You know that energy is an intangible thing. You can’t hold energy in the palm of your hand, because energy is to do with stress, which is akin to pressure, and you need a volume of stress to get the dimensionality right.
You know that mass is a tangible thing. You can hold an object in your hand and feel the mass of it. You even know that E=mc², and that the intangible thing called energy can be used to make the tangible thing called mass. But you don’t know how, because you don't know how simple it really is.
The answer is all down to motion. Or the lack of it. You have to get relative, because motion is relative, and you have to think in terms of momentum and inertia, because there's a symmetry between the two that depends on who you say is moving. A 10kg cannonball is a tangible thing, you can hold it in the palm of your hand and feel the mass of it. And if it’s travelling at 1m/s it’s “got” kinetic energy as quantified by KE=½mv2. It’s also “got” momentum, as quantified by p=mv, and it’s hard to stop. Brace yourself, then apply some constant braking force by catching it in the midriff. Ooof, and you feel the energy/momentum. Kinetic energy is looking at this in terms of stopping distance, whilst momentum is looking at it in terms of stopping time. The measure we call momentum is conserved in the collision because your gut and the cannonball shared a mutual force for the same period of time. The measure we call kinetic energy isn’t conserved, because some of the mass in motion is redirected into your deformation and heat and bruises, all of which involve mass in motion, but scattered motion instead of the tidy motion of a mass moving relative to you. Or you moving relative to it, because all the while you were never too sure whether it was you moving or the cannonball.
When we consider that the cannonball is stationary and it’s you moving at 1m/s, the cannonball hasn’t got any kinetic energy or momentum, all it’s got is inertia. You’re attempting to move it and accelerate it to your velocity of 1m/s, and that makes it harder to start, not harder to stop. In this respect momentum and inertia can be considered as two different aspects of the same thing, and that thing is energy.
You can get a better feel for this with a gyroscope. Waggle it back and forth. See how insubstantial it feels. Now wind the string round the spindle, grasp it tight, and pull. You give the gyroscope some circular motion, you are in effect pulling tension out, and since tension is negative stress, you are introducing stress. Your gyroscope is now humming, maybe precessing a little. When you try to waggle it you can feel the angular momentum working against you. It feels more substantial than before. You can feel more resistance to motion, more "inertia", and that’s the secret of E=mc². The gyroscope feels more “massive” because you have added energy.
You’re beginning to get a feel for mass, but to really understand it you have to stop thinking of momentum as something that a mass has got. A thing that’s got energy can have momentum without having any mass. Like a photon. A photon has energy/momentum but it has no mass. And a photon has a size but it has no surface. You might therefore think a photon is something really strange, but it isn’t. You’ve seen something similar down on the beach. You’re playing in the surf and along comes a massive wave. You know it’s a wave, a travelling stress, and you know it has no mass because the water has the mass. It's a pressure wave, you can see how big it is, and you know it has no surface because the ocean has the surface. But the wave does have energy/momentum, enough to knock you and your girlfriend flat on your back, laughing and screaming with salt water up your nose. It’s both insubstantial and substantial, both tangible and intangible. Because whilst you can’t grab hold of it, it can grab hold of you.
When we think about the photon some more, we know that it isn’t some miniature cannonball. It’s a transverse wave in space, it’s got a wavelength and a frequency instead of a mass, so instead of KE=½mv² and p=mv we express energy as E=hf and momentum as p=hf/c. The h is Planck’s constant of 6.63 x 10^-34 Joule-seconds, and is an “action”, which is energy multiplied by time, or alternatively momentum multiplied by distance. The f is frequency per second, and the speed of light c is distance divided by time, converting a stopping-distance measure of energy into a stopping-time measure of momentum.
The use of the word “action” is important here. It means what it says, and should doubly remind us that the photon is not some billiard-ball particle. It truly is an action. It's akin to a kick. In itself it’s intangible, but like a kick it delivers a quite tangible effect. We can see this in action via Compton scattering, which makes the sky look blue:
When a photon interacts with a free electron, the electron receives a “bump” and is sent recoiling off at an angle, while the photon is deflected and its wavelength is increased. The underlying reason for this is easy to grasp. Look at the central portion of the squiggle that is the incident photon in the picture above, turn it upside down, and imagine it’s a U-shaped steel bar. Now grasp it tight and unbend it a little. This increases the distance between the two ends and changes the angle between them. In similar vein the photon wavelength increases and the direction of propagation is altered whilst the energy decreases. Meanwhile the electron gains energy and receives an alteration to its velocity vector, or in other words: it moves.
Now remember your relativity: there is no absolute motion. Look again at the picture. Imagine you’re that target electron, but you’re not at rest. Imagine it’s you moving instead of the photon. Bump, and you’re sent flying off at an angle. It would feel like you hit something solid instead of a photon. It would feel like a bad flight with turbulence and so many air pockets it’s like riding over rocks. Instead of delivering a bump, the photon would be a bump. It wouldn’t feel like something intangible and insubstantial, it would feel like something tangible and substantial. It would feel like the photon had inertia instead of momentum. It would feel like the photon had mass.
