So in a way, yes, black objects do get heavier (since mass and energy are interchangeable). You are adding energy to the objects, which we’re calling “heat”.
Remember–particle mass is measured in “electron volts.”
I knew someone would bring up this nitpick.
In a practical sense, it’s not heavier. Put it on a lab scale and weigh it – you won’t see a difference.
Then in that same sense there’s no relativity.
So if a proof comes down to a=2a, you’ve proven that a=0? Cool…
Let a = b = 1
a[sup]2[/sup] = ab
a[sup]2[/sup] - b[sup]2[/sup] = ab - b[sup]2[/sup]
(a+b)(a-b) = b(a-b)
a+b = b
2a = a
Therefore a = 0, therefore 1 = 0.
In that sense, yeah. Do you add velocities relativistically out on the highway?
Slightly related Q:
in FRET, do they actually know how the energy of the photon is transferred from donor to acceptor? Or is it just, you know, “magic”?
Am I right in assuming that a photon acquires momentum from whatever emits it? - i.e. that not only is the spot projected by my laser pointer pressing ever so slightly against the wall, but the laser pointer itself is pressing ever so slightly against my hand in the opposite direction? It must be, from the POV of conservation of momentum, mustn’t it?
Yup.
In general, no – reflected light is the same color as the incident light. There are situations in which you do get a change in momentum. These cases are called inelastic scattering, and there are books and conferences devoted to it, because it is comparatively a rare phenomenon. In general, light being reflected or scattered has the same wavelength (and therefore the same momentum) as the incident light.
The same sort of reflection and scattering without momentum change can happen with electrons, too. Look up Mossbauer scattering and Mossbauer spectroscopy.
Correct.
Ahhh – misread your question.
You divided by 0 again (bolding mine)
To get from the first bold line to the second you divide by (a-b), but since a=b, (a-b)=0.
It most certainly is not a nitpick. The fact that a system of photons with a center of momentum frame has mass goes right to heart of reality as described by Special Relativity. The complete equation for relativistic energy is:
E[sup]2[/sup] = m[sup]2[/sup]c[sup]4[/sup] + p[sup]2[/sup]c[sup]2[/sup]
Setting c = 1 for simplification:
E[sup]2[/sup] = m[sup]2[/sup] + p[sup]2[/sup]
So if the system of photons has a center of momentum frame p equals zero and
E = m or m = E.
So the container has more mass even if the photons don’t get absorbed but just keep bouncing around.
Mass equals energy that cannot be transformed away. In fact with enough photon energy in a small enough volume you could form a black hole.
I know. I was just pointing out the flaw in the earlier “refutation” that claimed that getting down to a=2a proved that a=0.
Oh - never mind 
Three men are travelling through the desert, and one of them pisses off the other two. One of them poisons the first guy’s canteen while he sleeps, but later in the night the other one (independently) drills a hole in the canteen, letting all the (poisoned) water out. Who’s guilty of murder?