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Thread: Definitions

  1. #1 Definitions 
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    I recently ran into a discussion concerning tensors. When the assertion as made that tensors are purely algebraic entities I expained that tensors are classified a geometric objects. This didn't sit too well. I guess that some people have a different idea of what a geometric object is than how it's defined. I hazard to guess that is why the objection to the definition. I gotta tell ya, I'm real tired of discussing definitions so much. I mean it's pretty clear how things are defined in physic and math. Just look it up and you know the definition. Pretty simple. So why do people feel the need to argue about definitions so much? I mean is there a better way to spend your time than argue about your dislike of a definition?

    Just curious.


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  3. #2  
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    I would be inclined to view tensors as algebraic entities rather than geometric objects. Particular tensors such as the Riemann tensor can be considered as geometric objects, but the notion of a tensor itself isn't geometric. I suppose one ought to consider what it actually means to be "geometric". For example, is the manifold of a Lie group geometric?


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    The simplest way to view tensors is that they are a generalization of the concept of matrices and vectors. Vectors can be looked as singly indexed arrays, matrices are doubly indexed arrays, tensors are multiple indexed. In appearance, it looks like linear algebra, but the principle application is in differential geometry, which is used in relativity theory.
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  5. #4  
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    Quote Originally Posted by mathman View Post
    The simplest way to view tensors is that they are a generalization of the concept of matrices and vectors. Vectors can be looked as singly indexed arrays, matrices are doubly indexed arrays, tensors are multiple indexed. In appearance, it looks like linear algebra, but the principle application is in differential geometry, which is used in relativity theory.
    I disagree with this because it focuses on the wrong aspect of what a tensor is about. Instead of its algebraic structure, a tensor is about how it transforms under some given group of transformations. For example, the partial derivative of a vector is not a tensor and one defines covariant derivatives specifically to be tensors.
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    Quote Originally Posted by KJW View Post
    ...but the notion of a tensor itself isn't geometric. I suppose one ought to consider what it actually means to be "geometric". For example, is the manifold of a Lie group geometric?
    That can’t be said unless the definition of geometric object is first established. I think were probably confused by the fact that the definition of geometric object used in differential geometry is not the same thing as they learned in their was that it’s not the kind of thing one is introduced to in a highschool geometry course.

    In order to determine whether the notion of a tensor is geometric or not one must neccesarily first determine what is meant by the term geometric object. One simply cannot get around that all too simple fact. Only then can you determine whether something is a geometric object or not. If I were to post the definition of a geometrical object as it’s defined in, say, Differential Geometry by Erwin Kreysig, Dover Pub., (1991). If you’d like I could post the definition. Or you could do me the favor of going to Electronic library. Download books free. Finding books downloading it, turn to page 91 and post the definition for me? Save me the trouble and the pain of posting all that? Pretty please?

    It doesn't really matter anyway. I was glad to leave the place. Bad vibes. E.g. there’s some undergraduate know-it-all there who thinks he knows everything. Yet when it comes to simple things like the difference between a Lorentz scalar and a generalized scalar (aka just plain “scalar”) forget it! I really hate it when people have to argue about basic definitions. Why can’t People simply look up and accept what the definition of the term in question and be done with it, regardless of whether you find it to be suitable to your taste or not?

    They had some really screwy ideas of the relationship between tensors and coordinate systems too. Regarding the difference in nature of the geometric definition of tensor and they analytical definition (e.g. defined according to they way the components transform), the authors of Gravitation and Spacetime – 3rd Ed., Ohanian and Ruffini explain
    Furthermore, we must not forget that the physicist who wishes to measure a tensor has no choice but to set up a coordinate system and then measure the numerical values of the components. Thus, to carry out the comparison of theory and experiment, the physicist cannot ultimately avoid the language of components - only a pure mathematician can adhere exclusively to the abstract, coordinate-free language of differential forms.
    Their counter arguement is that one doesn't need a coordinate system but simply a set of basis vectors upon which they could project the components of the tensor onto. They didn't understand the basic fact that there is a close relationship between basis vectors, tensors and coordinate systems. People often forget the prototype of the vector, i.e. the position vector. The position vector is defined as the displacement of a given point from a point chosen as the origin of the coordinate system. Therefore if one claims to be able to specify tensors using basis vectors then one has to also be able to specify the displacement vector and that requires a reference point. Given a reference point and a basis that defines a coordinate system.

