turns out in fact that we cannot prove
geodesic completeness without establishing at the
same time sufficiently fast rates of decay to flatness
corresponding to at least some peeling.
The problem of stability of Minkowski space is
intimately related to that of describing the asympto-
tic properties of the gravitational field at large
distances from an isolated, weakly radiating physical
system. Precise laws of gravitational radiation can
be deduced from the assumption that the spacetime
(M, g) under consideration can be conformally
compactified by adding a boundary S, called skry,
to M so that an appropriate conformal rescaling of g
can be extended smoothly to the new manifold
(
^
M,
^
g) with boundary. In reality, the compactified
spacetime cannot be smooth at the particular point
i
0
corresponding to spacelike infinity. A spacetime
14 Stability of Minkowski Space
(M, g) is called asymptotically simple (AS) if its
conformal completion is smooth everywhere except
i
0
and every null geodesic intersects S at precisely
two endpoints. The AS assumption allows one to
derive precise decay asymptotic for various curvature
components of (M, g) along null geodesics which
are referred to as strong peeling. The obvious
questions raised by this procedure are: do there exist
nontrivial AS spacetimes and, if so, do they contain
a sufficiently large class of radiating spacetimes
including those which appear in all relevant
applications?
Clearly, the two problems mentione d above are
related but not equivalent. Asymptotically simple
spacetimes verify strong peeling, in particular they
are globally asymptotically flat, that is, their
curvature tensor tends to zero along all geodesics.
Yet, it is perfectly possible that arbitrarily small
perturbations of the Minkowski space are geodesi-
cally complete and globally asymptotically flat
without being asymptotically simple.
The first global stability result of the Minkowski
metric was proved by Christodoulou and Klainer-
man (1993). Their result proves sufficiently strong
peeling estimates to allow one to derive the most
important properties of gravitational radiation, such
as the Bondi mass-law formula, but not as strong as
those consistent with asymptotic simplicity. A
companion result was proved by Klainerman and
Nicolo` (2003). Recently, Rodnianski and Lindblad
(submitted) have obtained a surprising global
stability of Minkowski result for the Einstein vacuum
equations in the Lorentz gauge, which provides
considerable weaker peeling than Christodoulou and
Klainerman (1993) and Klainerman and Nicolo`
(1999) but is much easier to prove.
The goal of this article is to describe various results
obtained since the early 1980s concerning both
aspects of the problem of stability of Minkowski
mentioned above.
Initial Data Formulation
The proper mathematical context for the stability of
Minkowski is that provided by the initial-value
problem for vacuum solutions to the Einstein field
equations, that is, Ricci flat spacetimes (M, g),
R
= 0. We recall the following simple definitions:
Definition 1 An initial data set is a triplet (, g, k)
consisting of a three-di mensional complete Rieman-
nian manifold (, g) and a 2-covariant symmetric
tensor k on satisfying the constraint equations:
r
j
k
ij
r
i
tr
g
k ¼ 0; R jkj
2
þðtr kÞ
2
¼ 0
where r is the covariant derivative, R the scalar
curvature of (, g). An initial data set is said to be
maximal if tr
g
k = 0. This is a gauge condition which
can be imposed without loss of generality. For
simplicity we shall assume, throughout this article,
that all initial data sets we consider are maximal.
Definition 2 An initial data set is said to be flat, or
trivial, if it corresponds to a complete spacelike
hypersurface in Minkowski space with its induced
metric and second fundamental form. An initial data
set is said to be asymptotically flat if there exists a
system of coordi nates (x
1
, x
2
, x
3
) defined in a
neighborhood of infinity on , with
r =
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
(x
1
)
2
þ (x
2
)
2
þ (x
3
)
2
p
, relative to which the
metric g approaches the Euclidian metric and k
approaches zero as r !1. We assume, for simpli-
city, that has only one end. A neighborhood of
infinity means the complement of a sufficiently large
compact set on .
