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Tensors, the geometric tool that solved Einstein's relativity problem

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Re: Tensors, the geometric tool that solved Einstein's relativity problem

#51

Post by abdullahkhalids »

adrian_b wrote: post #44
mr_mitm wrote: post #38

Earlier quoted context omitted.

Yes, very simple, except that when physicists say "tensor", they mean tensor fields, on smooth, curved manifolds, in at least four dimensions, often with a Lorentz metric. Things stop being simple quickly.

It does not matter on what set a tensor field is defined. A tensor field is not a tensor, but the value of a tensor field at any point is a tensor, which satisfies the definition given above, exactly like the value of a vector field at any point is a vector. The "fields" are just functions. There are physics books that do not give the easier to understand definition given above, but they give an equivalent, but more…

> There are physics books that do not give the easier to understand definition given above, but they give an equivalent, but more obscure, definition of a tensor, by giving the transformation rules for its contravariant components and for its covariant components at a change of the reference system.

The definition of a tensor as linear maps, while simple to understand, has no content that is useful for doing physics. To do any physics, or for that matter, any geometry with tensors, you need to define the notion of covariance and contravariance.

Besides, the starting with the latter notions allow you to define tensors more naturally. You start with trying to understand how geometric objects transform under coordinate transformations, and you slowly but surely end up with tensors.

Re: Tensors, the geometric tool that solved Einstein's relativity problem

#52

Post by adrian_b »

will-burner wrote: post #6
senderista wrote: post #2

If you have any linear algebra background, then the definition of a tensor is straightforward: given a vector space V over a field K (in physics, K = R or C ), a tensor T is a multilinear (i.e. linear in each argument) function from vectors and dual vectors in V to numbers in K . That's it! A type (p, q) tensor T takes p vectors and q dual vectors as arguments ( p+q is often called the rank of T but is ambiguous comp…

The definition may be simple, but it's not very concrete and I'd argue that makes it not strait forward. While examples of vector spaces can be very concrete (think R, R^2, R^30), I struggle to think of a concrete example of a multilinear function from vectors and dual vectors in V to numbers in K. On top of that when working with tensors, you don't usually use the definition os a multilinear function at least as far…

In physics, the first and even now the most important application of multilinear functions, a.k.a. tensors, is in the properties of anisotropic solids.

A solid can be anisotropic, i.e. with properties that depend on the direction, either because it is crystalline or because there are certain external influences, like a force or an electric field or a magnetic field that are applied in a certain direction.

In (linear) anisotropic solids, a vector property that depends on another vector property is no longer collinear with the source, but it has another direction, so the output vector is a bilinear function of the input vector and of the crystal orientation, i.e. it is obtained by the multiplication with a matrix. This happens for various mechanical, optical, electric or magnetic properties.

When there are more complex effects, which connect properties from different domains, like piezoelectricity, which connects electric properties with mechanical properties, then the matrices that describe vector transformations, a.k.a. tensors of the second order, may depend on other such tensors of the second order, so the corresponding dependence is described by a tensor of the fourth order.

So the tensors really appear in physics as multilinear functions, which compute the answers to questions like "if I apply a voltage on the electrodes deposited on a crystal in this positions, which will be the direction and magnitude of the displacements of certain parts of the crystal". While in isotropic media you can have relationships between vectors that are described by scalars and relationships between scalars that are also described by scalars, the corresponding relationships for anisotropic media become much more complicated and the simple scalars are replaced everywhere by tensors of various orders.

What in an isotropic medium is a simple proportionality becomes a multilinear function in an anisotropic medium.

The distinction between vectors and dual vectors appears only when the coordinate system does not use orthogonal axes, which makes all computations much more complicated.

The anisotropic solids have become extremely important in modern technology. All the high-performance semiconductor devices are made with anisotropic semiconductor crystals.

Re: Tensors, the geometric tool that solved Einstein's relativity problem

#53

Post by twothreeone »

max_likelihood wrote: post #3

I've always thought the use of "Tensor" in the "TensorFlow" library is a misnomer. I'm not too familiar with ML/theory, is there a deeper geometric meaning to the multi-dimensional array of numbers we are multiplying or is "MatrixFlow" a more appropriate name?

I agree. Just like NumPy's Einsum. "Multi-Array Flow" doesn't sound sexy and associating your project with a renowned physicist's name gives your project that "we solve big science problems" vibe by association. Very pretentious, very predictable, and very cringe.

Re: Tensors, the geometric tool that solved Einstein's relativity problem

#54

Post by bjourne »

ogogmad wrote: post #24
MathMonkeyMan wrote: post #14

Earlier quoted context omitted.

The joke I learned in a Physics course is "a vector is something that transforms like a vector," and "a tensor is something that transforms like a tensor." It's true, though. The physicist's tensor is a matrix of functions of coordinates that transform in a prescribed way when the coordinates are transformed. It's a particular application of the chain rule from calculus. I don't know why the word "tensor" is used in…

The mathematical definition is 99% equivalent to the physical one. I find that the physical one helps to motivate the mathematical one by illustrating the numerical difference between the basis-change transformation for (1,0)- and (0,1)-tensors. The mathematical one is then simpler and more conceptual once you've understood that motivation. The concept of a tensor really belongs to linear algebra, but occurs mostly i…

The problem with the physicist's definition is that the larger the N the less the geometrical interpretation makes sense. For 1, 2, and even 3-dimensional tensors there is some connection to geometry, but eventually it loses all meaning. Physicist has to give up and "admit" that an N-dimensional tensor really just is a collection of N-1-dimensional tensors.

