- #1

jouvelot

- 53

- 2

Hi,

I stumbled upon an identity when studying tensor perturbations in cosmology. The formula states that

$$

\int d^2\hat{p} f(\hat{p}.\hat{q})\hat{p}_i \hat{p}_k e_{jk}(\hat{q}) = e_{ij}(\hat{q})/2 \int d^2\hat{p} f(\hat{p}.\hat{q})(1-(\hat{p}.\hat{q})^2),

$$ where ##e_{ij}## is a symmetric traceless matrix-valued function of ##\hat{q}## with ##\hat{q}_i e_{ij}(\hat{q}) = 0##, ##\hat{p}## is the unit vector parallel to ##p## and ##f## any function of the scalar product ##\hat{p}.\hat{q}##.

Is there a simpler way to get this formula than to proceed by using cumbersome spherical or euclidian coordinate expressions?

Thanks in advance.

Bye,

Pierre

I stumbled upon an identity when studying tensor perturbations in cosmology. The formula states that

$$

\int d^2\hat{p} f(\hat{p}.\hat{q})\hat{p}_i \hat{p}_k e_{jk}(\hat{q}) = e_{ij}(\hat{q})/2 \int d^2\hat{p} f(\hat{p}.\hat{q})(1-(\hat{p}.\hat{q})^2),

$$ where ##e_{ij}## is a symmetric traceless matrix-valued function of ##\hat{q}## with ##\hat{q}_i e_{ij}(\hat{q}) = 0##, ##\hat{p}## is the unit vector parallel to ##p## and ##f## any function of the scalar product ##\hat{p}.\hat{q}##.

Is there a simpler way to get this formula than to proceed by using cumbersome spherical or euclidian coordinate expressions?

Thanks in advance.

Bye,

Pierre

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