    When they started to claim that the authors, Ohanian and Ruffinit, didn't know what they are talking about I realized that it was time to abandon that place. When I hear comments like that made about experts as good as these gentlemen then I know that it's time to walk away anyway.
    Last edited by PhyMan; June 30th, 2013 at 04:35 PM.
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    Furthermore, we must not forget that the physicist who wishes to measure a tensor has no choice but to set up a coordinate system and then measure the numerical values of the components. Thus, to carry out the comparison of theory and experiment, the physicist cannot ultimately avoid the language of components - only a pure mathematician can adhere exclusively to the abstract, coordinate-free language of differential forms.
    I'm not a fan of the coordinate-free approach. While tensors (as they apply to physics) are about equations that are form-invariant with respect to coordinate transformations (and therefore independent of the coordinate system), I still feel that the notion of a coordinate system is important (but no preferred coordinate system).
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    Quote Originally Posted by mathman View Post
    The simplest way to view tensors is that they are a generalization of the concept of matrices and vectors. Vectors can be looked as singly indexed arrays, matrices are doubly indexed arrays, tensors are multiple indexed. In appearance, it looks like linear algebra, but the principle application is in differential geometry, which is used in relativity theory.
    I disagree. I think this gives the wrong impression of what a tensor or a geometrical object is. People often confuse matrices and ordered sets of numbers with tensors and vectors. Are you familiar with the text Introduction to Electrodynamics by David Griffiths? If you can get a hold of it look at the section labeled How Vectors Transform. It has a discussion of why it's wrong to define a vector as an ordered set.
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    Quote Originally Posted by KJW View Post
    I'm not a fan of the coordinate-free approach. While tensors (as they apply to physics) are about equations that are form-invariant with respect to coordinate transformations (and therefore independent of the coordinate system), I still feel that the notion of a coordinate system is important (but no preferred coordinate system).
    The way I prefer to avoid confusion is to define tensors as multilinear maps from the set of vectors and its dual space (1-forms) and then derive the transformation property or vise-versa. Have you ever done this? I highly recommend it. It helps when one is attempting to understand how the two ways of defining tensors are related to each other and how things like Lorentz tensors or Cartesian tensors fit into the mix.

    When you made the comments above regarding tensors not being geometric objects what is it you had in mind about what a geometric object is such that tensors don't fit that nottion? Thank you in advance.


    Note - Long story cut short - Ohanian and Ruffini are correct and those who thought they are wrong made a boo-boo!

    Also, all one has to do to verify that tensors are geometrical objects is do an internet search using Google. For example, please see

    http://beige.ucs.indiana.edu/P573/node106.html
    Vectors and forms are geometric objects, they have life of their own independent of systems of coordinates. Their coordinates, which is what you get in a column or a row, will vary depending on a choice of a system of coordinates, whereas the vector itself remains unchanged.
    http://en.wikipedia.org/wiki/Tensor
    Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors.
    http://webee.technion.ac.il/people/b...ntoTensors.pdf
    Like all tensors, it is a geometric object, invariant under change-of-basis transformations.
    http://www.ita.uni-heidelberg.de/~du...sor/tensor.pdf
    [quote]

    http://farside.ph.utexas.edu/teachin...es/node12.html
    Recall that tensors are geometric objects which possess the property that if a certain interrelationship holds between various tensors in one particular coordinate system, then the same interrelationship holds in any other coordinate system which is related to the first system by a certain class of transformations.
    http://samizdat.mines.edu/tensors/ShR6b.pdf
    (explains what geometric objects are)

    Whew! I think that's enough.

    No wonder they were so short tempered. If I was wrong as often as they are I'd be short tempered too [hey! who said that! ]


    Are we now clear on how the term geometric object is defined?
    Last edited by PhyMan; June 30th, 2013 at 10:21 PM.
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    Something's been bothering me about all of this. I think I understand now. I don't believe a basis can be constructed until a coordinate system is defined. I general, geometric objects are not dependant on a particular basis to be defined. However that can't be said for the basis vectors themselves since they are defined according to a coordinate system. E.g. the basis vectors that we all know and love i, j, k are defined as being parallel to the x, y and z axes respectively. So that person's claim that one doesn't need a coordinate system to make measurements against is false. E.g. for a Cartesian coordinate system you have to know the direction of the x, y and z axes in order to know what direction the basis vectors i, j, k are pointing.

    Okay. I'm happy now.
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  11. #10  
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    Quote Originally Posted by PhyMan View Post
    The way I prefer to avoid confusion is to define tensors as multilinear maps from the set of vectors and its dual space (1-forms) and then derive the transformation property or vise-versa. Have you ever done this?
    Not explicitly. I tend to start with the chain rules of multivariable calculus to establish the coordinate transformation laws for covariant and contravariant vectors ( and being the prototypical examples), extending this to tensors by means of the outer (tensor) product. By focussing on the transformation properties of mathematical objects, my approach to relativity is via the connection rather than the metric. I regard the various expressions obtained as systems of partial differential equations for which integrability conditions can be obtained, and these lead to the definition of curvature as well as the Bianchi identity for the curvature. Thus, while I am ultimately applying the notions of differential forms, exterior products, etc, I choose not to use the modern notation.

