Equations of Spacetime Invariant - Understanding the Difference

[Thevan]

TL;DR Summary

Spacetime Invariant - 2 Equations with signs changed.

The distance/difference between two points in spacetime can be written in two forms (as shown in attachment). Can anyone explain the difference in the two equations? I have read that the two equations are the same, but i don't understand the change in sign. Why is it written in two forms?

(Considering motion only along x-axis.)

 

[anuttarasammyak]

Obviously
$$d\tau^2=-ds^2$$
They are just two different sign convention of spacetime invariant. You should choose one of them, i.e. (time interval)^2 is accounted positive or (space interval)^2 is accounted positive.

 

[Ibix]

Some people define it one way, some the other way. Either is fine. It makes no difference to the physics being described, just means that you have minus signs in some places if you choose one convention and in other places if you choose the other.

 

[lomidrevo]

As said in previous posts, it is a matter of convention. Maybe it is just worth to highlight that square root of the first definition of the spacetime interval in you picture is infinitesimal interval of proper time $d\tau = \sqrt{dt^2 - dx^2}$, which is defined only when $d\tau^2 > 0$, ie. for time-like wordlines.

 

[robphy]

Possibly useful:
https://en.m.wikipedia.org/wiki/Sign_convention#Metric_signature

(More advanced (beyond I-level) discussion of possible interest:
https://www.math.columbia.edu/~woit/wordpress/?p=7773&cpage=1
https://physics.stackexchange.com/q...erent-metric-signatures-in-relativity-and-qft )