Good morning.(adsbygoogle = window.adsbygoogle || []).push({});

I would like to prove that the integral

[itex]h^{\mu \nu} (\vec{r},t) = \int d \zeta \int d^3 \vec{y} \frac{F^{\mu \nu} (\zeta,\tilde{\tau}) \delta^{(3)} (\vec{r} - \vec{x}(\zeta,\tilde{\tau}))}{|\vec{r}-\vec{y}|}[/itex]

where [itex]\tilde{\tau} = t - |\vec{r}-\vec{y}|[/itex], is equal to

[itex]\int d \zeta \frac{F^{\mu \nu} (\zeta,\tau)}{|\vec{r}-\vec{x}(\zeta,\tau)| (1-\hat{n} \cdot \dot{\vec{x}}(\zeta,\tau))}[/itex]

where [itex]\displaystyle \hat{n}= \frac{\vec{r}-\vec{x}(\zeta,\tau)}{|\vec{r}-\vec{x}(\zeta,\tau)|}[/itex], [itex]\tau = t - |\vec{r}-\vec{x}(\zeta,\tau)|[/itex] and [itex]\dot{\vec{x}}(\zeta,\tau)[/itex] is the derivative of [itex]\vec{x}[/itex] with respect to his second variable.

I would like to integrate with respect to [itex]y[/itex] using the Dirac function [itex]\delta^{(3)}[/itex] but I don't manadge to find the value of [itex]\vec{y}[/itex] such that [itex]\vec{r} - \vec{x}(\zeta,\tilde{\tau})[/itex] vanishes. I also tried to use

[itex]\delta^{(3)} ( \vec{x} - \vec{a}) = \frac{\delta^{(3)} ( \vec{\xi} - \vec{\alpha})}{|J|}[/itex]

where [itex]\vec{\xi} = \vec{\xi} (\vec{x})[/itex], [itex]\vec{\alpha} = \vec{\xi} (\vec{a})[/itex] and [itex]J[/itex] is the Jacobian of the transformation of [itex]\vec{x}[/itex] into [itex]\vec{\xi}[/itex] but without success.

Thank you in advance for your help.

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Calculation of an integral with retarded time

Loading...

Similar Threads - Calculation integral retarded | Date |
---|---|

A How to calculate the extrinsic curvature of boundary of AdS_2 | Jul 15, 2017 |

A Help to understand the derivation of the solution of this equation | May 1, 2017 |

I Gravitational time dilation calculation near a Black Hole | Mar 27, 2017 |

**Physics Forums - The Fusion of Science and Community**