136:. Indeed, physicists rarely imagine a universe containing a single star and nothing else when they construct an asymptotically flat model of a star. Rather, they are interested in modeling the interior of the star together with an exterior region in which gravitational effects due to the presence of other objects can be neglected. Since typical distances between astrophysical bodies tend to be much larger than the diameter of each body, we often can get away with this idealization, which usually helps to greatly simplify the construction and analysis of solutions.
22:
990:
used this to circumvent the tricky problem of suitably defining and evaluating suitable limits in formulating a truly coordinate-free definition of asymptotic flatness. In the new approach, once everything is properly set up, one need only evaluate functions on a locus in order to verify asymptotic
1022:
In metric theories of gravitation such as general relativity, it is usually not possible to give general definitions of important physical concepts such as mass and angular momentum; however, assuming asymptotical flatness allows one to employ convenient definitions which do make sense for
440:(the family of all stationary axisymmetric and asymptotically flat vacuum solutions). These families are given by the solution space of a much simplified family of partial differential equations, and their metric tensors can be written down in terms of an explicit
1205:. Version dated May 16, 2002. Roberts attempts to argue that the exterior solution in a model of a rotating star should be a perfect fluid or dust rather than a vacuum, and then argues that there exist no asymptotically flat rotating
490:, which far from the origin behaves much like a Cartesian chart on Minkowski spacetime, in the following sense. Write the metric tensor as the sum of a (physically unobservable) Minkowski background plus a perturbation tensor,
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771:
688:
121:
The condition of asymptotic flatness is analogous to similar conditions in mathematics and in other physical theories. Such conditions say that some physical field or mathematical function is
128:
In general relativity, an asymptotically flat vacuum solution models the exterior gravitational field of an isolated massive object. Therefore, such a spacetime can be considered as an
974:, and others began to study the general phenomenon of radiation from a compact source in general relativity, which requires more flexible definitions of asymptotic flatness. In 1963,
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616:
355:
A manifold is asymptotically flat if it is weakly asymptotically simple and asymptotically empty in the sense that its Ricci tensor vanishes in a neighbourhood of the boundary of
109:, as well as any matter or other fields which may be present, become negligible in magnitude at large distances from some region. In particular, in an asymptotically flat
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in which, roughly speaking, the curvature vanishes at large distances from some region, so that at large distances, the geometry becomes indistinguishable from that of
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113:, the gravitational field (curvature) becomes negligible at large distances from the source of the field (typically some isolated massive object such as a star).
266:
214:
162:
1143:
Mars, M. & Senovilla, J. M. M. (1998). "On the construction of global models describing rotating bodies; uniqueness of the exterior gravitational field".
452:
The simplest (and historically the first) way of defining an asymptotically flat spacetime assumes that we have a coordinate chart, with coordinates
1026:
While this is less obvious, it turns out that invoking asymptotic flatness allows physicists to import sophisticated mathematical concepts from
437:
862:
One reason why we require the partial derivatives of the perturbation to decay so quickly is that these conditions turn out to imply that the
1007:
Models of physical phenomena in general relativity (and allied physical theories) generally arise as the solution of appropriate systems of
1119:
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1303:
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1198:. This doesn't imply that no models of a rotating star exist, but it helps to explain why they seem to be hard to construct.
1103:
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1000:
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This is a short review by three leading experts of the current state-of-the-art on constructing exact solutions which model
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Only spacetimes which model an isolated object are asymptotically flat. Many other familiar exact solutions, such as the
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43:
36:
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On the other hand, there are important large families of solutions which are asymptotically flat, such as the AF
418:
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866:(to the extent that this somewhat nebulous notion makes sense in a metric theory of gravitation) decays like
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98:
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Since the latter excludes black holes, one defines a weakly asymptotically simple manifold as a manifold
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911:
165:
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which includes the well-known
Wahlquist fluid and Kerr-Newman electrovacuum solutions as special case.
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The authors argue that boundary value problems in general relativity, such as the problem matching a
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is also asymptotically flat. But another well known generalization of the
Schwarzschild vacuum, the
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The notion of asymptotic flatness is extremely useful as a technical condition in the study of
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8:
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Second order perturbations of rotating bodies in equilibrium: the exterior vacuum problem
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While this notion makes sense for any
Lorentzian manifold, it is most often applied to a
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105:. In this case, we can say that an asymptotically flat spacetime is one in which the
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Mark
Roberts is an occasional contributor to Knowledge, including this article.
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solution, which models a spherically symmetric massive object immersed in a
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is the conformal compactification of some asymptotically simple manifold.
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1271:
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406:
914:, the energy of the electromagnetic field of a point charge decays like
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1194:
perfect fluid interior to an asymptotically flat vacuum exterior, are
1157:
395:
94:
852:{\displaystyle \lim _{r\rightarrow \infty }h_{ab,pq}=O(1/r^{3})}
766:{\displaystyle \lim _{r\rightarrow \infty }h_{ab,p}=O(1/r^{2})}
1065:
1015:
which assist in setting up and even in solving the resulting
401:
A simple example of an asymptotically flat spacetime is the
1142:
1304:
Einstein's field equations and their physical implications
1003:
and allied theories. There are several reasons for this:
417:
asymptotically flat. An even simpler generalization, the
1203:
Spacetime
Exterior to a Star: Against Asymptotic Flatness
683:{\displaystyle \lim _{r\rightarrow \infty }h_{ab}=O(1/r)}
1034:
in order to define and study important features such as
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standing as a solution to the field equations of some
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for a discussion of asymptotically simple spacetimes.
