I’m not a mathematician so this will be sketchy at best.
Tensors are tied up in the idea of the measurement of distances between two points in a generalized coordinate system. In a rectangular coordinate system the distance from the origin, s, of the point p(x,y) is s[sup]2[/sup] = x[sup]2[/sup] + y[sup]2[/sup]. This is the familiar Pythagorean formula.
From Analytic Geomety the distance between the points p[sub]1/sub and p[sub]2/sub is:
For ease of writing them out we’ll call (x[sub]2[/sub] - x[sub]1[/sub]) dx and (y[sub]2[/sub] - y[sub]1[/sub]) dy.
Now a rectangular coordinate system isn’t the only one around. For instance there is the curvilinear coordinate system of latitude and longitude that is used to locate points on the surface of the earth.
So we can talk about a generalized coordinate system of a set of curves of any shape that intersect another set of curves. The two sets of curves need not be regularly spaced and they can intersect each other at any old angle. If we number the lines in the sets and call one set u[sub]1[/sub] and the other u[sub]2[/sub] (so that we don’t confuse them with x and y of the rectangular system) then the distance between any two points in this set will be:
s[sup]2[/sup] = du[sub]1[/sub][sup]2[/sup] + du[sub]2[/sub][sup]2[/sup], by analogy to the rectangular case. Once we assign a metric to the coordinate numbers we can compute a distance in that metric.
This is for a coordinate system in a plane. Carl Freidrich Gauss developed a metric analogous to the one dimentional case for distances between two points on a surface. I am unable to follow his method very well and the following is from Edna E. Kramer, Professor Emeritus of the Polytechnic Institute of Brooklyn in her book The Nature and Growth of Modern Mathematics.
The Gauss formula for distances between points on a surface is:
u[sub]1[/sub] and u[sub]2[/sub] are the two coordinates on the surface, for example latitude and longitude, and the g’s are constants.
Since the du’s are just furniture we can assume they are present and the metric is determined by the value of the g’s which can be put into a matrix.
(g[sub]11[/sub]) (g[sub]12[/sub])
(g[sub]12[/sub]) (g[sub]22[/sub])
When this matrix is accompanied by a law of transformation that allows transforming g into g’ in all permissible meshes or frames of reference it is called a tensor.
I’ll leave it to the mathematicians to explain what “all permissible frames of reference” is all about.
Well IANAM, but I know you can’t use a Lorentz transformation to transform a time-like interval into a space-like interval, for example. So while you and I might disagree on an object’s speed if we’re in different reference frames, you won’t have a situation where something’s moving slower than light from my perspective but faster than light from your perspective. I think this is related to the fact that no reference frame can be moving faster than light speed, so /[Gamma] = 1 / sqrt(1-v[sup]2[/sup]/c[sup]2[/sup]) (a factor that appears in Lorentz transforms involving boosts) is always positive. So I guess you could say an inertial reference frame that’s moving faster than light speed relative to another inertial frame is not “permissible.”
To revise my comment above, I guess it’s more correct to say that a tensor maps one kind of mathematical object to another, and the scalar, vector, matrix, what-have-you is a representation of this map. But, again, IANAM.
The standard application of tensors to differential geometry (and thus GR) is what you’re talking about. Specifically, these are tensors over the tangent vector space at a given point of a manifold (T[sub]p[/sub]). Changing coordinate systems acts in the tangent space as an invertible linear transformation: an element of GL(T[sub]p[/sub]) like I said before.
That’s because in the context of SR the relevant group action is SO(3,1), not GL(4).
This is an element of V?V[sup]*[/sup], not a tensor in general.
I hate to be so authoritatian here, but everyone who’s said that it has little or nothing to do with transformation properties is flat-out wrong. Tensor products are intimately connected with group (or generally: algebra; or more generally: category) actions. If the dual vector space doesn’t transform according to the contragrediant representation of the appropriate group it’s not a contravariant tensor. If the new vector space constructed from the two previous ones doesn’t transform via the action guaranteed by the coalgebra structure on C[G], it’s not the tensor product.
Yeah I know, but few of us understand the lingo. For example “tangent vecor space” and “manifold” are specialized mathematical usages that don’t have a lot of meaning to most of us.
Application to General Relativity is one use of tensors. What are some others? I think applications that give a sense of what tensors do are what rjk and those of us without advanced mathematical training could use.
For example, having read Dr. Kramer’s book I have a general feel for what tensors do in General Relativity by analogy with ordinary Analytic Geometry. Likewise the transformation of coordinates from, say, rectangular to polar gives me a small sense of the Lorenz transformation but I certainly can’t use it with any confidence.
I’m not a mathmetician or even a physicist, but I’ll have a shot at getting past “transform”, hopefully as far as “basis vectors”.
Physicists and non-physicists alike occasionally measure things, and express the measurement using numbers. Examples are temperature, altitude, weight, wind strength, whatever. A lot of these things can be expressed using only one number, e.g. temperature and altitude. They are single component quantities or scalar quantities.
