User:L33th4x0r/twin

Dear Wikipedians:

I've been trying to work out the details of some of the calculations found in arXiv:physics/0411233, I got very very close to one of the final results but seem to have gotten stuck on a small bit of detail that didn't add up, which I was hoping to get your help with punching through.

The general formula for calculating time elapsed (T) for the stay-at home twin purely from the proper time elapsed (${\displaystyle {\bar {\tau }}}$) for the traveling twin, and purely from the traveling twin's non-inertial frame of reference, was given in the paper as:

${\displaystyle T^{2}=\left[\int _{0}^{\bar {\tau }}e^{\int _{0}^{\tau }a(\tau '){\text{d}}\tau '}{\text{d}}\tau \right]\left[\int _{0}^{\bar {\tau }}e^{-\int _{0}^{\tau }a(\tau '){\text{d}}\tau '}{\text{d}}\tau \right]}$

(1)

In the extreme (and classic when talking about any traveling twin problems) case of the infinite acceleration happening in an instant mid-trip for the traveling twin, the paper modeled it using Dirac's Delta as:

${\displaystyle a(\tau )=(\operatorname {artanh} (-v)-\operatorname {artanh} v)\delta (\tau -{\tilde {\tau }})=-2\operatorname {artanh} v\delta (\tau -{\tilde {\tau }})}$

(2)

where ${\displaystyle \operatorname {artanh} (-v)}$ is the rapidity of the ending half of the journey, ${\displaystyle \operatorname {artanh} (v)}$ is the rapidity of the starting half of the journey, and ${\displaystyle \delta (\tau -{\tilde {\tau }})}$ is Dirac's Delta centered at ${\displaystyle {\tilde {\tau }}}$, where ${\displaystyle {\tilde {\tau }}}$ is the instant the infinite acceleration (so-called "turn-around") happened for the traveling twin.

According to the paper, plugging (2) into (1), and after some work with hyperbolic functions, one finds ${\displaystyle T={\bar {\tau }}/{\sqrt {1-v^{2}}}}$, which is one of the classical twin paradox results. I did get this result, but only when starting from the following expression:

Plugging (2) into (1), we have:

