50 research outputs found
Conservation of asymptotic charges from past to future null infinity: Supermomentum in general relativity
We show that the BMS-supertranslations and their associated supermomenta on
past null infinity can be related to those on future null infinity, proving the
conjecture of Strominger for a class of spacetimes which are
asymptotically-flat in the sense of Ashtekar and Hansen. Using a cylindrical
3-manifold of both null and spatial directions of approach towards spatial
infinity, we impose appropriate regularity conditions on the Weyl tensor near
spatial infinity along null directions. The asymptotic Einstein equations on
this 3-manifold and the regularity conditions imply that the relevant Weyl
tensor components on past null infinity are antipodally matched to those on
future null infinity. The subalgebra of totally fluxless supertranslations near
spatial infinity provides a natural isomorphism between the
BMS-supertranslations on past and future null infinity. This proves that the
flux of the supermomenta is conserved from past to future null infinity in a
classical gravitational scattering process provided additional suitable
conditions are satisfied at the timelike infinities.Comment: v2: corrected formula for epsilon in Eqs. A.4E and A.9 v1: (published
version in JHEP) 49 pages, 2 figures. arXiv admin note: substantial text
overlap with arXiv:1808.0786
The First Law of Black Hole Mechanics for Fields with Internal Gauge Freedom
We derive the first law of black hole mechanics for physical theories based
on a local, covariant and gauge-invariant Lagrangian where the dynamical fields
transform non-trivially under the action of internal gauge transformations. The
theories of interest include General Relativity formulated in terms of tetrads,
Einstein-Yang-Mills theory and Einstein-Dirac theory. Since the dynamical
fields of these theories have gauge freedom, we argue that there is no group
action of diffeomorphisms of spacetime on such dynamical fields. In general,
such fields cannot even be represented as smooth, globally well-defined tensor
fields on spacetime. Consequently the derivation of the first law by Iyer-Wald
cannot be used directly. We show how such theories can be formulated on a
principal bundle and that there is a natural action of automorphisms of the
bundle on the fields. These bundle automorphisms encode both spacetime
diffeomorphisms and gauge transformations. Using this reformulation we define
the Noether charge associated to an infinitesimal automorphism and the
corresponding notion of stationarity and axisymmetry of the dynamical fields.
We define certain potentials and charges at the horizon of a black hole so that
the potentials are constant on the bifurcate Killing horizon, giving a
generalised zeroth law for bifurcate Killing horizons. We identify the
gravitational potential and perturbed charge as the temperature and perturbed
entropy of the black hole which gives an explicit formula for the perturbed
entropy analogous to the Wald entropy formula. We obtain a general first law of
black hole mechanics for such theories. The first law relates the perturbed
Hamiltonians at spatial infinity and the horizon, and the horizon contributions
take the form of a potential times perturbed charge term. We also comment on
the ambiguities in defining a prescription for the total entropy for black
holes.Comment: v4: 69 pages, shorter, less pedagogical version, updated references
(published in CQG). v3: minor typographic corrections (PhD thesis version).
v2: 80 pages, 1 figure, significant changes to presentation and text; added
Thm 1 and Thm2. v1: 64 pages, 1 figur
Conservation of asymptotic charges from past to future null infinity: Maxwell fields
On any asymptotically-flat spacetime, we show that the asymptotic symmetries
and charges of Maxwell fields on past null infinity can be related to those on
future null infinity as recently proposed by Strominger. We extend the
covariant formalism of Ashtekar and Hansen by constructing a 3-manifold of both
null and spatial directions of approach to spatial infinity. This allows us to
systematically impose appropriate regularity conditions on the Maxwell fields
near spatial infinity along null directions. The Maxwell equations on this
3-manifold and the regularity conditions imply that the relevant field
quantities on past null infinity are antipodally matched to those on future
null infinity. Imposing the condition that in a scattering process the total
flux of charges through spatial infinity vanishes, we isolate the subalgebra of
totally fluxless symmetries near spatial infinity. This subalgebra provides a
natural isomorphism between the asymptotic symmetry algebras on past and future
null infinity, such that the corresponding charges are equal near spatial
infinity. This proves that the flux of charges is conserved from past to future
null infinity in a classical scattering process of Maxwell fields. We also
comment on possible extensions of our method to scattering in general
relativity.Comment: v3: M=Misner in ADM; author thanks D for this correction! v2:
references updated, minor typos fixed, few explanatory comments added
(published in JHEP) v1: 33 pages, 2 figure
Ricci flow of unwarped and warped product manifolds
We analyse Ricci flow (normalised/un-normalised) of product manifolds
--unwarped as well as warped, through a study of generic examples. First, we
investigate such flows for the unwarped scenario with manifolds of the type
, , and also, similar multiple products. We are able to
single out generic features such as singularity formation, isotropisation at
particular values of the flow parameter and evolution characteristics.
Subsequently, motivated by warped braneworlds and extra dimensions, we look at
Ricci flows of warped spacetimes. Here, we are able to find analytic solutions
for a special case by variable separation. For others we numerically solve the
equations (for both the forward and backward flow) and draw certain useful
inferences about the evolution of the warp factor, the scalar curvature as well
the occurence of singularities at finite values of the flow parameter. We also
investigate the dependence of the singularities of the flow on the inital
conditions. We expect our results to be useful in any physical/mathematical
context where such product manifolds may arise.Comment: 25 pages, 25 figures, some figures replace
Fields and fluids on curved non-relativistic spacetimes
We consider non-relativistic curved geometries and argue that the background
structure should be generalized from that considered in previous works. In this
approach the derivative operator is defined by a Galilean spin connection
valued in the Lie algebra of the Galilean group. This includes the usual spin
connection plus an additional "boost connection" which parameterizes the
freedom in the derivative operator not fixed by torsion or metric
compatibility. As an example we write down the most general theory of
dissipative fluids consistent with the second law in curved non-relativistic
geometries and find significant differences in the allowed transport
coefficients from those found previously. Kubo formulas for all response
coefficients are presented. Our approach also immediately generalizes to
systems with independent mass and charge currents as would arise in
multicomponent fluids. Along the way we also discuss how to write general
locally Galilean invariant non-relativistic actions for multiple particle
species at any order in derivatives. A detailed review of the geometry and its
relation to non-relativistic limits may be found in a companion paper
[arXiv:1503.02682].Comment: Reference added. 44 page