This is kinda like arguing about the high-energy particle emission from a warp drive. The warp drive (or time machine) does what the writers say it does (insofar as it does anything at all); you can only object if they fail to conform to a previously-established 'fact'. Since Gene's scientists obviously didn't fully understand what their device would do, and this was the first full test, there ain't no precedent for how the machine and its effects "should" work.With regard to C-14 dating of the artifact brought forward from Syracuse, I believe it would undergo a temporal as well as a spatial shift, i.e., it would age 2200-plus years during it's instantaneous transfer. The C-14 dating would be accurate. Unless tachyon transfer somehow obviates the temporal component.
Nice try. Tachyons are a little like FTL, you are free to invent your own. You could write it that way, of course, but you didn't. For one thing, the polished brass still brightly reflected sunlight. Funny things happen to brass and wood when they age, even in a vacuum. And since the mirrors of Archimedes were actually more likely to have been made of bronze, instead of brass, it wouldn't reflect any light at all after 2200 years. Archimedes was brilliant, but brass wasn't intentionally made until around 20 B.C..
Carbon uptake stops when the tree stops photosynthesis. The minute amount of [sup]14[/sup]C that makes-up all carbon just sits and cooks away with a known half-life from that point on. Nothing in the universe can make it decay faster or slower. Nothing.
Though there is something else to think about. The mirror that was sent back in time is made up of atoms that actually existed--somewhere in the universe--back 2200 years ago. Can the same matter exist in more than one place at the same time? And did the mirror that was 'swapped' un-age during the transfer?
Time travel stories are always full of questions and apparent paradoxes.
But they're sure fun to play with.
Gene may get into trouble saying (here, not in the story) that the Archimedean mirror aged 2200 years 'instantaneously'; however, what he is really saying is that the device moved the mirror (or its component subatomic particles) only in space -- the main effect was to keep those particles in their 2200-years-ago configuration and relocate them en masse to the lab from Syracuse (aside from those neutrons lost due to carbon-14 decay). Conversely, the materials in the modern mirror also already existed 2200 years in the past, albeit in a radically-different configuration...
Hence the atoms (and smaller bits) of both objects were sorta-kinda duplicated in their destination space-time frames, but the total mass (= energy) was conserved...
Now, suppose the twinned particles are linked on the quantum level (or whatever the correct terminology is)... 2200 years in the past (relative to the story) will the probably-still-in-ore-form metal atoms of the modern mirror be affected by anything that happens to the mirror-as-a-whole? And will the now-scattered component bits of the ancient mirror react when that mirror is manipulated in the 'now'? Discuss. Show your work. Bueller? What do you think will happen? Bueller?