We study Killing vector fields in asymptotically flat space–times. We prove the following result, implicitly assumed in the uniqueness theory of stationary black holes. If the conditions of the rigidity part of the positive energy theorem are met, then in such space–times there are no asymptotically null Killing vector fields, except if the initial data set can be embedded in Minkowski space–time. We also give a proof of the nonexistence of nonsingular (in an appropriate sense) asymptotically flat space–times that satisfy an energy condition and that have a null ADM four‐momentum, under conditions weaker than previously considered.
REFERENCES
1.
A.
Ashtekar
and A.
Magnon-Ashtekar
, “On conserved quantities in general relativity
,” J. Math. Phys.
20
, 793
–800
(1979
).2.
A.
Ashtekar
and B.
Schmidt
, “Null infinity and Killing fields
,” J. Math. Phys.
21
, 862
–867
(1978
).3.
A.
Ashtekar
and B. C.
Xanthopoulos
, “Isometries compatible with asymptotic flatness at null infinity: A complete description
,” J. Math. Phys.
19
, 2216
–2222
(1978
).4.
J.
Bičák
and B.
Schmidt
, “Isometries compatible with gravitational radiation
,” J. Math. Phys.
25
, 600
–606
(1984
).5.
P. T.
Chruściel
and R.
Wald
, “Maximal hypersurfaces in stationary asymptotically flat space-times
,” gr-qc/9304009, Commun. Math. Phys.
163
, 561
–604
(1994
).6.
P. T. Chruściel, “’No hair’ theorems—Folklore, conjectures, results,” gr-qc/9402032, Proceedings of the Joint AMS CMS Conference on Mathematical Physics and Differential Geometry, August 1993, Vancouver, edited by J. Beem and K. L. Duggal, Contemporary Math. 1994, pp. 23–49.
7.
P. T.
Chruściel
, “On completeness of orbits of Killing vector fields
,” gr-qc/9304029, Class. Quantum Grav.
10
, 2091
–2101
(1993
).8.
J. W. York, Jr., “Kinematics and dynamics of general relativity,” in Sources of Gravitational Radiation, edited by L. L. Smarr (Cambridge University, Cambridge 1979).
9.
D.
Christodoulou
and N.
O’Murchadha
, “The boost problem in general relativity
,” Commun. Math. Phys.
80
, 271
–300
(1981
).10.
N.
Meyers
, “An expansion about infinity for solutions of linear elliptic equations
,” J. Math. Mech.
12
, 247
–264
(1963
).11.
R.
Bartnik
, “The mass of an asymptotically flat manifold
,” Commun. Pure Appl. Math.
39
, 661
–693
(1986
).12.
P. T.
Chruściel
, “A-priori estimates in weighted Hoelder spaces for a class of scale-covariant second order elliptic operators
,” Ann. Fac. Sci. Toulouse
XI
, 21
–37
(1990
).13.
P. T.
Chruściel
, “A remark on the positive energy theorem
,” Class. Quantum Grav.
3
, L115
–L121
(1986
);14.
R.
Beig
, “Arnowitt–Deser–Misner energy and
,” Phys. Lett. A
69
, 153
–155
(1978
).15.
P. C. Aichelburg and R. U. Sexl, “On the gravitational field of a massless particle,” Gen. Relat. Grav. 303–312 (1971).
16.
H. W.
Brinkmann
, “Einstein spaces which are mapped conformally on each other
,” Math. Ann.
94
, 119
–145
(1925
).17.
R.
Schimming
, “Riemannsche Räume mit ebenfrontiger und mit ebener Symmetrie
,” Math. Nachr.
59
, 129
–162
(1974
).18.
P. T. Chruściel, “Boundary conditions at spatial infinity from a Hamiltonian point of view,” in Topological Properties and Global Structure of Space-Time, edited by P. Bergmann and V. de Sabbata (Plenum, New York, 1986), pp. 49–59.
19.
E.
Witten
, “A new proof of the positive energy theorem
,” Commun. Math. Phys.
80
, 381
–402
(1981
).20.
A.
Ashtekar
and G. T.
Horowitz
, “Energy-momentum of isolated systems cannot be null
,” Phys. Lett. A
89
, 181
–184
(1982
).21.
