How to calculate a trace of a tensor?

[Haorong Wu]

Homework Statement

Use the tensor approach to work out the legendre polynomials $P_4 \left ( cos \theta \right)=P_4 ^0 \left ( cos \theta \right)$.

Relevant Equations

None

The textbook gives some examples for $P_1 \left ( cos \theta \right)$, $P_2 \left ( cos \theta \right)$, and $P_3 \left ( cos \theta \right)$.

Consider the unit vector $V^i$. In spherical coordinates, $V^3=\hat z=cos \theta$, which is just $P_1 \left ( cos \theta \right)$.

Consider the symmetric traceless tensor $T^{ij}=V^i V^j -\frac 1 3 \delta ^{ij} \left |V \right |^2 = V^i V^j - \frac 1 3 \delta ^{ij}$. Then $T^{33}=cos^2 \theta - 1/3$, which is $P_2 \left ( cos \theta \right)$.

Consider the symmetric traceless 3-indexed tensor $T^{ijk}=V^i V^j V^k-\frac 1 5 \left ( \delta ^{ij} V^k+ \delta ^{jk} V^i+ \delta ^{ki} V^j \right )$. Then $T^{333}=cos^3 \theta -\frac 3 5 cos\theta$, which is $P_3 \left ( cos \theta \right)$.

The procedure is clear. The key to the problem is to find the symmetric traceless tensor for the proper rank.

I first construct a symmetric tensor $U^{ijkl}=V^i V^j V^k V^l$.

Then I am not sure how to determine its trace. There is no clear definition of the trace of a high rank tensor. I infered that a trace of a high rank tensor is given by contraction of indices from the example for $P_3 \left ( cos \theta \right)$ given above. Then I write the trace of $U^{ijkl}$ as $U^{kl}=\delta ^{ij} U^{ijkl}=V^i V^i V^k V^l =V^k V^l$ where the repeated index summation is implied. Next I would subtract the trace out of $U^{ijkl}$ to get a symmetric traceless tensor $T^{ijkl}$.

But I am wrong because in the answer, the tensor is given by $T^{ijkl}=V^i V^j V^k V^l - \frac 1 7 \left ( \delta ^{ij} V^k V^l +\delta ^{jk} V^l V^i +\delta ^{ki} V^j V^l +\delta ^{il} V^j V^k +\delta ^{jl} V^k V^i +\delta ^{kl} V^i V^j \right ) +\frac 1 {35} \left ( \delta ^{ij}\delta ^{kl}+\delta ^{jk}\delta ^{il}+\delta ^{ki}\delta ^{jl} \right )$.

I can get the terms following $- \frac 1 7$, but not the terms following $\frac 1 {35}$.

I believe the problem is at the calculation of the trace. Should I contract $U^{kl}$ further and subtract it from $U^{kl}$?

Thanks!

 

[Fred Wright]

Please see https://ocw.mit.edu/courses/physics...ii-fall-2012/lecture-notes/MIT8_07F12_ln9.pdf equ. 9.43.

 

[Haorong Wu]

Please see https://ocw.mit.edu/courses/physics...ii-fall-2012/lecture-notes/MIT8_07F12_ln9.pdf equ. 9.43.

Thanks, Fred Wright. It is just what I need!