Conformal Transformations

In this work, both special and general conformal relativity are developed, based on the symmetry group SO(4, 2), which extends the Poincaré transformations to include dilatations and special conformal transformations. An active interpretation of conformal symmetry is proposed, building on the concept of conformally invariant mass introduced by Barut and Haugen. To ensure consistency with the principle of relativity, it is assumed that the light-cone constraint η a η a = 0 among the six coordinates η a = (κx µ , κ, λ), a = 0, 1, 2, 3, 5, 6, does not generally hold, except in the case of the photon. The additional coordinate, the scale κ, is regarded as a physical degree of freedom of an object that determines its atomic size and emission spectrum (redshift or blueshift). This generalized six-dimensional framework, referred to as conformal relativity, is explored in detail. By extending the theory to curved space, gravity and the electromagnetic interaction in four dimensions can be described as components of the six-dimensional metric tensor, in a manner reminiscent of the Kaluza-Klein theory.

In this paper we study a generalization of the classical Rarita-Schwinger type operators and construct their fundamental solutions. We give some basic integral formulas related to these operators. We also establish that the projection operators appearing in the Rarita-Schwinger operators and the Rarita-Schwinger equations are conformally invariant. We further obtain the intertwining operators for other operators related to the Rarita-Schwinger operators under actions of the conformal group.

Two standard physics problems are solved in terms of the Lambert W  function, to show the applicability of this recently defined function to physics. Other applications of the function are cited, but not described. The problems solved concern Wien's displacement law and the fringing fields of a capacitor, the latter problem being representative of some problems solved using conformal transformations. The physical content of the solutions remains unchanged, but they gain a new elegance and convenience. PACS No.: 41.10F

In this paper we explore the physical consequences of assuming Weyl invariance of the laws of gravity from the classical standpoint exclusively. Actual Weyl invariance requires to replace the underlying Riemannian geometrical structure of the background spacetimes by Weyl integrable geometry (WIG). We show that gauge freedom, a distinctive feature of Weyl invariant theories of gravity, leads to very unusual consequences. For instance, within the cosmological setting in a WIG-based conformal invariant gravity theory, also known as conformal general relativity (CGR), a static universe is physically equivalent to a universe undergoing de Sitter expansion. It happens also that spherically symmetric black holes are physically equivalent to singularity-free wormholes. Another outstanding consequence of gauge freedom in the framework of CGR is that inflation is not required to explain the flatness, horizon and relict particle abundances, among other puzzles that arise in standard GR-based cosmology. Besides, the cosmological constant and the mass hierarchy problems do not arise neither in this setup.

This book provides advanced physics and mathematics students with an accessible yet detailed understanding of the fundamentals of differential geometry and symmetries in classical physics. Most of the topics covered in this book have previously only been scattered across many different sources of literature, therefore this is the first book to coherently present this treatment of topics in one comprehensive volume.

- - Key features:

-> Contains a modern, streamlined presentation of classical topics, which are normally taught separately.

-> Includes several advanced topics, such as the Belinfante energy-momentum tensor, the Weyl-Schouten theorem, the derivation of Noether currents for diffeomorphisms, and the definition of conserved integrals in general relativity.

-> Focuses on the clear presentation of the mathematical notions and calculational technique.


- - - Table of Contents:

- Chapter 1. Manifolds and Tensors.
- Chapter 2. Geometry and Integration on Manifolds.
- Chapter 3. Symmetries of Manifolds.
- Chapter 4. Newtonian Mechanics.
- Chapter 5. Lagrangian Methods and Symmetry.
- Chapter 6. Relativistic Mechanics.
- Chapter 7. Lie Groups.
- Chapter 8. Lie Algebras.
- Chapter 9. Representations.
- Chapter 10. Rotations and Euclidean Symmetry.
- Chapter 11. Boosts and Galilei Symmetry.
- Chapter 12. Lorentz Symmetry.
- Chapter 13. Poincare Symmetry.
- Chapter 14. Conformal Symmetry.
- Chapter 15. Lagrangians and Noether's Theorem.
- Chapter 16. Spacetime Symmetries of Fields.
- Chapter 17. Gauge Symmetry.
- Chapter 18. Connection and Geodesics.
- Chapter 19. Riemannian Curvature.
- Chapter 20. Symmetries of Riemannian Manifolds.
- Chapter 21. Einstein's Gravitation.
- Chapter 22. Lagrangian Formulation.
- Chapter 23. Conservation Laws and Further Symmetries.

- - Appendices
- A) Notation and Conventions.
- Physical Units and Dimensions.
- Mathematical Conventions.
- Abbreviations.
- B) Mathematical Tools.
- Tensor Algebra.
- Matrix Exponential.
- Pauli and Dirac Matrices.
- Dirac Delta Distribution.
- Poisson and Wave Equation.
- Variational Calculus.
- Volume Element and Hyperspheres.
- Hypersurface Elements.
- C) Weyl Rescaling Formulae.
- D) Spaces and Symmetry Groups.

