Dynamics of Fermat potentials in non-perturbative gravitational lensing

We present a framework, based on the null-surface formulation of general relativity, for discussing the dynamics of Fermat potentials for gravitational lensing in a generic situation without approximations of any kind. Additionally, we derive two lens equations: one for the case of thick compact lenses and the other one for lensing by gravitational waves. These equations in principle generalize the astrophysical scheme for lensing by removing the thin-lens approximation while retaining the weak fields.Comment: Accepted for publication in Phys. Rev.

On bending angles by gravitational lenses in motion

The bending of lightrays by the gravitational field of a ``lens'' that is moving relative to the observer is calculated within the approximation of weak fields, small angles and thin lenses. Up to first order in $v/c$ -- and assuming the acceleration to be much smaller than $v/c$ -- the bending angle, time delay and redshift of the images are found to be affected by the component of the speed of the deflector along the line of sight. The correction takes the form of an overall factor of $1+v/c$ accompanying the mass of the deflector, leading to an indeterminacy of the order of $v/c$ in the mass of the lens inferred on the basis of the separation of multiple images. The consequent correction to the microlensing lightcurve is pointed out, as well as scenarios where the correction is potentially relevant.Comment: 6 pages, to appear in MNRA

Aberration by gravitational lenses in motion

It is known that a fully relativistic integration of the null geodesics of a weak perturbation of flat spacetime leads to a correction of order $v/c$ to the bending angle and time delay due to a gravitational lens in slow motion with small acceleration. The existence of the $v/c$ correction was verified by the VLBI experiment of the bending of light by Jupiter on September 8, 2002. Here the $v/c$ correction is interpreted by means of standard aberration of light in an optically active medium with an effective index of refraction induced by the gravitational field of a lens in motion.Comment: 3 page

Status of Mexican Trucks in the United States: Frequently Asked Questions

In the North American Free Trade Agreement (NAFTA), which took effect in January 1994, the United States and Mexico agreed to allow each other’s trucks to carry goods across the border to make deliveries anywhere inside their respective countries. This provision was controversial in the United States, and a trial program begun in September 2007 by the George W. Bush Administration was defunded by Congress in March 2009. Mexico imposed tariffs on certain U.S. goods in response to the program’s termination, as permitted by NAFTA. After bilateral negotiations, the Obama Administration announced a new pilot program to allow long-haul Mexican trucks into the United States in April 2011. The first Mexican truck with long-haul operating authority crossed the border in October 2011. This report answers frequently asked questions about the pilot program permitting Mexican trucks into the United States

North American Free Trade Agreement (NAFTA) Implementation: The Future of Commercial Trucking Across the Mexican Border

