Tensors

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Introduction

As a retired K9 math teacher I said to myself I want to understand the math language used in the Einstein field equations. This led me to the issue tensors. In my math studies in Sweden we never studied more than vector calculus. I had to repeat vector calculus and learn about tensors. This page collect what I have learned so far and what sources I used. Maybe you will find it useful too.

This in one of several math pages. You find the others in the menu above under learning math.

Index

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Greek letters in math and science

This is a copy from the same section in my equation page.
I took this from Wiki  and adjusted it with comments that I find useful.

Αα Alpha Νν Nu
Ββ Beta Ξξ Xi
Γγ Gamma Οο Omicron

Curly d
In partial derivatives
 δ  Kronecker delta
Δ δ= Delta Ππ Pi
Εε Epsilon Ρρ Rho
Ζζ Zeta Σσς Sigma
Ηη Eta Ττ Tau
Θθ Theta Υυ Upsilon
Ιι Iota Φφ Phi

Line maybe curved as slash /

Κκ Kappa Χχ Chi
Λλ Lambda Ψψ Psi
Μμ Mu Ωω Omega
    Σσ
Sigma

not equal ~ About

Source: Wiki 

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Vector definition

I jumped into  Dan Fleisch youtube (see below under tensor basics) but discovered that I needed a basic vector update. I found that EigenChris offers a good one in this video.

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Tensors basics

I started this issue writing my page about Einsteins field equation of general relativity in http://www.kinberg.net/wordpress/stellan/einsteins-field-equations-of-general-relativity/
Tensors are “the facts of the universe” (Lillian Liebherr)

I have seen this video where Dan Fleisch  gives a very good explanation about tensors. You need to know about vectors as I do but Dan explains vectors to  in case you forgot about these.  Vectors is just one type (rank 1) of tensors in the tensor family. It has to be seen several times to understand the indexing system.

“Ax is the component (that pertains to and is a perfect representation of) the x-hat vector ( ed. x-unit vector)”

(In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or “hat”:  pronounced “i-hat”” (Source . Wiki )

Unit vectors are quite well explained in https://en.wikipedia.org/wiki/Unit_vector

like“the unit vectors in the direction of the x, y, and z axes of a three dimensional Cartesian coordinate system are”

“A student guide to vectors and tensors” is available as a ebook at books.google.it/books

There is a kind of tensor called metric tensor. I like the definition shared in mathworld.wolfram.com/MetricTensor.html

“Roughly speaking, the metric tensor g_(ij) is a function which tells how to compute the distance between any two points in a given space….,.” (Read more in wolfram.com )

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Tensor definition and calculus

How you can understand the rank of a tensor is well explained by eigenchris in the video below. In his channel you find a serie of tensor calculus videos.

When he continues with calculus it become very difficult if you never have done vector calculus. I have seen this video at least three times. 🙂
EigenChris shares two good definitions of tensors.

  1. Array definition: tensor is a multidimensional grid of numbers
  2. Coordinate definition: A tensor is an object that is invariant under a change of coordinates and has components that change in a predictable way under a change of coordinates.
  3. Abstract definition: Tensor is a collection of vectors and covectors combined together using a tensor product.
  4. Tensors as partial derivatives that transform with the Jacobian matrix  (Requires knowledge of matrix calculus and the Jacobian determinant )

I see that it is good to read the math world.wolfram.com definition of tensor in http://mathworld.wolfram.com/Tensor.html :

“An nth-rank tensor in m-dimensional space is a mathematical object that has n indices and m^n components and obeys certain transformation rules.”

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Tensors visualized

  1. V1=1.826 (with subscript for the index value) is a variable with co-variant components 
  2. V1=4  (with superscript for the index value) is a variable with a contra-variant components  

Multiplying P and V

  • the contra-components of a vector P
  • with the contra-variant components of a vector V

e get these matrix of 9 possible multiplications or tensors rank 2:

11= VP11  12= VP12 13= VP 13
T 21= VP21 22= VP22 23= VP 23
T 31= VP31 T 32= VP32 T 22= VP22

 

Multiplying P and V

  • the con-variant components of a vector p
  • with the contra-variant components of a vector Vwe get these tensors:
11 12 13
T 21 22 23
T 31 T 32 T 32=

 

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The stress tensor

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Forward transformation

 

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Backward transformation

Back ward transformation gives us the inverse of matrix F (F-1)

This can be checked by mutliplying F and B
If you multiply FB you get I (the identity matrix) So B is the inverse of F. F=B-1. Read more about inverse matrices in http://www.kinberg.net/wordpress/stellan/matrices/#issue2

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Chain rule

To write a equation for a 2 frame system, we need the chain rule that tells how to derivate a external function with a inner function.