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Mars introduces a rotating spacetime of Petrov type
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Mars, Marc (1998). "The
Wahlquist-Newman solution".
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294:isometric to a neighbourhood of the boundary of
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1066:Hawking, S. W. & Ellis, G. F. R. (1973).
1281:MacCallum, M. A. H.; Mars, M.; and Vera, R.
1011:, and assuming asymptotic flatness provides
429:spacetime which is not asymptotically flat.
910:, which would be physically sensible. (In
436:and their rotating generalizations, the AF
1216:
1345:
1248:
1230:
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1070:. Cambridge: Cambridge University Press.
116:
66:Learn how and when to remove this message
1339:
1098:. Chicago: University of Chicago Press.
545:{\displaystyle g_{ab}=\eta _{ab}+h_{ab}}
164:is asymptotically simple if it admits a
29:This article includes a list of general
1068:The Large Scale Structure of Space-Time
611:{\displaystyle r^{2}=x^{2}+y^{2}+z^{2}}
1360:
1340:Townsend, P. K (1997). "Black Holes".
1001:exact solutions in general relativity
982:the essential innovation, now called
139:
134:exterior influences can be neglected
15:
1209:solutions in general relativity. (
13:
864:gravitational field energy density
792:
709:
639:
35:it lacks sufficient corresponding
14:
1379:
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448:A coordinate-dependent definition
196:such that every null geodesic in
1038:which may or may not be present.
20:
994:
405:solution. More generally, the
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1023:asymptotically flat solutions.
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419:de Sitter-Schwarzschild metric
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1:
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390:Some examples and nonexamples
80:asymptotically flat spacetime
1124:Living Reviews in Relativity
962:A coordinate-free definition
377:{\displaystyle {\tilde {M}}}
345:{\displaystyle {\tilde {M}}}
316:{\displaystyle {\tilde {M}}}
238:{\displaystyle {\tilde {M}}}
189:{\displaystyle {\tilde {M}}}
99:metric theory of gravitation
7:
1042:
10:
1384:
1259:10.1103/PhysRevD.63.064022
984:conformal compactification
951:{\displaystyle O(1/r^{4})}
912:classical electromagnetism
903:{\displaystyle O(1/r^{4})}
287:{\displaystyle U\subset M}
166:conformal compactification
1289:rotating bodies (with an
1175:10.1142/S0217732398001583
1309:
1145:Modern Physics Letters A
1090:Wald, Robert M. (1984).
1054:Einstein field equations
123:asymptotically vanishing
483:{\displaystyle t,x,y,z}
50:more precise citations.
1017:boundary value problem
1009:differential equations
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425:, is an example of an
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117:Intuitive significance
1032:differential topology
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427:asymptotically simple
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125:in a suitable sense.
1368:Lorentzian manifolds
1130:on December 31, 2005
1120:"Conformal Infinity"
1118:Frauendiener, Jörg.
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618:. Then we require:
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403:Schwarzschild metric
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132:: a system in which
1291:asymptotically flat
1241:2001PhRvD..63f4022M
1167:1998MPLA...13.1509M
1013:boundary conditions
442:multipole expansion
107:gravitational field
88:Minkowski spacetime
84:Lorentzian manifold
1094:General Relativity
1028:algebraic geometry
980:algebraic geometry
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423:de Sitter universe
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140:Formal definitions
103:general relativity
1293:vacuum exterior).
1201:Mark D. Roberts,
1151:(19): 1509–1519.
1105:978-0-226-87033-5
1077:978-0-521-09906-6
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46:this article by
37:inline citations
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1196:overdetermined
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978:imported from
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988:Robert Geroch
985:
981:
977:
976:Roger Penrose
973:
969:
968:Hermann Bondi
966:Around 1962,
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438:Ernst vacuums
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56:November 2008
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1219:Phys. Rev. D
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1210:
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1148:
1144:
1132:. Retrieved
1128:the original
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1111:
1093:
1083:
1067:
998:
995:Applications
965:
863:
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451:
434:Weyl metrics
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133:
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62:
53:
34:
1134:January 23,
1084:Section 6.9
407:Kerr metric
398:, are not.
144:A manifold
48:introducing
1112:Chapter 11
1060:References
991:flatness.
552:, and set
396:FRW models
31:references
1324:"Physics"
1245:CiteSeerX
793:∞
790:→
710:∞
707:→
640:∞
637:→
515:η
369:~
337:~
308:~
279:⊂
230:~
181:~
95:spacetime
1362:Category
1287:isolated
1043:See also
323:, where
1267:1644106
1237:Bibcode
1183:5289048
1163:Bibcode
44:improve
1272:eprint
1265:
1247:
1188:eprint
1181:
1102:
1074:
33:, but
1342:arXiv
1327:(PDF)
1310:Notes
1263:S2CID
1227:arXiv
1211:Note:
1192:given
1179:S2CID
1153:arXiv
413:, is
82:is a
1136:2004
1110:See
1100:ISBN
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69:)
63:(
58:)
54:(
40:.
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