Special Relativity

Minkowski space

Definition (Minkowski Space)

Minkowski space is the real four-dimensional vector space $M={\mathbb{R}}^4$. The elements of Minkowski space are called spacetime points (or events). Each spacetime point

$$x=(x^{\mu }{)}_{\mu =0,1,2,3}=(x^0,x^1,x^2,x^3)$$

has one time coordinate $x^0$ and three spatial coordinates $x^1,x^2,x^3$. We also write

$$x=(x^0,\mathbf{x})\mathbf{x}=(x^i{)}_{i=1,2,3}=(x^1,x^2,x^3)$$

with the spatial coordinates combined in a three-vector $\mathbf{x}\in {\mathbb{R}}^3$.

By convention, greek letters are used for indices running through all spacetime coordinates, and latin letters are used for indices running just through the spatial coordinates.

Definition (Lorentz Bilinear Form)

The Lorentz bilinear form is the bilinear form on Minkowski space $M$ given by

$$M\times M\to \mathbb{R},(x,y)\mapsto x\cdot y=x^0{y}^0-x^1{y}^1-x^2{y}^2-x^3{y}^3$$

We have

$$x\cdot y=x^0{y}^0-\mathbf{x}.\mathbf{y},$$

where $\mathbf{x}.\mathbf{y}=x^1{y}^1+x^2{y}^2+x^3{y}^3$ denotes the the Euclidean scalar product of two three-vectors $\mathbf{x},\mathbf{y}\in {\mathbb{R}}^3$.

The Lorentz bilinear form is bilinear, symmetric and indefinite. It is not an inner product. Nevertheless, it is commonly called Lorentz inner product.

The Lorentz bilinear form classifies the four-vectors of Minkowski space as follows:

Definition (Timelike, Lightlike and Spacelike Vectors)

A vector $x$ in Minkowski space is called

• timelike if $x\cdot x>0$,
  • positive timelike if $x\cdot x>0$ and $x^0>0$,
  • negative timelike if $x\cdot x>0$ and $x^0<0$,
• lightlike or null if $x\cdot x=0$,
  • positive lightlike if $x\cdot x=0$ and $x^0>0$,
  • zero if $x=0$,
  • negative lightlike if $x\cdot x=0$ and $x^0<0$,
• spacelike if $x\cdot x<0$.

A vector $x$ in Minkowski space is

• timelike if and only if there exists an inertial coordinate system in which its spatial components vanish, i.e. if there is a Lorentz transformation $\Lambda$ such that $(\Lambda x{)}^i=0$ for $i=1,2,3$.
• spacelike if and only if there exists an inertial coordinate system in which its temporal component vanishes, i.e. if there is a Lorentz transformation $\Lambda$ such that $(\Lambda x{)}^0=0$.

Definition (Special Cones in Minkowski Space)

• $V_{+}=\left\{x\in M:x\cdot x>0,x^0>0\right\}$ open future cone (or open forward cone)
• ${\bar{V}}_{+}=\left\{x\in M:x\cdot x\ge 0,x^0\ge 0\right\}$ closed future cone (or closed forward cone)
• $V_{-}=\left\{x\in M:x\cdot x>0,x^0<0\right\}$ open past cone (or open backward cone)
• ${\bar{V}}_{-}=\left\{x\in M:x\cdot x\ge 0,x^0\le 0\right\}$ closed past cone (or closed backward cone)
• $\left\{x\in M:x\cdot x=0\right\}$ light cone
• $\left\{x\in M:x\cdot x<0\right\}$ side cone

Definition (Causal Complement)

The causal complement of a subset $S$ of Minkowski space $M$ is the set

$$S^{\prime }=\left\{x\in M:\text{x−y is spacelike for all y∈S}\right\}$$

The causal complement of $S$ consists of all spacetime points which are spacelike separated from all points in $S$.