- - Bibliography.
- - Index.

The knowledge of how the light will behave in a waveguide with a radius of curvature becomes more and more important because of the development of integrated photonics, which include ring micro-resonators, phasars, and other devices with a radius of curvature. This work presents a numerical calculation method to determine the eigenvalues and eigenvectors of curved waveguides. This method is a hybrid method which uses at first conform transformation of the complex plane generating a new and modified index profile that represent the curvature. Then, on this modified index profile, we use a matrix multilayer formalism; we divide the profile in virtual layers of various indices to calculate both the propagation constants (eigenvalues) and the optical field (eigenvectors) into each of such virtual layers.

Starting from the weak field limit, we discuss astrophysical applications of Extended Theories of Gravity where higher order curvature invariants and scalar fields are considered by generalizing the Hilbert-Einstein action linear in the Ricci curvature scalar R. Results are compared to General Relativity in the hypothesis that Dark Matter contributions to the dynamics can be neglected thanks to modified gravity. In particular, we consider stellar hydrostatic equilibrium, galactic rotation curves, and gravitational lensing. Finally, we discuss the weak field limit in the Jordan and Einstein frames pointing out how effective quantities, as gravitational potentials, transform from one frame to the other and the interpretation of results can completely change accordingly.

Conformal transformations play a widespread role in gravity theories in regard to their cosmological and other implications. In the pure metric theory of gravity, conformal transformations change the frame to a new one wherein one obtains a conformal-invariant scalar-tensor theory such that the scalar field, deriving from the conformal factor, is a ghost. In this work, conformal transformations and ghosts will be analyzed in the framework of the metric-affine theory of gravity. Within this framework, metric and connection are independent variables, and hence, transform independently under conformal transformations. It will be shown that, if affine connection is invariant under conformal transformations then the scalar field under concern is a non-ghost, non-dynamical field. It is an auxiliary field at the classical level, and might develop a kinetic term at the quantum level. Alternatively, if connection transforms additively with a structure similar to yet more general than that of the Levi-Civita connection, the resulting action describes the gravitational dynamics correctly, and more importantly, the scalar field becomes a dynamical non-ghost field. The equations of motion, for generic geometrical and matter-sector variables, do not reduce connection to the Levi-Civita connection, and hence, independence of connection from metric is maintained. Therefore, metric-affine gravity provides an arena in which ghosts arising from conformal factor are avoided thanks to the independence of connection from the metric.

It is known that probing gravity in the submillimeter-micrometer range is difficult due to the relative weakness of the gravitational force. We intend to overcome this challenge by using extreme temporal precision to monitor transient events in a gravitational field. We propose a compressed ultrafast photography system called T-CUP to serve this purpose. We show that the T-CUP’s precision of 10 trillion frames per second can allow us to better resolve gravity at short distances. We also show the feasibility of the setup in measuring Yukawa and power-law corrections to gravity which have substantial theoretical motivation.

Starting from the weak field limit, we discuss astrophysical applications of Extended Theories of Gravity where higher order curvature invariants and scalar fields are considered by generalizing the Hilbert-Einstein action linear in the Ricci curvature scalar R. Results are compared to General Relativity in the hypothesis that Dark Matter contributions to the dynamics can be neglected thanks to modified gravity. In particular, we consider stellar hydrostatic equilibrium, galactic rotation curves, and gravitational lensing. Finally, we discuss the weak field limit in the Jordan and Einstein frames pointing out how effective quantities, as gravitational potentials, transform from one frame to the other and the interpretation of results can completely change accordingly.

The recent claim by Grebenev et al. [J. Phys. A: Math. Theor. 50, 435502 (2017)] that the inviscid 2D Lundgren-Monin-Novikov (LMN) equations on a zero vorticity characteristic naturally would reveal local conformal invariance when only analyzing these by means of a classical Lie-group symmetry approach, is invalid and will be refuted in the present comment. To note is that within this comment the (possible) existence of conformal invariance in 2D turbulence is not questioned, only the conclusion as is given in Grebenev et al. (2017) and their approach how this invariance was derived is what is being criticized and refuted herein. In fact, the algebraic derivation for conformal invariance of the 2D LMN vorticity equations in Grebenev et al. (2017) is flawed. A key constraint of the LMN equations has been wrongly transformed. Providing the correct transformation instead will lead to a breaking of the proclaimed conformal group. The corrected version of Grebenev et al. (2017) just leads to a globally constant scaling in the fields and not to a local one as claimed. In consequence, since in Grebenev et al. (2017) only the first equation within the infinite and unclosed LMN chain is considered, also different Lie-group infinitesimals for the one-and two-point probability density functions (PDFs) will result from this correction, replacing thus the misleading ones proposed.