GENERAL RELATIVISTIC QUASI-LOCAL ANGULAR MOMENTUM CONTINUITY AND THE ...

87 Pages·2014·0.53 MB·English

GENERAL RELATIVISTIC QUASI-LOCAL ANGULAR MOMENTUM
CONTINUITY AND THE STABILITY OF STRONGLY ELLIPTIC EIGENVALUE
PROBLEMS
by
Shawn M. Wilder
A Dissertation Submitted to the Faculty of
The Charles E. Schmidt College of Science
in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
Florida Atlantic University
Boca Raton, Florida
August 2014
Copyright by Shawn M. Wilder 2014
ii

ACKNOWLEDGEMENTS
This material is based upon work supported by the National Science Foundation under Grant
No. DGE: 0638662. Any opinions, findings, and conclusions or recommendations expressed in this
material are those of the author and do not necessarily reflect the views of the National Science
Foundation.
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ABSTRACT
Author: Shawn M. Wilder
Title: General Relativistic Quasi-local Angular Momentum Continuity
and the Stability of Strongly Elliptic Eigenvalue Problems
Institution: Florida Atlantic University
Advisor: Christopher Beetle
Degree: Doctor of Philosophy
Year: 2014
In general relativity, angular momentum of the gravitational field in some volume bounded by
an axially symmetric sphere is well-defined as a boundary integral. The definition relies on the
symmetry generating vector field, a Killing field, of the boundary. When no such symmetry exists,
one defines angular momentum using an approximate Killing field. Contained in the literature
are various approximations that capture certain properties of metric preserving vector fields. We
explore the continuity of an angular momentum definition that employs an approximate Killing
field that is an eigenvector of a particular second-order differential operator. We find that the
eigenvector varies continuously in Hilbert space under smooth perturbations of a smooth boundary
geometry. Furthermore,wefindthatnotonlyistheapproximateKillingfieldcontinuousbutthatthe
eigenvalue problem which defines it is stable in the sense that all of its eigenvalues and eigenvectors
are continuous in Hilbert space. We conclude that the stability follows because the eigenvalue
problem is strongly elliptic. Additionally, we provide a practical introduction to the mathematical
theoryofstronglyellipticoperatorsandgeneralizetheabovestabilityresultsforalargeclassofsuch
operators.
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GENERAL RELATIVISTIC QUASI-LOCAL ANGULAR MOMENTUM
CONTINUITY AND THE STABILITY OF STRONGLY ELLIPTIC EIGENVALUE
PROBLEMS
CHAPTER1. GRAVITATIONAL ANGULAR MOMENTUM AND ITS STABILITY . . . . . . 1
1.1 QUASI-LOCAL OBSERVABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 APPROXIMATE KILLING FIELDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3 ON THE CONTINUITY OF QUASI-LOCAL ANGULAR MOMENTA . . . . . . . . 21
CHAPTER2. PROVING ANGULAR MOMENTUM CONTINUITY . . . . . . . . . . . . . . 25
2.1 LINEAR PERTURBATIONS OF THE KILLING LAPLACIAN . . . . . . . . . . . . 30
2.2 THE KILLING LAPLACIAN BOUNDS ITS PERTURBATIONS . . . . . . . . . . . . 33
2.2.1 PERTURBATIONS OF THE HILBERT INNER PRODUCT . . . . . . . . . . 35
2.2.2 ALGEBRAIC FACTORS IN THE PERTURBED KILLING LAPLACIAN . . 36
2.2.3 THE KILLING LAPLACIAN RELATIVELY BOUNDS THE HESSIAN . . . . 39
2.2.4 THE KILLING LAPLACIAN RELATIVELY BOUNDS ITS PERTURBATION41
2.3 QUASI-LOCAL ANGULAR MOMENTUM CONTINUITY . . . . . . . . . . . . . . 43
CHAPTER3. REGARDING STRONGLY ELLIPTIC OPERATORS . . . . . . . . . . . . . . 45
3.1 EXISTENCE OF INHOMOGENEOUS SOLUTIONS . . . . . . . . . . . . . . . . . . 46
3.1.1 COMMENTS ON COERCIVITY AND ELLIPTICITY . . . . . . . . . . . . . 54
3.1.2 SPECTRAL THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.1.3 COMMENTS ABOUT SOBOLEV SPACES . . . . . . . . . . . . . . . . . . . 59
3.2 REGULARITY AND STABILITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
CHAPTER4. SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
APP.E.N.D.IC.E.S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
APPENDIXA. A LOCAL ATTRIBUTE OF AXIAL KILLING FIELDS. . . . . . . . . . . 67
APPENDIXB. METRIC PERTURBATIONS ON A 2-SPHERE . . . . . . . . . . . . . . . 73
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
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CHAPTER 1
GRAVITATIONAL ANGULAR MOMENTUM AND ITS STABILITY
The gravitational field in general relativity does not have a point-wise stress-energy tensor. That is,
thefieldisinherentlynon-local[1]. Thenon-localitysharesanintimatelinkwiththefactthatmatter
fields evolve along geodesics and that experimental outcomes in inertial coordinates are described
by special relativity or, put another way, by the equivalence principle.