Remark 1 Because of the constraint equations, the
asymptotic behavior cannot be arbitrarily pre-
scribed. A precise definition of asymptotic flatness
has to involve the ADM mass of (, g). Taking the
mass into account, we write
g
ij
¼ 1 þ
2M
r
ij
þ oðr
1
Þ
According to the positive-mass theorem, M 0 and
M = 0 implies that the initial data set is flat.
Definition 3 We say that an initial data set is
strongly asymptotically flat if, for some 1=2,
relative to the coordinate system mentioned above,
g
ij
1 þ
2M
r
ij
¼ Oðr
1
Þ; k
ij
¼ Oðr
2
Þ
as r !1
Moreover, every derivative of g (1 þ 2M=r) and k
improves the asymptotics by one.
Definition 4 A Cauchy development of an initial
data set (, g, k) is a spacetime manifold (M, g)
satisfying the Einstein equations together with an
embedding i: ! M such that i
(g), i
(k) are the
first and second fundamental forms of i()inM.
A development is required to be also globally
hyperbolic (which means that i() is a Cauchy
hypersurface, i.e., each causal curve in M intersects
i() at precisely one point) in order to assure the
unique dependence of solut ions on the data. A
future development of (, g, k) consists of a globally
hyperbolic manifold (M, g) with boundary, satisfy-
ing the Einstein equations, and an embedding i as
before which identifies to the boundary of M.
Stability of Minkowski Space 15
The most primitive question asked about the
initial-value problem, solved in a satisfactory way,
for very large classes of evolution equations, is that of
local existence and uniqueness of solutions. For the
Einstein equations, this type of result was first
established by Bruhat (1952) with the help of wave
coordinates which allowed her to cast the Einstein
equations in the form of a system of nonlinear wave
equations to which one can apply the standard theory
of symmetric hyperbolic systems. A stronger result,
due to Hughes et al. (1976), states the following:
Theorem 1 Let (, g, k) be an initial data set for
the Einstein vacuum equations. Assume that can
be covered by a locally finite system of coordinate
charts U
related to each other by C
1
diffeomorph-
isms, such that (g, k) 2 H
s
loc
(U
) H
s1
loc
(U
) with
s > 5=2. Then there exists a unique (up to an
isometry) globally hyperbolic, Hausdorff, devel op-
ment (M, g) for which is a Cauchy hypersurface.
In Theorem 1, the uniqueness up to an isometry
requires additional regularity, s > (5=2) þ 1, on the
data. One has uniqueness, however, without addi-
tional regularity for the reduced Einstein equations
system in wave coordinates.
Remark 2 In the case of nonlinear systems of
differential equations, the local existence and
uniqueness result leads, through a straight forward
extension argument, to a global result. The formula-
tion of the same type of result for the Einstein
equations is a little more sub tle; it was done by
Bruhat and Geroch.
Theorem 2 (Bruhat–Geroch). For each smooth
initial data set, there exists a unique maxi mal future
development.
Thus, any construction, obtained by an evolution-
ary approach from a specific initial data set, must be
necessarily contained in its maximal development.
This may be said to solve the problem of global
existence and uniquene ss in general relativity. This is
of course misleading, for equations defined in a fixed
background global is a solution which exists for all
time. In genera l relativity, however, we have no such
background as the spacetime itself is the unknown.
The connection with the classical meaning of a global
solution requires a special discussion concerning the
proper time of timelike geodesics; all further ques-
tions may be said to concern the qualitative properties
of the maximal development. The central issue is that
of existence and character of singularities. First, we
can define a regular maximal development as one
which is complete in the sense that all future timelike
and null geodesics can be indefinitely extended
relative to their proper time (or affine parameter in
the case of null geodesics). If the initial data set is
sufficiently far off from the trivial one, the corre-
sponding future development may not be regular.
This is the content of the following well-known
theorem of Penrose (1979).
