Our understanding of the speed of light and many other things is predicated on the fact that light has no mass. As we can’t weight it directly like on a scale I am wondering (outside of mathematics) how we can test and prove this theory? Is it possible that light does have mass, it is just very very very small?
Further, if light has no mass, does it also have no energy? e=mc2 means energy for something massless would be 0. We know light has energy, so how does this equation work?
In: 42
Lets build some common ground. In special relativity we operate with 4D vector quantities that tell us about how an object moves through space and time. We can define the 4D equivalents of quantities like momentum or velocity. A path in spacetime is called a world line. The lenght of a path (we will call this s² for lenght squared) is a value for a spacetime path and its invariant for coordinate transmissions. Lets get started:
Well how do you define mass?
We need four-momentum for this which looks like this (E/c, p) where p is the 3D momentum with the relativistic correction we call the gamma factor. 1/sqrt(1-v²/c²)
Lets look at its lenght squared:
P²=(E/c)²-p² = (m²c²-m²v²)/(1-v²/c²) = m²c²
This is how we define mass. We can play around here a bit:
P² = E²/c² – p² = m²c²
E² = m²c²c² + p²c²
E² = (mc²)² + (pc)²
Cool little formula, it tells us that the energy of a thing comes from its mass and momentum.
Now lets calculate the length squared of a light path. Light in spacetime travels at a rate of r=ct e_r (e_r is a unit vector in the direction of r.) We can calculate s² which for a normal world line s²=(ct)²-r² same as we did with four-momentum. Now plug in r for light s²=(ct)²-(ct)² = 0 (as the lenght of a unit vector is 1).
Four-momentum has to be tangent to the world line if the world line has s²=0 in order for four-momentum to be tangent to the world line P²=0 must also be true. So now we get that 0=m²c² for light. Great m=0.
Momentum not necessarily. P=(E/c, px, py, pz) for a photon travelling in the x direction:
P² = E²/c² – px² = 0
px²=E²/c²
If we plug in m=0 for the result of our previous formula we get:
E²=p²c².
Wow same thing.
How can we know from experiment that this is how light behaves? Well this is the direct consequence of assuming c is constant in all reference frames. This idea came from an experiment that tried to measure our relative velocity to the medium in which light propagates. And we can also double check from general relativity that light travels one null geodesics. Like lets say we roughly know the mass of the Sun and see how much light gets bent around it during a solar eclipse. This experiment has shown how GR made perfect predictions and so if the behaviour of light is the same as predicted by GR the way we treat light in SR has to be correct and so anything we derive from it has to be correct. Or in simpler words you don’t have to weigh light, you can just test the axioms and if they are true everything that follows must also be true.
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