How to intuitively understand the formula S² = x² - c²t²

[PhysicsEnjoyer31415]

TL;DR Summary

So i watched a online video where this formula was shown with a graph of (x) vs (ct) and he explained how a apple 1 min in the future is farther away than a apple 1m away . I understood the concept but how to imagine/visualize any particle travelling on spacetime matrix ? Can anyone please cite a good source for spacetime physics visualisation . I am reading a book recommended by some users here called Taylor and wheeler spacetime physics and i think this might help the effort. Thank you!

I am on Chapter one and have not yet reached this formula yet but i wanted to know in advance because i was a bit curious
Also i noticed it written as (c²t²- x² ) in some places while (x² - c²t²) in some places ,is there a difference?

 

[PeterDonis]

I am reading a book recommended by some users here called Taylor and wheeler spacetime physics

That is a good source, yes. "An online video" is not. Stick with the book.

I am on Chapter one and have not yet reached this formula yet but i wanted to know in advance

Know what in advance? That the formula exists and is used in relativity? Yes, it is. For anything else you wold do better to read the book and then ask questions after you've read what it says about the formula.

Also i noticed it written as (c²t²- x² ) in some places while (x² - c²t²) in some places ,is there a difference?

Not physically, no. It's just two different sign conventions for squared intervals.

 

[PhysicsEnjoyer31415]

That is a good source, yes. "An online video" is not. Stick with the book.


Know what in advance? That the formula exists and is used in relativity? Yes, it is. For anything else you wold do better to read the book and then ask questions after you've read what it says about the formula.


Not physically, no. It's just two different sign conventions for squared intervals.

Ok i will complete the book first

 

[Ibix]

Taylor and Wheeler will explain it in context. The spoiler is that it's kind of a squared distance between two events separated by $x$ in space and $t$ in time. It's not the same as Pythagoras' Theorem (the minus sign is really important and has a lot of consequences) but it does much the same job in spacetime as Pythagoras' theorem does in space.

See how Taylor and Wheeler get to it and what they do with it.

 

[Dale]

TL;DR Summary: So i watched a online video where this formula was shown with a graph of (x) vs (ct) and he explained how a apple 1 min in the future is farther away than a apple 1m away . I understood the concept but how to imagine/visualize any particle travelling on spacetime matrix ? Can anyone please cite a good source for spacetime physics visualisation . I am reading a book recommended by some users here called Taylor and wheeler spacetime physics and i think this might help the effort. Thank you!

i noticed it written as (c²t²- x² ) in some places while (x² - c²t²) in some places ,is there a difference?

It is just a convention. I personally prefer the “space positive” convention. Also denoted as a (-+++) signature.

 

[PhysicsEnjoyer31415]

Thank you Dale and ibix

 

[PhysicsEnjoyer31415]

Yes i will read it in context of the book, that will clear any unnecessary doubts in my mind

 

[robphy]

The level hypersurfaces $x^2 +y^2 +z^2 -(ct)^2=Q$ (where $Q$ is real) are hyperboloids.
In a (1+1)-Minkowski spacetime, $x^2-(ct)^2=Q$ are hyperbolas.
($Q=0$ is the light-cone at the origin event, and
The future-branch of $Q=-1$ is a visualization of the metric tensor.)

As Minkowski suggests (see my post in orthogonality-in-minkowski-spacetime-meaning-visualization ), the hyperboloid plays the role of a "sphere" (a "circle" in 1+1) by providing a "metric" to assign numbers to displacements and to define "orthogonality" in that geometry, which encodes the speed of light principle and the relativity principle, as well as "effects" like time dilation, length contraction, and the relativity of simultaneity.

While thinking in terms of hyperbolas is important (in order to understand the "metric"),
I found that it is also helpful [in the (1+1)-case] to visualize and think in terms of
"causal diamonds" and "light-clock diamonds" (based on light-cone coordinates [the eigenbasis of the boost])
because the "area of the diamond (in units of light-clock diamonds)"
is equal to the "square-interval along its diagonal"
(see my PF Insights:
https://www.physicsforums.com/insights/spacetime-diagrams-light-clocks/
https://www.physicsforums.com/insights/relativity-rotated-graph-paper/
https://www.physicsforums.com/insights/relativity-on-rotated-graph-paper-a-graphical-motivation/
and
https://www.desmos.com/calculator/4jg0ipstya

described in https://www.physicsforums.com/threa...le-go-through-spacetime.1048187/#post-6832310 )