Photon energies do vary; they just don’t vary based on velocity. If you’re moving forward at high speed in some reference frame, and turn on your headlights, the photons from your headlights will, in fact, be higher energy (in that frame) than if you were standing still.
Absolutely!
It does add energy to the light. If you’re traveling at .99c and turn on the headlights, the light gets more energy. One way it could get more energy is to be traveling faster. But it can’t do that. So instead the light gets a higher frequency. It gets blue-shifted. And the opposite happens. As you’re traveling .99c and turn on the taillights, the taillights lose energy and get redshifted.
And so, objects traveling towards you will be blueshifted, objects traveling away will be redshifted. And when we observe distant galaxies they are all redshifted, which means they are all traveling away from us, which means in the past they were closer together, which leads back to the Big Bang.
But how does the light “know” that it should be higher energy, lower velocity?
See this is why I didn’t go into the sciences in college…
It knows it has higher energy for the same reason the baseball you throw from your moving car knows it has higher energy. It has higher energy because, as per your intuition, the energy added to the light from your .99c traveling spaceship can’t just disappear.
So, the kinetic energy of an object is mv^2/2, or mass, multiplied by velocity squared, divided by two. So when you have a ball weighing one kilogram and throw it at 10 meter/second you have a certain amount of kinetic energy. Throw it at the same speed from a car moving 10 meters/second, and it’s moving at 20 meters/second, and so it has more kinetic energy. Throw it backwards at 10 meters/second, and you find that it is moving at 0 meters/second, and has zero kinetic energy.
Same thing with the light, except you can’t make the light travel faster. So what happens instead is that the equivalent of the mass of the ball increases–in this case, the energy of the photon. So radio waves increase in energy to microwaves, microwaves increase to infra-red, red to yellow, yellow to blue, blue to ultraviolet, ultraviolet to x-rays, x-rays to gamma rays. All these photons travel at the same speed, but have higher and higher energies.
A quibble:
When galaxies move toward or away from us, their light is blue-shifted or red-shifted. However, the red shift we see for very distant galaxies is NOT caused by their relative velocity away from us. It’s caused by the expansion of space over the billions of years the light took to reach us. Essentially the light wave has been slowly stretched over the duration of its trip.