Theorem 3 If the manifold support of an initial
data set is noncompact and contains a closed
trapped surface, the corresponding maximal devel-
opment is incomplete.
Stability of Minkowski Space
At the opposite end of Penrose’s trapped-surface
condition, the problem of stability of Minkowski
space concerns the development of asymptotically
flat initial data sets which are sufficiently close to
the trivial one. Although it may be reasona ble to
expect the existence of a sufficiently small neighbor-
hood of the trivial initial data set, in an appropriate
topology, such that all corresponding developments
are geodesically complete and globally asymptoti-
cally flat, such a result was by no means preor-
dained. First, all known explicit asymptotically
flat solutions of the Einstein vacuum equations,
that is, the Kerr family, are singular. The attempts
to construct nonexplicit, dynamic, solutions based
on the conformal compactification method, due
to Penrose (1962), were obstructed by the irregular
behavior of initial data sets at i
0
. (The problem is
that the singularity at i
0
could propagate and thus
destroy the expected smoothness of scry. This
problem has been recently solved by constructing
initial data sets which are precisely stationary at
spacelike infinity.) Finally, the attempts, using
partial differential equation hyperbolic methods,
to extend the classical local result of Bruhat
ran into the usual difficulties of establishing global
in time existence to solutions of quasilinear hyper-
bolic systems. Indeed, as mentioned above, the
wave coordinate gauge allows one to express
the Einstein vacuum equations in the form of
a system of nonlinear wave equations which does
not satisfy Klainerman’s null condition (the null
condition (Klainerman 1983, 1986) identifies an
important class of quasilinear systems of wave
equations in four spacetime dimensions for which
one can prove global in time existence of small
solutions) and thus was sought to lead to formation
of singularities. (The conjectured singular behavior of
wave coordinates was sought, however, to reflect
onlytheinstabilityofthespecificchoiceofgauge
condition and not a true singularity of the equations.)
According to Bruhat (personal communication),
16 Stability of Minkowski Space
Einstein himself had reasons to believe that the
Minkowski space may not be stable. The problem
of stability of the Minkowski space was first settled
by Christodoulou and Klainerman (1990).
Theorem 4 (Global stability of Minkowski). Any
asymptotically flat initial data set which is suffi-
ciently close to the trivial one has a complete
maximal future development.
A related result (Theorem 5) proved recently by
Klainerman and Nicolo` (2003a), solves the problem
of radiation for arbitrary asymptotically flat initial
data sets: a proof the result below can also be
derived, indirectly, from Christodoulou and Klainer-
man (1993). The proof of Klainerman and Nicolo`
(2003a) avoids, however, a great deal of the
technical complications of this proof.
Theorem 5 For any, suitably defined, asymptoti-
cally flat initial data set (, g, k) with maximal
future development (M, g), one can find a suitable
domain
0
with compact closure in such that
the boundary D
þ
0
of its domain of influence C
þ
(
0
),
or causal future of , in M has complete null
geodesic generators with respect to the correspond-
ing affine parameters.
Both the results of Christodoulou–Klainerman and
Klainerman–Nicolo` prove in fact a lot more than
stated above. They provide a wealth of information
concerning the behavior of null hypersurfaces as well
as the rate at which various components of the
Riemann curvature tensor approach zero along time-
like and null geodesics. Here are more precise
versions for Theorems 4 and 5.
