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Einstein Notation - out of fashion?

Kiwi

Active member
Feb 7, 2012
109
Melbourne, Australia
I am studying Pavel Grinfeld's text "Introduction to tensor analysis and the calculus of moving surfaces" along with his you-tube lecture series on the same topic. I highly recommend both!

1.) Am I correct to understand that the Einstein tensor notation used throughout the book is out of fashion and doesn't get used anymore? I don't see any of it on mathhelpboards.

Here is the Riemann Christoffel tensor in Einstein notation:
\(\nabla_i \nabla_j T^k - \nabla_j \nabla_j T^k = R^k_{mij}T^m \) where \(R^k_{mij}= \frac{\partial \Gamma^k_{jm}}{\partial Z^i}-\frac{\partial \Gamma^k_{im}}{\partial Z^j}+\Gamma^k_{in} \Gamma^n_{jm} - \Gamma^k_{jn} \Gamma^n_{im} \)

2.) Now it is obvious to me that this is valid in a space of any dimension. Is the same true for modern notation without any special thought or is it necessary in each dimension to come up with a new definition for the symbols in each space?

3.) As a simpler example in the tensor notation it is not necessary to define the cross product. What we know as the cross product in 3 dimensions can easily be expressed in Tensor notation and then extended to any dimension without much thought. Can the same be said of the modern notation?

4.) Is there a name for the modern notation?

Cheers


Dave
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,632
Leiden
Hi Dave,

What is this modern notation that you speak of?
I am not aware of Einstein's notation conventions as being out of fashion. Quite the opposite.
A quick search does not reveal anything either.
I found a copy of the book you mentioned. It has copyright 2013.
That looks like quite a nice book btw.
It seems to just build up to Einstein's conventions as expected with no mention of it being out of fashion, nor can I find any mention of a new modern notation.

What am I missing?
 

Kiwi

Active member
Feb 7, 2012
109
Melbourne, Australia
Thanks Klass, I'm glad to hear that it is not out of fashion. My guess that it could be out of fashion was partly motivated by Pavel's comment:

"This particular textbook is meant for advanced undergraduate and graduate audiences. It envisions a time when tensor calculus, once championed by Einstein is once again a common language among scientists.

From a quick look at this area of MHB I don't see many indical equations that are recognisable to me.

Also when I look at an equation like:
$(A\times B)\cdot(C \times D)=(A \cdot C)(B\cdot D )-(B \cdot C )(A \cdot D )$

Here A, B, C and D are vectors and there are no indices to be seen. This kind of equation, which is what I think of when I say modern, needs to be remembered nowadays but is completely unnecessary using Einstein notation? It is also unclear to me how that equation would be interpreted (or if it would be valid) if the system dimension was different from 3.

This equation is the kind of thing we were shown in Engineering school (quite a few years ago). It includes separate concepts of dot product and cross product that aren't really needed (or at least can be thought of as secondary) in Einstein notation.
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,632
Leiden
Well, I think tensor calculus is kind of reserved for an undergraduate math/physics audience and up, just because the accompanying level of abstraction is otherwise too high.
Your example of the use of the cross product is what all other audiences are typically taught, although that formula is already more complicated than anything they will ever encounter.
It's not new - it's classic and predates tensor calculus.

When we are limiting ourselves to 3D, and when we are merely applying rules, I do consider the cross product easier to work with than the Levi-Civita symbol. Not to mention that students have to be careful when they write all those indices...
Oh, and yes, the cross product specifically only applies to 3 dimensions, just like the curl operator only applies to 3 dimensions.

Then again, perhaps in some future generation, tensor notation will become the standard to be taught in high school. After all, it simplifies the various calculation rules and is more generally applicable.

As for this area of MHB, I do see a number of questions about tensor calculus.
However, I think tensor notation is typically only relevant if we are talking about specific coordinate systems.
Theorems that are independent of coordinate systems transcend it.
 

Kiwi

Active member
Feb 7, 2012
109
Melbourne, Australia
Thanks again. I think my confusion has probably come from here, where you say:

"However, I think tensor notation is typically only relevant if we are talking about specific coordinate systems."

It seems that Pravel's particular quirk might be that he disagrees with this statement. He goes to a lot of trouble to say (in his opinion) that Tensor notation is applicable when we are not talking about a particular coordinate system, that this is a strength of the notation, and that the coordinate system can be considered at the end if required. Clearly an opinion that could be a bit contentious.

This begs new questions:
1. How would we write the Riemann-Christoffel tensor in a coordinate free notation?
2. Is it still called a tensor if it is not in the Einstein notation?
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,632
Leiden
Ah, I see now that Pavel mentions in his introduction:
A popular alternative to tensors is the so-called modern language of differential geometry.

Well, at least I believe that answers your question how it is called.
It is called the "modern language of differential geometry".
Searching for that specific string gives me a number of hits.
Suffice to say that I am not familiar with it at this time.

1. How would we write the Riemann-Christoffel tensor in a coordinate free notation?
Well, as I see it, that is just $R(u,v)$ without specifying a tensor formula for it, which would otherwise bind it to a coordinate system.

2. Is it still called a tensor if it is not in the Einstein notation?
A tensor is the representation in some coordinate system of a point or a tangent vector in an abstract manifold.
So we would for instance talk about a tangent vector $v$ at a point $x$ in manifold $M$.
This is indeed not a tensor. Instead it has a representation as a tensor $v^i$ in some coordinate system.

In essence it seems that Pavel prefers that we always use $v^i$ instead of $v$ (or $\mathbf v$) to refer to the vector with the understanding that it is with respect to some coordinate system, but without specifying what that coordinate system is.