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Vincent Rodgers © A Very Brief Intro to Tensor Calculus Two important concepts: Covariant Derivatives and Tensors Familiar objects but dressed up a little differently

Vincent Rodgers © What are tensors? Two important objects in elementary Calculus Derivative Operators Differentials Recall Calculus 101

Vincent Rodgers © START WITH COORDINATE TRANSFORMATION: Functions transforms as: Then the derivative operator transforms like: The differentials transform as: COVARIANT TRANSFORMATION CONTRAVARIANT SCALAR 1

Vincent Rodgers © We can build coordinate invariants by using the covariant and contravariant tensors. This is invariant under coordinate transformations. This is called a scalar or tensor of rank zero. contravariant covariant scalar

Vincent Rodgers © Transforms covariantly Transforms contravariantly Transformations in more than one dimension: Transforms covariantly Transforms contravariantly Inverses of each other Einstein Implied Sum Rule is Used. Also we always use these definition to define the fundamental raised and lowered indices.

Vincent Rodgers © Simple Example: Consider a rectangular to polar coordinate transformation where Notation

Vincent Rodgers © x y r Length of A

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Vincent Rodgers © So motion in two dimensions is independent of the coordinate chart. Between the different coordinate systems there is a “dictionary” the transformation laws, that tell one observer how a different observer perceives some event. Physics should be independent of the coordinate system. GENERAL COORDINATE INVARIANCE Build physical theories out of quantities that can be translated to another coordinate without depending on a particular coordinate system. This is the essence of Tensors.

Vincent Rodgers © SOME COMMON EXAMPLES OF TENSORS Covariant Contravariant Mixed Tensor Product

Vincent Rodgers © Example: Stress-Strain Tensor Stress Tensor and Strain Tensor Stress-Strain relationship represents how a body is distorted in the y direction (say) due to a force applied in the x direction (say). dx dy

Vincent Rodgers © r Length of A The Metric Tensor: used to measure distance and to map contravariant tensors into covariant tensors x y In the (x,y) coordinate system In the (r, ) coordinate system THE METRIC AS WELL AS ALL TENSORS HAVE MEANING INDEPENDENTLY OF A COORDINATE SYSTEM. THE COORDINATE SYSTEM IS ONLY REPRESENTING THE METRIC! ds

Vincent Rodgers © Not all metrics are the same. Here are two metrics that cannot be related by a smooth coordinate transformation A metric on a flat sheet of paper A metric on a basketball

Vincent Rodgers © Two other favorites but in four dimensions Minkowski Space metric using Cartesian coordinates A black hole metric using spherical coordinates

Vincent Rodgers © HOW DO WE TAKE DERIVATIVES OF TENSORS? BIG PROBLEM! Ordinary derivative of tensors are not tensors! Derivatives are supposed to measure the difference between the tops of the tensors but here the tails are not at the same place. We need to figure out how to get the tails to touch. x x + dx x

Vincent Rodgers © INTRODUCE COVARIANT DERIVATIVES tensornontensor Christoffel Symbol

Vincent Rodgers © Christoffel Symbol parallel transports the tails of tensors together! x + dx x Tails are together so now we can compute the difference.

Vincent Rodgers © GEOMETRP Distances are determined by the metric. Here it is clearly different on these two dimensional surfaces PARALLELL TRANSPORT IDEA ON A BALL AND FLAT SHEET OF PAPER

Vincent Rodgers © Metric gives us the geometry through the Covariant derivative. Little loops can be measured through the commutator of the Covariant derivative operator. Riemann Curvature Tensor Ricci Tensors Ricci Scalar Metric tensor

Vincent Rodgers © Explicitly “geometry” is defined by the Riemann Curvature Tensor. Explicitly we write:

Vincent Rodgers © GEOMETRY R=0 These manifolds have different Ricci scalars that is how we know they have Different geometry and not just look different because of a choice of coordinates. Recall gauge symmetry in Electrodynamics, different potentials give same E and B if they are related by a gauge transformation. You get different E and B if they are not. R=1/r 2

Vincent Rodgers © THE GEODESIC EQUATION AS A FORCE LAW Velocities (the time components are the same gamma factors seen in special relativity). Acceleration Force Proper time Shortest Distance is defined by the differential equation called the Geodesic Equation.

Vincent Rodgers © Maxwell Theory in terms of Tensors Start with the Vector Potential Scalar potential Vector Potential

Vincent Rodgers © Gauge transformations These Change These Remain the same

Vincent Rodgers © The Covariant Relationship to E and B ANTISYMMETRIC RANK 2 TENSOR

Vincent Rodgers © The Maxwell Field Tensor and its Dual Tensor

Vincent Rodgers © MAXWELL’S EQUATIONS IN A COVARIANT NUTSHELL (SI units)

Vincent Rodgers © THE COVARIANT DERIVATIVE Curvature of a gauge theory!

Vincent Rodgers © Electricity and Magnetism’s Energy-Momentum Tensor A Conservation Law