Theorem 4 (Expanded version). Assume that
(, g, k) is maximal and strong asymptotically
flat, g (1 þ 2M=r) = 0(r
3=2
), k = 0(r
5=2
) plus
an appropriate global smallness assumption. We can
constructcompletespacetime(M, g) together with a
maximal foliation
t
given by the level hypersurfaces
of a time function t and null foliation C
u
, given by the
level hypersurfaces of an outgoing optical function u
such that relative to an adapted null frame e
4
= L,
e
3
= L, and (e
a
)
a = 1, 2
we have, along the null hyper-
surfaces C
u
the weak peeling decay,
ab
¼ RðL; e
a
; L; e
b
Þ¼Oðr
7=2
Þ
2
a
¼ RðL; L; L; e
a
Þ¼Oðr
7=2
Þ
4 ¼ RðL;
L; L; LÞ¼Oðr
3
Þ
4 ¼
RðL; L; L; LÞ¼Oðr
3
Þ
2
a
¼ RðL; L; L; e
a
Þ¼Oðr
2
Þ
ab
¼ RðL; e
a
; L; e
b
Þ¼Oðr
1
Þ
½1
as r !1 with 4r
2
= Area( S
t, u
=
t
\ C
u
). Also,
, = O(r
7=2
), with
the average of over the
compact 2-surfaces S
t, u
=
t
\ C
u
.
Three poin ts are noteworthy. (1) The outgoing
optical solution refers to the solution of the Eikonal
equation g
@
a
u@
u = 0 whose level hypersurfaces
C
u
intersect
t
in expanding wave fronts for
increasing t; (2) the generators L and
L are given
by: L = g
@
u@
, the null geodesic genera tor of
C
u
; L is then the null conjugate of L, perpendicular
to S
t, u
= C
u
\
t
; and (3) e
a
is an orthonormal frame
on S
t, u
.
Theorem 5 (Expanded version). For any asympto-
tically flat initial data sets (, g, k), verifying the same
asymptotically flat conditions as in Theorem 4 one
can find a suitable domain
0
with compact
closure in such that its future domain of influence
C
þ
(
0
) can be foliated by two null foliations; one
outgoing C(u) whose leaves are complete towards the
future and the second one
C(u) which is incoming.
Let S(u,
u) = C(u) \ C(u) denote the compact
2-surfaces of intersection between the outgoing and
incoming null hypersurfaces, whose area is denoted
by 4r
2
, and consider an adapted null frame (that is,
L is a the geodesic null generator of C(u),
Litsnull
conjugate perpendicular to S(u,
u), and e
a
an ortho-
normal frame on S(u,
u)) L, L,(e
a
)
a = 1, 2
at every
point along an outgoing null cone C(u). Then,
denoting by , , , ,
, the null components of
the curvature tensor, as in Theorem 5, we have, along
C(u) as r !1,
; ;
; ¼ Oðr
7=2
Þ; ¼ Oðr
2
Þ;
¼ Oðr
1
Þ
½2
Observe that the rates of decay in [1] and [2] are
the same. This will be referred to as weak peeling to
distinguish from the rates of decay compatible with
asymptotic simplicity, that is,
¼ Oðr
5
Þ;¼ Oðr
4
Þ
; ¼ Oðr
3
Þ; ¼ Oðr
2
Þ; ¼ Oðr
1
Þ
½3
to which we shall refer as strong peeling. We shall
discuss more about these in the next section,
following a review, of a recent result of Lindblad–
Rodnianski.
Even the expanded forms of Theorems 4 and 5
stated here do not exhaust, all the information
provided by global stability results in Christodoulou
and Klainerman (1993) and Klainerman and Nicolo`
(2003a). Of particular interest are the main
asymptotic conclusions which can be derived
with the help of these information, the most
Stability of Minkowski Space 17
important being the Bondi mass-law formula which
calculates the gravitational energy radiated at null
infinity.
