Minkowski Metric and Lorentz Metric

[wpan]

I am currently studying special relativity on my own and I am looking into space time and space time diagrams. While reading through various sources I came across what seemed to be two methods to describe space time.

X₀, X₁, X₂, X₃ (ct, x,y,z) -> Lorentz Metric

X₁, X₂, X₃, X₄ (x,y,z,ict) -> Minkowski Metric

The two "metrics" are confusing me. I understand that both systems were formed so that all 4 dimensions have the same units. However, I don't know which one to use and I don't really understand how each metric system was formed (the thought process behind their creation). The Minkowski metric is especially confusing for me due to the imaginary "ict" factor. I think the differences in the metric system also has to do with the signs of each term. For example i see both these notations floating around (-+++) and (+---). Any help would be greatly appreciated.

 

[Bill_K]

The two "metrics" are confusing me.

They're called signatures. The ict notation is an especially bad idea and hardly ever used any more, but the (+---) and (-+++) are widespread, and you have to keep alert for that. Both of these conventions are vigorously advocated by those who favor them.

 

[yuiop]

The (x,y,z,ict) metric is an attempt to have a signature of the form (++++) so that the time dimension appears to be just like the space dimensions. In other words, $s^2 = x^2+y^2+z^2+(ict)^2$. This notation did not really catch on. Mathematically $s^2 = x^2+y^2+z^2+(ict)^2$ is equivalent to $s^2 = x^2+y^2+z^2-(ct)^2$ because $i = \sqrt{-1}$ and $i^2 =-1$.

The more common notation is (ct, x,y,z). Here you have a choice of signature convention of (-+++) or (+---) where the time dimension is the odd one out and is a reminder that the time dimension is not exactly like the space dimensions. The signature convention used should be obvious in the context.

 

[bcrowell]

If a book uses a Minkowski-style ict, that's a sign that it's extremely old and out of date.

 

[robphy]

If a book uses a Minkowski-style ict, that's a sign that it's extremely old and out of date.

One peculiar exception is 't Hooft's relativity text
www.staff.science.uu.nl/~hooft101/lectures/genrel_2013.pdf

(The prologue tries to justify his use of that signature... in the beginning.)

 

[PeterDonis]

One peculiar exception is 't Hooft's relativity text
www.staff.science.uu.nl/~hooft101/lectures/genrel_2013.pdf

(The prologue tries to justify his use of that signature... in the beginning.)

By "in the beginning" you mean "only for the first 12 pages of the text, which discuss SR, but not for the rest of the text, which discusses GR", correct? The argument he gives is interesting to me because MTW makes the opposite argument: since "ict" can't be used in GR (which t'Hooft apparently agrees with), it shouldn't be used in SR either.

I suspect one key factor here is that t'Hooft is really a quantum field theorist, not a relativist. Quantum field theorists like "ict" for the reason t'Hooft gives: it means you don't have to worry about signs any more. But quantum field theorists, practically speaking, never need to deal with curved spacetime, so SR is all they need.