Physics 616

  • Prof. Andrew W. Steiner
    (or Andrew or "Dr. Steiner")
  • Office hour: 103 South College, Thursday 11am
  • Email:
  • Homework: Electronically as .pdf
  • You may work with each other on the homework, but you must write the solution in your own words
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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 $$ \frac{d}{d \lambda} \vec{V} = \nabla_{\vec{U}} \vec{V} = 0 $$


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