The total energy, momentum, angular momentum or center-of-mass of the composition of both
the gravitational field and matter sources is well-defined however in any finite spacetime region
admitting a timelike, translational, axial or boost symmetry generating field, respectively, i.e., an
appropriate Killing field. Each definition relies on Noether’s theorem, which implies that every
conservedquantityisassociatedwithsomeunderlyingsymmetry,andidentifiestherelevantphysical
observablewiththephase-spacegeneratorofageometricsymmetry. Thesegravitationalobservables,
specifically, are integrals over closed 2-surfaces (surfaces) S. They are quasi-local integrals. The
overall picture is reminiscent of Gauss’s law in electrostatics where one replicates total volume
measurements from field data at the boundary of the volume. Each quasi-local integral depends
explicitly on a particular Killing vector field.
Typically, spacetimeisnotsymmetricandnonon-trivialKillingfieldexistsonwhichtobasethe
definitionofaquasi-localobservable. Ontheotherhand,certainlythephysicalangularmomentum,
for example, of an isolated astrophysical body, exists even if that body is not exactly axisymmetric.
Accordingly,theefforttoextendthedefinitionofquasi-localobservablesfromthestrictlysymmetric,
Noetherian context to more general spacetimes is now an area of active research; see [2, 3] for
reviews. The principal challenge in this effort is to identify a vector field on a generic 2-sphere S to
play the role formerly played by the Killing field. Simply using an arbitrary vector field in this role
introduces an enormous ambiguity that does not occur in the symmetric case since, whereas any
finite-dimensional manifold has at most a finite number of symmetries, the space of all vector fields
on S is infinite-dimensional. Many schemes that aim to extend the domain of validity of a quasi-
local observable’s definition away from the exact symmetry realm therefore hinge on the problem of
1
selecting a preferred vector field on a general surface to play the role of the Killing field.
We hold that any physically reasonable prescription to select a preferred vector field needs to
possess, at least, the following properties:
1. Theschememustselectauniquevectorfield,oratmostafinite-dimensionalspaceofsuchvector
fields, for any arbitrary 2-surface S.
2. Whenever a spacetime does admit a symmetry, the vector field(s) selected must agree with the
Killing field(s) generating that symmetry.
3. Small changes in a spacetime’s geometry, at or near S, must produce only small changes in the
vector field or, at least, in any quasi-local observable constructed from it.
Someauthorsrequireadditionalpropertiesthata“correct”fieldshouldexhibitbutanysatisfactory
candidate ought to have at least this minimal set of characteristics.
One natural way to select a vector field to define an observable is to seek a best approximation
to a Killing field when no actual symmetry exists. There are several schemes to do this in the
literature [4, 5, 6, 7, 8]. One key feature of all of them is that the approximate Killing field does not
depend on a manifold’s local, or point-wise, geometry. This is because even small but short length
scale metric perturbations of an exactly symmetric geometry potentially introduce large curvature
perturbations, albeit only on short length scales. If the objective is to recover an approximation to
theexactsymmetryofanunderlying,unperturbedmanifoldthenonemustseekthatapproximation
globally.
The global nature of an approximate Killing field helps satisfy condition 2 because an exact
Killing field itself is a global feature of a symmetric geometry. But, a global character complicates
the satisfaction of condition 3. Considering the global, rather than the local, geometry means that
determiningifcondition3issatisfiedrequiresglobalfunctionalanalysisandnotjustlocalfunctional
considerations. This question has not received much attention prior to the work [9], which forms
the foundation of this thesis.
