Key result in general relativity
The
positive energy theorem
(also known as the
positive mass theorem
) refers to a collection of foundational results in
general relativity
and
differential geometry
. Its standard form, broadly speaking, asserts that the gravitational energy of an isolated system is nonnegative, and can only be zero when the system has no gravitating objects. Although these statements are often thought of as being primarily physical in nature, they can be formalized as
mathematical theorems
which can be proven using techniques of
differential geometry
,
partial differential equations
, and
geometric measure theory
.
Richard Schoen
and
Shing-Tung Yau
, in 1979 and 1981, were the first to give proofs of the positive mass theorem.
Edward Witten
, in 1982, gave the outlines of an alternative proof, which were later filled in rigorously by mathematicians. Witten and Yau were awarded the
Fields medal
in mathematics in part for their work on this topic.
An imprecise formulation of the Schoen-Yau / Witten positive energy theorem states the following:
Given an asymptotically flat initial data set, one can define the energy-momentum of each infinite region as an element of
Minkowski space
. Provided that the initial data set is
geodesically complete
and satisfies the
dominant energy condition
, each such element must be in the
causal future
of the origin. If any infinite region has null energy-momentum, then the initial data set is trivial in the sense that it can be geometrically embedded in Minkowski space.
The meaning of these terms is discussed below. There are alternative and non-equivalent formulations for different notions of energy-momentum and for different classes of initial data sets. Not all of these formulations have been rigorously proven, and it is currently an
open problem
whether the above formulation holds for initial data sets of arbitrary dimension.
Historical overview
[
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]
The original proof of the theorem for
ADM mass
was provided by
Richard Schoen
and
Shing-Tung Yau
in 1979 using
variational methods
and
minimal surfaces
.
Edward Witten
gave another proof in 1981 based on the use of
spinors
, inspired by positive energy theorems in the context of
supergravity
. An extension of the theorem for the
Bondi mass
was given by
Ludvigsen
and James Vickers, Gary Horowitz and
Malcolm Perry
, and Schoen and Yau.
Gary Gibbons
,
Stephen Hawking
, Horowitz and Perry proved extensions of the theorem to asymptotically
anti-de Sitter spacetimes
and to
Einstein?Maxwell theory
. The mass of an asymptotically anti-de Sitter spacetime is non-negative and only equal to zero for anti-de Sitter spacetime. In Einstein?Maxwell theory, for a spacetime with
electric charge
and
magnetic charge
, the mass of the spacetime satisfies (in
Gaussian units
)
with equality for the
Majumdar
?
Papapetrou
extremal black hole
solutions.
Initial data sets
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]
An
initial data set
consists of a
Riemannian manifold
(
M
,
g
)
and a symmetric 2-tensor field
k
on
M
. One says that an initial data set
(
M
,
g
,
k
)
:
- is
time-symmetric
if
k
is zero
- is
maximal
if
tr
g
k
= 0
[1]
- satisfies the
dominant energy condition
if
- where
R
g
denotes the
scalar curvature
of
g
.
[2]
Note that a time-symmetric initial data set
(
M
,
g
, 0)
satisfies the dominant energy condition if and only if the scalar curvature of
g
is nonnegative. One says that a Lorentzian manifold
(
M
,
g
)
is a
development
of an initial data set
(
M
,
g
,
k
)
if there is a (necessarily spacelike) hypersurface embedding of
M
into
M
, together with a continuous unit normal vector field, such that the induced metric is
g
and the
second fundamental form
with respect to the given unit normal is
k
.
This definition is motivated from
Lorentzian geometry
. Given a Lorentzian manifold
(
M
,
g
)
of dimension
n
+ 1
and a spacelike immersion
f
from a connected
n
-dimensional manifold
M
into
M
which has a trivial normal bundle, one may consider the induced Riemannian metric
g
=
f
*
g
as well as the
second fundamental form
k
of
f
with respect to either of the two choices of continuous unit normal vector field along
f
. The triple
(
M
,
g
,
k
)
is an initial data set. According to the
Gauss-Codazzi equations
, one has
where
G
denotes the
Einstein tensor
Ric
g
-
1
/
2
R
g
g
of
g
and
ν
denotes the continuous unit normal vector field along
f
used to define
k
. So the dominant energy condition as given above is, in this Lorentzian context, identical to the assertion that
G
(
ν
, ?)
, when viewed as a vector field along
f
, is timelike or null and is oriented in the same direction as
ν
.
[3]
The ends of asymptotically flat initial data sets
[
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]
In the literature there are several different notions of "asymptotically flat" which are not mutually equivalent. Usually it is defined in terms of weighted Holder spaces or weighted Sobolev spaces.
However, there are some features which are common to virtually all approaches. One considers an initial data set
(
M
,
g
,
k
)
which may or may not have a boundary; let
n
denote its dimension. One requires that there is a compact subset
K
of
M
such that each connected component of the complement
M
?
K
is diffeomorphic to the complement of a closed ball in Euclidean space
?
n
. Such connected components are called the
ends
of
M
.
