Einstein Summation Convention / Lorentz Boost

In summary, the Einstein Summation Convention is a rule that states that whenever an expression contains one index as a superscript and the same one as a subscript, a summation over those values is implied. In the context of Lorentz transformations, this rule means that the \Lambda^{\mu'}_\mu tensor can be represented by a matrix with \mu' representing the row number and \mu representing the column number. This allows for a simpler and more concise notation when dealing with tensor equations.
  • #1
raintrek
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Einstein Summation Convention / Lorentz "Boost"

Homework Statement



I'm struggling to understand the Einstein Summation Convention - it's the first time I've used it. Would someone be able to explain it in the following context?

Lorentz transformations and rotations can be expressed in matrix notation as

[tex]x^{\mu'} = \Lambda^{\mu'}\!_{\mu}\:x^{\mu}[/tex]


Coordinates are defined by [tex]x^{\mu}[/tex] with [tex]\mu = 0,1,2,3[/tex], such that [tex](x^{0}, x^{1}, x^{2}, x^{3}) = (ct, x, y, z)[/tex]

I'm seeking clarification on the meanings of the various [tex]\mu, \mu'[/tex] indices in the matrix notation equation. Any help would be massively appreciated!
 
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  • #2
Summation rule

[tex]\mu, \acute{\mu}[/tex] means that these are all different indices which can have 0,1,2,3 and cannot contract with each other. I mean for example, while [tex]\mu = 0[/tex], the other one can be [tex]\acute{\mu} = 3[/tex].
Einstein summation rule is that you have to sum the terms with same indices. So you should scan all the values of the indices.
So
[tex]x^{\acute{\mu}} = \Lambda^{\acute{\mu}}\!_{\mu}\:x^{\mu}[/tex] means that

[tex]x^0 = \Lambda^{0}\!_{0}\:x^{0}+\Lambda^{0}\!_{1}\:x^{1}+\Lambda^{0}\!_{2}\:x^{2}+\Lambda^{0}\!_{3}\:x^{3}[/tex]
[tex]x^1 = \Lambda^{1}\!_{0}\:x^{0}+\Lambda^{1}\!_{1}\:x^{1}+\Lambda^{1}\!_{2}\:x^{2}+\Lambda^{1}\!_{3}\:x^{3}[/tex]
[tex]x^2 = \Lambda^{2}\!_{0}\:x^{0}+\Lambda^{2}\!_{1}\:x^{1}+\Lambda^{2}\!_{2}\:x^{2}+\Lambda^{2}\!_{3}\:x^{3}[/tex]
[tex]x^3 = \Lambda^{3}\!_{0}\:x^{0}+\Lambda^{3}\!_{1}\:x^{1}+\Lambda^{3}\!_{2}\:x^{2}+\Lambda^{3}\!_{3}\:x^{3}[/tex]

Don't forget that each term is actually a matrix element. So write the values of the matrix elements above and obtain the Lorentz transformation equations.
 
  • #3
Ah, so am I correct in thinking that whenever an expression contains one index as a superscript and the same one as a subscript, a summation over those values is implied?

Also, does the [tex]\Lambda^{\mu'}\!_{\mu}[/tex] just mean a matrix with [tex]\mu'[/tex] representing the row number and [tex]\mu[/tex] representing the column number?

so, in essence,

[tex]\Lambda^{0'}\!_{0} \Lambda^{0'}\!_{1} \Lambda^{0'}\!_{2} \Lambda^{0'}\!_{3}[/tex]
[tex]\Lambda^{1'}\!_{0} \Lambda^{1'}\!_{1} \Lambda^{1'}\!_{2} \Lambda^{1'}\!_{3}[/tex]
[tex]\Lambda^{2'}\!_{0} \Lambda^{2'}\!_{1} \Lambda^{2'}\!_{2} \Lambda^{2'}\!_{3}[/tex]
[tex]\Lambda^{3'}\!_{0} \Lambda^{3'}\!_{1} \Lambda^{3'}\!_{2} \Lambda^{3'}\!_{3}[/tex]

is the [tex]\Lambda^{\mu'}\!_{\mu}[/tex] matrix produced from your equations?
 
Last edited:
  • #4
Yes you'r right.
But
[tex]\Lambda^{\musingle-quote}\!_{\mu}[/tex] matrix cannot produce from above equation. We'r talking about a new representation of Lorentz transforms.

You can find the [tex]\Lambda^{\musingle-quote}\!_{\mu}[/tex] matrix here --> http://en.wikipedia.org/wiki/Lorentz_transformation
 
  • #5
raintrek said:
Ah, so am I correct in thinking that whenever an expression contains one index as a superscript and the same one as a subscript, a summation over those values is implied?

Also, does the [tex]\Lambda^{\mu'}\!_{\mu}[/tex] just mean a matrix with [tex]\mu'[/tex] representing the row number and [tex]\mu[/tex] representing the column number?

Yes.
 
  • #6
More correctly [itex]\Lambda^{\mu'}_\mu[/itex] means a tensor that can, in a given coordinates system, be represented by such a matrix.
 

1. What is the Einstein Summation Convention?

The Einstein Summation Convention is a mathematical convention used to simplify the writing and understanding of equations in tensor calculus. It states that when an index appears twice in a single term, it is assumed to be summed over all possible values, unless otherwise specified.

2. Why is the Einstein Summation Convention important?

The Einstein Summation Convention allows for the concise and compact representation of complex equations in tensor calculus. This not only makes calculations easier, but also helps to reveal the underlying symmetry and structure of equations.

3. What is the Lorentz Boost?

The Lorentz Boost is a mathematical transformation that describes how the measurements of space and time change when observed from different reference frames. It is an essential concept in the theory of relativity.

4. How does the Lorentz Boost affect the Einstein Summation Convention?

The Lorentz Boost is particularly useful in the context of the Einstein Summation Convention as it allows for the simplification of equations involving four-vectors and tensors. This is because the transformation properties of these quantities are consistent with the Lorentz Boost.

5. What are some real-world applications of the Einstein Summation Convention and Lorentz Boost?

The Einstein Summation Convention and Lorentz Boost have numerous applications in fields such as physics, engineering, and mathematics. They are used in the development of theories such as relativity and quantum mechanics, as well as in practical applications such as GPS systems and particle accelerators.

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