[Excerpt] NAFTA set forth a schedule for implementing its trucking provisions that would have opened the border states to cross-border trucking competition in 1995 and all of North America in 2000, but full implementation has been stalled because of concern with the safety of Mexican trucks. Congress first addressed these concerns in the FY2002 Department of Transportation Appropriations Act (P.L. 107-87) which set 22 safety-related preconditions for opening the border to long-haul Mexican trucks. In November 2002, the U.S. Department of Transportation announced that all the preconditions had been met and began processing Mexican applications for U.S. long-haul authority. However, a suit over environmental compliance delayed implementation further. After the suit was resolved, in February 2007, the U.S. and Mexican Secretaries of Transportation announced a demonstration project to implement the NAFTA trucking provisions. The purpose of the project was to demonstrate the ability of Mexico-based motor carriers to operate safely in the United States beyond the border commercial zones. Up to 100 Mexico-domiciled carriers would be allowed to operate throughout the United States for one year and Mexico would allow the same for up to 100 U.S.-based carriers. With passage of the U.S. Troop Readiness, Veteran’s Care, Katrina Recovery, and Iraq Accountability Appropriations Act, 2007 (P.L. 110-28), Congress mandated additional requirements before the project could begin. After failing to defund the demonstration project in the FY2008 Consolidated Appropriations Act (P.L. 110-161), Congress succeeded in terminating the demonstration project through a provision in the FY2009 Omnibus Appropriations Act (P.L. 111-8). Subsequently, Mexico announced it would retaliate by increasing import duties on 90 U.S. products. The Obama Administration has indicated it intends to propose a revamped program that will address the concerns of Congress. The FY2010 Consolidated Appropriations Act (P.L. 111-117) passed in December 2009 did not preclude funds from being spent on a long-haul Mexican truck pilot program, provided the terms and conditions stipulated in section 350 of P.L. 107-87 and section 6901 of P.L. 110-28 were satisfied. One truck safety statistic, “out-of-service” rates, indicates that Mexican trucks operating in the United States are now safer than they were a decade ago. The data indicate that Mexican trucks and drivers have a comparable safety record to U.S. truckers. Another study indicates that the truck driver is usually the more critical factor in causing accidents than a safety defect with the truck itself. Service characteristics of long-haul trucking suggest that substandard carriers would likely not succeed in this market. As shipment distance increases, the relative cost of trucking compared to rail increases, and thus shippers utilizing long-haul trucking are willing to pay more because they require premium service, such as precise delivery windows or cargo refrigeration. These exacting service requirements would seem to disqualify truckers with unreliable equipment or incompetent drivers. In contrast, the short-haul “drayage” carriers that Mexican long-haul carriers would displace, typically use older equipment because of the many hours spent idling awaiting customs processing at the border. If Mexican carriers do eventually receive long-haul authority, the short term impact is expected to be gradual as Mexican firms deal with a number of stumbling blocks, including lack of prearranged back hauls and higher insurance and capital costs, in addition to the customs processing delays. In the long run, use of drayage companies is likely to decline as they lose part of their market share to Mexican long-haul carriers. The most common trips for these carriers will probably be from the Mexican interior to warehouse facilities on the U.S. side of the border or to nearby cities in the border states

The Theory of Caustics and Wavefront Singularities with Physical Applications

This is intended as an introduction to and review of the work of V, Arnold and his collaborators on the theory of Lagrangian and Legendrian submanifolds and their associated maps. The theory is illustrated by applications to Hamilton-Jacobi theory and the eikonal equation, with an emphasis on null surfaces and wavefronts and their associated caustics and singularities.Comment: Figs. not include

Dual Representation of Quasiconvex Conditional Maps

We provide a dual representation of quasiconvex maps between two lattices of random variables in terms of conditional expectations. This generalizes the dual representation of quasiconvex real valued functions and the dual representation of conditional convex maps.Comment: Date changed Added one remark on assumption (c), page

On the Newtonian Limit of General Relativity

We find a choice of variables for the 3+1 formulation of general relativity which casts the evolution equations into (flux-conservative) symmetric-hyperbolic first order form for arbitrary lapse and shift, for the first time. We redefine the lapse function in terms of the determinant of the 3-metric and a free function U which embodies the lapse freedom. By rescaling the variables with appropriate factors of 1/c, the system is shown to have a smooth Newtonian limit when the redefined lapse U and the shift are fixed by means of elliptic equations to be satisfied on each time slice. We give a prescription for the choice of appropriate initial data with controlled extra-radiation content, based on the theory of problems with different time-scales. Our results are local, in the sense that we are not concerned with the treatment of asymptotic regions. On the other hand, this local theory is all what is needed for most problems of practical numerical computation.Comment: 16 pages, uses REVTe

On the super replication price of unbounded claims

In an incomplete market the price of a claim f in general cannot be uniquely identified by no arbitrage arguments. However, the ``classical'' super replication price is a sensible indicator of the (maximum selling) value of the claim. When f satisfies certain pointwise conditions (e.g., f is bounded from below), the super replication price is equal to sup_QE_Q[f], where Q varies on the whole set of pricing measures. Unfortunately, this price is often too high: a typical situation is here discussed in the examples. We thus define the less expensive weak super replication price and we relax the requirements on f by asking just for ``enough'' integrability conditions. By building up a proper duality theory, we show its economic meaning and its relation with the investor's preferences. Indeed, it turns out that the weak super replication price of f coincides with sup_{Q\in M_{\Phi}}E_Q[f], where M_{\Phi} is the class of pricing measures with finite generalized entropy (i.e., E[\Phi (\frac{dQ}{dP})]<\infty) and where \Phi is the convex conjugate of the utility function of the investor.Comment: Published at http://dx.doi.org/10.1214/105051604000000459 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org