P. F.
Yip
, “A strictly-positive mass theorem
,” Commun. Math. Phys.
108
, 653
–665
(1987
).22.
T.
Parker
and C. H.
Taubes
, “On Witten’s proof of the positive energy theorem
,” Commun. Math. Phys.
84
, 223
–238
(1982
).23.
R.
Schoen
and S. T.
Yau
, “Proof of the positive mass conjecture. II
,” Commun. Math. Phys.
79
, 231
–260
(1981
).24.
R.
Schoen
and S. T.
Yau
, “Proof that the Bondi mass is positive
,” Phys. Rev. Lett.
48
, 369
–374
(1982
).25.
M.
Ludvigsen
and J. A. G.
Vickers
, “An inequality relating mass and electric charge in general relativity
,” J. Phys. A
16
, 1169
–1174
(1983
).26.
J.
Jezierski
, “Positivity of mass for certain spacetimes with horizons
,” Class. Quantum Grav.
6
, 1535
–1539
(1989
).27.
J. Jezierski and J. Kijowski, “A simple argument for positivity of gravitational energy,” in Differential Geometric Methods in Theoretical Physics, Proceedings of the 15th International Conference, Clausthal (World Scientific, Singapore, 1987).
28.
J.
Jezierski
and J.
Kijowski
, “Positivity of total energy in general relativity
,” Phys. Rev. D
36
, 1041
–1044
(1987
).29.
G. W.
Gibbons
, S. W.
Hawking
, G. T.
Horowitz
, and M. J.
Perry
, “Positive mass theorem for black holes
,” Commun. Math. Phys.
99
, 285
–308
(1983
).30.
Y. Choquet-Bruhat, “Positive energy theorems,” in Relativity, Groups and Topology II, Proceedings of the Les Houches Summer School, June–August 1983, edited by B. S. DeWitt and R. Stora (North-Holland, Amsterdam, 1984).
31.
O.
Reula
, “Existence theorem for solutions of Witten’s equation and non-negativity of total mass
,” J. Math. Phys.
23
, 810
–814
(1982
).32.
O.
Reula
and P.
Tod
, “Positivity of the Bondi energy
,” J. Math. Phys.
25
, 1004
–1008
(1984
).33.
P.
Bizon
and E.
Malec
, “On Witten’s positive-energy proof for weakly asymptotically flat space–times
,” Class. Quantum Grav.
3
, L123
–128
(1986
).34.
R.
Penrose
, “Light rays near : A new mass-positivity theorem
,” Twistor Newslett.
30
, 1
–5
(1990
).35.
R. Penrose, R. D. Sorkin, and E. Woolgar, “Positive mass theorem based on the focusing and retardation of null geodesics,” Cambridge preprint, 1993, gr-qc/9301015.
36.
G. T. Horowitz, “The positive energy theorem and its extensions,” in Asymptotic Behavior of Mass and Spacetime Geometry, edited by F. Flaherty, Springer Lecture Notes in Physics, 1984, p. 202.
37.
J. A. Wolf, Spaces of Constant Curvature (McGraw-Hill, New York, 1967).
38.
P. T.
Chruściel
, “On the invariant mass conjecture in general relativity
,” Commun. Math. Phys.
120
, 233
–248
(1988
).39.
S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time (Cambridge University, Cambridge, 1973).
40.
A.
Sen
, “On the existence of neutrino ’zero-modes’ in vacuum spacetimes
,” J. Math. Phys.
22
, 1781
–1786
(1981
).41.
R. Penrose and W. Rindler, Spinors and Spacetime (Cambridge University, Cambridge, 1984), Vol. 1.
42.
K. P.
Tod
, “All metrics admitting super-covariantly constant spinors
,” Phys. Lett. B
121
, 241
–244
(1983
).43.
A.
Taub
, “Space-times admitting a covariantly constant spinor field
,” Ann. Inst. Henri Poincaré
41
, 227
–236
(1984
).44.
A. Ashtekar, Lectures on Non-Perturbative Canonical Gravity (World Scientific, Singapore, 1991).
This content is only available via PDF.
© 1996 American Institute of Physics.
1996
American Institute of Physics
You do not currently have access to this content.