The simplest gauge condition in which the
hyperbolic character of the Einstein field equations
are easiest to exhibit is the wave coordinate
condition; that is, one solves the Einstein vacuum
equations relative to a special system of coordinates
x
which satisfy the equation
&
gx
= 0. Then,
denoting by h
= g
m
with m the standard
Minkowski metric, we obtain the following system
of quasilinear wave equations in h,
g
@
@
h ¼ Nðh;@hÞ½4
with N(h, @h) a nonlinear term, quadratic in @h,
which can be exhibited explicitly. This form of the
Einstein field equations, called the wave coordinates
reduced Einstein equations, is precisely the one
which allowed Bruhat (1952) to prove the first
local existence result. Later, she also pointed out
that the first nontrivia l iterate of [4] behaves like
t
1
log t rather than t
1
as expected from the decay
properties of solutions to
&
h = 0 in Minkowski
space. This seems to indicate that the wave
coordinates may not be suitable to study the long-
time behavior of solutions to the Einstein field
equations. This negative conclusion is also consis-
tent with the fact that the eqns [4] do not verify
Klainerman’s null condition. (Klainerman’s null
condition (Klainerman 1983) is an algebraic condi-
tion on systems of nonlinear wave equations in
(1 þ 3) dimensions, similar to [4], which allows one
to extend all local solutions, corresponding to small
initial data, for all time. Moreover, these solutions
decay at the rate of t
1
as t !1consistent to the
decay of free waves.) Lindblad and Rodnianski
(2003) were able to isolate a new condition, which
they call the weak null condition, verified by the
wave coordinates reduced Einstein eqns [4], for
which one can prove a small data global existence
result consistent with the weaker decay rates
suggested by the linear asymptotic analysis of
Bruhat. Although the new result provides far
weaker peeling information than [1], it is much
simpler to prove than both Theorems 4 and 5.
Moreover, the result seems to apply to a broader
class of initial data than in Theorems 4 and 5.It
remains an intriguing open problem whether the
result of Lindblad–Rodnianski can be used as a
stepping stone towards the more complete results of
Theorems 4 and 5; that it is once a complete
solution, with limited peeling, is known to exist
whether one can improve, using the more precise
techniques employed in Theorems 4 and 5 minus an
important part of their technical complications, the
weak peeling properties of [1].
Strong Peeling
The weak peeling properties [1] derived in Theorems
4 and 5 are consistent, from a scaling point of view,
with the SAF condition. To derive strong peeling,
see [3], one needs stronger asymptotic conditions.
Recently, Corvino–Schoen and Chrus
´
ciel and Delay
(2002) have proved the existence of a large class of
asymptotically flat initial data sets (, g, k) which
are precisely stationary (here g
kerr
, k
kerr
are the initial
data of the a Kerr solution in standard coordinates)
g = g
kerr
, k = k
kerr
outside a sufficiently large com-
pact set. Moreover, they have proved the existence
of sufficiently small solutions in this class which
satisfy the requirements needed in Friedrich’s con-
formal compactification method (see Friedrich
(2002) and the references within) to produce
asymptotically simple spacetimes, that is, spacetimes
satisfying Penrose’s regular compactification condi-
tion (Penrose 1962). Simultaneously, Klainerman
and Nicolo` (1999) were able to refine the methods
used in the proof of Theorem 5 to prove the
following:
Theorem 6 Assume that the initial data set (, g, k)
of Theorem 5 satisfies the stronger assumption,
g g
S
¼ Oð r
ð3=2þÞ
Þ; k ¼ðr
ð5=2þÞ
Þ½5
for some >3=2. Here
g
S
¼ 1 2
M
r
1
dr
2
þ r
2
ðd
2
þ sin
2
d
2
Þ
denotes the restriction of the Schwarzschild to t = 0
in standard polar coordinates. Then, in addition to
the results reported in Theorem 5, we have the
strong peeling estimates,
¼ Oðr
5
Þ;¼ Oðr
4
Þ
as r !1 along the outgoing null leaves C(u).
Moreover, the same conclusions hold true if [5] is
replaced by
g g
kerr
¼Oðr
ð3=2þÞ
Þ; k k
kerr
¼ðr
ð5=2þÞ
Þ½6
for some >5=2.
The first part of the theorem was proved in
Klainerman and Nicolo` (2003b). The second part is
work in progress by Klainerman and Nicolo` . The
existence of initial conditions of the type required in
Theorem 6 was established in the works of Corvino
(2000) and Chrus
´
ciel and Delay (2002).
18 Stability of Minkowski Space