Physics 616

  • Prof. Andrew W. Steiner
    (or Andrew or "Dr. Steiner")
  • Office hour: 103 South College, Thursday 11am
  • Email: awsteiner@utk.edu
  • Homework: Electronically as .pdf
  • You may work with each other on the homework, but you must write the solution in your own words
 
 
 
Use down and up arrows to proceed to the next or previous slide.

Outline

Taking a Step Back

Parallel Transport

  • The vector \( \vec{U} = d \vec{x} / d \lambda \) is tangent to the curve
  • If the vector \( \vec{V} \) is parallel transported at some point, then $$ \frac{d V^{\alpha}}{d \lambda} = U^{\beta} {V^{\alpha}}_{,\beta} = U^{\beta} {V^{\alpha}}_{;\beta} = 0 $$
  • The last form is frame invariant, thus we define it as the parallel transport $$ \frac{d}{d \lambda} \vec{V} = \nabla_{\vec{U}} \vec{V} = 0 $$

Geodesic

Euler-Lagrange formalism, etc.

 

Affine transformations

  • Affine transformations is a transformation which preserves points, lines, and planes
  • Affine transformations on affine spaces (generalization of a Euclidean space) are just linear transformations
  • If the geodesic equation is satisfied, then \( \lambda \) is an affine parameter
  • The proper distance is an affine parameter
  • If \( \lambda \) is an affine parameter, then so is \( \phi = a \lambda + b \)

Curvature Tensor

Connection to Covariant Differentiation, Symmetries

Geodesic Deviation and Bianchi Identities

Ricci and Einstein Tensors

Group Work