The weight of light

It is a fairly commonly known fact that the photon has no mass. While this is true for a single photon it is not necessarily true for collections of photons. Lets take a step back for a moment and say what we mean when we say "rest mass". In both Newtonian dynamics and relativity the energy of an object is a function only of its velocity. In Newtonian dynamics the zero for the energy scale is determined arbitrarily and only changes in energy have meaning. In other words it doesn't matter whether you say a baseball going 2m/s has an energy of 4 joules or -10 joules all that matters is that when the ball goes from 2m/s to 4m/s the energy of the ball goes from 4 to 16 or from -10 to 2. Once a zero was picked for the energy scale you could talk about changes in energy and that was all that mattered. Relativity changed all that because in relativity there is a more absolute meaning to energy. I don't remember the derivation but the end of the story is that the equation E^2 = (pc)^2 + (mc^2)^2 holds. Here p is momentum m is mass E is energy and of course c it goes without saying is the speed of light. Solving this equation for mass you get m = c^-2 * sqrt(E^2 - (pc)^2) Just like for any other particle light obeys the relation p = h/L where L is its wavelength. Also E = hf and f = 1/T = c/L so
E = hc/L plugging this in to the first equation we get that m = c^-2*sqrt((hc/L)^2 - (h/L * c)^2 = 0 and all is well. So you might think that the mass of a system of 2 photons would also be massless 0 + 0 = 0 right? Well... maybe and maybe not. A single photon had no mass because it just so happened that for a photon E = pc so E^2 - (pc)^2 = 0. But when you consider a system of 2 photons while its energy is just the sum of the energies of the two particles the momentum of the system is the vector sum of the two momenta. So if you have two photons of the same wavelength moving exactly opposite of each other then you get 0 momentum for the system. Now you find that the system of 2 photons does indeed have mass!! m = 2*h/(cL).

So here is the big idea I want to know what happens when two photons traveling perfectly parallel to one another with one in front of the other by a little distance hit a mirror. At first they are traveling with each other so their momenta add and we have a system with 0 mass. But after the first photon hits the mirror and gets reflected we now have a system with mass because now their momenta cancel. This is where an awesome experimental setup comes into play. The LIGO setup is built in order to be an incredibly sensitive instrument for the direct detection of gravity waves. This is done by making a gigantic perfectly tuned michelson inferometer. You watch the fringe on the inferometer and look for changes. Theoretically if there are no gravity waves there shouldn't be any changes to the fringe but of course because the thing is so damn sensitive it actually picks up vibrations from people walking around a mile away and stuff like that. Fortunately the changes in the fringe from local vibration and what you would see from a gravity wave are different. Now the thing is that we still haven't detected a gravity wave with the LIGO. So here is the deal why not try and make a gravity wave right smack on top of the damn thing? The idea is to shove an extra little laser pulse on top of the continuous beam already circulating in the inferometer. every time the laser pulse passes we add a little bit to it. The laser pulse would have to be extremely brief and extremely powerful in order for it to make a measurable g-wave. I don't really have the necessary knowledge to know exactly how powerful it would have to be to be detectable but as the wave began to hit a mirror its mass would increase to a peak as half the pulse has been reflected and then go back to zero as the rest of it got reflected. The mass generated by the pulse even at its peak would be absolutely minute but it would go from 0 to full mass in the tiniest fraction of a second. I don't know if that violence would be enough to make a measurable gravity wave either. The point is it is a totally freaking awesome idea. Use the device built to detect gravity waves to make them!!!!

To put things in perspective the gravity wave would be extremely extremely weak. Even assuming we could get the total energy in the burst to be somewhere around the order of a petajoule the mass generated by the pulse would be about a ten thousandth of a gram. So a bounce event at one of the mirrors would be something like a mote of dust appearing and vanishing in the tiniest fraction of a second. Now on top of the absolutely tiny mass we are faced with an additional 11 orders of magnitude or so coming from the gravitational constant. So even a petawatt burst would probably not be enough.

Actually I just realized that I've been ignoring the fact that the momentum from the light being reflected goes into the mirror so when you consider the mirror and photons as a system the total rest mass is unchanged. so the whole idea is dead in the water. Still though it was an interesting thought.