# 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

• Finish Ch. 2
• One-forms, gradients, and tensors
Schutz Ch. 3

## Momentum Conservation and the "Center-of-momentum" frame

• Total four-momentum $$\vec{p}_{\mathrm{tot}} = \sum_i \vec{p}_i$$
• Every inertial observer has the same $\vec{p}_{\mathrm{tot}}$
• The CM frame is the inertial frame where $\vec{p}_{\mathrm{tot}} = (E_{\mathrm{tot}},0,0,0)$

## Vectors and one-forms

• The metric tensor defines the dot product $$\vec{A} \cdot \vec{B} = A^{\alpha} B^{\beta} \eta_{\alpha \beta}$$ and $$\eta_{\alpha \beta} = \vec{e}_{\alpha} \cdot \vec{e}_{\beta}$$
• The metric is defined with lower indices (it's inverse has upper indices)
• A covariant vector, row vector, one-form, $x_{\alpha}$ (lower-index), $\tilde{x}$, or
$\left( \begin{array}{c} 0 \\ 1 \end{array} \right)$ tensor, transform as $A_{\bar{\alpha}} = \Lambda^{\beta}_{~~\bar{\alpha}} A_{\beta}$
• A "vector", a column vector, contravariant vector, $x^{\alpha}$ (upper-index), $\vec{x}$ or
$\left( \begin{array}{c} 1\\ 0 \end{array} \right)$ tensor, transform as $A^{\bar{\alpha}} = \Lambda^{\bar{\alpha}}_{~~\beta} A^{\beta}$
• Basis vectors have lower-indices and transform in the same way as one-forms, but are indicated with an arrow.

## One Forms

• We think of a dot product as something like $$A^{\alpha} B^{\beta} \eta_{\alpha \beta}$$ or $$A^{\alpha} B_{\alpha}$$ $$\vec{A} \cdot \vec{B}$$
• From Schutz, one-forms are also functions that turn vectors into real numbers $$\tilde{B}(\vec{A}) = A^{\alpha} B_{\alpha}$$
• Thus the basis vectors are functions which turn vectors into a number (they just pick out the component in one direction)

## One Forms and Gradients

• The idea of one-forms is useful, e.g. because the gradient of a function is a one-form
• Consider a function $\phi(\vec{x})$, (where here $x$ is a contravariant vector)
• The gradient $\tilde{d}\phi$ is a one-form $$\frac{\partial \phi}{\partial x^{\bar{\alpha}}} = \frac{\partial x^{\beta}}{\partial x^{\bar{\alpha}}} \frac{\partial \phi}{\partial x^{\beta} } \quad \mathrm{or} \quad \left( \tilde{d} \phi \right)_{\bar{\alpha}} = \frac{\partial x^{\beta}}{\partial x^{\bar{\alpha}}} \left( \tilde{d} \phi \right)_{\beta}$$

## Gradients and Lorentz Transformations

• The gradient $\tilde{d}\phi$ one-form transforms like a covariant vector $$\left( \tilde{d} \phi \right)_{\bar{\alpha}} = \frac{\partial x^{\beta}}{\partial x^{\bar{\alpha}}} \left( \tilde{d} \phi \right)_{\beta} \Rightarrow \left( \tilde{d} \phi \right)_{\bar{\alpha}} = \Lambda^{\beta}_{~~\bar{\alpha}} \left( \tilde{d} \phi \right)_{\beta}$$ similar to $$x^{\beta} = \Lambda^{\beta}_{~~\bar{\alpha}} x^{\bar{\alpha}} \Rightarrow \partial x^{\beta} / \partial x^{\bar{\alpha}} = \Lambda^{\beta}_{~~\bar{\alpha}}$$

## Derivative Notation and Matrix Warning

• We denote $$\phi_{,x} \equiv \frac{\partial \phi}{\partial x}$$ and $$\phi_{,\alpha} \equiv \frac{\partial \phi}{\partial x^{\alpha}}$$
• What is $x^{\alpha}_{~~,\beta}$ ?
• The quantity $\Lambda^{\alpha}_{~~\beta}$ is appropriately written as a matrix, but the metric tensor $\eta_{\alpha \beta}$ has two covariant indices
• The matrix representation does not make clear the properties under a Lorentz transformation

## Larger Tensors

• A $\left( \begin{array}{c} 0 \\ 2 \end{array} \right)$ tensor is a function of two vector arguments
• The outer product of two one-forms, $\tilde{p} \otimes \tilde{q}$, is a $\left( \begin{array}{c} 0 \\ 2 \end{array} \right)$ tensor (but not all such tensors are equal to outer products)
• Outer products do not commute
• One can decompose tensors into symmetric and antisymmetric parts $$h_{\alpha \beta} = \frac{1}{2} \left( h_{\alpha \beta} + h_{\beta \alpha} \right) + \frac{1}{2} \left( h_{\alpha \beta} - h_{\beta \alpha} \right) \equiv h_{(\alpha \beta)} + h_{[\alpha \beta]}$$
• The metric tensor is a symmetric $\left( \begin{array}{c} 0 \\ 2 \end{array} \right)$ tensor

## Inner Products of One Forms

• The inner product of a one form $$\tilde{p}^2 = \eta^{\alpha \mu} p_{\mu} p_{\alpha}$$
• Requires the inverse of the metric tensor (but this is just equal to the metric tensor)
• One can also form the quantity $$\eta^{\alpha}_{~~\beta} = \eta^{\alpha \mu} \eta_{\mu \beta} = \delta^{\alpha}_{~~\beta}$$ just the identity matrix

## Generic tensors

• A $\left( \begin{array}{c} M \\ N \end{array} \right)$ tensor is a function of $M$ vectors and $N$ one-forms into the real numbers
• $M$ upper and $N$ lower indices, "M-times contravariant" and "N-times covariant".
• Differentiation with respect to a vector gives a $\left( \begin{array}{c} M \\ N+1 \end{array} \right)$ tensor (we get an extra index from the denominator)

## Group Work

• Pick any problem from HW 1 to do