TheSkySearchers

Thanks to @Graeme1858 this is a very nice discussion of Emmy Noether.

https://www.bbc.co.uk/programmes/m00025bw

Note that in the discussion one of the panel notes that the troubles with energy (and other) conservation in General Relativity arise from GR having "too many" symmetries: invariance under any coordinate transformation. This is the core issue that causes folks to make the error that GR does not conserve energy. This is why the stress energy for gravity can only be represented by a pseudo tensor. Too much symmetry is built into the theory. So if you have an emotional issue (like Erwin Schrödinger) about pseudo tensors you'll deny global conservation of energy in GR. If you've done therapy to get over that (like Noether did for Einstein, Hilbert, Landau, and Lifschitz), you'll realize that though the stress energy might be a pseudo tensor the divergence of the stress energy pseudo tensor is a tensor which combines with the divergence of stress energy tensor of matter to give a tensor conservation of energy law.

Thanks Graeme.

You're welcome.

Don't suppose you could do a stress energy pseudo tensor explanation for dummies?

Regards

Graeme

Graeme1858 wrote: ↑Tue Jul 06, 2021 9:32 pm You're welcome.

Don't suppose you could do a stress energy pseudo tensor explanation for dummies?

Regards

Graeme

That's an interesting proposition! I think it necessary to make clearer why I find Dr H's views (and too many others) on energy non conservation in GR so repugnant. (See viewtopic.php?f=74&t=19138) I'll post it here or there. I think it might be possible to do it by emphasizing the geometry......

Thanks to the both of you!

I think I've found a geometric path to explaining the difference between a pseudo tensor and a tensor that makes clear that balking at pseudo tensors is a silly quibble. I'll write it up and post it here.

BTW, it's easy to remember how her surname is written: No-ether.

I’m having some heartburn with my idea that I could make a simple post describing the difference between a tensor and a pseudotensor. It was a happy delusion while it lasted. Here is an attempt to fulfill the promised post. You are certainly welcome to point fingers and giggle at my abject failure compared to what I thought I could do.

The proper mathematical definition of a tensor is already ‘too hard’, for instance. But if we restrict ourselves to a shortcut that is found in Gravitation by Misner, Thorne and Wheeler we can say that a tensor of rank r in a space of dimension d is a square array of dr numbers and a multilinear multiplication rule that can convert r vectors into a number. I think that it’s best to proceed with examples.

Example #1
A number is just a tensor of rank 0 in a 1-dimensional space and the multiplication is the usual one. Multilinearity just means, taking a, b, c to be scalars: a(b+c)=ab+ac.

Example #2
I will assume that the reader knows what a vector is in rectangular Cartesian coordinates. If not there is:

https://en.wikipedia.org/wiki/Vector_no ... ar_vectors

I will also assume that the reader is familiar with the dot (or scalar product) of two vectors

https://en.wikipedia.org/wiki/Dot_product

In a 3-dimensional space a vector is a tensor of rank 1 and the multiplication rule is given by the dot (or scalar) product. Taking x, y, and z to be vectors and • to be the dot product multilinearity just means x•(y+z) = x•y+ x•z.

Now the difference a tensor and a pseudotensor is simply this: the form of equations involving tensors is completely invariant under any coordinate transformation. An equation among pseudotensors is only invariant under coordinate transformations that preserve the “handedness” of the coordinate system. If the handedness changes, there is a sign change. We can now give some examples of a pseudotensor once the handedness of the coordinate system is established.

Most coordinate systems obey the right-hand rule:

https://en.wikipedia.org/wiki/Right-hand_rule

in which the index finger is the direction of the 1st coordinate, the 2nd finger is in the direction of the 2nd coordinate and the thumb is in the direction of the 3rd coordinate.

Example #3
With the right-hand rule one can define the cross product of two vectors:

https://en.wikipedia.org/wiki/Cross_product

Now let a, b, and c be three noncolinear vectors in a 3-dimensional space. The link to geometry and pseudovectors can then be illustrated.

a•b = |a||b|cos(θ)

where θ is the angle between the two vectors in the plane they define. Positive angles are, by convention, angles which obey the right-hand rule.

