Lorentz invariance

Last updated Dec 18, 2022

• Rework

At large energies or velocities a symmetry manifests: Lorentz symmetry Lorentz symmetry is the invariance under transformations of the Lorentz group. The Lorentz group consists of rotations and boosts.

Simple rotations of vectors through a rotation matrix $$x_i \\ to R_{ij}x_j$$

1. Since rotation matrices are othorgonal: $R^TR=1$
2. Since they preserve lengths: $detR=1$ These properties define rotations. Rotations in $N$ dimensions are described by the $SO(N)$ group. An alternative definition of rotations on $\Reals^n$ is through the invariance of the norm $x^ix_i$.

Lorentz transformations preserve $s^2 = x^\mux_\mu = t^2 - x^2 - y^2 - z^2$. While rotations leave e.g. $x^2+y^2$ invariant (so they need $\cos^2 + \sin^2 =1$), boosts preserve e.g. $t^2 - z^2$ (so they need $\cosh^2 - \sinh^2 = 1$) Since $x^\mu x_\mu = g_{\mu\nu}x^\mu x^\nu$, they are defined as all transformations that leave the minkowski metric g invariant $$\Lambda^T g \Lambda = g$$

The matrix representations of rotations in $x,y,z$ are $$ \begin{pmatrix} 1 & & & \\ & 1 & & \\ & & \cos\Theta_x & \sin\Theta_x \\ & & -\sin\Theta_x & \cos\Theta_x \\ \end{pmatrix} \begin{pmatrix} 1 & & & \\ & \cos\Theta_y & & \sin\Theta_y\\ & & 1 & \\ & -\sin\Theta_y & & \cos\Theta_y \\ \end{pmatrix} \begin{pmatrix} 1 & & & \\ & \cos\Theta_z & \sin\Theta_z & \\ & -\sin\Theta_z & \cos\Theta_z & \\ & & & 1 \\ \end{pmatrix} $$ with the usual angels $0 \leq \Theta_i \leq 2\pi$

The matrix representations of boosts in $x,y,z$ are $$ \begin{pmatrix} \cosh\beta_x & \sinh\beta_x & & \\ \sinh\beta_x & \cosh\beta_x & & \\ & & 1 & \\ & & & 1\\ \end{pmatrix} \begin{pmatrix} \cos\beta_y & & \sinh\beta_y & \\ & 1 & & \\ \sin\Theta_y & & \cosh\beta_y & \\ & & & 1\\ \end{pmatrix} \begin{pmatrix} \cos\beta_y & & & \sinh\beta_y \\ & 1 & & \\ & & 1 & \\ \sin\Theta_y & & & \cosh\beta_y\\ \end{pmatrix} $$ with rapididties $-\infty \leq \beta_i \leq \infty$. We have $$\cosh\beta_i = \frac{1}{\sqrt{1-v^2}}, \sinh\beta_i = \frac{v}{\sqrt{1-v^2}}$$

A Scalar field $\phi(x)$ is a function of space time that is invariant under Lorentz transformation $$\phi(x)\to\phi(x)$$

Four vectors $V_\mu$ transform under lorentz transformation like $V^\mu \to \Lambda^\mu_\nuV^\nu$. If $V^\mu = V^\mu(x)$ depends on $x$ it is a vector field.

Tensors $T^{\mu\nu}$ transform as $\T^{\mu\nu}\to\Lambda^\mu_\alpha\Lambda^\nu_\beta T^{\alpha\beta}$ The number of indices is the rank of the tensor. Contractions are always lorentz invariant since the metric is invariant and cancels the tranformations.

• Objects that do not depend on the Lorentz frame are called lorentz invariant, e.g. $$x^\mux_\mu, \phi, 1, \partial_mu V^\mu$$
• Objects with open lorentz indices on the otherhand are called lorentz covariant, as they change in different frames, precisely as defined by the lorentz transformation: $$V_\mu, F_{\mu\nu}, \partial_\mu, x_\mu$$ There are also objects that do change under Lorentz transformation but are not covariant in the sense that they are a component of a tensor

There are discrete transformations that leave the product $$s^2 = x^\mux_\mu = t^2 - x^2 - y^2 - z^2$$ invariant.

Parity: The parity transformation is the transformation $P: (t,x,y,z) \to (t,-x,-y,-z)$

Time reversal: The time reversal transformation is the transformation $T: (t,x,y,z) \to (-t,x,y,z)$

These transformation leave $s^2$ and therefore the metric invariant, but cannot be obtained from a product of rotations and boosts.

Timelike: A vector $x^\mu$ is called timelike if $x^\mu x_\mu >0$

Spacelike: A vector $x^\mu$ is called spacelike if $x^\mu x_\mu < 0$

Lightlike: A vector $x^\mu$ is called Lightlike if $x^\mu x_\mu = 0$ The name stems from $p^2 = m^2$ for massless photons

This property is preserved under Lorentz transformations, since they preserve the norm.