Bianchi Identities and Weak Gravitational Lensing Brian Keith, Mentor: Thomas Kling, Department of Physics, Bridgewater

1. 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 don’t 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: • 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. Ricci tensor in spin coefficient formalism 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: volumeI&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.