negative distance

[ajb]

Something like;

 

[math]P^{i} = \epsilon_{0}\chi^{i}_{\:\: j}E^{j}[/math]?

 

(In Euclidean space we don't need to worry too much about upper and lower indices).

[Pete]

Something like;

 

[math]P^{i} = \epsilon_{0}\chi^{i}_{\:\: j}E^{j}[/math]?

 

(In Euclidean space we don't need to worry too much about upper and lower indices).

Yep. That's it!

 

By the way. I completely screwed up bad! I made the mistake in believing that the spacetime interval does not remain invariant for an arbitrary coordinate transformation.

 

I had a bad feeling about making that statement so I worked it out again and this time I realized that I had somehow made a mistake when I tried this month two ago. I can't understand how I made such a dumb mistake since this is a basic fact of tensor analysis!

 

I'm glad I pressed this point because I'd otherwise still not understand in what sense a generalized coordinate transformation is used in GR.

 

Oh well. As they say, live and learn, right?

That's exactly right.

Where did you learn this from? Did you learn it from a particular text? A number of texts etc?

 

What you are thinking of as the, presumably, "correct" definition of a tensor is really the definition of a certain type of tensor, namely what is known as a general tensor. They get their name from the type of allowable transformations they allow, i.e. generalized coordinate transformations. There are other kinds of tensors such as Lorentz tensors and of course Cartesian tensors. The former is defined in terms of Lorentz transformations while the other is defined in terms of orthogonal transformations in R³. An affine tensor is defined in terms of orthogonal transformations in Rⁿ.

 

Therefore it can only be said that a Cartesian tensor is not a general tensor (the term general does not refer to something which is more applicable, it only refers to the fact that the class of transformations are general). But is still a tensor, in this case a Cartesian tensor. After all it must be a type of tensor by definition. It can also be said that a Lorentz tensor is not a Cartesian tensor as that a Cartesian tensor is not a Lorentz tensor.

 

This definition is given in most of the really good math/physics texts including

 

Mathematical Methods in the Physical Sciences - Third Ed., by Mary L. Boas

Mathematical Methods of Physics, by Mathews and Walker

Classical Mechanics - Third Edition, by Goldstein, Safko and Poole

Classical Electrodynamics - Third Edition by J.D. Jackson

 

Given the fact that such well known physicists refer to Cartesian tensors a tensors make you rethink your assumption at least a little?

 

Pete

Edited July 26, 2008 by Pete
multiple post merged

[D H]

Where did you learn this from? Did you learn it from a particular text? A number of texts etc?

A number of texts, but the one I in front of me right now is F. Byron and R. Fuller, "Mathematics of Classical and Quantum Mechanics", which goes so far as to call them Cartesian-tensors rather than tensors because the name "Cartesian tensor" is misleading. All tensors are "Cartesian tensors", but the converse is not true. Cartesian tensors are not tensors.

 

There are other kinds of tensors such as Lorentz tensors and of course Cartesian tensors.

These are not other kinds of tensors. They are generalizations of the concept of a tensor. Then again, the Minkowski metric is a generalization of the concept of distance.

[Pete]

A number of texts, but the one I in front of me right now is F. Byron and R. Fuller, "Mathematics of Classical and Quantum Mechanics", which goes so far as to call them Cartesian-tensors rather than tensors because the name "Cartesian tensor" is misleading. All tensors are "Cartesian tensors", but the converse is not true. Cartesian tensors are not tensors.

Any decent text will refer them as Cartesian tensors rather than simply tensors and for a very good reason. But most texts will qualify it with "Cartesian" first and then drop the qualifier when it is agreed upon to be undrestood. The class of tensors which are defined in terms of generalized coordinate transformations are most often called tensors, i.e. tensor is shorthand for general tensor. It makes life easier if we agree on what the qualified is and then simply not use it.

These are not other kinds of tensors. They are generalizations of the concept of a tensor. Then again, the Minkowski metric is a generalization of the concept of distance.

Unless a particular definition is agreed on it will be impossible to get past these semantics. You don't honestly believe that someone can succesfully argue a point by merely repeating it, do you? I mean I can keep repeating that the term "tensor" refers to a multi linear map from vectors and dual vectors to real numbers and whether it is a general tensor or a Cartesian tensor will depent on what the domain of this map is, i.e. whether it maps Cartesian vectors and 1-forms or does it map general vectors and general 1-forms. Of course all this will do is to divert our attention from the definition of tensors to the definition of vector and their duals.

 

Can you please quote the definition of tensor the Byron and Fuller give in the text you referrenced? Thank you. If you could scan and and E-mail the scanned page then that would be even better.

 

I'm curious as to why you didn't respond to the question I posed above. I'll repeat my question - Doesn't the fact that such well known physicists and physics texts refer to Cartesian tensors being tensors give pause and/or make you wonder about whether you should rethink your position at least a little?

 

Let me clarify the position I have stated in more detail. Surely you're aware of the idea that there are two different but equivalent definitions of tensor in the literature. That definition refers to tensors as multilinear maps from vectors and dual vectors to real numbers. The matrix coefficients in A defined by the transformation dx' = A dx determine what kind of vector one is dealing with. The well-known change in coordinates which defines a tensor is actually what appears in the transfomation equation of tensors. I.e. to obtain the transformation of the components of a tensor all one needs is the transformation coeffcients of the basis vectors which change the old basis vectors to the new ones. This is how the elements of A come into play in the transformation law for tensors.