WeshowthatoneapproximateKillingfielddefinition,whichisbasedonthesolutionofapartic-
ular eigenvalue problem on a Riemannian geometry, depends continuously on the global geometry
of that manifold in a precise sense. Formally, proving such continuity involves ideas from operator
theory on manifolds. However, a direct approach, rather than relying on general theorems, is more
immediatelyaccessible. Wepresentadirectargumentinchapter2,andthenfollowitbyhighlighting
some connections to general results from operator theory in chapter 3.
2
Before delving into differential equation theory in chapter 3, we lay some groundwork for the
physical problem in this introductory chapter. We review/motivate quasi-local observables, and
specifically quasi-local angular momenta, in section 1.1; see [2, 3] for more comprehensive reviews.
We describe three schemes, including our preferred scheme, that approximate symmetry generating
vectorfieldsinsection1.2. Weconcludeinsection1.3bymotivatingthefunctionalanalyticmethods
that we require in chapter 2 to demonstrate gravitational angular momentum continuity.
1.1 QUASI-LOCAL OBSERVABLES
We start by considering how to characterize the flow of stress-energy-momentum (stress-energy)
(cid:0) (cid:1)
through the hypersurface boundary ∂U of some region U in a given spacetime M,g composed
ab
of the manifold M and the Lorentzian metric g ; lowercase roman letters, unless otherwise stated,
ab
representabstractindices. Considerthestress-energytensorT ,itsmatter-current(current)density
ab
T aum :=gnaT um
m mn
asmeasuredbythefieldua—henceforward, weusethemetricanditsinversegab tofreelylowerand
raise indices—and its divergence ∇ (cid:0)T bua(cid:1) with respect to the Levi-Civita connection
b a
∇ defined such that ∇ g :={0} ;
a a bc abc
{0} is the zero (0,3)-tensor which we write similarly for other tensor spaces. We construct a
abc
hypersurface-flux law for the total flow of the stress-energy current density
 (cid:90)
(cid:90)  (cid:15) Tab∇bua
(cid:15) ∇b(Tabua)= (cid:73)U (1.1)
U  dV nˆbTabua
∂U
where (cid:15) and dV are the volume forms over U and its boundary ∂U, which is normal to the
unit-normalized field nˆa, respectively; the top equation follows because the stress-energy tensor
is divergence-free, i.e.
∇ T b ={0} ,
b a a
and the bottom equation from Stokes’ theorem. Observe, if ua has a purely antisymmetric gradient
1(cid:0) (cid:1)
∇ u = ∇ u −∇ u =:∇ u
a b 2 a b b a [a b]
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within U or, similarly, if its symmetric gradient vanishes
2∇ u :=∇ u +∇ u ={0} ,
(a b) a b b a ab
then
∇ (cid:0)T bua(cid:1)=0
b a
andsothetotalvolume-fluxofstress-energythrough∂U is0. Thatis,thestress-energydensityisev-
eryconservedinU. Putanotherway,thestress-energycurrentdensityisexpansion-freethroughout
the neighborhood U.
Vector fields that have entirely antisymmetric gradients in torsion-free spacetimes preserve the
metric along their orbits, i.e.
L g ={0} (1.2)
ξ ab ab
where L , the Lie derivative, generates infinitesimal coordinate transformations along, in this case,
ξ
the field ξa. This expression is Killing’s equation and its solutions ξa are Killing fields. We express
the Lie derivative in terms of our covariant derivative as
L g :=ξc∇ g +(cid:0)g ∇ +g ∇ (cid:1)ξc =2∇ ξ .
ξ ab c ab cb a ca b (a b)
The mappings of a manifold to itself along the orbits generated by ξa are isometric because they do
not alter the metric. To whit, Killing fields generate the geometric symmetries of a given geometry.
RecognizethatKilling’sequationonagivenspacetimerepresents10first-orderdifferentialequations.
GeometriesthatadmitsolutionsforeachindependentcomponentofKilling’sequationaremaximally
symmetric. Maximallysymmetricgeometriesexhibitthesamescalar(Ricci)curvatureateverypoint
xµ ∈ M; greek indices are concrete and run the dimension of the manifold. Arbitrary geometries
need not admit even a single solution to Killing’s equation.
Minkowski spacetime
(cid:0)M,g (cid:1):=(cid:0)R1,3,η (cid:1),
ab ab
where
R1,3 :=R×R3
and η is flat, is maximally symmetric. Consider an orthogonal, inertial basis {b a} which spans
ab µ
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