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.

This entry was posted in Mathematics, Physics, Relativity, Special Relativity. Bookmark the permalink.

9 Responses to Spontaneous creation of matter and antimatter.

  1. katie says:

    yeah nice

  2. Pingback: Ultimate Speed of A Material Particle – Denying the Concept of Infinite mass – Photons and More | Gaurav Happy Tiwari

  3. Pingback: Antigravity Could Replace Dark Energy as Cause of Universe’s Expansion « Talesfromthelou's Blog

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  7. paul says:

    if the spontaneous creation of matter can happen why can it only happen once?? (big bang)

    • Stephen Snell says:

      I’ve seen a few studies claiming it happens all the time. Just usually in insanely small quantities. Small enough that the tiny bit of extra matter forming compared to anti-matter is irrelevant. Also, if you look at space as jnfinite, then it would go with out saying there would be an infinite number of these events happening. We being inside one, can (at least atm) only see the edge of our matter generating event (big bang). There would be ‘universes’ of all different shapes and sizes, but if you were to try to draw a scaled down map of them all, it would be nearly impossible. Since as big as a universe is, it would be nothing compared to all of space. For every ‘big bang’ that happens, I’d think there’d be distances thousands of times longer of just empty space. Lol I’m not a scientist obviously, but if it has happened at least once (the fact I’m typing shows it has) it must be a naturally occurring event.

    • Not once – it will happen again – yet the total lifetime of our universe is not over…

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