In calculus, the chain rule is a formula to compute the derivative of a composite function. That is, if f and g are differentiable functions, then the chain rule expresses the derivative of their compositef ∘ g — the function which maps x to — in terms of the derivatives of f and g and the product of functions as follows:

Alternatively, by letting F = f ∘ g  F(x) = f(g(x))for all x, one can also write the chain rule in Lagrange’s notation, as follows:

The chain rule may also be rewritten in Leibniz’s notation in the following way. If a variable Φ depends on the variable x, which itself depends on the variable y (i.e., y and z are dependent variables), thenΦ, via the intermediate variable of x, depends on y as well. In which case, the chain rule states that

(Source: Wiki  )

I found 3 applications of this rule: from  NancyPi Youtube

1 example from Khan

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Chain rule with Leibniz’s notation

 is explained by rootmath with this video: 

here is the printscreen with the derivation of y=(3x+1)2 solution withLeibniz’s notation.

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Metric tensors in EFE

This part is Actually a copy taken from my equation page.

This is Einsteins field equation (EFE) in simplified form. It is called Einstein Field equations as it is really 16 equations  depending on the value of the indeces (μν) for g and T in the equation. 6 are duplicates. (read more below)

Source: Physicist page Facebook )

(Source: Dr Physics A Youtube )

This is a very good video that I recommend where . DrPhysicsA explains some basics and derives the equation.

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Dr PhysicsA Lesson content

  • How you can feel gravitation and acceleration force
  • How light bends in gravitation fields.
  • 5:33 Proven at a solar eclipse
  • 5:50 Why Newtons gravitation law does not apply to (0 mass) photons.
  •  6:42 All forms of movement are represented by movement in curved spacetime
  • Trampoline metaphore
  • 9:00 What is spacetime?
  • 11:00 Not only one equation as it looks like.
    The indeces μν represent the spacetime 4×4. 6 are duplicates = 10 EFE
  • 13:15. “Everything on the left hand of EFE refers to curvature of spacetime. Everything on the right hand side is to do with mass and energy. What Einstein field equation basically say is that
    mass on (ed. on the right side)
    tells  space-time how to curve
    and curved space-time (ed. on the left side) tells mass how to move'”

    Describing the metric tensor in EFE.

    • 15:15 A  bumpy field
    • 16;13 How does my height change
    • 17:40 gradient (dϕ/dx)
    • 23;23 Critical step 1. Partial derivative
    • 24:24 Relabeling the coordinates xyz
    • 14:50 metric tensor
    • 24:35 Critical step 2. relabeling the coordinates.
    • 29:47 Critical step 3. The chain rule. (see above)
    • 27:38 Length contraction – Time dilation
    • 32:05 Tensor – relationship between two vectors.
    • 34:31 Vector transformation
    • 42:17 Critical step 4. Contravariant and covariant transformation (see above)
    • 46:15 Kronecker delta
    • 49:45 Metric tensor – correct pythagoras equation on a curved space.
    • 53:14 Christophel symbol

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Newtons learning method

Newton studied alone Descartes Geometry. Newtons biographer John Conduitt, described how he did.  When he got stuck (reavher a critical point) he restarted, reading from the beginning until he got stuck again. He advanced a few pages after every rereading. (source: “Newton, the math genious”, Italian version, Jose Muñoz Santonja,  page 25). I did something similar.
I share in my notes belowe my own critical issues/steps.

  • I followed the teacher until I got stuck for some reason.
  • I got stuck e.g. when he came to using the chain rule.
  • I started then to learn about the chain rule from another teacher as you can see in the notes below
  • Then I scrolled back to minute 17.:40 to restart from there.
    Everytime , I got stuck I restarted from somewhere.

Good luck with following DrPhysicsA. He is a very good teacher:

Derivation of the metric tensor

Walking in x or y direction

dΦx is the change in height if you move in the x-direction.
dΦ/dx is the gradient in x-direction
These two equations can be used to calculate dΦs for the resulting vector s as shown below.

Pythagoras and vectors

I were confused of his using ϕ (capital phi) in the same way as dh where h=height in the field. I understand that dϕxis a movement in the field that is  parallell to the x axes.

With Pythagoras we get a formula for 2-dimensional vectors like ds in the image above. 

  • Φ/∂x is the gradient in the x-direction
  • Φ/∂y is the gradient in the y-direction

 

Critical step 1. Partial derivative.

The term “Partial derivative” has confused me with its new symbol ∂(curly d).

Let say  you have  a function  that depends on  e.g. 2 variables. If you want derivate this function, then you have to make 2 partial derivations. Every partial derivation is a partial derivative that is ddnoted with a curly d that has the  character (Unicode: U+2202). It is a stylized cursive/curly d.

