# Where did the inertia go in anti-matter reaction?

1. Aug 22, 2011

### skeleton

This question is about momentum in a system when an electron and positron annihilate each other.

SITUATION
Consider a system of 3 particles. They are in motion (which I will explain with Earth as the geometric frame).

The electron (#1) is moving, coming from the north, moving towards the center.
The positron (#2) is moving, coming from the south, moving towards the center.
The neutron (#3) is in the equatorial plane, rotating in orbit around the system's center.
(#1 and #2 are collinear in their trajectory (sort of), while #3 is perpendicular).

As the electron and positron move towards each other, they actually have a minor spiral trajectory because of the neutron's circular rotation. Their mutual precession is opposite to the orbital orientation of the neutron. This minor feature should be immaterial to my later point. The neutron is able to orbit around the system's center because of the presence of the electron and positron (which offer a gravitational field). While the neutron has non-trivial rotational inertia (from the reference point of the first two particles), the system as a whole can have zero rotational inertia.

OUTCOME
Upon eventual impact, the electron and positron annihilate, with only photons emerging.
e− + e+ → γ + γ

The energy is converted as, E=m*c^2 => E=h*v. The mass of the two particles is converted to energy. But mass is now gone. Since the former pair of particles can no longer offer a gravitation field, then the remaining neutron can no longer orbit - there is no mass particles for it to interact with. So, mathematically the neutron's rotational inertia must go to zero. That is OK, since the system as a whole originally had zero rotational inertia, and now the photons also have zero rotational inertia. So far, so good regarding rotational inertia.

But, how does the neutron itself transition from having some angular momentum (on its associated orbital movement) to having no rotational inertia?

QUANDARY
1) Does this mean the neutron flies off on a tangential linear path? I hope not, because this would introduce a linear momentum that the original system never had.

2) Does this mean the resulting two photons have an equivalent virtual (or maybe real) linear momentum which together is opposite to the ensuing linear momentum of the neutron?

3) Does the annihilation reaction impart some of its mutual energy (via work) onto the neutron in reducing the neutron's angular momentum to zero? If so, then the gamma radiation would have reduced energy, where some of it was given to the neutron.
Force, F = delta(r X p)/delta(t)
Energy, E = F*r

Last edited: Aug 22, 2011
2. Aug 22, 2011

### PAllen

Matter and energy equally contribute as gravitational source. The photons have the same 'invariant mass' as a system as the original electron/positron pair did. Thus, the same strength of gravitational source.

In GR, the source of gravity is called the stress/energy tensor for a reason.

Forget the neutron for a moment: the annihilation conserves energy, momentum, and angular momentum. A pair of photons definitely can have angular momentum.

3. Aug 23, 2011

### JeffKoch

I honestly can't parse what you are trying to say. Are you talking about spherical coordinates when you say "Earth as a geometrical frame"? What is "center", the origin or the equator? Collinear, sort of? Rotating in orbit? Gravitational field of isolated electrons and positrons orbiting some third point in space and then somehow finding a way to collide - how did you manage to arrange this? Speaking precisely is key to thinking precisely in physics, and also to communicating physics questions.

4. Aug 24, 2011

### skeleton

If I was fully precise in my question, then any answer would immediately follow. So your recognition of my imprecision suggests the need for my question. (Enough of philosophy.)

Let me try again: imagine earth as a reference frame although the planet itself does not exist in my scenario. One electron starts at the surface of the north pole and travels towards the center of the earth. The positron has the opposite trajectory: it starts at the south pole and is traveling also towards the center. The neutron is orbiting around the circumference of the equator.

5. Aug 24, 2011

### skeleton

PAllen,

(I forgot that photons are bent by a gravitational field.)

Thanks for your elegant explanation. All clear now.