Einstein’s Field Equations of General Relativity

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Introduction

I am fighting to reach the next level of math, the one used by Einstein’s general relativity. It is said that no one really understands the equation. But I want at least to have a basic understanding of the language used in the equation. I understand it is about multidimensional tensors. I have to let my brain get used to it. And it takes time, having to listen several times to the same lesson

I use in this page EFE as an acronym for  Einstein field equations”

To understand EFE and especially its derivation, it is important to have good knowledge about:

Index

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EFE presentation

 

Source: Physicist page Facebook )

(Source: Dr Physics A Youtube )

This is Einstein’s field equation (EFE) in simplified form. It is called Einstein Field equations as it is really 16 equations depending on the value of the indices (μν) for g and T in the equation. 6 are duplicates. (read more below)

Source: Physicist page Facebook )

As is correctly explained in Wikipedia

mu (μ) and nu (ν) represent the dimensions
of space-time 0, 1, 2, 3:
  • 0 is time
  • 1 is the x axis
  • 2 is the y axis
  • 3  is the z axis

so mu (μ) and nu (ν) can each take four values between 0 and 3.

The combination of mu (μ) and nu (ν) means that
there are 16 variations of this equation with mu (μ) and nu (ν) having numbers between 0 and 3. You might therefore think that there are 16 equations in four-dimensional space-time but in fact 6 of them are duplicates so in fact, it reduces to 10 so you are looking at 10
Einstein field equations.
 
“The whole point about this equation is that it balances two things:
  1. everything on the left-hand side refers to the curvature of space-time
  2. everything on the right-hand side is to do with mass and energy

and as some people have some sometimes helpfully said that:

Einstein field equations basically say:

“mass tells space-time how to curve
and curved space-time tells mass how to move “

I found this youtube that explains the general relativity field equation. I like it because of the good graphic representations.

As with the tensor youtube made by Dan Fleisch, this made by Eugene Khutoryansky shows graphically how a tensor change depending on curvature of space.

Eugene Khutoryansky shares another youtube with amazing graphics.

Einstein followed his friends Grossman’s advice in 1912 to study Riemann geometry (“Albert Einstein” written by Vincenzo Barone, page 61). It looks like I have to deal with Riemann´s geometry and math too to get closer to the math in the field equation. I have started a post about this issue. Click here.

I follow in this page DrPhysicsA lesson. He recommends listening to Stanford Leonard Susskind’s Lectures. Here is his lecture 1 that he calls
“Quantum physics for old people”

 

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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.
    As i wrote above:
    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 says 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

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

DrPhysicsA recommends to listen to Stanford Leonard Susskinds Lectures. Her is lecture 1 he calls
“Quantum physics for old people”

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The metric tensor (gμν ) derivation

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

 

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 denoted with a curly d that has the  character (Unicode: U+2202). It is a stylized cursive/curly d.

read more in my separate page “Equations

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.

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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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. Read more about this in my equation page.

“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?

The chain rule is used as this screenshot shows:

As we must express y1 position as  x1 and x2 coordinates we must add two chainrule operations, the first described as x1 -coordinate and the second described as x2 -coordinate.
It looks like he writes the interior derivation first and then the interior and not as usual.

32:01 Tensors

I wrote about tensors in another page. But as a repetition, I share here Dr Physics explanation:“Let me start off by talking about a scalar. A scalar is something that has magnitude but no direction. Temperature would be a good example of a scalar. It just has a value 32 degrees centigrade.
A scalar is
what’s called a tensor of Rank 0.
Now
consider a vector. A vector of course has magnitude and direction so it has a length and it has a direction which is often described as the angle to one of the axes in this case to the x-axis. A vector is described as a tensor of Rank 1 “

( Source DrPhysicsA min 31:30 )

34:31 Vector transformation

We take the function of Φ

and make a slight adjustment to it. We 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.

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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