# 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

• End of ch5 and start of ch6 in Schutz
• 1.5, 2.20, 5.7, and 5.8

## Tangent Vectors

• Consider a curve, $\xi = f(s), \eta = g(s), a \leq s \leq b$
• The derivative of a scalar field along this curve is $$\frac{d \phi}{d s} = \left< \tilde{d} \phi , \vec{V} \right>$$ where $\vec{V}$ is the vector $( d \xi/d s, d \eta/ d s )$
• We call $\vec{V}$ the tangent vector
• Schutz distinguishes "paths" and "curves", where in the latter a parameterization has been specified

## Christoffel symbols and the metric

• We left off at covariant differentiation...
• Christoffel symbols are symmetric in their two lower indices, ${\Gamma^{\mu}}_{\alpha \beta} = {\Gamma^{\mu}}_{\beta \alpha}$
• The covariant derivative of the metric is zero in all coordinate systems $$g_{\alpha \mu ; \beta} = 0$$
• The Christoffel symbols can be computed from the metric $$\frac{1}{2} g^{\alpha \gamma} \left( g_{\alpha \beta, \mu} + g_{\alpha \mu,\beta} - g_{\beta \mu,\alpha} \right) = {\Gamma^{\gamma}}_{\beta \mu}$$

## Noncoordinate bases

• Last class we generated a coordinate system and used that coordinate transformation to generate a metric
• Then, we can use that metric to obtain the Christoffel symbols
• However, some metrics do not correspond to any coordinate transformation: a noncoordinate basis
• In a noncoordinate basis $$\phi_{,\mu} \neq \frac{\partial \phi}{\partial x^{\mu}}$$
• Schutz shows that the "polar unit" basis (distinct from normal polar coordinates) is a noncoordinate basis
• This is the origin of the distinction between eq. 5.56: $${V^{\alpha}}_{; \alpha} = \frac{1}{r} \frac{\partial}{\partial r} (r V^{r}) + \frac{\partial}{\partial \theta} V^{\theta}$$ and eq. 5.87: $$\nabla \cdot \mathbf{V} = \frac{1}{r} \frac{\partial}{\partial r} (r V^{\hat{r}}) + \frac{1}{r} \frac{\partial}{\partial \theta} V^{\hat{\theta}}$$

## Tensor algebra

• Tensor fields
• Vectors and one-forms are linear operators on each other
• Tensors are linear operators on multiple vectors and one-forms
• Tensors can be equal (if they are equal everywhere)
• There are some operations which create new tensors (in the general sense): scalar multiplication, addition, outer products, covariant differentiation, and contraction (including inner products).
• The aforementioned operations lead to results which are independent of basis (because of linearity)
• "Normal" coordinate differentiation does not

## Manifolds

• Some subspace of a larger Euclidean space
• Sometimes (continuous and) differentiable
• E.g. one-dimensional curve on a two-dimensional plane
• Parameterized by coordinates unrelated to the coordinates of the original space
• Thus, can have their own length and (positive-definite) metric "Riemannian manifold"
• Metric determines the curvature of the manifold
• Such manifolds are locally flat
• This is important because it means we can "flatten out" small elements of space-time

## Integration on manifolds

• Length $$\ell = \int_{\lambda_0}^{\lambda_1} \left| \vec{V} \cdot \vec{V} \right|^{1/2} = \int_{\lambda_0}^{\lambda_1} \left| g_{\alpha \beta} \frac{d x^{\alpha}}{d \lambda} \frac{d x^{\beta}}{d \lambda} \right|^{1/2}~d \lambda$$
• "Proper" volume element $$\left\{ dx^{\mu} \right\} = \left[ - \mathrm{det}(g_{\bar{\alpha}\bar{\beta}})\right]^{1/2} \left\{ dx^{\bar{\mu}} \right\}$$
• And, a generalized divergence theorem $$\int {V^{\alpha}}_{;\alpha} \sqrt{-g} d^4 x = \oint V^{\alpha} n_{\alpha} \sqrt{-g} d^3 S$$ where the "proper surface element" appears

## Group Work

• Complete 5.5, 5.7, and 5.8 in Schutz