 

For this reason a tensor is classified as Cartesian, Lorentzian, Affine etc. according to the domain of the vectors and 1-forms. If a map takes Lorentz vectors and Lorentz 1-forms to a real number then that tensor is called a "Lorentz tensor". For that reason a tensor is classified according to its domain. I explained all this a web page I created which is located at

 

http://www.geocities.com/physics_world/gr_ma/tensor_via_geometric.htm

 

Unless we agree on a particular definition then it will be impossible to get past semantics.

 

Best wishes

 

Pete

 

Please note: I wanted to rewrite my last post to (1) make it more accurate and (2) more polite. Its hard to write something regarding a disagreement without comming across in way which is contrary to how it is intended (or perhaps its just me.

 

Is there a moderator/administrator who can delete my previous attempt at this post? Thanks.

 

A number of texts, but the one I in front of me right now is F. Byron and R. Fuller, "Mathematics of Classical and Quantum Mechanics", which goes so far as to call them Cartesian-tensors rather than tensors because the name "Cartesian tensor" is misleading. All tensors are "Cartesian tensors", but the converse is not true. Cartesian tensors are not tensors.

Any decent text will refer them as Cartesian tensors rather than simply tensors and for a very good reason. But most texts will qualify it with "Cartesian" first and then drop the qualifier when it is agreed upon to be implicitly understood. The class of tensors which are defined in terms of generalized coordinate transformations are most often called general tensors. When it is understood that the tensor is a general tensor then qualifier general is dropped and the general tensor is then simply called a tensor. It makes life easier if we agree on what the qualified is then simply call it a tensor.

 

From how it appears that what you are literally saying (although perhaps you are merely not implying it) is that a Cartesian tensor is not a tensor only in the sense that it is not a general tensor in that the only transformation applicable is an orthogonal transformation in E³. But since the term general tensor is sometimes meant shortened to mean tensor then it is not a tensor in that sense of the term. But to be precise that reads the same as

 

A Cartesian tensor is not a general tensor.

 

This has a different meaning than the statement

 

A Cartesian tensor is not a tensor.

 

in that the later implies something else. In such case the qualfier general should not be dropped since the resultant sentance has a different meaning than it does befoe it is dropped. In the end I believe we can both agree to the fact that the following statement is accurate

 

A Cartesian tensor is not a general tensor.

These are not other kinds of tensors. They are generalizations of the concept of a tensor. Then again, the Minkowski metric is a generalization of the concept of distance.

Your assertion These are not other kinds of tensors has a different meaning than A Cartesian tensor is not a general tensor where the later is the more precise statement. Qualifiers should never be dropped when their inclusion in a sentance can change its meaning and interpretation.

 

Also the Minkowski metric denotes a special conceptualization of the concept a metric in that the concept of distance takes on a new meaning than it does in Euclidean geometry. Here is another example of when qualifiers shouldn't be dropped.

 

Unless a particular definition and its meaning is agreed on it is impossible to get past semantics. A point cannot be succesfully argued merely by repeating it as I'm sure you know all too well. E.g. one can keep repeating that the term "tensor" refers to a multi linear map from vectors and dual vectors to real numbers. Whether it is a general tensor or not a Cartesian tensor will depend on what the domain of the map is, i.e. whether it maps Cartesian vectors and Cartesian 1-forms to real numbers or whether it map general vectors and general 1-forms to real numbers. The map itself has not changed and the map itself is the tensor. Of course all this will do is to divert our attention from the definition of tensors to the definition of vector and their duals and we are back to the same problem.

 

I was wondering if you could do me a favor? Can you either quote the definition of tensor used in Byron and Fuller's text (or in any other text that you believe agrees with how you are interepreting all of this) or scan the relavent pages into my computer and E-mail the scanned pages? Thank you. If you could make them available to me otherwise (e.g. upload it unto a web site and I can simply download it from there myself) then that would be even better. Thank you.

 

Let me clarify the position I have stated in more detail. Surely you're aware of the idea that there are two different but equivalent definitions of tensor in the literature, right? By this I am referring to the definition of tensor as it refers to a multilinear map from vectors and dual vectors to real numbers versus how a tensor is defined according to how its components transform from one basis to another. Let dx' = A dx define the matrix coefficients A_jk. Then the A_jk determines what kind of object/map we are discussing.

 

The well-known change in coordinates which defines a tensor is actually what appears in the transfomation equation of tensors. I.e. to obtain the expression for the transformation of the components of a tensor all one needs is the coeffcients of the old basis vectors which are used to express the new basis vectors. This is how the elements of A come into play in the transformation law for the components of tensors.

 

For this reason a tensor is classified as Cartesian, Lorentzian, Affine etc. according to the domain/classification of the vectors and 1-forms. If a map takes Lorentz vectors and Lorentz 1-forms to a real number then that tensor is called a "Lorentz tensor". For that reason a tensor is classified according to its domain. I explained this a web page I created which is located at

 

http://www.geocities.com/physics_world/gr_ma/tensor_via_geometric.htm

 

Unless we agree on a particular definition then it will be impossible to get past all this semantics.

 

Best wishes

 

Pete

Edited July 27, 2008 by Pete