Einstein’s postulates

• All laws of nature are the same for all inertial observers
• The speed of light is the same for all inertial observers

• Lorentz transformations:

  $$ \begin{aligned} ct'&=\gamma(ct-\beta x) \\ x'&=\gamma(x-\beta ct) \\ y'&=y \\ z'&=z \end{aligned} $$

  where $\beta=v/c$ and $\gamma=(1-\beta^2)^{-1/2}$.

• Lorentz invariant properties:

  $$ c^2t^2-\vec x^2=c^2t^2-x^2-y^2-z^2 $$

Consequences:

• Moving clocks run slower by a factor of $\gamma$, time dilation

• Moving objects look shorter by a factor of $\gamma$ along the direction of motion known as length contraction

• basic quantities in mechanics are modified as high speeds, for example:

  $$ E=\gamma mc^2 \quad ; \quad \vec p=\gamma m\vec v \quad ; \quad K=(\gamma-1)mc^2 $$

• Electricity and magnetism are unified in electromagnetism

• The standard relative velocity law is modified to

  $$ V=\frac{v_1+v_2}{1+\frac{v_1v_2}{c^2}} $$

  which becomes $V=v_1+v_2$ when $v_1$ and $v_2\ll c$

Group theory

🧑‍🎤 Vocabulary:

$$ \forall \,= \text{for all} \quad ; \quad \in\,=\text{is a member of} \quad ; \quad \exist\,=\text{there exists} \\ \therefore\,=\text{therefore} $$

Consider a group $(G,\bullet)$ which is a set of elements $G=\{a,b,c,\cdots\}$ and a composition (or operation) $\bullet$ which have the following properties

• Closure: $\forall a,b\in G, c=a\bullet b\in G$
• Associativity: $\forall a,b,c\in G, a\bullet(b\bullet c)=(a\bullet b)\bullet c$
• Identity: $\exist e\in G \text{ such that }e\bullet a=a\bullet e=a\forall a\in G$
• Inverse: $\forall a\in G\exists a^{-1}\text{ such that } a^{-1}\bullet a=a\bullet a^{-1}=e$

A group is said to be Abelian if it is commutative, that is: $\forall a,b \in G,a\bullet b=b\bullet a$.

💰 Example: The real numbers $\R$ and the complex numbers $\Complex$. Define $\R\backslash\{0\}$ and $\Complex\backslash\{0\}$ to be the sets without the zero element. We see that $(\R,+)(\R\backslash\{0\},\times)$ and $(\Complex,+)(\Complex\backslash\{0\},\times)$ are Abelian groups

🗒️ Note: sets of matrices can be grouped, but they are not Abelian since matrix multiplication is not commutative.

• We define the General Linear group over the real numbers covering all $n$-dimensional linear transformations

  $$ \mathbb{GL}(n,\R)=\{\text{Real } n\times n\text{ matrice with determinant }\ne 0\} $$

  Subsets of this group:

  • $O(n)=\{M\in\mathbb{GL}(n,\R):MM^T=M^TM=I\}$ - the group of orthogonal $n\times n$ matrices
  • $SO(n)=\{M\in O(n):\det (M)=1\}$ - the special orthogonal group which are the rotation matrices

  💰 Example:

  $$ SO(2)=\left \{ R(\theta)=\begin{pmatrix} \cos\theta & - \sin \theta \\ \sin\theta & \cos\theta \end{pmatrix} :0\le \theta \le2\pi \right \} $$

  🗒️ Note: it is easy to show that $R(\theta)R(\phi)=R(\theta+\phi)$ hence matrix multiplication is associative. $R(0)=I$ and $R(\theta)R(-\theta)=I$ implying that there is an identity and an inverse for each component so we have checked the conditions for it to be a group

Orthogonal matrices

Orthogonal matrices maintain the inner product of two vectors and hence the length of vectors and the angle between them.

💰 Example: consider 2 $n$-dimensional vectors $\vec a$ and $\vec b$ which are rotated into $\vec a'$ and $\vec b'$ by a matrix $M\in O(n)$

$$ \vec a'=M\vec a \quad \vec b'=M\vec b $$

• Thus we can show that

  $$ \vec a'\cdot \vec b'=\vec a'^T\vec b'=\vec a^TM^TM\vec b=\vec a^T\vec b=\vec a\cdot\vec b \\ \vec a'\cdot \vec b'=a'ib'i=M{ij}a{j}M_{ik}b_{k}=\delta_{jk}a_jb_k=a_jb_j=\vec a\cdot\vec b $$

  since $M_ijM_{ik}=\delta_{jk}$ or $M^TM=I$

  🗒️ Note: both ways of writing are equivalent first is in matrix form second is using summation convention