Today we audit mainstream physics and find it wanting. It is delicious. Some moron establishes the establishment view:
This answer makes no sense to me pic.twitter.com/kd2M1ecfYR— Subliminal Porte (@simplic10) December 27, 2015
No, you idiot.
@simplic10 However, if you try to use this immense power to heat it to hotter than the sun, all you get is metaphorical backscatter.— Alrenous (@Alrenous) December 27, 2015
Spoiler: forget all the fancy stuff. Get a mirror to reflect sunlight onto the sun. Less energy is escaping, so it has to heat up. What temperature you optically 'see' is a red herring at best. Yes, you can absolutely use mirrors and lenses to make a solar furnace hotter than the sun.
Joules/area/second is (part of) the definition of heat. The above moron is simply contradicting themselves. This is probably why academics love academese so much - if you say something stupid in clear language, then the stupidity is clear. We all say stupid things sometimes, but some of us are more willing to admit it than others.
To think clearly enough to devalue experiment, it's necessary to consider all the factors. For collecting all the considerations, highly mistaken experts are perfectly adequate. I'm only disappointed that "brightness theorem" doesn't seem to have a Wikipedia page, so I can't determine if it's pre-war or post-war science. (I predict it's post-war, I don't think prewar scientists made such glaring mistakes.) For those who prefer optimism to cynicism, it's time to hope the theorem is confined to the lower-class physicists who must resort to writing textbooks, which is why it has no official entry.
1. Are optics really passive?
2. Is area a factor or a non-factor?
3. What's with joule-per-second rates?
4. Can you in fact build a perpetual motion machine with it?
5. The sun has finite size and is difficult to focus onto a small image.
6. Is there an effective temperature limit, which the optics 'see'?
As it turns out, the xkcd forum-goers lgw and Xanthir are correct. Area is a factor, and you can trade radiative intensity for radiative area. Minerva, Quaanol, elasto, Taas, are rationalizing beliefs handed down by sacred, unalterable authority. There's some other people, but they seem too confused to even reliably categorize as on-topic.
If we surround Sol with an ideal ellipsoid Dyson mirror with one focus at Sol and the other at Mercury, removing all the annoying debris, then all of Sol's light will be focused on Mercury, and vice-versa, and they'll reach thermal equilibrium. However, Mercury, necessarily, will be emitting more photons/area/second. Or, equivalently, higher-energy photons. The only way to do this is if Mercury is at a higher temperature. Seems we're done here, but let's try to destructively test it. Some mistaken lines of thought: what if I get it to emit more photons by having more higher-energy molecules, without going over? T is average kinetic energy, so that's a contradiction - I have posited it's hotter while staying cold. Maybe molecules stop absorbing photons, becoming reflective? First, they don't, second, reflection still involves a transfer of energy: Newton 3. One non-mistaken line of thought: if we have two Sol-equivalent lens targets and layer them over each other, the power must be higher, meaning the target has to dump more energy at equilibrium than a target hit by only one.
It is true that both Sol and Mercury have finite size. It is possible that the inefficiency of the Dyson mirror at, say, Oort cloud-radius would somehow misplace enough photons from Sol to prevent Mercury from getting too hot. After all, it's focused at Sol's core, it can't be focused at all the various points of Sol's surface. But, it seems unlikely. While there's a finite size that Sol's image can be resolved down to, I've resolved such an image and it was smaller than Mercury. Nevertheless, to be certain, I'd have to do math, and I'm lazy. Let's put the mirror at infinity instead, so Sol appears to be a point source. Problem: solved. Sure I've now removed the entire universe as 'annoying debris' and cancelled the latest season of space expansion, but you can do that for cheap in gedankenland.
There are two ideal things preventing this from being a perpetual motion machine. First, to optically focus Mercury's higher-temperature light back onto Sol requires not simply an ideal lens, but a magic lens. No matter where it is, the lens will disrupt the mirror's focus. It would have to be a daemonic lens that dodges into hyperspace when it sees solar photons but comes back when it sees mercurial photons. Second, we can only heat objects smaller than Mercury to something that's hotter than Mercury. Maybe we could heat a small circle of Sol's surface with our daemon lenses, but then we'd have to heat an even smaller circle of Mercury with the resulting increased radiation. We cannot use Mercury to in turn heat Sol hotter than Sol.
That said, because we can so manipulate temperature, we don't need a full Dyson mirror. The temperature of the smaller objects will be a direct function of the angular coverage of the mirror and an inverse function of the area of the object, meaning if we have a smaller mirror or fewer lenses, to get superhot we only need to choose a smaller target. Half efficiency path, half area target. Though notably a half-Dyson-ellipsoid or half-silvered Dyson would be roughly quarter efficiency, since we lose half of the energy on the way in and then half again on the way back. Nevertheless, we're talking engineering here. It's most probably entirely feasible to make a +6000K solar furnace right now with real budgets and real materials.
There's two further non-idealisms. When we focus Sol onto Mercury, entropy must increase - we end up with more photons flying around in the space between them, each with an energy budget. To oversimplify, we're heating the vacuum. For bonus perspective, imagine a non-energy-generating heat blob, alone except surrounded by a perfect mirror. The system won't lose energy because mirror, but its equilibrium temperature will be lower than T0, since the blob has to fill the space between it and the mirror with photons. Even ideal optics aren't passive, they're an implicit heat pump. Non-ideal optics are even worse, since the mirror will heat up and radiate out the back, wasting energy that could be making Mercury hot. Lenses are no better - the idea lens has a thickness of zero atoms. Good luck building that. Even if I had worked out that we could focus light from Sol onto a target and then back at Sol to break things with ideal lenses, it would just mean that real lenses would scatter and absorb more light than we were supposed to be getting out.
Second, to get a perpetual motion machine, not only do we have to get useful work by using Mercury as a heat source and Sol as a sink, we have to get enough useful work out to fission an alpha particle back into four protons, two electrons, and negative two neutrinos. This would require Sol to emit energy while being an infinite-sized 0K heat sink. Slightly impossible. Unlike our passive heat blob, Sol would increase in temperature if surrounded by mirrors. Seeing the sun's surface is not like seeing the hot surface of a passive object, which may be confusing our hidebound expert physicists. If we have to model Sol as an optical 'temperature' to be 'seen,' then the correct quantity depends somehow on the fusion reaction, which is not only hotter than the surface, but hotter than the core. Sadly can't find the wag who noted that you can divide total fusion-to-iron energy in Sol by total system mass. Spoiler: you get a very large number, not 6000K. Thinking about trying to hook a Carnot engine between Sol and Mercury hurts my brain, I used a match as a model instead. As long as I got more work out than radiative energy the match puts in, I can ignore the convection, and declare perpetual motion. (That said the mirror may have to be slightly magic, to avoid soot.) However, I didn't get much work, it was simply a slightly less abstract version of the above. Further, any real engine will have friction losses on the piston.
Finally, there's some points to make about power.
In normal heat conduction, more conductive materials (copper?) will deliver more power. This doesn't mean the target gets hotter, it means it gets hot faster. This is what the brightness theorem imagines would happen with a solar furnace. Simply put, light is not heat. In those terms, a real shocker. Heat doesn't travel in waves. There's no mirror or lens for heat, since there's no wave to guide. There's no heat double-slit experiment. Heat cannot destructively interfere, whereas laser cooling is a thing. Even if you could lens heat, since it's essentially velocity with no net direction, it would fail to be net focused. It would chill in the lens, having a cold one.