When applicable, we present the following definitions in more precise mathematical language, as we will refer to them throughout the book. A more English-friendly description follows each definition block. A n-dimensional manifold is a topological space M for which every point x∈M has a neighborhood homeomorphic to Euclidean space Rn In a non-mathematical summary, a manifold is a surface that may or may not be curved. For instance, the surface of a sphere is a manifold, the surface of a doughnut is a manifold, and the surface of a sheet of paper is a manifold. There are exceptions about how the surface can warp, but these are good examples to relay the concept of a manifold. In addition, for these surfaces to be a Riemannian manifold, they must adhere to the requirement that at any location on the surface, a local area can use Euclidean (often termed flat space) geometry. Even if only at an infinitesimal point. Thus, the sphere-like shape of the Earth fits within the definition of a manifold. It is a curved surface, and at every local location (say the place on Earth where you are reading this), it appears flat, at least flat enough on your desk that it can sit level, and the surface does not appear curved. This is why an engineer can square a building. All the angles add up to 360 degrees. But, if the building was large enough, say the size of a county or state in the United States, it could not be squared using Euclidean geometry, as it followed the curvature of the Earth. Thus, local use of Euclidean geometry and measurements depends on the curvature of the surface at the local point where the measurement is being made.
Let M be a topological space and U⊆M an open set. Let V⊆R be open. A homeomorphism ϕ:U→V,ϕ(u)=(x1(u),…,xn(u)) is called a coordinate system on U. Coordinate systems are used in everyday life. Some formal, like the GPS use of longitude and latitude. Another less formal way is when giving directions, one may say go straight two blocks, turn left, then go three blocks, and then on your right-hand side will be your destination. Each is a type of coordinate placed on a manifold (a surface) to give a measurement. Algebra’s most commonly used coordinate system in teaching is the XY coordinate system. The ordered pair of (x,y) gives a location on the graph. The x values directs one to follow the x direction, starting at zero and following the axis until you reach x grid marks, the same for the y direction. A manifold can have an infinite number of coordinate systems applied to it. Sometimes, there is an advantage to using one coordinate system over another. If the surface is a sphere, it is easier (mathematically) to use spherical coordinates rather than Euclidean XY coordinates. It should be noted that using different coordinate systems never changes the physical world. The distance between two points on the surface remains invariant between the coordinate systems. Consider using two different maps showing the route between two cities. One is in English units, the other in metric units. Regardless of which “coordinate” system you use (English or Metric), the physical distance between the cities is invariant (unchanged). It is only the “language” used to map the distance that changes, not the real world. This is the same case with coordinate systems used on a manifold. The math may change to describe the physical world, but the physical world does not change. Thus, if one develops a coordinate system that represents the physical world changing because they chose a particular coordinate system, then by common sense, it is the coordinate system that is in error, not the physical world around us! One cannot simply change the coordinate system and shorten the distance between two cities. This sounds obvious (and it is), but often, this is confusing when one is in the depths of working with mathematical objects that remain invariant between coordinate systems. The coordinate system pair (U,ϕ) is called a chart on a n-dimensional manifold M. The chart allows a coordinate system to map an area or region of the manifold. Often, a single coordinate system cannot map an entire manifold because of the manifold’s curvature. However, a coordinate system can map a local area of the manifold. This area that a coordinate system can properly map is called a chart. Sometimes the choice of a coordinate system can cause artificial effects that are artifacts of the coordinate system and not the manifold. Consider the Earth; if one desires to use North, South, East, and West to define locations, what happens when they are at the North Pole and told to go west (or east)? What direction do they go? Move the north pole to any location on the Earth, and one will have the same issue. Thus, it is not the manifold (the surface of the Earth) that has this issue. Instead, it is the choice of the coordinate system used to define measurements over the surface. This follows up the point that coordinate systems do not change the physical world. By moving the origin of coordinates, one does not make east or west physically disappear. Instead, it is an artifact of the