Toy Universes

Dear Readers!

If you have been reading my latest Twitter posts, you may know that I have been quite busy preparing a complete Christmas dinner for my family and relatives. Also, in the next days until New Year’s Day or so, my attention will continue to be focused on being a host. So it may be a little quiet around here. Also, I will be travelling in January. In Brazil, it’s holliday time. I will be back from vacation in February, and hopefully my blog will return to full power. In the meantime, if I find a window in my time frame, I may end up succeeding in writing a post. I have a lot of plans for my new blog and I am very motivated in developing them here. I hope you will enjoy them as well, as they become realized with time, and I count with your participation.

Best wishes,

Christine

To my surprise, while accessing this blog via my iPhone, an ad by Google entitled “Einstein was Wrong” popped-up just below the title of one of my posts. I use the free server services by WordPress to host this blog. I have searched a bit about the issue and found the answer here. Since I use the free services, I have no control over such ads. This post is just to make clear to readers that I do not directly support the contents of such ads, but since I use the free services, I will have to accept the situation. Eventually, I may choose to pay for removing the ads, but for now, I and my readers will have to live with them.

Simple Spaces.

We start our understanding of tensors by exploring them in a particular way. We will later generalize this vision.

The ‘particular way’ that we now have in mind regards the kind of ‘space’ that we will be first using to define tensors. Indeed, tensors do not float around in nothingness, they need to be locally defined, that is, they need to be associated with a point in a mathematical space. Tensors take objects defined locally and operate on them, as we will see. But even before we address the kind of space that we will need, we must first clear up what we mathematically mean by such spaces.

We start mentioning the most natural and intuitive space, first rationally devised by the Greek geometers of the past, namely, the Euclidean space. We denote this space by, whereis the dimension of the space.  Evidently, for the Greeks,could only make sense ranging from 1 to 3, since the Euclidean space was an abstraction of our usual physical surroundings, and these are constrained by up to 3 degrees of freedom, as experience dictates us. But the Euclidean space can be easily generalized to any dimension.

Well, you may immediately consider thatis nothing but the set of ordered real n-tuples,But this is not exactly correct. That set is actually another space,. Undoubtedly, that can be a source of confusion. What is the difference betweenandanyway?

A model.

Well,can be used as a mathematical model forThe main point here is that when we talk aboutwe mean a space endowed with certain properties. We could attach these properties toand define it as our usual ‘Euclidean space’. But let us first make a clear separation.

The properties that we are talking about are those that the Greek geometers first explored in great detail. They did that from just a small set of postulates or axioms, and derived various theorems, from which they were able to deduce innumerous relationships between objects (lines, angles, circles, solids, etc.). We are evidently talking about Euclidean geometry. ‘Geometry’ here more or less means the set of those relationships.

So the point is that one can use the spaceto express those Euclidean geometrical relationships by framing these relationships in terms of relations among  coordinates — the so-called Cartesian coordinates. In other words, geometrical objects and their relationships ‘exist’  independently of any coordinatization of the space they live (and this can be proved, as the Greeks did). But once you choose one coordinatization, you can express all those same relationships and properties of Euclidean geometry by a clever mapping between its objects and their algebraic representations, a mapping that involves the principles of algebra and analysis — analytic or cartesian geometry is then born.

Adding structure.

As already mentioned, you will find that in the literaturemay be given a richer structure than simply that of the set of ordered real n-tuplesthat composes it. Then, some authors defineas a space with that richer structure. For instance, some authors may consideras an n-dimensional vector space (more on that some other time).

For the moment, we will focus on just attaching the Euclidean metric toA metric, for our present purposes, is simply a way to calculate distances in a given space. Let us, for concreteness, fix for a moment the dimension of the space to 3. Then, the 3D Euclidean metricis a function that grabs two points in the space,and [where the triple is attached to and the triple is attached to ], and delivers a single real, positive number resulting from the well-known Pythagorean formula:

.

That function can be easily generalized to n dimensions.

So that is what we mean byhere, unless otherwise stated. It is a set of ordered real n-tupleswith the Euclidean metric defined in it. You could ask, then: is it not simply the Euclidean space itself? It is not. Notice thathas the ‘zero point’ (the origin of coordinates) defined in it. But in the Euclidean space, no origin can be defined. It is an affine space.

Different metric functions can be attached to spaces. And as already pointed out, other structures may be attached to spaces. In fact, if we go back to more and more primitive concepts, we may start defining a space by simply a set. Then we may define other notions like an open neighbourhood of a point, and then a topology, and we may end up with a topological space. We may define a manifold as a space that locally resembles the Euclidean space. We may add more and more structure to spaces like the concept of differentiability, even for non-Euclidean spaces and general manifolds. And do calculus on it. We will address these notions more rigorously in future posts, since we will need to generalize tensors living in arbitrary manifolds, like the curved spaces of general relativity.

But for the moment, tensors will be explored in the simple Euclidean space. We will call them Euclidean tensors. More on them opportunely.

Teaching new tricks to an old dog.

Learning tensors recalls me of the first time that I went into contact with object-oriented programming (OO), after decades using procedural languages, like Fortran, Pascal, etc. It was quite baffling for me, specially when OO programming could do that acrid job by the use a small trick somewhere in there. But for those who learned OO languages first (specially when being their only programming experience), the tricks often were not all that baffling.

