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.
- Greek letters in math and science
- Vector basics
- Tensor basics
- Tensor definition and calculus
- tensors visualized
- Stress tensor
- forward transformation
- backward transformation
- Chain rule
- Metric tensor (EFE)
- Lesson content
- Newton Autodidact Method
- Derivation of the metric tensor
- critical step 1. relabeling coordinates
- Critical step 3 – Two (2) frame system
- Critical step 4. DrPhysicsA application of the Chain rule.
Greek letters in math and science
In partial derivatives
Line maybe curved as slash /
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.
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 is a function which tells how to compute the distance between any two points in a given space….,.” (Read more in wolfram.com )
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.
- Array definition: tensor is a multidimensional grid of numbers
- 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.
- Abstract definition: Tensor is a collection of vectors and covectors combined together using a tensor product.
- 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 th-rank tensor in -dimensional space is a mathematical object that has indices and components and obeys certain transformation rules.”
- V1=1.826 (with subscript for the index value) is a variable with co-variant components
- 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:
|T 11= VP11||T 12= VP12||T 13= VP 13|
|T 21= VP21||T 22= VP22||T 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:
|T 11||T 12||T 13|
|T 21||T 22||T 23|
|T 31||T 32||T 32=|
The stress tensor
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
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
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.
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.
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
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.
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 y1 coordinate then you get a different value of x1 and x2
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:
- the gradient (derivative) for x1 (∂Φ/∂x1)
- 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:
- f(y) —> external function
- 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
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:
45:58 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
dx12 = dx1*dx1
With Kronecker you get
*δm n = dx1*dx2 * 0 = 0
*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.
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.