Three symbols i can never understand in relativity tesbooks

In summary, the symbols in relativity textbooks that the protagonist has trouble understanding are the Einstein summation convention, the kronecker delta, and the Levi-Civita symbol. After understanding these symbols, it becomes clear. The protagonist would benefit from learning more about tensors and the tensor product before asking more specific questions.
  • #1
Terilien
140
0
There are three symbols in relativity textbooks that I've never encoutered before and need lots of help with.

1.Einstein summation convetion : Though not really a symbol i still don't quite understand what is meant by it.

2. Upper case lambda with super scripts and subscripts: It seems to be some sort of linear transformation, but I still don't quite understand it.

3. The great demon, the kronecker delta: I really cannot understand what is meant by this.

The reason I have such a hard time with this notation is because I have very little formal education.

However after the symbols are understood, it becomes very lucid.

Can someone please give me a CLEAR and informal description of what this means with examples relating to relativity and tensor analysis?

I'm very sorry.
 
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  • #2
I kinda tried http://www.mathphyswiki.com/index.php?title=Tensors" [Broken]. If it's too brief, at least you have their names & you can look them up on google & wikipedia. I also included some texts on the main relativity page if you want to search through that.

#2 is the lorentz transform. I'd recommend waiting on that till you have the fundamentals down.
 
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  • #3
the kronecker delta is easy. even us bonehead enjuneers get that one. it's the dirac delta that's a b1tch.
 
  • #4
The summation convention works like this. Whenever you see a lower/upper pair of indexes with the same letter, expand over the dimensions like so -

[tex] F^{\mu}F_{\mu} = F^{0}F_{0} + F^{1}F_{1} + F^{2}F_{2} + F^{3}F_{3} [/tex]

The Kroenecker delta is just the unit matrix written a different way.
In the unit matrix the elements I(i,j) are zero if i<>j and 1 if i=j.
So [tex] \delta_{ij} [/tex] is an element from the unit matrix.

I'm surprised you didn't ask about the Levi-Civita symbol [tex] \epsilon_{ijkl} [/tex]
 
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  • #5
The [itex]\Lambda[/itex] will take components in one frame to the components in another. The Einstein summation convention is to write a repeated upper and lower index out as a sum over the number of dimensions (as Mentz114 has said). I shall use both in the example below:

[tex]p^{a'} = \Lambda^{a'}\mbox{}_{a} p^{a} \equiv \sum_{a=0}^{a=3} \Lambda^{a'}\mbox{}_{a} p^{a} = \Lambda^{a'}\mbox{}_{0} p^{0} + \Lambda^{a'}\mbox{}_{1} p^{1} + \Lambda^{a'}\mbox{}_{2} p^{2} + \Lambda^{a'}\mbox{}_{3} p^{3}[/tex]

Assuming the transformation is along the x-axis of a velocity [itex]v[/itex] such that [itex]\beta = v/c, \gamma = (1-\beta^2)^{\frac{1}{2}}[/itex] the components of the transformation are

[tex]\Lambda^{a'}\mbox{}_{a}=\begin{bmatrix} \gamma&-\beta \gamma&0&0\\ -\beta \gamma&\gamma&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{bmatrix}[/tex]

The Kronecker delta is the tensor form of the identity matrix:

[tex]\delta^a_b = \begin{cases} 1 & \mbox{if } a = b, \\ 0 & \mbox{if } a \ne b. \end{cases}[/tex]

As in: if you think of the indices labelling rows and columns of a matrix, only the entries along the main diagonal will be 1, and the off diagonal elements will all be 0.
 
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  • #6
There's one more thing I don't quite understand. what is the tensor product? why do we have it? why can EVERY m, l tensor be formed with it? That's what I've been having a real problem with. Everything else is pretty clear.

look I know I'm extremely stupid. no one has to rub it in.

EDIT: The rest makes a lot of sense now, but the darn tensor [product is just annoying.

It seems these days people don't care to list motivations for certain things.
 
  • #7
Think what you can do with two vector spaces. You can add two vector spaces to make a new one, whose dimension is just the sum of the dimensions of the first two. The direct or tensor product is much richer.
When we multiply vector spaces, each element of a vector from the first field has associated with it an entire vector space, each being a copy of the second space in the product.
So the dimension is mxn, and the resulting object has 2 indexes.


http://en.wikipedia.org/wiki/Tensor_product
 
  • #8
Small hint, Terilien. If you want better answers, google 'tensor product' or whatever you're stuck on, read a little & make an effort to understand it ... then ask more specific questions. People are more likely to help you with a difficult problem if they see you're making an effort.
 

1. What are the three symbols that are commonly used in relativity textbooks?

The three symbols commonly used in relativity textbooks are: c, G, and Λ. C is the symbol for the speed of light, G is the symbol for the gravitational constant, and Λ is the symbol for the cosmological constant.

2. Why are these symbols important in relativity?

These symbols are important in relativity because they are used to represent fundamental constants and concepts that are central to understanding the theory of relativity.

3. What is the significance of the speed of light, represented by the symbol c, in relativity?

The speed of light, represented by the symbol c, is a fundamental constant in relativity. It is the maximum speed at which all objects and information can travel in the universe, and it plays a crucial role in the equations of relativity.

4. How does the gravitational constant, represented by the symbol G, relate to relativity?

The gravitational constant, represented by the symbol G, is a fundamental constant in physics that is used to quantify the strength of the gravitational force between two objects. It is an important component in Einstein's theory of general relativity, which describes the relationship between gravity and the curvature of spacetime.

5. What is the significance of the cosmological constant, represented by the symbol Λ, in relativity?

The cosmological constant, represented by the symbol Λ, is a term that was added to Einstein's equations of general relativity to account for the observed expansion of the universe. Its value is still a topic of debate in modern physics and cosmology, as it has implications for the nature of dark energy and the fate of the universe.

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