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
>(outside of mathematics)
“Mathematics” *is* the answer though.
“e=mc^(2)” is just the short version. The full equation is “e = sqrt(m^(2)c^(4) + p^(2)c^(2))”, where p is momentum. Thus, light has energy from momentum even when mass is 0.
E=mc^2 is only valid for objects in rest. The complete formula is E= sqrt(m²c⁴+p²c²), where p is the momentum. And as light has an momentum it has an energy.
But it has no rest mass (the mass an object has at rest), as according to special relativity it then would not be able to move at speed of light. If it were, it would need infinite energy which is not possible.
There are a bunch of things that move at the speed of light by default. That is to say that they do not slow down or speed up. A photon goes at c from its creation to its absorption.
e=mc^2 is not the complete equation, which is why you’re getting e=0.
If light had mass a whole bunch of our physics wouldn’t work. You are looking at it a bit the wrong way around; it isn’t that so many things are predicated on light having no mass, but that all those things are evidence that light has no mass.
> Further, if light has no mass, does it also have no energy? e=mc2 means energy for something massless would be 0.
*E = mc^(2)* in its simplest form comes from Special Relativity, which doesn’t apply to massless things. The slightly more general version of this is a bit more complicated:
> E^2 = (pc)^2 + (mc^(2))^2
where *p* is the object’s momentum. Mass is just one expression of energy, momentum is another way energy can be expressed. So a normal object has some energy from just existing (it’s “mass” energy), and some from moving (its “momentum” energy).
If something is “at rest”, with no momentum, that equation simplifies down to the normal *E = mc^(2)* that we all know.
Photons, or light, can never be at rest, so we can never simplify it down that way. For a massless object we get:
> E = pc
which is perfectly fine. If the photon is moving it will have momentum, so will have energy.
This has some fun results. If we take the photon energy of *E = hf* and plug that in, with a bit of algebra we get:
> p = h / λ
where λ is the wavelength of our light, and h is Plank’s constant. This encouraged de Broglie to wonder if anything with momentum also had a corresponding wavelength, and that led him down the path of thinking about matter waves, and wave-particle duality; that all things we think of as “particles” have a wavelength based on their momentum, and behave in wave-like ways sometimes, just as how wave-like things (like light) behave in particle-like ways sometimes.
>Further, if light has no mass, does it also have no energy? e=mc2
E=mc^2 only applies to objects at rest, light moves so it is not enough.
For moving objects the formulate is
e^(2=) (mc^(2)2) + (pc)^(2) where p is th momentum of the object.
If the mass m=0 then (mc^(2)2) =0 and the formula is rescue to e^(2=(pc)2) => e= pc
So energy is momentum times the speed of light for a massless object. Light has momentum so it has energy. There are many ways do demonstrate this, a practical example is light sails. That is sails that work by reflecting sunlight, it has practical application that has been demonstrated with experimental spacecraft [https://en.wikipedia.org/wiki/Solar_sail](https://en.wikipedia.org/wiki/Solar_sail) There are lots of other measurements of this too.
A probe with mass is it increases when speed increases and would be infinite at the speed of light. So even if a photon had a minuscule mass at rest it would have an infinite mass at the speed of light
We can just look at the momentum light imposes in a light source, it is clearly not infinite, It was a flashlight that would get accelerated to the speed of light in the other direction.
The speed of light is also the same of all observers regardless of how they move the [Michelson–Morley experiment](https://en.wikipedia.org/wiki/Michelson%E2%80%93Morley_experiment) shows that.
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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.
Get a box with a flashlight inside and put it on a scale. If light has weight, it would weigh more when the light is on. It does not.
Can you actually have a photon at rest? Is it possible to have light that does not move?
PhD physicist here who survived Jackson E&M. The answer is we aren’t 100% sure that light has zero mass, but there have been experiments that place an upper bound on said mass to incredibly small values. I’ll refer you to the intro chapter of Jackson for more details.
Also, it turns out we’re not 100% sure of a lot of things we usually take for certain (because we might as well). People still look for Pauli exclusion violation, for example.
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