In any case, there are many ways to implement an algorithm. It is often a question of naturalness, efficiency, applicability or other sensible criteria. Sometimes it is just a question of familiarity and practicability.

If you are a pessimist, sticking to one way of doing or seeing things is just a matter that one cannot teach new tricks to an old dog. Understanding the many faces of a tensor requires an optimist.

Different views.

The physics or engineer student will probably approach the subject of tensors by the classical, coordinate (or index) notation. Tensor analysis is summarized as an intense exercise in index gymnastics. Tensors for them are a set of quantities that transform in a [insert here: a long series of derivatives and indices]…, well, certain way.

On the other hand, the mathematics student will probably approach the matter geometrically, using index-free notation, the language of modern differential geometry.

For the physicist or engineer that (creepy for some) tensor gymnastics allows them to make concrete calculations. For the mathematician, that (creepy for some) abstract view of tensors offers them a unifying, powerful vision of mathematics as a whole.

The clash.

All is fine and well when each tribe is confined to their own communities. However, at some point, specially beginning at a graduate level, the theoretical/mathematical physicist will notice that he/she needs to explore new lands. The need for a more abstract conceptualization will come one way or another, and he/she will search for a more formal understanding of tensors in order to express their theories in a more rigorous way.

Most may end up trying to understand why, oh why, they were not warned that tensors could be something completely different from what they were being taught, specially during their undergraduate courses. Many may get confused to digest the abstract view at first, to the point of thinking what really a tensor is after all.

On the other hand, those exposed early to modern differential geometry cannot see how one could ever attempt to express everything in component form, but at the same time, has never felt the need for concrete calculations using a specific coordinate frame. Until he/she gets interested in physics.

Then, the clash.

The agreement.

In this blog, as we explore various matters in gravity, we will make use of index notation and index-free notation of tensors, which are needed specially in general relativity theory. It will be important to expose these many faces of tensors and be able to navigate through them as smoothly as possible. This will be explored in future posts.

Some references that help that navigation are, for instance: [Dod91], [MTW73], and [Sch80].

We start by stating a few facts about Mach’s Principle, without going into details. These will be explored later, in future posts.

Mach’s Principle as an initial guide to General Relativity

The initial inputs that guided Einstein to conceive his General Theory of Relativity (GR) were, basically:

• Mach’s Principle;
• The Principle of Equivalence;
• The Principle of General Covariance.

Although the last two principles of the above list are relatively well understood, the first one is not.

The blurred Mach’s Principle

Mach’s Principle is on people’s mind as basically a general statement on the origin of inertia. This is often understood by many as a problem that has been solved by Einstein’s GR, although Einstein himself disregarded Mach’s Principle later on.

From time to time, you may find obscure papers by obscure people on Mach’s Principle, but also authoritative works written by well-known experts of various areas of physics. This curious phenomenon serves as a testimony to the enthusiasm and reverence to a subtle and deep matter, even if stated so imprecisely.

But what exactly is Mach’s Principle?

The problem is the term “exactly”.  There is no consensus on what Ernst Mach really meant by the principle which carries his name. In fact, the term “Mach’s Principle” was coined by Einstein. But, there is no clear statement of the principle in Mach’s book, “The Science of Mechanics” [Mach10], as cited by Einstein as one of the books which impressed him mostly while constructing his ideas towards GR.

If you want to know all there is to know about Mach’s Principle, you should start by reading a collection of contributions to the Tübingen meeting (1993) entirely dedicated to Mach’s Principle, edited by Barbour and Pfister [Barb95]. This collection is extremely interesting and includes transcriptions of the discussion sections as well.

The various interpretations of Mach’s Principle

Here I merely list the various interpretations identified on that meeting. Hopefully, we will be able to go into each of them (or some of them) in more detail in future posts.

1. As the problem of defining velocity (motion).
2. As mere redescription of Newtonian Mechanics.
3. As determination of inertial frames from relative motions of masses.
4. As mechanical interaction of masses, in particular through an interactive Machian ‘kinetic energy’.
5. As generation of inertial forces in any body accelerated with respect to distant masses.
6. As induction of inertial forces by accelerated masses by analogy with electromagnetic induction.
7. As requirement that the metric tensor be completely determined by matter.
8. The same, but by matter and gravitation (geometrical) degrees of freedom.
9. As prediction of the future given relational initial data (initial-value approach).
10. As cosmic derivation of inertial mass.
11. As generalization of the special principle of relativity to nonuniform motions.
12. As requirement of general covariance of the laws of nature.
13. As need for generally covariant boundary conditions.
14. As need for nonexistence of boundary conditions.
15. There should be no matter-free, singularity-free solutions of GR.
16. As an appeal to the principle of sufficient reason (observable facts must have observable causes).
17. As expectation there will be dragging effects.
18. As explanation of nonrotation of compass of inertia relative to distant masses.
19. As proposition that the universe at large influences local physics (Dicke-type approach).
20. As selection principle of solutions of GR that are intuitively ‘Machian’.
21. As requirement that dynamics should not contain absolute elements.

“It is a mathematical fact that the casting of a pebble from my hand alters the centre of gravity of the universe.”

— Thomas Carlyle

[In James Wood, Dictionary of Quotations from Ancient and Modern, English and Foreign Sources (1893), 190:1.]

Welcome to Toy Universes!

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I am very glad to start this new project, and I hope that it will be an interesting and productive journey.