Re: Tensors, the geometric tool that solved Einstein's relativity problem

#55

Post by eigenket »

bjourne wrote: post #49
senderista wrote: post #2

If you have any linear algebra background, then the definition of a tensor is straightforward: given a vector space V over a field K (in physics, K = R or C ), a tensor T is a multilinear (i.e. linear in each argument) function from vectors and dual vectors in V to numbers in K . That's it! A type (p, q) tensor T takes p vectors and q dual vectors as arguments ( p+q is often called the rank of T but is ambiguous comp…

Well, I can write a definition that is both easier to understand and shorter than yours: A tensor is a multi-dimensional array. :)

This only works in finite dimensions, which for mathematicians excludes pretty much all of the interesting cases.

Re: Tensors, the geometric tool that solved Einstein's relativity problem

#56

Post by ithinkso »

bjourne wrote: post #49
senderista wrote: post #2

If you have any linear algebra background, then the definition of a tensor is straightforward: given a vector space V over a field K (in physics, K = R or C ), a tensor T is a multilinear (i.e. linear in each argument) function from vectors and dual vectors in V to numbers in K . That's it! A type (p, q) tensor T takes p vectors and q dual vectors as arguments ( p+q is often called the rank of T but is ambiguous comp…

Well, I can write a definition that is both easier to understand and shorter than yours: A tensor is a multi-dimensional array. :)

This is actually a harmful definition, both (1,1) and (0,2) tensors can be written as a matrix but they are very different. It's like calling vector an array but vectors require vector space and arrays are just arrays. It doesn't help that std::vector is very common in CS but 'pushing back' to a mathematical vector just doesn't make any sense

Re: Tensors, the geometric tool that solved Einstein's relativity problem

#57

Post by adrian_b »

max_likelihood wrote: post #3

I've always thought the use of "Tensor" in the "TensorFlow" library is a misnomer. I'm not too familiar with ML/theory, is there a deeper geometric meaning to the multi-dimensional array of numbers we are multiplying or is "MatrixFlow" a more appropriate name?

Since the beginning of computer technology, "array" is the term that has been used for any multi-dimensional array, with "vectors" and "matrices" being special kinds of arrays. An exception was COBOL, which had a completely different terminology in comparison with the other programming languages of that time. Among the long list of differences between COBOL and the rest were e.g. "class" instead of "type" and "table" instead of "array". Some of the COBOL terminology has been inherited by languages like SQL or Simula 67 (hence the use of "class" in OOP languages).

A "tensor", as used in mathematics in physics is not any array, but it is a special kind of array, which is associated with a certain coordinate system and which is transformed by special rules whenever the coordinate system is changed.

The "tensor" in TensorFlow is a fancy name for what should be called just "array". When an array is bidimensional, "matrix" is an appropriate name for it.

Re: Tensors, the geometric tool that solved Einstein's relativity problem

#58

Post by eigenket »

nyrikki wrote: post #5

I would argue that today, geometric algebra/Clifford calculus and space time algebra are more intuitive and useful. Gibbs/Heavysides vectors were more popular at the time. At least for me.

Nah, at least in the context of General Relativity tensors and tensor fields are everything. Geometric algebra doesn't add much.

Re: Tensors, the geometric tool that solved Einstein's relativity problem

#59

Post by mvaliente2001 »

The idea of tensors as "a matrix of numbers" or the example of a cube with vectors on every face never clicked for me. It was this (NASA paper)[https://www.grc.nasa.gov/www/k-12/Numbers/Math/documents/Ten...] what finally brought me clarity. The main idea, as others already commented, is that a tensor or rank n is a function that can be applied up to n vector, reducing its rank by one for each vector it consumes.

Re: Tensors, the geometric tool that solved Einstein's relativity problem

#60

Post by tacomonstrous »

abdullahkhalids wrote: post #51
adrian_b wrote: post #44

Earlier quoted context omitted.

It does not matter on what set a tensor field is defined. A tensor field is not a tensor, but the value of a tensor field at any point is a tensor, which satisfies the definition given above, exactly like the value of a vector field at any point is a vector. The "fields" are just functions. There are physics books that do not give the easier to understand definition given above, but they give an equivalent, but more…

> There are physics books that do not give the easier to understand definition given above, but they give an equivalent, but more obscure, definition of a tensor, by giving the transformation rules for its contravariant components and for its covariant components at a change of the reference system. The definition of a tensor as linear maps, while simple to understand, has no content that is useful for doing physics.…

Covariance and contravariance are mathematical notions, and have to do with whether each multiplicative constituent or the tensor is a vector in your given vector space (covariant) or a linear functional on this space (contravariant). There is no inherent physical meaning to either concept.
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