# If I proposed that two parting photons have mass, how would I quantify that mass?

While an individual photon has no rest frame, when two photons move apart, it makes sense to ask where their centre of mass is, as described here.

If I proposed that while a photon has no mass, a system of two parting photons does have mass, how would I quantify that mass, in a manner consistent with the prevailing theories of physics?

• I do not believe that involving some symmetry changes something. Imagine that supernova has exploded and say all it's rest mass was converted into photons which fly apart evenly in a spherical fashion. Now we have exact static center reference point, which was ex-star barycenter, but this doesn't mean that somehow magically star energy converts back into rest mass just to please you that you know it's COM. Commented Jan 6 at 12:40
• See this Q&A: physics.stackexchange.com/questions/10612/… Commented Jan 6 at 13:21
• do you know that the pi0 decays into two photons? The pi0 has a fixed mass en.wikipedia.org/wiki/Pion#Neutral_pion_decays Commented Jan 6 at 14:36
• @AgniusVasiliauskas the part where you wrote "just to please you that you know its COM" seems unscientific. Just to be clear what you are saying: Your claim is that the system of explanding light, centered at the rest frame of the ex-star barycenter has mass 0 grammes, correct? Commented Jan 7 at 9:38
• Yes, correct. Rest mass is zero, but energy - does not. Commented Jan 7 at 9:53

## 2 Answers

In particle physics, it is common to define the invariant mass $$M$$ of a system of $$n$$ particles with four-momenta $$p_i =(E_i, \vec{p}_i), \quad (i=1, \ldots n, \; p_i^2=E_i^2-\vec{p}_i^2=m_i^2),$$ by the Lorentz invariant quantity $$M^2:= \left(p_1+\ldots +p_n\right)^2=(E_1+\ldots +E_n)^2-(\vec{p}_1+\ldots+\vec{p}_n)^2.$$ In the case of two photons (where $$E_1= |\vec{p}_1|$$, $$E_2=|\vec{p}_2|$$), the invariant mass of this two-particle system is given by $$M^2=(p_1+p_2)^2=(|\vec{p}_1|+|\vec{p}_2|)^2-(\vec{p}_1+\vec{p}_2)^2.$$ In the reference frame where $$\vec{p}_1 + \vec{p}_2=\vec{0}$$ (the so-called center-of-mass frame), the invariant mass (squared) of the two-photon system becomes $$M^2=4E_{\rm CM}^2$$ with $$E_{\rm CM}:= |\vec{p}_1|=|\vec{p}_2|$$.

Mass is rest energy divided by c$$^2$$, so you could define the mass as hf$$_1$$+hf$$_2$$. However, for it to be useful as a concept in physical analysis, mass should be constant and localized. In the case of two free photons parting in opposite directions, I don't see a use for the concept of total mass. In the case of an opaque box containing light and with infinite Q, it might be useful.

• Thank-you. What's $Q$? Where you write "for it to be useful as a concept..." Would it be of use in order to account for the gravitational mass of this photon pair, as calculated around their centre of mass? Commented Jan 8 at 14:31
• $Q$ is the quality factor of a cavity. ccrma.stanford.edu/~jos/fp/…. Commented Jan 8 at 14:38