For the cross product of b and c we write bxc and note that if φ is the angle in the plane between b and c that the magnitude of bxc is

|bxc| = |b||c||sin(φ)|

which is just the area of the parallelogram defined by b and c. The thing to notice is that

bxc = - cxb

In each case the object resulting from bxc and cxb has components perpendicular to the plane defined by b and c but the two different orders of the product give results that point in the opposite directions, hence the minus sign. The different orders have opposite interpretations under the right-hand rule.

Now consider the volume interior to a parallelepiped defined by the three vectors, a, b, and c. It turns out the volume of this parallelepiped is proportional to this dot and cross product of the three vectors.

a•(bxc) = +/- volume.

Thus a•b is a scalar; a•(bxc) is a pseudoscalar; a, b, and c are vectors; but the cross product of any two, say, bxc is a pseudovector.

The result of a pure tensor equation does not depend on the orientation of coordinates but a pseudo tensor equation depends by a sign on the coordinate orientation.

notFritzArgelander wrote: ↑Sun Jul 11, 2021 8:24 pm You are certainly welcome to point fingers and giggle at my abject failure compared to what I thought I could do.

Or not..... This is simply me telling a joke at my own expense as to how far I fell short of my goal....

Many of the ordinary quantities of mechanics and electrodynamics have tensor or pseudo tensor properties:

Mechanics:
position, velocity, acceleration force and momentum are vectors.
torque and angular momentum are pseudo vectors
kinetic and potential energy are scalars

Electromagnetics:
charge density is a scalar
current density is a vector
the electric field, polarization and displacement are all vectors
the magnetic induction, magnetization, and field are all pseudo vectors
the Poynting vector is a vector
the Maxwell stress energy tensor is a tensor

So I think throwing up one's hands in dismay that the gravitational stress energy is a pseudo tensor is an emotional reaction. All those magnetic pseudo vectors are pseudo tensors of rank 1 and so no hysteria is needed.

notFritzArgelander wrote: ↑Sun Jul 11, 2021 8:24 pm ....

The proper mathematical definition of a tensor is already ‘too hard’, for instance. But if we restrict ourselves to a shortcut that is found in Gravitation by Misner, Thorne and Wheeler we can say that a tensor of rank r in a space of dimension d is a square array of dr numbers and a multilinear multiplication rule that can convert r vectors into a number. I think that it’s best to proceed with examples.
....

The bold is a typo. What was intended is

a tensor of rank r in a space of dimension d is a square array of d^r numbers and a multilinear multiplication rule that can convert r vectors into a number

d^r not dr.....

notFritzArgelander wrote: ↑Mon Jul 12, 2021 4:24 am

notFritzArgelander wrote: ↑Sun Jul 11, 2021 8:24 pm ....

The proper mathematical definition of a tensor is already ‘too hard’, for instance. But if we restrict ourselves to a shortcut that is found in Gravitation by Misner, Thorne and Wheeler we can say that a tensor of rank r in a space of dimension d is a square array of dr numbers and a multilinear multiplication rule that can convert r vectors into a number. I think that it’s best to proceed with examples.
....

The bold is a typo. What was intended is

a tensor of rank r in a space of dimension d is a square array of d^r numbers and a multilinear multiplication rule that can convert r vectors into a number

d^r not dr.....

Thanks, I think I get the idea now. As I've said before, I usually start skipping ahead the moment I see the term tensor in an article. I'll try to persevere in the future.

GCoyote wrote: ↑Mon Jul 12, 2021 12:59 pm

notFritzArgelander wrote: ↑Mon Jul 12, 2021 4:24 am

notFritzArgelander wrote: ↑Sun Jul 11, 2021 8:24 pm ....

The proper mathematical definition of a tensor is already ‘too hard’, for instance. But if we restrict ourselves to a shortcut that is found in Gravitation by Misner, Thorne and Wheeler we can say that a tensor of rank r in a space of dimension d is a square array of dr numbers and a multilinear multiplication rule that can convert r vectors into a number. I think that it’s best to proceed with examples.
....

The bold is a typo. What was intended is

a tensor of rank r in a space of dimension d is a square array of d^r numbers and a multilinear multiplication rule that can convert r vectors into a number

d^r not dr.....

Thanks, I think I get the idea now. As I've said before, I usually start skipping ahead the moment I see the term tensor in an article. I'll try to persevere in the future.

