Division with Einstein summation convention

  • #1

Homework Statement



I have the following equation

Aab= c ua ub

Where Aab is a rank 2 tensor and ua is a vector and c is a scalar and a,b = {0,1,2,3}. I know both Aab , ua and ua
I want to find c explicitly but I don't know how to interpret or calculate

c = Aab / ( ua ub )

Does anyone know how I should proceed?

Homework Equations


Aab= c ua ub
3. The Attempt at a Solution

c = Aab / ( ua ub )
 

Answers and Replies

  • #2
2,792
594
Actually [itex] A^{ab}=c u^a u^b [/itex], in matrix form becomes the following:

[itex]
A=c\left[ \begin{array}{ccc} u^1 u^1 \quad u^1 u^2 \quad u^1 u^3 \\ u^2 u^1 \quad u^2 u^2 \quad u^2 u^3 \\ u^3 u^1 \quad u^3 u^2 \quad u^3 u^3 \end{array} \right]
[/itex].

Now if you write [itex] A^{ab} [/itex] in its matrix form too, you will find your answer.
 
  • #3
Thank you. I got tunnel vision on getting getting some sort of contraction going. Just to be clear, did you mean that I can get c by
c = A00 / (u0)2 = A11 / (u1)2 = ... etc?
Is it even possible to evaluate Aab / (ua ub)?
 
  • #4
2,792
594
Thank you. I got tunnel vision on getting getting some sort of contraction going. Just to be clear, did you mean that I can get c by
c = A00 / (u0)2 = A11 / (u1)2 = ... etc?
Is it even possible to evaluate Aab / (ua ub)?
Yeah, that's correct.
But your last sentence tells me that you don't know enough about what you're doing. [itex] A^{ab} [/itex] is not a tensor, its only the (a,b) component of the tensor A. So [itex] A^{ab} [/itex] is only a number. When you work with tensors like that, you're working in the component notation. So [itex] c=\frac{A^{ab}}{u^a u^b} [/itex] is just the division of two real numbers and nothing more. Also it has nothing to do with Einstein summation convention. But I can tell you how it can be related. At first we should bring down one of the indices so that we have one index up and one index down to sum over. Then we should contract the indices:
[itex] A^{ab}=cu^a u^b \Rightarrow g_{ca}A^{ab}=c g_{ca} u^a u^b \Rightarrow A_c^{\ b}=c u_c u^b \Rightarrow A_b^{\ b}=c u_b u^b \Rightarrow c=\frac{A_b^{\ b}}{u_b u^b}[/itex]
Now [itex] A_b^{\ b} [/itex] and [itex] u_b u^b [/itex] are just numbers, not components of a tensor.
 
  • #5
That's a neat trick and clears up alot of confusion I have. Thank you for the help.
 

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