147:. 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.
33:
1001:
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
1033:
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
451:(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
1216:. 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
501:, 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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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
139:
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
985:, 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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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
120:, 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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124:, the gravitational field (curvature) becomes negligible at large distances from the source of the field (typically some isolated massive object such as a star).
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Mars, M. & Senovilla, J. M. M. (1998). "On the construction of global models describing rotating bodies; uniqueness of the exterior gravitational field".
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The simplest (and historically the first) way of defining an asymptotically flat spacetime assumes that we have a coordinate chart, with coordinates
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While this is less obvious, it turns out that invoking asymptotic flatness allows physicists to import sophisticated mathematical concepts from
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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
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Models of physical phenomena in general relativity (and allied physical theories) generally arise as the solution of appropriate systems of
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1209:. 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.
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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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On the other hand, there are important large families of solutions which are asymptotically flat, such as the AF
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877:(to the extent that this somewhat nebulous notion makes sense in a metric theory of gravitation) decays like
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Since the latter excludes black holes, one defines a weakly asymptotically simple manifold as a manifold
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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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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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116:. 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 (XXG), 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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perfect fluid interior to an asymptotically flat vacuum exterior, are
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863:{\displaystyle \lim _{r\rightarrow \infty }h_{ab,pq}=O(1/r^{3})}
777:{\displaystyle \lim _{r\rightarrow \infty }h_{ab,p}=O(1/r^{2})}
26:
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which assist in setting up and even in solving the resulting
412:
A simple example of an asymptotically flat spacetime is the
1315:
Einstein's field equations and their physical implications
1014:
and allied theories. There are several reasons for this:
428:
asymptotically flat. An even simpler generalization, the
1214:
Spacetime
Exterior to a Star: Against Asymptotic Flatness
694:{\displaystyle \lim _{r\rightarrow \infty }h_{ab}=O(1/r)}
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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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1292:MacCallum, M. A. H.; Mars, M.; and Vera, R.
1022:, and assuming asymptotic flatness provides
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921:, which would be physically sensible. (In
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77:Learn how and when to remove this message
1109:. Chicago: University of Chicago Press.
556:{\displaystyle g_{ab}=\eta _{ab}+h_{ab}}
175:is asymptotically simple if it admits a
40:This article includes a list of general
1326:
1079:The Large Scale Structure of Space-Time
622:{\displaystyle r^{2}=x^{2}+y^{2}+z^{2}}
1351:Townsend, P. K (1997). "Black Holes".
1012:exact solutions in general relativity
993:the essential innovation, now called
7:
145:exterior influences can be neglected
1220:solutions in general relativity. (
875:gravitational field energy density
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46:it lacks sufficient corresponding
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459:A coordinate-dependent definition
207:such that every null geodesic in
1049:which may or may not be present.
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416:solution. More generally, the
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401:Some examples and nonexamples
91:asymptotically flat spacetime
1135:Living Reviews in Relativity
973:A coordinate-free definition
388:{\displaystyle {\tilde {M}}}
356:{\displaystyle {\tilde {M}}}
327:{\displaystyle {\tilde {M}}}
249:{\displaystyle {\tilde {M}}}
200:{\displaystyle {\tilde {M}}}
110:metric theory of gravitation
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1270:10.1103/PhysRevD.63.064022
995:conformal compactification
962:{\displaystyle O(1/r^{4})}
923:classical electromagnetism
914:{\displaystyle O(1/r^{4})}
298:{\displaystyle U\subset M}
177:conformal compactification
1300:rotating bodies (with an
1186:10.1142/S0217732398001583
1156:Modern Physics Letters A
1101:Wald, Robert M. (1984).
1065:Einstein field equations
134:asymptotically vanishing
494:{\displaystyle t,x,y,z}
61:more precise citations.
1028:boundary value problem
1020:differential equations
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128:Intuitive significance
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136:in a suitable sense.
1379:Lorentzian manifolds
1141:on December 31, 2005
1131:"Conformal Infinity"
1129:Frauendiener, Jörg.
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629:. Then we require:
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143:: a system in which
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1252:2001PhRvD..63f4022M
1178:1998MPLA...13.1509M
1024:boundary conditions
453:multipole expansion
118:gravitational field
99:Minkowski spacetime
95:Lorentzian manifold
18:Asymptotically flat
1105:General Relativity
1039:algebraic geometry
991:algebraic geometry
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151:Formal definitions
114:general relativity
1304:vacuum exterior).
1212:Mark D. Roberts,
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1143:. Retrieved
1139:the original
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1006:Applications
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445:Weyl metrics
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1145:January 23,
1095:Section 6.9
418:Kerr metric
409:, are not.
155:A manifold
59:introducing
1123:Chapter 11
1071:References
1002:flatness.
563:, and set
407:FRW models
42:references
1335:"Physics"
1256:CiteSeerX
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526:η
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106:spacetime
1373:Category
1298:isolated
1054:See also
334:, where
1278:1644106
1248:Bibcode
1194:5289048
1174:Bibcode
55:improve
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1199:eprint
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1353:arXiv
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1321:Notes
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474:,
471:t
377:M
345:M
316:M
293:M
287:U
267:M
238:M
215:M
189:M
163:M
80:)
74:(
69:)
65:(
51:.
20:)
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