"Absolutely"???
The painting is an off-axis bump on a slowly rotating roughly spherical planet; consequently it sheds gravitational radiation. The painting is effectively a rigid extended object, and thus the gravitational radiation has a corresponding characteristic mode depending on the painting's position and orientation. The mode propagates to infinity. Sure, it would take an ultra-super-hyper-advanced gravitational wave detector to pick it out at interstellar range, but it is possible in principle.
Before forgery: one Mona Lisa mode; after forgery: two Mona Lisa modes. The modes will differ because of different position and/or orientation on Earth. The position and orientation of the "original" has changed over time, pretty substantially; the latter's position and orientation is likely to change over time too. Anyone observing the relevant bits of Earth's gravitational radiation spectrum (over time, if you want to think of it that way) would be able to distinguish the two. Trouble if the buyers are already accomplished gravitational wave astronomers (we've just barely started) and keep detailed records.
I think it's easier to handwave an extremely tuned gravitational wave detector than an exact atomic duplicator or a device which determines that a pair of ~ 8kg paintings are atomically identical (especially one that does not render them atomically non-identical).
Well, lets look at the current state of the art with respect to each.
We can make non-destructive 3-D scans of objects, and we can image and manipulate individual atoms. We can fabricate 3-D structures with feature sizes a few atoms wide. That's not quite at the level of a 3-D atomic duplicator, but it seems to me to be well on the way. The machines that do these things are not yet household items, but they are not uncommon, and the technology is improving all the time.
By way of contrast, the state of the art in gravity wave detection lets us see nothing less energetic than the collision of two black holes, and there are two devices that can do this on the entire planet.
So a 3-D atomic duplicator seems much more plausible than a device that can track an 8kg object through a gravitational field at interstellar distances.
Of their surfaces. What about the internal components?
The Mona Lisa is many things but it is not very thin or flat.
There are pigments and bindings involved that are sensitive to protein structure, as well; it's not just the count of atoms (and subatomic components; there are metals involved too, and isotope ratios are fingerprints), or even their relative position sensu stricto that matters.
I'm not sure how you could go about non-destructively determining (for example) potassium radioisotope differences in the original painting's layers between the optical surface and the canvas.
> state of the art in gravity wave detection ... black holes
We can detect gravitational waves of much closer much smaller systems just fine, and did so about 105 years before Poincaré proposed them formally. It was almost eighty years later that they were cast into the current perturbative formalism though.
The amplitude isn't really the critical factor with LIGO. It's sensitive to a particular frequency band (~ Hz - kHz) that is far from that of a small off-axis bump on a planet (~ microHz). It would totally pick up a set of deadlift weights spun around by a dental drill, if you could make that work in practice. (The amplitude difference between the sites would exclude it as an astronomical signal; the final inspiral of binary black holes and neutron stars have enormous amplitude, but that falls off linearly with distance, and they are very very far away, so the amplitude difference between sites is basically nil).
> track an 8kg object through a gravitational field
Given that one knows a number of things about the object including rough position and its surrounding environment (notably one wants to know its background metric exactly, or at least the exact model metric for Earth and some subleading-order non-Mona-Lisa contributions to perturbations thereof), it gets much easier. For a painting it's not something we could do today even if we had a spaceborne observatory sensitive to the relevant bits of Earth's GW spectrum, but it is plausible that we could observe the GWs shed by Olympus Mons with current technology, for example. (It'd be expensive to build).
We have good lunar masscon data so can watch the GWs shed by those as the moon relaxes into a more spherically symmetric shape; we have decent masscon data for Earth so can in principle with today's technology watch seismic events using gravitational wave Earth observation (as opposed to astronomy) (earthquakes make Earth rounder and spin faster, so the remaining masscons shed even more gravitational radiation).
https://en.wikipedia.org/wiki/X-ray_microtomography
> We can detect gravitational waves of much closer much smaller systems just fine
So what? We're talking about interstellar commerce here.
> today's technology watch seismic events using gravitational wave Earth observation
Seismic events are many, many orders of magnitude more energetic than a painting being moved around.
