in the ultra-relativistic approximation. This approximation is usually done in the literature since the error introduced is very small. In order to estimate the error due to the ultra-relativistic approximation, we expand in series the energies $E_i$ of Eq.  in the ultra-relativistic regime ($pc>>mc^{2}$). The energy difference, at the second order, becomes: $$E_{i}^{\left(k\right)}-E_{i}^{\left(j\right)}\simeq\frac{m_{k}^{2}c^{4}-m_{j}^{2}c^{4}}{2p_{i}c}-\frac{1}{8}\left(\frac{m_{k}^{4}c^{8}-m_{j}^{4}c^{8}}{p_{i}^{3}c^{3}}\right)\simeq\frac{\triangle m^{2}}{2E}-\frac{\triangle m^{2}}{8E^{3}}\left(m_{k}^{2}c^{4}+m_{j}^{2}c^{4}\right)$$ where we introduced $\triangle m^{2}=m_{k}^{2}c^{4}-m_{j}^{2}c^{4}$ and we approximated, only for the denominator, $E_{i}^{\left(k\right)}\simeq E_{i}^{\left(j\right)}\simeq p_{i}c:=E$. The first term is the oscillation frequency usually considered in the literature [@deco1]. The other one is the most relevant correction. Even taking the upper value for neutrinos masses of order of 2.2 eV [@upper] and considering the case of solar neutrinos, those having a lower energy ($E=10^{6}$ eV), the second term on the right hand side of the above equation is 12 order of magnitude smaller than the first term. This shows that the ultra-relativistic approximation is very good in general.

The error in the oscillation formula due to the ultra-relativistic approximation is: $$\frac{\triangle m^{2}}{8E^{3}}\left(m_{k}^{2}c^{4}+m_{j}^{2}c^{4}\right)\frac{t}{\hbar}=\left(1.01\times10^{10}\;\textrm{s}^{-1}\textrm{eV}^{3}\right)\frac{t}{E^{3}}.$$ Using the data in table I, the effect is $\sim10^{-29}$ for cosmogenic neutrinos, $\sim10^{-6}$ for solar neutrinos and $\sim10^{-22}$ for laboratory neutrinos. Compared with the effect due to the collapse, reported in the last 