Formal statements
[
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]
Schoen and Yau (1979)
[
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]
Let
(
M
,
g
, 0)
be a time-symmetric initial data set satisfying the dominant energy condition. Suppose that
(
M
,
g
)
is an oriented three-dimensional smooth Riemannian manifold-with-boundary, and that each boundary component has positive mean curvature. Suppose that it has one end, and it is
asymptotically Schwarzschild
in the following sense:
Suppose that
K
is an open precompact subset of
M
such that there is a diffeomorphism
Φ : ?
3
?
B
1
(0) →
M
?
K
, and suppose that there is a number
m
such that the symmetric 2-tensor
on
?
3
?
B
1
(0)
is such that for any
i
,
j
,
p
,
q
, the functions
and
are all bounded.
Schoen and Yau's theorem asserts that
m
must be nonnegative. If, in addition, the functions
and
are bounded for any
then
m
must be positive unless the boundary of
M
is empty and
(
M
,
g
)
is isometric to
?
3
with its standard Riemannian metric.
Note that the conditions on
h
are asserting that
h
, together with some of its derivatives, are small when
x
is large. Since
h
is measuring the defect between
g
in the coordinates
Φ
and the standard representation of the
t
= constant
slice of the
Schwarzschild metric
, these conditions are a quantification of the term "asymptotically Schwarzschild". This can be interpreted in a purely mathematical sense as a strong form of "asymptotically flat", where the coefficient of the
|
x
|
?1
part of the expansion of the metric is declared to be a constant multiple of the Euclidean metric, as opposed to a general symmetric 2-tensor.
Note also that Schoen and Yau's theorem, as stated above, is actually (despite appearances) a strong form of the "multiple ends" case. If
(
M
,
g
)
is a complete Riemannian manifold with multiple ends, then the above result applies to any single end, provided that there is a positive mean curvature sphere in every other end. This is guaranteed, for instance, if each end is asymptotically flat in the above sense; one can choose a large coordinate sphere as a boundary, and remove the corresponding remainder of each end until one has a Riemannian manifold-with-boundary with a single end.
Schoen and Yau (1981)
[
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]
Let
(
M
,
g
,
k
)
be an initial data set satisfying the dominant energy condition. Suppose that
(
M
,
g
)
is an oriented three-dimensional smooth complete Riemannian manifold (without boundary); suppose that it has finitely many ends, each of which is asymptotically flat in the following sense.
Suppose that
is an open precompact subset such that
has finitely many connected components
and for each
there is a diffeomorphism
such that the symmetric 2-tensor
satisfies the following conditions:
- and
are bounded for all
Also suppose that
- and
are bounded for any
- and
for any
- is bounded.
The conclusion is that the ADM energy of each
defined as
is nonnegative. Furthermore, supposing in addition that
- and
are bounded for any
the assumption that
for some
implies that
n
= 1
, that
M
is diffeomorphic to
?
3
, and that Minkowski space
?
3,1
is a development of the initial data set
(
M
,
g
,
k
)
.
Witten (1981)
[
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]
Let
be an oriented three-dimensional smooth complete Riemannian manifold (without boundary). Let
be a smooth symmetric 2-tensor on
such that
Suppose that
is an open precompact subset such that
has finitely many connected components
and for each
there is a diffeomorphism
such that the symmetric 2-tensor
satisfies the following conditions:
- and
are bounded for all
- and
are bounded for all
For each
define the ADM energy and linear momentum by
For each
consider this as a vector
in Minkowski space. Witten's conclusion is that for each
it is necessarily a future-pointing non-spacelike vector. If this vector is zero for any
then
is diffeomorphic to
and the maximal globally hyperbolic development of the initial data set
has zero curvature.
Extensions and remarks
[
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]
According to the above statements, Witten's conclusion is stronger than Schoen and Yau's. However, a third paper by Schoen and Yau
[4]
shows that their 1981 result implies Witten's, retaining only the extra assumption that
and
are bounded for any
It also must be noted that Schoen and Yau's 1981 result relies on their
1979 result, which is proved by contradiction; therefore their extension of their 1981 result is also by contradiction. By contrast, Witten's proof is logically direct, exhibiting the ADM energy directly as a nonnegative quantity. Furthermore, Witten's proof in the case
can be extended without much effort to higher-dimensional manifolds, under the topological condition that the manifold admits a spin structure.
[5]
Schoen and Yau's 1979 result and proof can be extended to the case of any dimension less than eight.
[6]
More recently, Witten's result, using Schoen and Yau (1981)'s methods, has been extended to the same context.
[7]
In summary: following Schoen and Yau's methods, the positive energy theorem has been proven in dimension less than eight, while following Witten, it has been proven in any dimension but with a restriction to the setting of spin manifolds.
As of April 2017, Schoen and Yau have released a preprint which proves the general higher-dimensional case in the special case
without any restriction on dimension or topology. However, it has not yet (as of May 2020) appeared in an academic journal.
