60 research outputs found

### The Vacuum Einstein Equations via Holonomy around Closed Loops on Characteristic Surfaces

We reformulate the standard local equations of general relativity for
asymptotically flat spacetimes in terms of two non-local quantities, the
holonomy $H$ around certain closed null loops on characteristic surfaces and
the light cone cut function $Z$, which describes the intersection of the future
null cones from arbitrary spacetime points, with future null infinity. We
obtain a set of differential equations for $H$ and $Z$ equivalent to the vacuum
Einstein equations. By finding an algebraic relation between $H$ and $Z$ this
set of equations is reduced to just two coupled equations: an
integro-differential equation for $Z$ which yields the conformal structure of
the underlying spacetime and a linear differential equation for the ``vacuum''
conformal factor. These equations, which apply to all vacuum asymptotically
flat spacetimes, are however lengthy and complicated and we do not yet know of
any solution generating technique. They nevertheless are amenable to an
attractive perturbative scheme which has Minkowski space as a zeroth order
solution.Comment: 28 pages, RevTeX, 3 PostScript figure

### Asymptotically Shear-free and Twist-free Null Geodesic Congruences

We show that, though they are rare, there are asymptotically flat space-times
that possess null geodesic congruences that are both asymptotically shear- free
and twist-free (surface forming). In particular, we display the class of
space-times that possess this property and demonstrate how these congruences
can be found. A special case within this class are the Robinson- Trautman
space-times. In addition, we show that in each case the congruence is isolated
in the sense that there are no other neighboring congruences with this dual
property.Comment: 10 page

### Electromagnetic Dipole Radiation Fields, Shear-Free Congruences and Complex Center of Charge World Lines

We show that for asymptotically vanishing Maxwell fields in Minkowski space
with non-vanishing total charge, one can find a unique geometric structure, a
null direction field, at null infinity. From this structure a unique complex
analytic world-line in complex Minkowski space that can be found and then
identified as the complex center of charge. By ''sitting'' - in an imaginary
sense, on this world-line both the (intrinsic) electric and magnetic dipole
moments vanish. The (intrinsic) magnetic dipole moment is (in some sense)
obtained from the `distance' the complex the world line is from the real space
(times the charge). This point of view unifies the asymptotic treatment of the
dipole moments For electromagnetic fields with vanishing magnetic dipole
moments the world line is real and defines the real (ordinary center of
charge). We illustrate these ideas with the Lienard-Wiechert Maxwell field. In
the conclusion we discuss its generalization to general relativity where the
complex center of charge world-line has its analogue in a complex center of
mass allowing a definition of the spin and orbital angular momentum - the
analogues of the magnetic and electric dipole moments.Comment: 17 page

### Center of Mass and spin for isolated sources of gravitational radiation

We define the center of mass and spin of an isolated system in General
Relativity. The resulting relationships between these variables and the total
linear and angular momentum of the gravitational system are remarkably similar
to their Newtonian counterparts, though only variables at the null boundary of
an asymptotically flat spacetime are used for their definition. We also derive
equations of motion linking their time evolution to the emitted gravitational
radiation. The results are then compared to other approaches. In particular one
obtains unexpected similarities as well as some differences with results
obtained in the Post Newtonian literature . These equations of motion should be
useful when describing the radiation emitted by compact sources such as
coalescing binaries capable of producing gravitational kicks, supernovas, or
scattering of compact objects.Comment: 16 pages. Accepted for publication in Phys. Rev.

### Astrophysical limits on quantum gravity motivated birefringence

We obtain observational upper bounds on a class of quantum gravity related
birefringence effects, by analyzing the presence of linear polarization in the
optical and ultraviolet spectrum of some distant sources. In the notation of
Gambini and Pullin we find $\chi < 10^{-3}$.Comment: 4 pages, submitted to Phys. Rev. Let

### Null Surfaces and Legendre Submanifolds

It is shown that the main variable Z of the Null Surface Formulation of GR is
the generating function of a constrained Lagrange submanifold that lives on the
energy surface H=0 and that its level surfaces Z=const. are Legendre
submanifolds on that energy surface.
The behaviour of the variable Z at the caustic points is analysed and a
genralization of this variable is discussed.Comment: 28 pages, 7 figure

### Differential Geometry from Differential Equations

We first show how, from the general 3rd order ODE of the form
z'''=F(z,z',z'',s), one can construct a natural Lorentzian conformal metric on
the four-dimensional space (z,z',z'',s). When the function F(z,z',z'',s)
satisfies a special differential condition of the form, U[F]=0, the conformal
metric possesses a conformal Killing field, xi = partial with respect to s,
which in turn, allows the conformal metric to be mapped into a three
dimensional Lorentzian metric on the space (z,z',z'') or equivalently, on the
space of solutions of the original differential equation. This construction is
then generalized to the pair of differential equations, z_ss =
S(z,z_s,z_t,z_st,s,t) and z_tt = T(z,z_s,z_t,z_st,s,t), with z_s and z_t, the
derivatives of z with respect to s and t. In this case, from S and T, one can
again, in a natural manner, construct a Lorentzian conformal metric on the six
dimensional space (z,z_s,z_t,z_st,s,t). When the S and T satisfy equations
analogous to U[F]=0, namely equations of the form M[S,T]=0, the 6-space then
possesses a pair of conformal Killing fields, xi =partial with respect to s and
eta =partial with respect to t which allows, via the mapping to the four-space
of z, z_s, z_t, z_st and a choice of conformal factor, the construction of a
four-dimensional Lorentzian metric. In fact all four-dimensional Lorentzian
metrics can be constructed in this manner. This construction, with further
conditions on S and T, thus includes all (local) solutions of the Einstein
equations.Comment: 37 pages, revised version with clarification

### Superselection Sectors in Asymptotic Quantization of Gravity

Using the continuity of the scalar $\Psi_2$ (the mass aspect) at null
infinity through $i_o$ we show that the space of radiative solutions of general
relativity can be thought of a fibered space where the value of $\Psi_2$ at
$i_o$ plays the role of the base space. We also show that the restriction of
the available symplectic form to each ``fiber'' is degenerate. By finding the
orbit manifold of this degenerate direction we obtain the reduced phase space
for the radiation data. This reduced phase space posses a global structure,
i.e., it does not distinguishes between future or past null infinity. Thus, it
can be used as the space of quantum gravitons. Moreover, a Hilbert space can be
constructed on each ``fiber'' if an appropriate definition of scalar product is
provided. Since there is no natural correspondence between the Hilbert spaces
of different foliations they define superselection sectors on the space of
asymptotic quantum states.Comment: 22 pages, revtex fil

### Linearized Einstein theory via null surfaces

Recently there has been developed a reformulation of General Relativity -
referred to as {\it the null surface version of GR} - where instead of the
metric field as the basic variable of the theory, families of three-surfaces in
a four-manifold become basic. From these surfaces themselves, a conformal
metric, conformal to an Einstein metric, can be constructed. A choice of
conformal factor turns them into Einstein metrics. The surfaces are then
automatically characteristic surfaces of this metric. In the present paper we
explore the linearization of this {\it null surface theory} and compare it with
the standard linear GR. This allows a better understanding of many of the
subtle mathematical issues and sheds light on some of the obscure points of the
null surface theory. It furthermore permits a very simple solution generating
scheme for the linear theory and the beginning of a perturbation scheme for the
full theory.Comment: 22 page

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