Minkowski spacetime ↩

aliases: [Minkowski space]
shorthands: {"M": "\mathbb{M}", "gmat": "\begin{pmatrix}1 &  &  &  \\  & 1 &  &  \\  &  & 1 &  \\  &  &  & -1\end{pmatrix}"}
created: 2022-02-22 13:43:26
modified: 2022-02-22 14:05:30

Convention

In this document, the following convention is used:

Where is the speed of light, which is in natural units. So here, is the time dimension.

Definition

is a Minkowski space if is a set isomorphic to () and is a bilinear form (analogous to the dot product of Euclidean vector space).

1. is non-degenerate: If
2. is bilinear: and the same for the other argument as well
3. is symmetric:
4. is of index :

Where , are the basis vectors of . The shown tensor is also called the metric tensor.

Bases in

The basis vectors are with

A vector can be expressed like this (with the Einstein notation):

Motivation for change of bases in Minkowski space time

expressed with the metric tensor

Let's express with the basis vectors and use the metric tensor:

Where is the metric tensor:

Intuition

• Points in describe events: space and time coordinates 4D vector space
• A reference frame has corresponding bases in Minkowski space
• Switching between frames of references linear transformations in
• Invariance: certain quantities do not change when switching between reference frames