Why does time have to be a complex (Minkowski metric)?

[feryopo]

I am studying special relativity, and I found that you have to work with a four dimensional space, where time is a complex variable. If you do so, you end up with the Minkowski metric, were the time component is negative and space components are positive (or vice versa). My questions are, why do we have to work with a complex time component, what is the physical meaning of time being complex and mathematically how can you deduct the Minkowski metric?
Thank you very much in advance.

 

[Simon Bridge]

Welcome to PF;
You will learn to discard this notation and use an object called the metric tensor instead.
Minkovski coordinates uses a complex time coordinate because of work done by Poincare, that the 4D euclidean metric had to subtract time-squared, where the usual pythagorean approach would suggest adding time squared. This makes lorentz transformaton look like ordinary rotations as opposed to something weird and mysterious.

There is no special physical significance to the time component being represented as a complex number - the convention is exploiting the properties of complex numbers to shortcut some maths. The physics is the Lorentz transformation.

 

[robphy]

The Bondi k-calculus
https://en.m.wikipedia.org/wiki/Bondi_k-calculus
is the best way to see how the signature arises.

 

[dextercioby]

##x_4 = ict ## was the invention of Hermann Minkowski in 1907 (published in 1908). I wonder why people still teach it nowadays.

 

[robphy]

##x_4 = ict ## was the invention of Hermann Minkowski in 1907 (published in 1908). I wonder why people still teach it nowadays.

The Law of Relativistic Inertia?