SOURCE FOR CONTENT: General Relativity: An Introduction for Physicists, Hobson, M.P., Efsttathiou, G., and Lasenby, A.N., 2006. Cambridge University Press.
D-Dimensional Hypersphere and Gamma Function: Introduction to Thermal Physics, Schroeder D.V. 2000. Addison-Wesley-Longmann.
IMAGE CREDIT: NASA/JPL.
The intended purpose of the post is to introduce the concept of manifolds in the context of physics (mathematicians beware!). Furthermore, I will discuss the concepts of Riemannian and pseudo-Riemannian manifolds before moving on towards tensors. This will be the first post in this topic of the series. In order to properly discuss the concepts of general relativity, I will have to break up this part of the series into smaller posts.
Part I. In this part of the series, I will discuss the concept of a tensor, and then discuss the introductory topics of manifolds.
A topic that has long eluded a conceptual understanding on my part is a tensor. In the first post of the series we saw a very technical and quite frustrating definition of a tensor. I have read numerous treatments and watched a number of lectures and videos and based on everything I have encountered, here is my understanding of a tensor as of writing this post…
Tensors are geometric objects that can be viewed in a similar way as one views matrices, whose elements are components of the tensor, and will have an overall value. More specifically, a tensor will take two geometric objects as inputs and will give you a scalar (a real number). Furthermore, under a transformation or in a different reference frame, the scalar that the tensor outputs will remain the same in all frames. The objects that change are, in fact, the components of the tensor. These components must obey specific transformation equations so as to preserve the scalar quantity of produced by the tensor. In more mathematical sense, a tensor is a mapping of a number of vectors (including 1-forms and the like) into the real number ordered field.
(**This is the best definition that I could come up with in order to define in a more satisfactory way a tensor.**)
According to the aforementioned reference, a manifold in its most general sense, is any set that one can describe by specifying parameters continuously. In the context of physics, we deal with differentiable manifolds.
A differentiable manifold is a continuous collection of points where each point is differentiable. This definition isn’t any better than the initial definition of a tensor. So, to elaborate a bit, we shall define the concept of continuity: a manifold is continuous if in the local region of a point , there exists points whose difference relative to is .
From this, we can say that a differentiable manifold is a manifold for which we can ascribe to it a scalar field containing points at which it is possible to take derivatives of all orders.
Some examples include: 3-Dimensional Euclidean Space and Phase Space.
This differentiable manifold requires 3 coordinates (parameters) to specify a single point in the space. Since it requires three parameters the dimension of this particular manifold is 3. Mathematicians sometimes call this 3-space.
This is a manifold that one encounters more often in physics. I came across this manifold (although I did not refer to it as such) while taking my thermodynamics and statistical mechanics course. I found that this manifold requires 6 parameters in order to specify any point. Typically, these parameters include positions (or one radius vector) and velocities or momenta.
A submanifold or surface that I found applicable to phase space would be the -dimensional hypersphere whose surface area is given by
where is the gamma function given by
To be more precise, the surface area (Eq.1) of this hypersphere is technically the volume of momentum space, but I am including to present a more concrete example of a manifold.
The next post will make the concepts mentioned here a bit more quantitative. This post was really just to introduce a more conceptual understanding.