But a photon doesn't have mass. A photon isn’t just sitting in one place, it always travels at c. You can’t nail it down like you can nail down a ripple in a rubber mat. So how do you keep a travelling volume of stressed space in the same place? It’s easy when you know how. Imagine you’ve got a couple of “free electron” table tennis bats, and you’re good at topspin. If you bat that photon just right, you can change its direction and give it some more energy/momentum. It’s called an Inverse Compton, like the picture of Compton scattering but with the arrows going the other way. Then you hit the photon with the other bat to change its direction again. Repeat in rapid succession until you’ve got a kind of hexagon going, then bat even faster until you’ve got a miniature electromagnetic swirl that is your photon going round in circles like this O. Now keep chopping away, but close your eyes, like you might close your eyes when you’re feeling the repulsion between a couple of magnets. You can feel something there between your bats. What you can feel is basically mass. You've “stopped” the photon and converted the momentum into inertia. You’ve made a “mass”.
OK, you haven’t really stopped the photon because it’s still travelling at c, but it’s close enough. Bah, I hear you say, there are no table tennis bats in particle physics. And a photon always travels at 299,792,458 m/s. You can’t really "stop" a photon. Oh yes you can. It’s simple. You use something called pair production:
In pair production, a gamma photon of a little over 1,022 thousand electron volts (keV) is effectively "broken" over a nucleus to create an electron and a positron of 511 keV apiece. Apart from some essential losses associated with the motion and the separation of the particles, the energy/momentum is reconfigured as mass. Pair production converts travelling kinetic energy or relativistic mass into non-travelling energy or rest mass. In effect the momentum is “stopped” and now appears as inertia, and the electron and the positron are like two stable eddies spinning off in opposite directions. Each can be modelled as a self-contained vortex, or vorton. Each is a photon going round and round in a circle, tied into something called a toroidal soliton. It isn’t just a circle, it's a circle with a twist, and it needs the twist to stay tied up. This twist makes it a knot, what’s called a “trivial knot”, which is the simplest knot you can get. An electron twists and turns one way, the positron twists and turns the other, so they're like mirror images with different "handedness" or chirality. You can think in terms of a möbius doughnut, as depicted on the left below. Note the similarity between the dark line as compared to the balloon knot on the right.
It’s rather like a möbius strip. The electron’s component photon is going round twice whilst twisting round once to return to its starting position and orientation. The electron is therefore a spin ½ particle exhibiting a jitter or zitterbewegung. It's both a transverse wave and a particle, and hence it's a soliton and hence it exhibits wave/particle duality. It is however somewhat difficult to depict. It isn't a solid object, and whilst we can draw a representation of it with contour lines, it has no actual surface. It has as much surface as an ocean wave, remembering that the ocean has the surface. The electron has as much surface as a light beam. The only “surface” to it is like the repulsion you’d feel between a pair of magnets. However this is quite enough to create what we perceive as a solid object. The massless photon now exhibits mass, because it’s going round and round circling at the speed of light. It’s tied in one location, and whilst not idle, it’s no longer going anywhere with respect to you. We could say it’s going nowhere fast. So when you hit it, it’s you hitting the photon instead of the photon hitting you. The photon had momentum, and now it’s got inertia. It’s got mass. But we don’t call it a photon any more. We call it an electron.
All we had to do to create mass was “stop” the momentum of the photon. It isn't really stopped because it's still going round so now it's angular momentum. But now the photon isn't moving in aggregate with respect to you. So now it’s an electron, and the travelling energy is going nowhere fast so it’s "rest energy". That’s what mass really is. They call it rest mass, only it’s rest energy, and it’s not actually at rest. When all’s said and done, if it's the photon moving it's got momentum, but if it's you moving it's got inertia. That’s mass. It just depends on who's moving. Because momentum and inertia are the same thing. There's a symmetry between them. Because motion is relative, and that's what relativity is all about.
We can destroy mass just as easily by bringing an electron and positron together to reverse the process. The electron and the positron are just two opposite knots which cancel one another and annihilate, resulting in two 511 keV photons. It’s like the electron is a twist in your tight taut fishing line, and the positron is the opposite twist. Slide them together and twang, gone. The knots are no more, and the photons are off like a shot. And we do destroy mass easily. We do it every day. Annihilation is routine, we take pictures with it. It's called a PET scan, and if you're lucky there's one in a hospital near you:
It really is that simple. Energy is usually a travelling stress like a photon, rippling through the volume of space like a pressure-pulse shooting through a ghostly block of transparent rubber. Mass is just how you measure it when it's tied in a knot so it's going nowhere fast. Or in other words:
Mass is a measure of the amount of energy that is not moving in aggregate with respect to the observer.