Even if you had the technology, all it would take to foil it would be for the copy to move close to the original at any point in time.
(And I have to emphasize here that the point here is not that it would be easy to execute a deception. The point is that no sensible being could possibly care whether or not they have "the original" if they have to resort to such extreme measures to figure it out.)
Neat! But, can it work in principle to pick out the structure and arrangement and composition of molecules? (I'd guess yes, but am not sure).
> We're talking about interstellar commerce here
Interstellar distances are ~ pc distances; the GWs we are looking for with LIGO are >> Mpc or << pc with GRAIL/GOCE/GRACE and their follow-ons. These observatories are sensitive to sources at radically different distances, but since such sources are not what's under study (and in particular it is hard to correlate their GW signals with other observables), their signals are excluded.
One could build a GW observatory to study an exoplanet in pretty fine detail (down to whatever the appropriate word for its seismology would be); there are a lot of unknowns about exoplanets, but their mass quadrupoles are determinable independently, and GW observatories essentially just look for changes in mass quadrupoles. What's responsible for any given change is the hard part, after one has built the (distributed) observatory.
GRAIL II would have done exactly that for the moon and (it was hoped) for a couple of other objects in the solar system. GRAIL itself was an approximation of this broad technique for the moon.
> Seismic events are many, many orders of magnitude more energetic than a painting being moved around
Sure, amplitude-wise. They have very distinct frequencies though. The argument you want to advance is that there are lots of similar objects (8kg rectangles, roughly) that generate a lot of noise, and worse, some of them are in the same art gallery (and with similar orientation) as the subject of observation. Worst of all, picking it out if it never leaves the Louvre wall is implausibly difficult; but if it's moved around the gallery, and in particular, it's back does not always face the gallery wall it normally hangs on, that opens a lot of "technology will improve" arguments, such as you use with the "perfect copier".
> all it would take to foil it would be for the copy to move close to the original at any point in time
The emissions from a bump on a rotating sphere is strongly determined by the bump's position. I happily admit that you can contrive ever-greater screening, though, but the cat would be out of the bag that you were deliberately trying to frustrate distinguishing between the original and the copy.
Which rather raises the question, why? Why not just hang the copy and ship the original to the aliens as agreed?
> no sensible being could possibly care ... if they have to resort to such extreme measures
I think you have the ordering wrong; they might resort to such extreme measures because they care. Their valuation function might not be identical to yours (why the Mona Lisa rather than any other Earth painting? or any painting they could make locally, in the style of an arbitrary Earthling?).
Or, more pointedly, maybe when 'all you have to do is to try to define what "the same thing" actually means', you, other people, and aliens' efforts might lead to different definitions.
Well, yeah, they might. But I'm assuming the aliens understand physics, in which case only one definition makes sense. You can snooker the irrational and the ignorant into paying for anything regardless of what species they are.
On which note, I have this nifty bridge you might be interested in? It's available at a great price. And it is the original!
> ... It had been believed to have been painted between 1503 and 1506; however, Leonardo may have continued working on it as late as 1517. Recent academic work suggests that it would not have been started before 1513. It was acquired by King Francis I of France ...
So what we rely on here is custody records, all the way back to King Francis I. Also, why limit the discussion to modern copies? Again, from Wikipedia:
> A version of Mona Lisa known as Mujer de mano de Leonardo Abince ("Leonardo da Vinci's handy-woman") held in Madrid's Museo del Prado was for centuries considered to be a work by Leonardo. However, since its restoration in 2012 it is considered to have been executed by one of Leonardo's pupils in his studio at the same time as Mona Lisa was being painted.
Basically, it was a copy, painted at about the same time. Now imagine that King Francis I liked the Mona Lisa so much that he had one or more copies created. How would we know now which was the original? Only via records.
Of course we do. We can, for example, carbon-date the wood that the original is painted on. We can do chemical analysis of the pigments. There are a lot of ways to distinguish a real Rennaissance painting from a modern copy. But (and this is the main point) only because the technology we have for making copies of paintings is not yet perfected. If we could make an atomically perfect copy of the Mona Lisa, it would be absolutely indistinguishable from the original.