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Holographic dualities are at the forefront of contemporary physics research: peering into the fundamental nature of our universe and providing best attempt answers to humankind's bold questions about basic physical phenomena. Yet, the concepts, ideas and mathematical rigors associated with these dualities have long been reserved for the specific field researchers and experts. In this book, we attempt to shatter this long held paradigm by bringing several aspects of holography research into the classroom, starting at the college-physics level and moving up from there. There are numerous examples of holographic dualities in physics, with the AdS/CFT correspondence of string theory being the most famous. Of particular interest to our work, and pedagogical implementation, are the holographic scaling properties of black holes and their gravitational fields. Exploiting these specific properties will allow us to introduce topics traditionally regarded as complicated, such as ADM mass, into the undergraduate classroom.
1 Introduction
1.1 The Four Forces & The Geometric Universe
1.2 QFT + GR
1.3 Cosmological Evolution
1.4 Black Holes and Holographic Scaling
2 ADM Mass and Holographic scaling in General Physics
2.1 Vector Notation and Coordinates
2.2 Vectorized Area
2.3 Gaussian Surfaces
2.4 Mass in College Physics
2.5 The Gravielectric Duality
2.6 Mass in University Physics
2.7 On Black Holes in College/University Physics
2.8 Schwarzschild Black Hole
2.9 Rotating/Kerr Black Hole
2.10 Effective Newtonian Black Hole Potentials and ADM Mass
3 Nöther’s Theorem: E&M and Gravity
3.1 Nöther’s Theorem: E&M
3.2 Nöther’s Theorem: Newtonian Gravity
3.3 Nöther’s Theorem and Gravity: The Metric Tensor
4 Tensor Calculus on Manifolds
4.1 Index Notation
4.2 Some Point Set Topology & Manifolds
4.3 Vectors, Tensors & Fields
4.4 The Lie Derivative
4.5 Exterior Algebra
4.6 Exterior Derivative
4.7 Orientation
4.8 Stoke’s Theorem
4.9 Riemannian Mainfolds
4.10 Covariant Differentiation
4.11 Geodesic Equation
4.12 Curvature
5 Lagrangian Field Theory
5.1 E&M
5.2 Energy Momentum Tensor
5.3 General Relativity
6 Black Hole Thermodynamics and Holographic Scaling
6.1 Black Holes
6.2 Killing Vectors and Horizons
6.3 Surface Element for Null Generators
6.4 The Laws of Black Hole Mechanics
6.5 Black Hole Thermodynaimcs
6.6 2D Holographic Representations of 4D Black Holes
6.7 On ADM Mass in General Relativity
6.8 4D ADM Mass and Holographic
6.9 Holographic Black Hole Thermodynamics
6.10 Concluding Remarks
A Gravitational Gauge Transformation as a Change in Coordinates
B Basic Group Theory
B.1 Homomorphisms & Isomorphisms
B.2 Automorphisms and Normal Subgroups
B.3 Normal Subgroups
B.4 Group Action