${\displaystyle {\begin{aligned}T^{2}&=\left[\int _{0}^{\bar {\tau }}e^{\int _{0}^{\tau }-2\operatorname {artanh} v\delta \left(\tau '-{\frac {\bar {\tau }}{2}}\right){\text{d}}\tau '}{\text{d}}\tau \right]\left[\int _{0}^{\bar {\tau }}e^{-\int _{0}^{\tau }-2\operatorname {artanh} v\delta \left(\tau '-{\frac {\bar {\tau }}{2}}\right){\text{d}}\tau '}{\text{d}}\tau \right]\\&=\left[\int _{0}^{\frac {\bar {\tau }}{2}}e^{\int _{0}^{\tau }-2\operatorname {artanh} v\delta \left(\tau '-{\frac {\bar {\tau }}{2}}\right){\text{d}}\tau '}{\text{d}}\tau +\int _{\frac {\bar {\tau }}{2}}^{\bar {\tau }}e^{\int _{0}^{\tau }-2\operatorname {artanh} v\delta \left(\tau '-{\frac {\bar {\tau }}{2}}\right){\text{d}}\tau '}{\text{d}}\tau \right]\left[\int _{0}^{\frac {\bar {\tau }}{2}}e^{-\int _{0}^{\tau }-2\operatorname {artanh} v\delta \left(\tau '-{\frac {\bar {\tau }}{2}}\right){\text{d}}\tau '}{\text{d}}\tau +\int _{\frac {\bar {\tau }}{2}}^{\bar {\tau }}e^{-\int _{0}^{\tau }-2\operatorname {artanh} v\delta \left(\tau '-{\frac {\bar {\tau }}{2}}\right){\text{d}}\tau '}{\text{d}}\tau \right]\\&=\left[\int _{0}^{\frac {\bar {\tau }}{2}}e^{-2\operatorname {artanh} v\int _{0}^{\tau }\delta \left(\tau '-{\frac {\bar {\tau }}{2}}\right){\text{d}}\tau '}{\text{d}}\tau +\int _{\frac {\bar {\tau }}{2}}^{\bar {\tau }}e^{\int _{0}^{\tau }-2\operatorname {artanh} v\delta \left(\tau '-{\frac {\bar {\tau }}{2}}\right){\text{d}}\tau '}{\text{d}}\tau \right]\left[\int _{0}^{\frac {\bar {\tau }}{2}}e^{-(-2\operatorname {artanh} v)\int _{0}^{\tau }\delta \left(\tau '-{\frac {\bar {\tau }}{2}}\right){\text{d}}\tau '}{\text{d}}\tau +\int _{\frac {\bar {\tau }}{2}}^{\bar {\tau }}e^{-\int _{0}^{\tau }-2\operatorname {artanh} v\delta \left(\tau '-{\frac {\bar {\tau }}{2}}\right){\text{d}}\tau '}{\text{d}}\tau \right]\\&=\left[\int _{0}^{\frac {\bar {\tau }}{2}}e^{-2\operatorname {artanh} v(0)}{\text{d}}\tau +\int _{\frac {\bar {\tau }}{2}}^{\bar {\tau }}e^{\int _{0}^{\tau }-2\operatorname {artanh} v\delta \left(\tau '-{\frac {\bar {\tau }}{2}}\right){\text{d}}\tau '}{\text{d}}\tau \right]\left[\int _{0}^{\frac {\bar {\tau }}{2}}e^{-(-2\operatorname {artanh} v)(0)}{\text{d}}\tau +\int _{\frac {\bar {\tau }}{2}}^{\bar {\tau }}e^{-\int _{0}^{\tau }-2\operatorname {artanh} v\delta \left(\tau '-{\frac {\bar {\tau }}{2}}\right){\text{d}}\tau '}{\text{d}}\tau \right]\\&=\left[\int _{0}^{\frac {\bar {\tau }}{2}}e^{0}{\text{d}}\tau +\int _{\frac {\bar {\tau }}{2}}^{\bar {\tau }}e^{\int _{0}^{\tau }-2\operatorname {artanh} v\delta \left(\tau '-{\frac {\bar {\tau }}{2}}\right){\text{d}}\tau '}{\text{d}}\tau \right]\left[\int _{0}^{\frac {\bar {\tau }}{2}}e^{0}{\text{d}}\tau +\int _{\frac {\bar {\tau }}{2}}^{\bar {\tau }}e^{-\int _{0}^{\tau }-2\operatorname {artanh} v\delta \left(\tau '-{\frac {\bar {\tau }}{2}}\right){\text{d}}\tau '}{\text{d}}\tau \right]\\&=\left[\int _{0}^{\frac {\bar {\tau }}{2}}(1){\text{d}}\tau +\int _{\frac {\bar {\tau }}{2}}^{\bar {\tau }}e^{-2\operatorname {artanh} v\int _{0}^{\tau }\delta \left(\tau '-{\frac {\bar {\tau }}{2}}\right){\text{d}}\tau '}{\text{d}}\tau \right]\left[\int _{0}^{\frac {\bar {\tau }}{2}}(1){\text{d}}\tau +\int _{\frac {\bar {\tau }}{2}}^{\bar {\tau }}e^{-(-2\operatorname {artanh} v)\int _{0}^{\tau }\delta \left(\tau '-{\frac {\bar {\tau }}{2}}\right){\text{d}}\tau '}{\text{d}}\tau \right]\\&=\left[\int _{0}^{\frac {\bar {\tau }}{2}}{\text{d}}\tau +\int _{\frac {\bar {\tau }}{2}}^{\bar {\tau }}e^{-2\operatorname {artanh} v(1)}{\text{d}}\tau \right]\left[\int _{0}^{\frac {\bar {\tau }}{2}}{\text{d}}\tau +\int _{\frac {\bar {\tau }}{2}}^{\bar {\tau }}e^{2\operatorname {artanh} v(1)}{\text{d}}\tau \right]\\&=\left[\int _{0}^{\frac {\bar {\tau }}{2}}{\text{d}}\tau +\int _{\frac {\bar {\tau }}{2}}^{\bar {\tau }}e^{-2\operatorname {artanh} v}{\text{d}}\tau \right]\left[\int _{0}^{\frac {\bar {\tau }}{2}}{\text{d}}\tau +\int _{\frac {\bar {\tau }}{2}}^{\bar {\tau }}e^{2\operatorname {artanh} v}{\text{d}}\tau \right]\\&=\left[\tau {\Bigg |}_{0}^{\frac {\bar {\tau }}{2}}+\int _{\frac {\bar {\tau }}{2}}^{\bar {\tau }}e^{-2\left({\frac {1}{2}}\ln {\frac {1+v}{1-v}}\right)}{\text{d}}\tau \right]\left[\tau {\Bigg |}_{0}^{\frac {\bar {\tau }}{2}}+\int _{\frac {\bar {\tau }}{2}}^{\bar {\tau }}e^{2\left({\frac {1}{2}}\ln {\frac {1+v}{1-v}}\right)}{\text{d}}\tau \right]\\&=\left[{\frac {\bar {\tau }}{2}}+\int _{\frac {\bar {\tau }}{2}}^{\bar {\tau }}e^{-\ln {\frac {1+v}{1-v}}}{\text{d}}\tau \right]\left[{\frac {\bar {\tau }}{2}}+\int _{\frac {\bar {\tau }}{2}}^{\bar {\tau }}e^{\ln {\frac {1+v}{1-v}}}{\text{d}}\tau \right]\\&=\left[{\frac {\bar {\tau }}{2}}+\int _{\frac {\bar {\tau }}{2}}^{\bar {\tau }}\left(e^{\ln {\frac {1+v}{1-v}}}\right)^{-1}{\text{d}}\tau \right]\left[{\frac {\bar {\tau }}{2}}+\int _{\frac {\bar {\tau }}{2}}^{\bar {\tau }}e^{\ln {\frac {1+v}{1-v}}}{\text{d}}\tau \right]\\&=\left[{\frac {\bar {\tau }}{2}}+\int _{\frac {\bar {\tau }}{2}}^{\bar {\tau }}\left({\frac {1+v}{1-v}}\right)^{-1}{\text{d}}\tau \right]\left[{\frac {\bar {\tau }}{2}}+\int _{\frac {\bar {\tau }}{2}}^{\bar {\tau }}{\frac {1+v}{1-v}}{\text{d}}\tau \right]\\&=\left[{\frac {\bar {\tau }}{2}}+\int _{\frac {\bar {\tau }}{2}}^{\bar {\tau }}{\frac {1-v}{1+v}}{\text{d}}\tau \right]\left[{\frac {\bar {\tau }}{2}}+\int _{\frac {\bar {\tau }}{2}}^{\bar {\tau }}{\frac {1+v}{1-v}}{\text{d}}\tau \right]\\&=\left[{\frac {\bar {\tau }}{2}}+{\frac {1-v}{1+v}}\int _{\frac {\bar {\tau }}{2}}^{\bar {\tau }}{\text{d}}\tau \right]\left[{\frac {\bar {\tau }}{2}}+{\frac {1+v}{1-v}}\int _{\frac {\bar {\tau }}{2}}^{\bar {\tau }}{\text{d}}\tau \right]\\&=\left[{\frac {\bar {\tau }}{2}}+{\frac {1-v}{1+v}}\tau {\Bigg |}_{\frac {\bar {\tau }}{2}}^{\bar {\tau }}\right]\left[{\frac {\bar {\tau }}{2}}+{\frac {1+v}{1-v}}\tau {\Bigg |}_{\frac {\bar {\tau }}{2}}^{\bar {\tau }}\right]\\&=\left[{\frac {\bar {\tau }}{2}}+{\frac {1-v}{1+v}}{\frac {\bar {\tau }}{2}}\right]\left[{\frac {\bar {\tau }}{2}}+{\frac {1+v}{1-v}}{\frac {\bar {\tau }}{2}}\right]\\&={\frac {\bar {\tau }}{2}}\left[1+{\frac {1-v}{1+v}}\right]{\frac {\bar {\tau }}{2}}\left[1+{\frac {1+v}{1-v}}\right]\\&={\frac {\bar {\tau }}{2}}\left[{\frac {1+v}{1+v}}+{\frac {1-v}{1+v}}\right]{\frac {\bar {\tau }}{2}}\left[{\frac {1-v}{1-v}}+{\frac {1+v}{1-v}}\right]\\&={\frac {\bar {\tau }}{2}}{\frac {\bar {\tau }}{2}}{\frac {1+v+1-v}{1+v}}{\frac {1-v+1+v}{1-v}}\\&={\frac {\bar {\tau }}{2}}{\frac {\bar {\tau }}{2}}{\frac {2}{1+v}}{\frac {2}{1-v}}\\&={\frac {\bar {\tau }}{\cancel {2}}}{\frac {\bar {\tau }}{\cancel {2}}}{\frac {\cancel {2}}{1+v}}{\frac {\cancel {2}}{1-v}}\\&={\frac {{\bar {\tau }}{\bar {\tau }}}{(1+v)(1-v)}}\\\end{aligned}}}$

This is based on the fact that

${\displaystyle \left\{{\begin{array}{lll}\int _{0}^{\tau }\delta \left(\tau '-{\frac {\bar {\tau }}{2}}\right){\textrm {d}}\tau '=0&{\text{if}}&0\leq \tau <{\frac {\bar {\tau }}{2}}\\\int _{0}^{\tau }\delta \left(\tau '-{\frac {\bar {\tau }}{2}}\right){\textrm {d}}\tau '=1&{\text{if}}&{\frac {\bar {\tau }}{2}}\leq \tau \leq {\bar {\tau }}\\\end{array}}\right.}$