A First Course in String Theory/Invariant Interval/Metric

In summary, the conversation discusses the use of symbols and the mathematics of special relativity and Minkowski spacetime in chapter 02 of the book "A First Course in String Theory" by Barton Zwiebach. The concept of the invariant interval is defined as -ds[2]=η[μν]dx[μ]dx[ν] and there is a question about the minus sign and its relationship with the positive-definite condition. The conversation also touches on the difference between the Minkowski metric and the Riemannian metric, and how they are used to calculate the length of a parametric curve in Minkowski space.
  • #1
Cosmology2015
31
1
Hello,
Before starting, I would like to apologize for any errors in the use of symbols. This is my first time :sorry:.
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 :smile:.
 
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  • #2
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 :smile:.
 
  • #3
In your case you have not the Riemannian metric, but the Minkovski metric. In writing any metric as a sum [itex]\sum_{ij}\,g_{ij}\ dx^i\otimes dx^j[/itex] the differentials [itex]dx^i[/itex] and [itex]dx^j[/itex] are formal symbols. The interval notation [itex]ds[/itex] or [itex]ds^2[/itex] is used if you want to calculate the length of a parametric curve [itex]x^i=x^i(\theta)[/itex], [itex]\theta\in [0,1][/itex], in the Minkovski space. In this case you write [itex]dx^i=(x^i)'_\theta\,d\theta[/itex] and then integrate [tex]s=\int^1_0 ds.[/tex]
 

1. What is String Theory?

String theory is a theoretical framework in physics that seeks to explain the fundamental nature of particles and their interactions. It proposes that particles are not point-like objects, but rather tiny, vibrating strings. It is a highly mathematical and complex theory that attempts to unify the four fundamental forces of nature (gravity, electromagnetism, strong and weak nuclear forces) into one single theory.

2. What is a First Course in String Theory?

A First Course in String Theory is an introductory course that covers the basic concepts and principles of string theory. It is usually taught at the undergraduate level and assumes a basic understanding of quantum mechanics and special relativity. This course provides a foundation for further study and research in string theory.

3. What is an Invariant Interval in String Theory?

Invariant interval, also known as proper time, is a concept in string theory that measures the distance between two points in space-time. It is a fundamental concept in special relativity and plays a crucial role in understanding the behavior of objects moving at high speeds. In string theory, the invariant interval is used to calculate the behavior of strings in curved space-time.

4. What is a Metric in String Theory?

A metric in string theory is a mathematical tool used to describe the geometric properties of space-time. It is a set of mathematical equations that define the distance between points in a multi-dimensional space. In string theory, the metric is used to describe the behavior of strings in various space-time configurations and to calculate the effects of gravity on strings.

5. What are the applications of String Theory?

String theory has many potential applications in physics, including attempts to unify the four fundamental forces, explain the behavior of black holes, and provide a framework for understanding the early universe. It also has implications in other areas such as mathematics, computer science, and philosophy. However, as it is still a developing theory, many of its applications are still being explored and debated.

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