# 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

• Ch6 in Schutz

## Taking a Step Back

• Special relativity: combine space and time
• General coordinate invariance: physical laws should not depend on the chosen coordinate system
• Covariant differentiation, metrics and their appearance in volume elements
• All of these ingredients appear in QFT without any reference to gravity
• General relativity: space-time is intrinsically curved by matter and energy
• (not the same as "curvilinear" coordinates which can have zero intrinsic curvature)

## 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

• Generalization of straight lines in a curved space
• The vector $\vec{U}$ is tangent to a geodesic iff $\nabla_{\vec{U}} \vec{U} = 0$
• Writing this out $$U^{\beta}{U^{\alpha}}_{,\beta} + {\Gamma^{\alpha}}_{\mu \beta} U^{\mu} U^{\beta} = 0$$
• If $\lambda$ is the parameter which describes the curve, and using again $$U^{\alpha}= \frac{d x^{\alpha}}{d \lambda} \quad \mathrm{and} \quad \frac{d}{d \lambda} = U^{\beta} \frac{\partial}{\partial x^{\beta}}$$ then (the geodesic equation) $$\frac{d}{d \lambda} \left( \frac{d x^{\alpha}}{d \lambda} \right) + {\Gamma^{\alpha}}_{\mu \beta} \frac{d x^{\mu}}{d \lambda} \frac{d x^{\beta}}{d \lambda} = 0$$
• Geodesics in GR describe the paths of unaccelerated particles
• Geodesics are the shortest paths

## Euler-Lagrange formalism, etc.

• The geodesic equation can be obtained as the Euler-Lagrange equation for the Lagrangian $${\cal L} = g_{\mu \nu} \frac{d x^{\mu}}{d \lambda} \frac{d x^{\nu} }{ d \lambda}$$ where $\lambda$ is the affine parameter
• Alternatively, one can define a Hamiltonian $${\cal H} = \frac{1}{2} g^{\mu \nu} p_{\mu} p_{\nu}$$ where $p_{\mu} \equiv g_{\mu \nu} \dot{x}^{\nu}$ and minimize the energy

• Can a one-dimensional manifold have an intrinsic curvature?

## 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

• Parallel transport around a "square" loop $$\delta V^{\alpha} = \delta a \delta b \left[ {\Gamma^{\alpha}}_{\mu \sigma,\lambda} - {\Gamma^{\alpha}}_{\mu \lambda,\sigma} + {\Gamma^{\alpha}}_{\nu \lambda} {\Gamma^{\nu}}_{\mu \sigma} - {\Gamma^{\alpha}}_{\nu \sigma} {\Gamma^{\nu}}_{\mu \lambda} \right] V^{\mu}$$
• If this quantity is zero, then there is no intrinsic curvature
• Define the Riemann curvature tensor $${R^{\alpha}}_{\beta \mu \nu} = {\Gamma^{\alpha}}_{\beta \nu,\mu} - {\Gamma^{\alpha}}_{\beta \mu,\nu} + {\Gamma^{\alpha}}_{\sigma \mu} {\Gamma^{\sigma}}_{\beta \nu} - {\Gamma^{\alpha}}_{\sigma \nu} {\Gamma^{\sigma}}_{\beta \mu}$$

## Connection to Covariant Differentiation, Symmetries

• One can show that $$\left[ \nabla_{\alpha},\nabla_{\beta} \right] V^{\mu} = {R^{\mu}}_{\nu \alpha \beta} V^{\nu}$$
• There are similar relations for commutators of covariant derivatives of higher rank tensors
• Similar to single covariant derivatives
• Also define $$R_{\alpha \beta \mu \nu} \equiv g_{\alpha \lambda} {R^{\lambda}}_{\beta \mu \nu}$$ and an expression involving second derivatives of the metric
• This latter tensor is asymmetric on the first and second pairs, and symmetric on the exchange of the two pairs

## Geodesic Deviation and Bianchi Identities

• Given a vector $\vec{\xi}$ which connects two geodesics, how does it change?
• The first derivative is trivial because of coordinate invariance, but the second derivative is $$\nabla_{V} \nabla_{V} \xi^{\alpha} = {R^{\alpha}}_{\mu \nu \beta} V^{\mu} V^{\nu} \xi^{\beta}$$
• This produces tidal forces (which come from curvature)
• One can show that $$R_{\alpha \beta \mu \nu ; \lambda } + R_{\alpha \beta \lambda \mu ; \nu } + R_{\alpha \beta \nu \lambda ; \mu } = 0$$ the Bianchi identities.

## Ricci and Einstein Tensors

• Ricci tensor is $$R_{\alpha \beta} \equiv {R^{\mu}}_{\alpha \mu \beta}$$ which is symmetric
• The Ricci scalar is $R \equiv g^{\mu \nu} R_{\mu \nu}$
• The Einstein tensor (also symmetric) is $$G^{\alpha \beta} \equiv R^{\alpha \beta} - \frac{1}{2} g^{\alpha \beta} R$$
• Thus finally the Einstein field equations are $$G^{\alpha \beta} = 8 \pi T^{\alpha \beta}$$ and the Bianchi identities then imply energy momentum conservation, ${G^{\alpha \beta}}_{; \beta} = {T^{\alpha \beta}}_{; \beta}= 0$

## Group Work

• Complete 6.4 and 6.6 in Schutz