Wind strength is slightly different - it has both a magnitude (the wind speed) and a direction. Generally we say stuff like “wind speed 10mph in a north east direction”. Quantities with magnitude and direction are called vector quantities.
To specify vector quantities, we need a co-ordinate system which includes directions. We have such a system in common use - the compass directions, N, S, E and W, plus UP and DOWN. These are our reference directions, and we can express any general direction in terms of them. (6 feet DOWN and 3 inches NORTH for every mile WEST is an example of a “general direction”).
The reference directions N, S, E, W, UP and DOWN are the set of basis vectors we use to express direction. In fact, since we can use negative numbers, we can reduce them to N, E and UP, and express S, W and DOWN as “-N”, “-E” and “-UP”. So my previous " general direction" becomes “-6 feet UP and 3 inches NORTH for every minus-mile EAST”.
To express wind strength properly in terms of our basis vectors, we need to use a minimum of three numbers - “wind speed 10mph in a north-east direction” becomes “wind speeds 7.07mph N, 7.07mph E, 0mph UP”. This is the case for all vector quantities in our three dimensional universe - you can reduce their expression to three numbers, but no fewer. You would normally write such a vector as (7.07, 7.07, 0).
Now, on to transformations of basis vectors! N, E and UP are all very well if you’re looking at the weather map or going sailing, but they’re not very convenient if you’re trying to land a plane on a north-east runway. You’d prefer have the wind strength expressed as TAILWIND, CROSSWIND (left to right) and DOWNDRAFT.
Remember “different components transform(ing) among each other subject to certain relations with a change of basis vectors”? In this case it is simply the three numbers of the vector changing as you switch from using N, E, UP to TAIL, CROSS and DDRAFT. Wind speed 10mph in a north east direction may be (7.07, 7.07, 0) using N, E and UP but it becomes (10, 0, 0) using TAIL, CROSS and DDRAFT for a NE runway as your basis vectors.
Now, it is obvious that the transformation of the three numbers of the Wind Strength vector from one set of basis vectors to the other is a matter of geometry. In fact, if you remember how to do matrix algebra, there is a 3 x 3 matrix that you can multiply the Wind Strength vector by to transform one to the other. That matrix is the transformation tensor and it relates the Wind Strength expressed using two different sets of basis vectors.
Revisiting measurements and quantities for a moment:
scalars are single component quantities,and we give them a single number, which is technically a 1 x 1 matrix. We don’t normally bother to think about it as a matrix, just as we don’t normally bother to write 11.6 as 11.60000000… with the zeros going on forever, but in this context it is a matrix.
Vectors are three-component quantities, and we give them three numbers, making a 3 x 1 matrix.
But there are other quantities that have even more components. Stress measured at a point in a material has NINE components - three plain stress components and six shear stress components, and it is described with nine numbers in a 3 x 3 matrix. Quantities with more than three components are often referred to as “tensor quantities”, kind of a step up from scalar and vector quantities. This is a misnomer really - scalars and vectors are tensor quantities as well.
So - a tensor is a quantity with components. (Even a scalar, with only one component.) The components can be written as a matrix. Each component expresses a magnitude in terms of basis vectors. (A scalar technically has ONE basis vector - the “number line”.) If you change the basis vectors, then all the values of the components of the tensor change in some well-defined way e.g. as Euclidian geometry dictates. The change from one set of basis vectors to another can be described by a matrix, a “transformation tensor”. I’m not too delighted by the use of the word “tensor” both for the type of quantity and the matrix used for transformations, but it seems quite common.
Mathematicians of course have no truck with this wishy-washy, stuck-in-the-real-world type of thinking. They are quite happy to imagine tensors in spaces with more than three dimensions, tensor transformations using non-Euclidian geometry, and all kinds of general cases of which real-world applications are only a tiny subset. I’m glad they’re around to think about that stuff, so I don’t have to!
Excellent, when considering the GL(V) action on V. The only thing I’d add is the emphasis that the tensor measures something that is there in the absence of a coordinate system, while the components are artifacts of the choice of coordinates. Changing the basis changes the components, but since there’s something “really there” independant of the coordinate system the way the coordinates change is fixed by the nature of the object.
Maybe in physics or engineering usage. To a mathematician it’s the matrix representation of the element of GL(V) in the basis chosen, nothing more.
Of course. My attempt at an illustration of the use of tensors was in response to rjk. I think Complex Conjugate has had enough math that I can be of no help whatsoever to him.
IAAP(physicist) We would not call the transformation matrix a transformation tensor for the simple reason that it is not one. Its elements transform under change of basis but do not transform as a tensor’s would. Otherwise, Matt, that was a good explanation.
First, thanks to all for the explanations. I think I’m starting to understand!
My first reaction to this (as for many math posts) was “Say whaaa?” but a little thought leads me to yet another question. (I assume we’re in abstract math here, or at least back to GR.) Do you mean that the tensor defines the transformation rules in a general way, no matter what coordinate system we might choose later?
My brain hurts. I must be a masochist! :rolleyes: (I do like the name, BTW.)