Applications
[
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]
References
[
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]
- ^
In local coordinates, this says
g
ij
k
ij
= 0
- ^
In local coordinates, this says
R
-
g
ik
g
jl
k
ij
k
kl
+ (
g
ij
k
ij
)
2
≥ 2(
g
pq
(
g
ij
k
pi
;
j
- (
g
ij
k
ij
)
;
p
)(
g
kl
k
qk
;
l
- (
g
kl
k
kl
)
;
q
))
1/2
or, in the usual "raised and lowered index" notation, this says
R
-
k
ij
k
ij
+ (
k
i
i
)
2
≥ 2((
k
pi
;
i
- (
k
i
i
)
;
p
)(
k
pj
;
j
- (
k
j
j
)
;
p
))
1/2
- ^
It is typical to assume
M
to be time-oriented and for
ν
to be then specifically defined as the future-pointing unit normal vector field along
f
; in this case the dominant energy condition as given above for an initial data set arising from a spacelike immersion into
M
is automatically true if the dominant energy condition in its
usual spacetime form
is assumed.
- ^
Schoen, Richard; Yau, Shing Tung (1981).
"The energy and the linear momentum of space-times in general relativity"
(PDF)
.
Comm. Math. Phys
.
79
(1): 47?51.
Bibcode
:
1981CMaPh..79...47S
.
doi
:
10.1007/BF01208285
.
S2CID
120151656
.
- ^
Bartnik, Robert (1986). "The mass of an asymptotically flat manifold".
Comm. Pure Appl. Math
.
39
(5): 661?693.
CiteSeerX
10.1.1.625.6978
.
doi
:
10.1002/cpa.3160390505
.
- ^
Schoen, Richard M. (1989). "Variational theory for the total scalar curvature functional for Riemannian metrics and related topics".
Topics in calculus of variations (Montecatini Terme, 1987)
. Lecture Notes in Mathematics. Vol. 1365. Berlin: Springer. pp. 120?154.
- ^
Eichmair, Michael;
Huang, Lan-Hsuan
; Lee, Dan A.;
Schoen, Richard
(2016).
"The spacetime positive mass theorem in dimensions less than eight"
.
Journal of the European Mathematical Society
.
18
(1): 83?121.
arXiv
:
1110.2087
.
doi
:
10.4171/JEMS/584
.
S2CID
119633794
.
- Schoen, Richard; Yau, Shing-Tung (1979).
"On the proof of the positive mass conjecture in general relativity"
.
Communications in Mathematical Physics
.
65
(1): 45?76.
Bibcode
:
1979CMaPh..65...45S
.
doi
:
10.1007/bf01940959
.
ISSN
0010-3616
.
S2CID
54217085
.
- Schoen, Richard; Yau, Shing-Tung (1981).
"Proof of the positive mass theorem. II"
.
Communications in Mathematical Physics
.
79
(2): 231?260.
Bibcode
:
1981CMaPh..79..231S
.
doi
:
10.1007/bf01942062
.
ISSN
0010-3616
.
S2CID
59473203
.
- Witten, Edward (1981).
"A new proof of the positive energy theorem"
.
Communications in Mathematical Physics
.
80
(3): 381?402.
Bibcode
:
1981CMaPh..80..381W
.
doi
:
10.1007/bf01208277
.
ISSN
0010-3616
.
S2CID
1035111
.
- Ludvigsen, M; Vickers, J A G (1981-10-01). "The positivity of the Bondi mass".
Journal of Physics A: Mathematical and General
.
14
(10): L389?L391.
Bibcode
:
1981JPhA...14L.389L
.
doi
:
10.1088/0305-4470/14/10/002
.
ISSN
0305-4470
.
- Horowitz, Gary T.; Perry, Malcolm J. (1982-02-08). "Gravitational Energy Cannot Become Negative".
Physical Review Letters
.
48
(6): 371?374.
Bibcode
:
1982PhRvL..48..371H
.
doi
:
10.1103/physrevlett.48.371
.
ISSN
0031-9007
.
- Schoen, Richard; Yau, Shing Tung (1982-02-08). "Proof That the Bondi Mass is Positive".
Physical Review Letters
.
48
(6): 369?371.
Bibcode
:
1982PhRvL..48..369S
.
doi
:
10.1103/physrevlett.48.369
.
ISSN
0031-9007
.
- Gibbons, G. W.; Hawking, S. W.; Horowitz, G. T.; Perry, M. J. (1983).
"Positive mass theorems for black holes"
.
Communications in Mathematical Physics
.
88
(3): 295?308.
Bibcode
:
1983CMaPh..88..295G
.
doi
:
10.1007/BF01213209
.
MR
0701918
.
S2CID
121580771
.
Textbooks
- Choquet-Bruhat, Yvonne.
General relativity and the Einstein equations.
Oxford Mathematical Monographs. Oxford University Press, Oxford, 2009. xxvi+785 pp.
ISBN
978-0-19-923072-3
- Wald, Robert M.
General relativity.
University of Chicago Press, Chicago, IL, 1984. xiii+491 pp.
ISBN
0-226-87032-4