Jorge Pinochet published General Relativity in a nutshell in two chapters. Here we are going to review the first one. This chapter was published in 2023, still it is highly recommended for beginners who are just starting to explore the writing and reading journey of physics research papers.
In this paper Jorge discussed it in the following order.
- Abstract
- Introduction
- Reimann Geometry and the concept of metric
- Curvature and geodesics: An intuitive Look
- The matric of a non-Euclidean manifold: A simple example
- Riemannian Geometry and the Minkowski Metric of flat space-time
We will discuss every single one of these headings.
Let’s start with the Abstract;
In abstract, Jorge claims that;
“Einstein’s general relativity is the best available theory of gravity”.
There was a time when Newton’s & Kepler’s gravitational theory was popular for centuries, but now Einstein’s general relativity is being considered as the best available theory of gravity. Einstein's General relativity has been proven multiple times and awakened the interest of going further in the investigation of unfolded mysteries of our universe.
Jorge also confirmed that in this paper he is only introducing the geometrical concepts that constitute the basis of Einstein’s theory.
Introduction
While introducing the General Relativity (GR) Jorge explains that GR is experiencing its second golden era. Kip Throne a theoretical physicist called the era of 1960s a golden age of GR and now after 50 years, three collaborations, “first detection of gravitational wave by LIGO in 2015, the first image of black hole obtained by the EHT in 2019 and the spectacular image of black hole obtained in 2022” are suggesting that this is the second golden age of General Relativity.
So, it’s safe to say that it’s high time to introduce young scientists to the GR. Breaking down the geometrical concepts of GR in this paper Jorge Pinochet paving the way for them.
Riemann Geometry and the concept of metric
Under the Riemann Geometry heading, Jorge starts with the explanation of Euclidian and non-Euclidian geometry. It briefly explains how an abstract investigation in Euclid's fifth postulate, also known as the parallel postulate,
"if a straight line intersects two other lines so that the interior angles on the same side sum to less than 180°, the two lines will eventually intersect on that side if extended infinitely."
ultimately led the way to the introduction of non-Euclidian geometry. Later this non-Euclidian geometry is generalized in the Riemann Geometry.
It also discusses how this Riemann geometry became the mathematical language of Einstein’s spacetime curvature.
The Riemann geometry, that is basically the geometry of curvature, is the mathematical foundation of GR. One of the fundamental principles of this geometry is that "in a small neighborhood of a point, non-Euclidian manifolds agree with Euclidian geometry".
Jorge establishes this claim in cartesian and polar coordinates of both 2D and 3D. In the process of establishment, he used the amazing concepts of manifold, metric and metric tensor. The explanation and derivation were so concise and simplified that as a first-time reader, I found no difficulty in understanding the arguments. A second look makes it feel like common sense to the reader.
Curvature and Geodesics: An intuitive look
Let’s define the geodesic first,
“A geodesic is the curve in a manifold that locally (and globally, when possible) gives the shortest distance between two points. It is the generalization of a “straight line” to curved spaces.
He starts this section with, “The geometric properties of a manifold are invariant under coordinate transformations. Curvature and Geodesics are two central properties of a manifold that are invariant”. Then he successfully provides a detailed derivation of this claim. The derivations show how the behavior of geodesics decides the nature of the curvature. He also points out that we require at least two geodesics in order to determine the nature of the curvature. In the end, the entire concept of determining the nature of a curvature was encapsulated in the understanding of Geodesic deviation.
This simple figure shows the geodesic deviation. If the geodesics are not deviating, we will call it zero curvature (Flat Geometry). If the deviation is decreasing, it will be called the positive curvature (Spherical Geometry) and if the deviation is increasing, it will be called negative curvature (Hyperbolic Geometry).
This simple understanding makes it much easier to identify the Euclidian and non-Euclidian manifolds.
The metric and non-Euclidean manifold
In this section, he specified three results of Euclidean geometry to disagree with in order to establish whether a manifold is non-Euclidean.
The specified results are;
- Sum of interior angles of a triangle
- Perimeter of a circle
- Surface area of a sphere
If a manifold is deviating from these results, then it is non-Euclidean for sure.
These results are the simplified form of “Geodesic Deviation” we discussed in the previous section. Fundamentally in all these results we are measuring the Geodesic deviations in different coordinate systems.
Riemannian geometry and the Minkowski Metric of flat space-time
In this section, Jorge starts his discussion with the establishment of spacetime concepts. He explains how time behaves like a fourth coordinate in a spacetime manifold. Then he steps into the next level of the concept building and gently warns that Riemannian and spacetime manifolds are not the exact same.
He also explains how we need Minkowski Metric to deal with the complexity of spacetime manifold a four-dimensional coordinate system.
The Minkowski metric is a symmetric rank-2 tensor that defines the spacetime interval between two infinitesimally separated events.
In further discussion, he derives and explains how this Minkowski Metric and tensor work for spacetime. He ends his discussion with the preferable signature (+, -, -, -) he is going to work with in his next chapter. From now on, will find this signature of Minkowski tensor every time he will be discussing about Minkowski Metric and spacetime coordinates.
Final Comments
In this first chapter of “General Relativity in a Nutshell”, I personally enjoyed going through the stepwise discussion and mathematical derivations. Whoever is trying to start his journey in theoretical physics, this paper can be his first step. Go through the concepts, derive all the mathematical establishments on your own, I am pretty sure you will enjoy it.
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