## Spontaneous creation of matter and antimatter.

In this post I will write something about the spontaneous creation of matter and antimatter. Let us consider matter as an elementairy particle. This elemenrairty particle has some mass and several quantum-numbers. We assume that charge itself is also a quantum number. Let the mass be denoted as $m$ and the quantum-numbers be denoted as $q_k$. The particle can be denoted as $\{m_0,q_k\}$ and the antiparticle can be denoted as $\{\overline{m}_0,\overline{q}_k\}$. Consider a time-line such that at $t=0$ the particle and the anti-particle are spontaneous created. Before $t=0$ there is no energy, no mass and no quantum-numbers; so after $t=0$ the total mass-energy must be zero as well as the total quantum-numbers. We assume that the mass of both the particle and the antiparticle are the same – but that the quantum-numbers are the opposite. Thus we have $\{m_0,q_k\}$ for the particle and $\{m_0,-q_k\}$ for the anti-particle. For simplicity the point of creation is given by $(x,y,z)=(0,0,0)$ and the particle moves in the positive $x$ direction and the anti-particle moves in the negative $x$ direction. The distance between the particle and the origin is latex $r$ and that is also the distance between the anti-particle and the origin. The mass-energy of each particle is given by $\displaystyle \frac{m_0 c^2}{\sqrt{1 - v^2/c^2}}$ – and the static energy is given by $\displaystyle -\frac{\phi}{r}$ – where $\displaystyle \phi = G m^2 + \sum_k \kappa_k q_k^2$. For example if we consider only mass and charge then $\displaystyle \phi = Gm^2 + \frac{q^2}{4\pi\varepsilon_0}$. Since the total energy is zero before $t=0$ we obtain the equation

$\displaystyle \frac{2 m_0 c^2}{\sqrt{1 - v^2/c^2}} - \frac{\phi}{r} = 0$

We ‘solve’ this equation by proposing the solution of the form $\displaystyle r = \frac{c}{\omega} \sin(\omega t)$ – then it is clear that $v = c \cos(\omega t)$ – so we obtain

$\displaystyle \frac{2 m_0 c^2}{\sin(\omega t)} - \frac{\phi\omega}{c\sin(\omega t)} = 0$

Thus $\displaystyle \omega = \frac{2m_0 c^3}{\phi}$. So

$\displaystyle r = \frac{\phi}{2m_0 c^2} \sin\left( \frac{2 m_0 c^3}{\phi} t\right)$

The maximum distance between the particle and the antiparticle is given by $\displaystyle r_\textit{max} = \frac{\phi}{m_0 c^2}$ and the life-time is given $\displaystyle \frac{\pi \phi}{m_0 c^3} = \frac{\pi r_\textit{max}}{c}$. So we can also write

$\displaystyle r_\textit{max} \sin \left( \frac{ct}{r_\textit{max}} \right)$

In case of gravity alone we have $\phi = G m_0^2$ – so the maximum distance is given by $\displaystyle \frac{G m_0}{c^2}$ which is half the Schwarzschild radius $r_s$ so for elementairy particles this is very small – the life time is even smaller as it is half the Schwarzschild radius divided by the speed of light. The case considered is based on creation in an empty space and based on Special Relativity. Special Relativity does allow creation of matter and antimatter – but it holds for only a very small time.

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### 9 Responses to Spontaneous creation of matter and antimatter.

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