I’m glad it’s excellent, but to be honest I don’t understand the terminology you use! I was introduced to tensors as a concept purely because I’m a materials scientist and we had to use them to describe stress. That’s a very specific “real world” application - any generalisation or mathematical depth was avoided like the plague.
I wish I’d said that. I probably will, somewhere else.
I wish I could blame my engineering textbooks and my lecturers, but the fault may well lie with my own capricious memory…
Before I do anything else I’ll put a basic explanation of what a tensor is (or at least how I like to define a tensor) for anyone whose interested (e.g. matt):
A vector space, simply put is just a set of objects called vectors (and a field of numbers like for example the real numbers) that satisfies certain axioms (the concept of a ‘vector’ in this instance is a complete abstraction, so don’t think they have to be vectors corresponding to physical quantities, but an example of a vector space is the set of all radius vectors in real space).
A one-form (aka co-vector, covariant vector and several other names) is a (linear) function that acts on a vector and produces a number. The set of all one-forms for a given vector space also satisfies the axioms of a vector space, so we call vector space of vectors and the vector space of one-forms ‘dual vector spaces’. For each vector there is a corresponding one-form, the two making up what is called a pair of dual vectors. Given a vector A and a one-form B~( ) (I’ve put the empty brackets to signify the fact that it is a fucntion of a vector) we can obtain some number B~(A), this number is called the dot product of the vector A and the vector B (BB~( ) form a pair of dual vectors). A one-form is a tensor of type (0,1).
A tensor of type (r,s) is a (multilinear) function that take r vectors and s one-forms and produces a number. To give an example, the metric tensor g( , ) (a tensor of type (0,2) is a very important tensor. Given two vectors A, B the metric tensor will produce a real number g(A,B) this number is called the dot product of the vector A and the vector B. Comparing this to the one-form:
g(A,B) = B~(A
g(B, ) = B~( )
Infact vectors themsleves are tensors of type (1,0) threfore they can take a one-form a produce a real number:
[a]A**(B~) = B~(A).
What this physically means in n-diemnsional space is that a tensor of type (r,s) has n[sup]r + s[/sup] componets so for example a tensor of rank 2 (r + s =2) in some (recatasngular) basis in three dimensional space has 9 componets: xx, xy, xz, yx, yy, yz, zx, zy (normal matrix notation does not work for tensors of rank 3 and higher). Under a transformation the components of a tensor will tranform in a way that depends on their type.
I would like to see some elaboration on this point, if someone is willing. What are the possible ways a tensor can transform under change of basis, and how does the transformation matrix transform? I’ve seen this stuff before, but I need a refresher.
If my understanding is correct, a tensor quantity is one that can be described by a matrix, and the elements of that matrix depend on the co-ordinate system being used. The matrix itself is referred to as a tensor.
Unitless quantities, such as the number of M & Ms in a jar, or the ratio of a circle’s circumference to its diameter, are not tensor quantities - there is no co-ordinate system, although they are real things and measurable.
Scalar quantities e.g. altitude, with units such as feet, are tensor quantities, but their matrices contain only one component. Of course, you don’t tend to think of a single value as a one-element matrix, nor should you have to.
The “co-ordinate system” of a scalar is the unit of measurent, and you can transform from one unit to another with a conversion factor, e.g. from feet to metres. The conversion factor is the single component of the transformation matrix between the two units.
This is the same as switching to a basis vector with a different magnitude. In my previous example with wind strength I neglected the possibility of changing the magnitude of the basis vectors, dealing only with their orientation.
Vector quantities in three-dimensional space have to have three components, which will depend on the co-ordinate system chosen. I used the compass directions because this is a co-ordinate system that everyone is familiar with, and the example of a skewed runway to show why you might want to transform to some other co-ordinate system.
The point ** Mathochist** was making is that the steadily blowing wind is a real thing - the tensor (the matrix of three values) used to describe it depends on the co-ordinate system chosen. The co-ordinate system includes both the directions and magnitudes of the basis vectors. in my previous example I made the implicit assumption that the basis vectors were a “mile per hour” long.
Transformation rules from one co-ordinate system to another depend on the relationship between the co-ordinate systems, and have nothing to do with the real thing being measured. For example, if you’re using N, E and UP as your basis vectors but want to convert to basis vectors a “kilometre per hour” long, the relationship between the two co-ordinate systems is simply a multiplication of the basis vector magnitudes by 1.6. The transformation is described by a 3x3 matrix with 1.6 down the top-left to bottom-right diagonal and the rest filled with zeros - a diagonal matrix.
If you wanted to convert to basis vectors of km/h for N and E but feet per second for the UP direction, you can. There is no reason why your basis vectors need to be of the same magnitude. They don’t even have to be at right angles to each other - you can still describe arbitary vectors with them. The co-ordinate system is up to you, the components of the resulting tensor describing the real thing depends on what you choose.
I can’t do complete abstraction at all. I hit things with hammers. It would take me a few days and a lot of reading and thinking just to get to grips with the idea of vector space, or a field of numbers for that matter! I don’t have the concepts in place to follow what you’re saying.
This is good for me. It reminds me to have patience with people who are bewildered by the terminology I use on a daily basis.