coordinate system chosen to represent the mapping of the manifold. An atlas on M is a collection of charts Uα,ϕα such that Uα covers M A collection of locally mapped charts can be “sewn” together (mathematically, this is performed by Christoffel symbols, a way to perform calculus over a curved surface) to form an atlas. The atlas then covers the entire manifold, thus successfully mapping the entire surface. A manifold M is a smooth manifold if all transition maps are C∞ diffeomorphisms. Transition maps are the “thread” that “sews” the charts together. In loose terms, if these transitions are infinitely differential across the whole of the manifold, then the manifold is said to be smooth. For instance, if there is a transition map that requires a divide by zero scenario, then at that point on the manifold, there is a singularity, and it is not infinitely differential. Thus, it is not smooth. The manifold may be smooth between preset boundaries (say, everywhere but the singularity), but this does not qualify the manifold as a smooth manifold unless all the transition maps are infinitely differential. Again, it should be noted that sometimes an artificial singularity can exist because of one’s choice of coordinate system (or metric), a singularity that does not exist outside of a particular coordinate system. This is the confusing part discussed previously when distinguishing between an artifact caused by the choice of coordinates and the physical world. In math, the physical representation is not always present. Thus, it is challenging to understand an artifact of the coordinate system versus an attribute of the manifold in a purely mathematical scenario. The Metric tensor is a function in which one may compute the distance between two points in a given space (or manifold). The metric tensor acts as a mechanism to measure the surface of the manifold (like a distance formula). This provides a way to perform other functions, such as algebra or calculus. Metric tensors are often used in calculating inner products, a method in which spaces (like a vector space) are used to define a distance. The metric is not constrained to using the common concept of measurement (like a yardstick). Often, the manifold is over more than three dimensions, and the metric must incorporate other dimensions (like time). Thus, the metric may produce results that are mathematically correct but seem foreign to or have no direct physical analogy. A Riemannian Metric requires for every x⊆M an inner product is defined on a tangent space. The collection of all inner products is called the Riemannian metric. Remember, a local space on the surface of the manifold must be able to use Euclidean geometry. A tangent space is a mathematical way of describing the local space on the manifold. Thus, if for every tangent space (i.e., local space) there is an inner product defined (a method to measure a distance), then the collection of all local spaces’ inner products is called the Riemannian metric. The inner product of a real vector space must satisfy the following properties. Let u,v,w be vectors and α be a scalar.
- <u+v,w>=<u,w>+<v,w>
- <αv,w>=α<v,w>
- <v,w>=<w,v>
- <v,v>≥0 and equal if and only if v=0
This rigorous definition is given as a future reference for the inner product. To be an inner product, these rules must be enforced. If the inner product cannot meet these requirements, it cannot be an inner product. Thus, if there is no formal inner product, then there can be no formal satisfying of the Riemannian metric requirement. The Riemannian metric depends upon the inner product. A Riemannian metric in which the requirement of positive definiteness is removed is called a Pseudo Riemannian metric. The Riemannian positive definitive requirement arises from the inner product requirement <v,v>≥0 and is equal if and only if v=0 This definition is given to a metric over the manifold that does not satisfy the Riemannian metric because of the positive definitive requirement of the inner product. In this special case, the metric is termed a pseudo-Riemannian metric (or false, non-genuine, spurious version of a Riemannian metric). This metric plays a pivotal role in Einstein’s theory of General Relativity. It will be examined in detail as to whether it was appropriate for Einstein to use this form of metric and retain the illusion of having a fully qualified Riemannian manifold and all the benefits it provides. A geodesic on a Riemannian manifold is a distance-minimizing path between two points on the manifold. There can be an infinite number of paths between two points on the manifold that are geodesics. Geodesics are mathematical ways of determining the shortest distance between two points on the manifold. In Euclidean geometry, this is a straight line, and there is only one path that determines the shortest distance. But a geodesic over a manifold may (and often does) have more than one shortest path. For instance, imagine a sphere with a point on “top” and a point on “bottom”; there are infinitely many ways from one point to the other that are the shortest path between the two points.