Do we move through 4D spacetime at the speed of light or do we all move at
the speed of light times the square root of minus one?

This article says we move through the temporal component of 4D spacetime at
the speed of light times the square root of minus one (aka c times i).
<https://math.ucr.edu/home/baez/physics/Relativity/GR/gravity.html>

However, all these say we move through 4D spacetime at the speed of light.
<https://medium.com/predict/we-all-travel-through-spacetime-at-the-speed-of-light-d60cb389dfc2>
<http://backreaction.blogspot.com/2020/08/do-we-really-travel-through-time-with.html>
<https://youtu.be/k73psdcmzEY>
<https://youtu.be/au0QJYISe4c>
<https://youtu.be/xYEcs9fOZy4>
<https://youtu.be/msVuCEs8Ydo>
<https://www.quora.com/What-speed-do-we-travel-through-spacetime>
<https://www.quora.com/Does-everything-travel-at-the-speed-of-light-in-spacetime>
etc

Notice there is no mention of the square root of minus one in those above.

Do we move through the temporal component of 4d spacetime at the speed of
light or do we move at the speed of light times the square root of -1?
--
Doveryay, no proveryay.

On Saturday, 14 August 2021 at 19:40:36 UTC+2, Ilya Kraskov wrote:

> This article says we move through the temporal component of 4D spacetime at
> the speed of light times the square root of minus one (aka c times i).
> <https://math.ucr.edu/home/baez/physics/Relativity/GR/gravity.html>

The metric signature in 1+1 Euclidean spacetime is (1,1) and the (squared) line element is d^2 = t^2 + x^2.

The metric signature in 1+1 Minkowski spacetime is (-1,1) and the (squared) line element is d^2 = -t^2 + x^2.

But we can also recover the Minkowski metric from the Euclidean signature if we take time to be imaginary, hence some i*t for t real: then the line element is d^2 = (i*t)^2 + x^2, which does equate d^2 = -t^2 + x^2.

Anyway, I understand these to just be mathematical alternatives to express the same geometric content. If you are interested in how complex numbers (and quaternions, etc.) play or do not play a significant role in physics, I'd suggest you have a look at Geometric Algebra: indeed, there are some amazing results there...

HTH,

Julio