# A First Course in String Theory/Invariant Interval/Metric

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1. Aug 8, 2015

### Cosmology2015

Hello,
Before starting, I would like to apologize for any errors in the use of symbols. This is my first time .
I am studying the wonderful book of Barton Zwiebach, "A First Course in StringTheory".
In chapter 02, I am experiencing for the first time with the mathematics of special relativity (Minkowski Spacetime).
My question is on the definition of invariant interval ds[2]. By definition, the invariant interval is given by -ds[2]=η[μν]dx[μ]dx[ν]
I am not able to understand the minus sign on ds[2]. Is there any relationship with the idea of positive-definite condition? Others books use only ds[2] for the invariant interval. Is there any advantage in using this convention?
Another question would be about the invariant interval -ds[2.]. The definition of the invariant interval is very similar to the definition of Riemannian metric (metric tensor) g[ij].
(a) invariant interval → -ds[2]=η[μν]dx[μ]dx[ν]
(b) Riemannian metric → g=∑g[ij]dx⊗dx[j]
Is there any direct relationship? What is the difference between them?
I sincerely thank any reply .

Last edited: Aug 8, 2015
2. Aug 9, 2015

### Cosmology2015

Hello,
I would like to apologize for the errors in the use of symbols. As I told before, it was my first time, and I am still learning how to use the resources of this forum.
I sincerely thanks any reply .

3. Aug 9, 2015

### Ruslan_Sharipov

In your case you have not the Riemannian metric, but the Minkovski metric. In writing any metric as a sum $\sum_{ij}\,g_{ij}\ dx^i\otimes dx^j$ the differentials $dx^i$ and $dx^j$ are formal symbols. The interval notation $ds$ or $ds^2$ is used if you want to calculate the length of a parametric curve $x^i=x^i(\theta)$, $\theta\in [0,1]$, in the Minkovski space. In this case you write $dx^i=(x^i)'_\theta\,d\theta$ and then integrate $$s=\int^1_0 ds.$$