Bianchi%20Identities%20and%20Weak%20Gravitational%20Lensing

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Bianchi Identities and Weak Gravitational Lensing
Brian Keith, Mentor Thomas Kling, Department of
Physics,Bridgewater State College, Bridgewater,
MA 02325
Abstract
Goals Objectives
Gravitational lensing is the bending of light
rays due to the gravitational attraction of
massive objects such as galaxies. Weak
gravitational lensing, the distortion of the
shapes of light rays, and general relativity, our
modern theory of gravity, have had divergent
paths. Astronomers who study weak lensing dont
rely on the principles of general relativity but
use approximations to understand their
observations. However, general relativity can be
used as a medium to explain weak lensing and thus
provide an ab initio understanding of it. The
research was done in the null tetrad and spin
coefficient formalism which hinges on the
properties of light rays. The Bianchi identities,
which come out of the theory of relativity, were
found to be the fundamental equations of weak
lensing. The ATP program made this summer
research possible.
Results
Final form of 1st Bianchi identity
The goal of the project is to find a relationship
between the Weyl tensor, or shear ( ), and
the Ricci tensor, or the matter distribution (
), using spin coefficient formalism.

• We use this equation to find the matter
  distribution, or the Ricci tensor.
• The same result that Astrophysicists get but we
  have started with an equation from general
  relativity.

1st Bianchi identity
Photo of weak lensing

• Yellow blobs are galaxies.
• Light is sheared around the main cluster of
  galaxies.
• The goal is to determine the mass density from
  the sheared images.

What does this mean? The breakdown is
Discussion
Derivative operators
Weyl tensor in spin coefficient formalism

• An integral relationship between the Ricci and
  Weyl tensors have already been found

Ricci tensor in spin coefficient formalism

• This is the relation used by astrophysicists,
  but it does not have a basis in general
  relativity.
• To prove that the Bianchi identity is the
  fundamental equation of weak lensing, we must
  derive the integral relation from the
  differential relation.
• Preliminary work indicates that one may be able
  to use the Green's functions of John Porter for
  the edth (similar to ? ) derivative operator to
  prove this relation.

Spin coefficients
Weak lensing phenomena
Calculations

• Light is bent by
• massive objects.
• The fake images we see as a result are
  elliptical.
• Astrophysicists can repiece the image but they
  do not use relativistic principles.
• It is the goal of a physicist to find the
  fundamental equations of physical phenomena.

Weyl and Ricci tensor components in terms of a
weak perturbation of a Minkowski spacetime.
References

• Jeremy Bernstein, Paul M. Fishbane, Stephen
  Gasiorowicz. Modern Physics. (Prentice Hall, New
  Jersey, 2000).
• Albert Einstein. Relativity. (Crown Publishers
  Inc., New York 1961).
• James B. Hartle. Gravity An introduction to
  Einstein's general relativity. (Pearson Education
  Inc., San Francisco, 2003).
• Steven R. Lay. Analysis with an Introduction to
  Proof. (Upper Saddle River, New Jersey Prentice
  Hall Inc., 1999).
• H. A. Lorentz, H. Weyl, H. Minkowski. Notes by
  A. Sommerfield. The Principle of Relativity.
  (General Publishing Company, Toronto 1952).
• E. T. Newman, K. P. Tod. Asymptotically Flat
  Space-Times from General relativity and
  Gravitation, Vol. 2. (Picnum publishing
  Cororation, 1980).
• James Peacock. Cosmological Physics. (Cambridge
  University press, Cambridge 1999).
• R. Penrose, W. Rindler. Spinors and spacetime
  volume I II. (Cambridge University Press,
  Cambridge 1984).
• Robert Wald. General Relativity. (University of
  Chicago press, Chicago 1984).

Null tetrad and spin coefficient formalism
Argument

• The calculations and the choice of tetrad and
  flat space for spin coefficients yields

• Integrating out over all of the z direction
  constrains the lens to a plane. This sets

and
The null tetrad allows tracking of light rays
using 4 vectors, 1 in the direction of the light
ray, 1 perpendicular, and 2 complex ones that
curl around the light ray in opposite directions.
12 complex spin coefficients follow from the
null tetrad and describe light ray properties
including the degree that light rays come
together and shearing.
Where the subscript L denotes the value of the
object in the lens plane.