    Quote Originally Posted by PhyMan View Post
    When you made the comments above regarding tensors not being geometric objects what is it you had in mind about what a geometric object is such that tensors don't fit that nottion?
    As I mentioned above, I would regard the Riemann tensor as a geometric object. I would tend to limit the notion of geometric objects to those objects that are associated with a geometric space. Personally, I prefer to downplay the whole notion of geometry even within physical space, and regard everything in terms of algebra and analysis. I feel that the modern approach, particularly its jargon, tends to promote an excessively geometric picture. Also, I feel that the modern approach tends to overcomplicate what are really fairly basic notions.
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  12. #11  
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    Quote Originally Posted by PhyMan View Post
    Something's been bothering me about all of this. I think I understand now. I don't believe a basis can be constructed until a coordinate system is defined. I general, geometric objects are not dependant on a particular basis to be defined. However that can't be said for the basis vectors themselves since they are defined according to a coordinate system. E.g. the basis vectors that we all know and love i, j, k are defined as being parallel to the x, y and z axes respectively. So that person's claim that one doesn't need a coordinate system to make measurements against is false. E.g. for a Cartesian coordinate system you have to know the direction of the x, y and z axes in order to know what direction the basis vectors i, j, k are pointing.

    Okay. I'm happy now.
    In the orthonormal tetrad (vierbein) formalism of GR, the orthonormal tetrad is a geometric object that generally doesn't correspond to the basis of any coordinate system. They act as invariant rulers and clocks by which objects can be compared. As such, they better match how physics is done in practice. By expressing the physical objects in terms of standard physical objects, one does away with the need for a coordinate system, and highlights the relativity in physics.
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  13. #12  
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    Quote Originally Posted by PhyMan View Post
    Quote Originally Posted by mathman View Post
    The simplest way to view tensors is that they are a generalization of the concept of matrices and vectors. Vectors can be looked as singly indexed arrays, matrices are doubly indexed arrays, tensors are multiple indexed. In appearance, it looks like linear algebra, but the principle application is in differential geometry, which is used in relativity theory.
    I disagree. I think this gives the wrong impression of what a tensor or a geometrical object is. People often confuse matrices and ordered sets of numbers with tensors and vectors. Are you familiar with the text Introduction to Electrodynamics by David Griffiths? If you can get a hold of it look at the section labeled How Vectors Transform. It has a discussion of why it's wrong to define a vector as an ordered set.
    I suppose that the difference of opinion reflects the difference in where the definition comes from. My exposure to tensors was in a course of differential geometry (a pure math course) - yours aapears to have come from a physics text.
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    Quote Originally Posted by mathman View Post
    I suppose that the difference of opinion reflects the difference in where the definition comes from. My exposure to tensors was in a course of differential geometry (a pure math course) - yours aapears to have come from a physics text.
    I chose to learn it from many different sources of math and physics texts so as to get a broad viewpoint.

    I think that problem with that other person(s) was that they either didn't understand or chose not to try to understand what I was trying to explain to them. Ohanian was refering to the fact that in a lab one makes measurements and records them. The data has a certain meaning and one has to understand how the data was recorded to understnd the meaning of the data/

    For example; Let's say that you want to record the motion of a particle and then analyze it later. How is that done? If you want to record and then analyze the trajectory then you must record it. That's true by definition. To record it you first must measure it. That amounts to measuring the position vector as a function of time. How is that measured and recorded? I could use optics and record the image on a CCD and then to analyze it I take the CCD data and try to extract the trajectory from that. The data on the CCD is a set consisting of an array of pixels which records a certain level of light data on them. The array is basically a Cartesian coorinate system. So we eventually ended up with a Cartesian coordinate system. Or perhaps there was a more direct route where we were able to record the position vector directly by recording the components of a particular basis. However all that is is just a coordinate system. The projection of the position vector on the ijk basis is simply a Cartesian coordinate system, by definition. I don't think the people I was speaking to quite grasped that. They claimed that no coordinate system was needed and that all one needs to do is measure projections onto a basis, i.e. the components of vector on a particular basis. When they told me that I knew that they really didn't understand what a coordinate system was since what they just described was the equivalent of a coordinate system. I started out by saying that measuring the components of a tensor on a basis was equivalent to measuring the tensor on a coordinate system.

