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## Homework Statement

**28.**Calculate the quadrupole tensor [itex]I_{jk}[/itex] and its traceless counterpart [itex]\overline{I}_{jk}[/itex] (Eq. (9.78)) for the following mass distributions.

(a) A spherical star whose density is [itex]\rho(r, t)[/itex]. Take the origin of the coordinates in Eq. (9.73) to be the center of the star.

## Homework Equations

(9.78)

[tex]I^{jk}=\int T^{00}x^ix^jd^3x[/tex]

(9.73)

[tex]\overline{I}_{jk}=I_{jk}-\frac{1}{3}\delta_{jk}I^l_l[/tex]

## The Attempt at a Solution

Well, first of all [itex]T^{00}=\rho[/itex]

so 9.78 becomes

[tex]I^{jk}=\int \rho(r,t)x^ix^jd^3x[/tex]

and then it should just be a simple case of integrating using spherical polars....right?

so for [itex]I^{11}[/itex] I should have

[tex]I^{11}=\int^r_0\int^{2\pi}_0\int^{\pi}_0 \rho(r,t)r^2 r^2sin\theta d\theta d\phi dr[/tex]

[tex]=\int^r_0\int^{2\pi}_0 2 \rho(r,t)r^4 d\phi dr[/tex]

[tex]=\int^r_0 4\pi \rho(r,t)r^4dr[/tex]

[tex]=4\pi \int^r_0 \rho(r,t)r^4 dr[/tex]

and to me all seems to have gone well, but the answer in the back of the book is

[tex]=\frac{4\pi}{3} \delta^{ij} \int^r_0 \rho(r,t)r^4 dr[/tex]

which is close, but not the same. If I look at other components of I they are nothing like this at all! even the components with different j & k don't vanish!

I'm obviously completely misunderstanding something here....