Essentially a tensor of rank r in d dimensions is a function that has r d-dimensional vectors as arguments and gives a number as a value which is independent of the coordinates used. A pseudo tensor is a function that has r d-dimensional vectors and gives either a number or its negative depending on the orientation of the vectors in the arguments. Switch 2 vector arguments and you get a sign change.

notFritzArgelander wrote: ↑Mon Jul 12, 2021 7:17 pm Essentially a tensor of rank r in d dimensions is a function that has r d-dimensional vectors as arguments and gives a number as a value which is independent of the coordinates used. A pseudo tensor is a function that has r d-dimensional vectors and gives either a number or its negative depending on the orientation of the vectors in the arguments. Switch 2 vector arguments and you get a sign change.

Yeah, this kind of reminds me of how I didn't really progress my maths studies after coming up against Laplace Transforms in electrical principles. It did however, make me smile when I discovered years later that Laplace was an astronomer too!

Regards

Graeme

nFA,

here is a book about tensors:

http://www.danfleisch.com/sgvt/

and a video:

chasmanian wrote: ↑Wed Jul 14, 2021 2:21 am nFA,

here is a book about tensors:

http://www.danfleisch.com/sgvt/

and a video:

The video is completely correct and simple enough that it cannot be made simpler. The emphasis on components and basis vectors for vectors and tensors, while perfectly correct, is slightly old fashioned. It was cutting edge in mathematical physics in the 19th and early 20th century. This approach is algebraic and not geometric and emphasizes manipulating components. I've looked at samples of the book on line and find that it holds true of the book also.

For most of my career, certainly through undergrad school, I was perfectly happy with that. Then I took a course in graduate school on General Relativity which had as texts Weinberg (Gravitation and Cosmology, which is written from the older algebraic POV of tensors) and Misner, Thorne, and Wheeler (Gravitation, which is written from the more modern geometric view of tensors).

MTW caused the scales to fall from my eyes and I underwent a profound conversion to the geometric POV. Tensors were no longer coordinates which transformed in a way that along with basis vectors. They were objects that when multiplied by a sufficient number of vectors produced a scalar that was invariant of any coordinate system.

Perhaps after one has mastered the material in the book you recommend one might try the geometric POV pioneered by Cartan? To that end I recommend https://www.amazon.com/Geometrical-Meth ... 0521298873

I recommend mastering the older algebraic approach first. I know of one would be mathematical physics person who was permanently damaged by skipping that step.....

PS (edit) The damage that the person experienced was that they were unable to compute a number with any facility having bypassed the compont algebra approach. My recommended book Bernard Schutz's Geometrical Methods of Mathematical Physics is an excellent remedy against that.

Thank you again nFA.
Enjoying the subject and these comparative ways to build up to a new concept really work for me.

I wish I'd had you to teach me math in Univ.

chasmanian wrote: ↑Wed Jul 14, 2021 2:21 am nFA,

here is a book about tensors:

http://www.danfleisch.com/sgvt/

and a video:

Thanks, that was actually pretty easy to follow.

SparWeb wrote: ↑Wed Jul 14, 2021 1:15 pm Thank you again nFA.
Enjoying the subject and these comparative ways to build up to a new concept really work for me.

I wish I'd had you to teach me math in Univ.

Agreed. My inadequate high school math preparation was further distorted by undergrad courses taught by grad students who were only doing it to avoid washing dishes for money. Their distain for us 'normals' made the entire experience frustrating.

GCoyote wrote: ↑Wed Jul 14, 2021 2:27 pm

SparWeb wrote: ↑Wed Jul 14, 2021 1:15 pm Thank you again nFA.
Enjoying the subject and these comparative ways to build up to a new concept really work for me.

I wish I'd had you to teach me math in Univ.

Agreed. My inadequate high school math preparation was further distorted by undergrad courses taught by grad students who were only doing it to avoid washing dishes for money. Their distain for us 'normals' made the entire experience frustrating.

Agreed here too. There's a severe cultural problem in most Anglo-American maths departments that IMO amounts to dereliction of duty. The instruction is often provided along a "straight and narrow" path. Exploring alternate routes which might be more congenial to many more students is discouraged.

I happen to be one of those examples where I seem o be good at mathematics but I do not learn by rote obedience as some schools teach it. Confronted with those teachers and my learning suffered.
I was once actually told to stop asking about the real applications of calculus (grad and curl) as if I was expected to master a tool with no material to work it on.
Many times I struggled until I could grasp a real use of the math being taught.
Eventually I developed ways to teach myself the methods, and remastered differentials that way.