    After that I decided that it was a waste of time explaining it. It occured to me that what they were doing was attempting to find a way not to admit that they made a mistake, i.e. admit that I was rigtht. Some people have a problem with that kind of thing and I decided after that to not make anymore attempts to explain things like that. I just have to learn the difference between not understanding and not willing to admit being wrong and having to learn something. Since that comes into play mostly in regards to definitions I decided to stay away from discussions about definitions as much as I can. At least with a few people. Who that is will take time tol figure out. I'll do it eventually but I won't mention names. That's not my style.
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    I edited this because of guitarists complaint.

    I was explaining something to someone and believe it's worthwhile to describe here.

    I'm leaving in a misquote to demonstrate how one needs to pay attention to what they read.

    The following comment in Gravitation and Spacetime by Ohanian and Ruffini which appears on page 223 was mentioned. The authors write
    We must not forget that the physicist who wishes to measure a tensor has no choice but to set up a coordinate system, and then measure the numerical values of the components. Thus, to carry out the comparison of theory and experiment, the physicist cannot ultimately avoid the language of components; only a pure mathematician can adhere exclusively to the abstract coordinate-free language of differential forms...
    The bogus counter argument presented is as follow arguement
    Ohanian and Ruffini are wrong on this point ("no choice but to set up a coordinate system"). You can set up a set of basis vectors without setting up a coordinate system. Components are then contractions with one of the basis vectors, which is still coordinate-free. So just because you are dealing with components doesn't mean that you are dealing with coordinates. Their conclusion ("no choice ...") doesn't follow from their argument ("cannot ultimately avoid ... components").

    Coordinates always imply a unique vector field called the coordinate basis, but a basis does not imply a unique coordinate system.
    This is incorrect so I explained
    Ummmm, nope! I'm afraid that you missed a very fundamental fact here. Ohanian and Ruffini are absolutely correct. They're experts in their field and know precisely what they're talking about. Setting up a system of basis vectors is identically the same thing as setting up a coordinate system. Contracting components on a basis is just another name for measuring components. Dealing with components is identical to dealing with a coordinate system. From your response it appear as if you might have a flawed notion of what a coordinate system is.
    which is quite correct.

    Here how a mistake was made. What I wrote was misconstrued as follows
    That example was a response to Phy_man's assertion that a coordinate system and a set of basis vectors are "identically the same thing". They are clearly not the same thing.
    Do you see the problem here? Nobody made such a claim.

    Here's the reason I said what I said; walk into a room where there is a static electromagnetic field present. The walls form a box. I want to set up a basis in which I can measure all possible geometric objects in the space. So I choose three basis vectors i, j and k and orient them so that the basis vectors are mutually orthogonal and each unit vector is parallel to two of the normal vectors of two walls. The two vectors i and j span a plane parallel to the floor and the two basis vectors i and k span one wall and likewise with j and k which spans the other wall. So now I have the orientation of the ijk basis. Recall that I said that I want to be able to express all possible geometric objects with this basis. That means that I must include the prototype of vectors which is the position vector (or in spacetime the 4-position 4-vector) The position vector is the displacement vector between a point on a curve and a reference point which you choose, probably for convenience. So you must choose a reference point or you'll bhe unable to describe all possible geometric objects.
    So now you have the i,j,k basis and a reference point. Now you can describe the motion and record the results quantittively. What you do is project the displacement vector onto each basis vector. The results, which I'll label x, y and z defined are as follows

    x = Projection onto i = R*i

    y = Projection onto k = R*j

    z = Projection onto k = R*k

    Do you recognize these quantities? They are, by definition, the Cartesian coordinates of the system. That's why I said that setting up one means setting up the other. Once you create a basis itís not enough to describe the motion entirely. You need a reference point. I think they didnít understand that fact. There are other ways to set up a basis as I'm sure you can guess. You can choose a reference point and a set of axes to reference angles from and then use spherical coordinates using a similar process.

    Final note on math/physics before I leave for good - All this was in regards to flat spacetime and Euclidean space. I also applies only to only holonomic bases.

    Guitarist - You're one poster I couldn't miss. Plus you lack the gland which allows other humans to admit their mistakes. This forum would be much better if you and the other postes who've flamed me weren't here.

    Goodbye and good luck all. I'll miss many of you.
    Last edited by PhyMan; July 2nd, 2013 at 04:45 PM.
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  16. #15  
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    Mod note: PhyMan, this forum is not the place to air your grievances against members and/or moderators on another forum - they are no interest to us. While not strictly against forum "rules", this thread is in very bad taste. Please desist.

    PS For what is is worth, I agree with your interlocutor there, but I decline to be drawn into a discussion about it in this thread.

    PPS look out for a PM from me
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    Quote Originally Posted by Guitarist
    PS For what is is worth, ..
    Exactly nothing. You're a very rude person for whom I have no respect. Why would I, or anybody for that matter, care what you think? Don't answer. It's a rhetorical question.
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