  • Critical step 2. Relabeling the coordinates xyz

This is a difficult move. having been used to the coordinates xyz it was confusing to relabel xyz. It took me a few weeks to get used to this new coordinate system.  As Newton did, I had to scroll back a few times.

x0 is used for time coordinate.

Relabeling xyz  is necessary to be able to have more coordinates as in e.g. a 11 dimension system .If you are not used to it, you get rather confused with the following calculations. So I keep this screenshot here as a memory backup.

With the new coordinate system this equation

can be rewritten as

where n is the indeces for the x coordinates.

Critical step 3 – Two (2) frame system

IMPORTANT! The y-coordinate will still be used but for another set of coordinates/frame of reference.

See the image below where a point P can be described

  • in the x-frame of reference: with the coordinates x1 and x2
  • in the y-frame of reference: with y1 and y2

If you want to describe P with the ycoordinate then you get a different value of  x1 and x2

 

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Critical step 4. DrPhysicsA application of the Chain rule.

If we have 2 reference systems xn and yn, then we have to describe P with both reference systems.

To calculate all the gradients in the y-frame reference, we must  us to use the chain rule with Leibniz’s notation.

“If we want to express the  y1 position of P in terms of x-coordinates, we must express y1 position as  x1 and x2 coordinates.

find all the gradients in the y frame of reference

“If we know all the gradients in the x frame of reference, how can we find all the gradients in the y frame of reference. How can we relate the two?

With the chain rule we get the gradient (derivative)∂Φ/y1 that is the sum of the gradients of x1 and x2. (ed. The left partial derivative in every addition term is the gradient in the x1 direction.) 

I have understood this:

As we have  a 2 frame system, to get the gradient (derivative) ∂Φ/y1. we must use:

  1. the gradient (derivative) for       x1 (∂Φ/∂x1)
  2. and the gradient (derivative) for  x2 (∂Φ/∂x2

To see how the chain rule for the composite function f(g(x)), has been applied, I understand that we have two functions:

  1. f(y)       —> external function
    and
  2. y=g(xn)   —> internal function (=gradient of xn )

In the standard application for the chain rule, as explained above,
you derive the external function first and then the internal
that is you do [  f'(g(xn)) * g'(xn))  ]

( Source DrPhysicsA min 31:30 )

  • I understand that ∂Φ/∂x1 is the derivation of external function f(g(y1
  • and ∂x1/∂y1 is the derivation of internal function g(y1)

So DrPhysicsA  makes the multiplication  f'(g(y1) * g'(y1)

 

and generalize for n=1-2 with f'(g(xn)) * g'(xn))

I hope I understood this right.

34:31 Vector transformation

We take the function of Φ

and make a slight adjustment to it. Se get

A vector in the x-frame reference Vxm with 2 coordinates (1 and 2 as we talk about a 2 dimension system)  transforms to a vector  in the y frame reference by multiplying that vector with the  2 gradients in the y-frame reference.

This sounds like a forward transformation as i described above but I am not sure about that.

41:43 Tensor combination

 

AxrBms=Txrs

Txrs is the tensor in the x-frame of reference.

This is the contravariant transformation, that is the

transformation from the y frame reference to the x frame reference explained in the backward transformation section above.

The covariant transformation is made with this equation:

45:14

45:58 In Pythagoras you have (x1)2 and (x2)2. There is no x1+x2
So a fourth term is added that is δmn that is called Kronecker delta

δ that is called Kronecker delta

  • If m=n δm n = 1
  • If m≠n δm n = 0

dx1= dx1*dx1

dx12≠ dx1*dx2

With Kronecker you get

dx1*dx2*δm n = dx1*dx2* 0 = 0

so Σmndx1*dx2*δmn = 0

Pythagoras can now be rewritten as

49:21 With gradients together we get the Metric tensor Gmn

So we get

ds2 = gmn dyrdys

The kronecker delta δmn (that is 1 if r=s) in the metric tensor gmn makes the metric tensor become a device that makes correction to pythagoras so the equation becomes valid in flat space as well in curved space.

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Conclusions

I am not ready with this page yet. I have to go back to the calculus part. When I am ready with that I can go back to my Euqation pagewhere I got stuck with  the derivation of EFE with that English teacher tensor derivations

In EFE you have gμν instead ofgmn as time is included in the metric tensor.

In the metric tensor μ can be =(zero, the time component), 1, 2,3 (the space component)

53:20 Christoffel symbol (Γr)

1:12:30 This used in the Ricci curvature tensor.

 

A agnostic pluralist seeker

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