Cats can certainly hear a lot of things we can't-- about an octave lower and two octaves higher than your typical adult human. It could be that there's an infra- or ultrasonic standing wave getting the cats' attention. These can be created by air-handling equipment or any other moving, vibrating machinery, or even by wind currents and lots of other things. Certain infrasonic frequencies are known to induce that "haunted" feeling and even hallucinations in humans, and at least one famous haunting has been solved by looking for standing waves (20 hz in this case).
Cats also might pay attention to sounds that don't interest us. A slightly squeaky hinge or house-settling noise might sound like prey to them.
Cats may also see things we don't, not because those things are invisible, but because we're not looking for them. Someone else suggested a small bug. It could be that, or anything that moves, like a swirl of dust, or even anything that looks like a bug or anything else that might interest a cat-- even if it doesn't look that way to you. For instance, the other week, my cat was staring intently at the bathroom door, his hair standing practically on end. There was nothing there, and no one in the bathroom, and no water running or anything. But he'd reach out a paw as if to probe something and then jerk it back as if shocked, making those weird bird-hunting-type vocalizations all the while. I spoke to him and then reached out to him, and he jumped backward, spitting and hissing, so suddenly and violently that he opened a big cut on my wrist. I think he'd been investigating a corner of a Kleenex that was sticking under the door. He'd decided it was evil and dangerous and thrilling.
People have always explained curious phenomena by reference to supernatural forces, sometimes I think because of a lack of imagination, other times because of a will to believe in the survival of bodily death. When rational explanations surface for their observations, they seldom discard the theory but instead reject the evidence or seek out new evidence that supports their conclusion. Starting with your conclusion and working backward to the evidence is sometimes called "a priori reasoning," and it signals a need to believe.
If I were going to evaluate a claim that there was a ghost around, and that the cat or others could perceive it, I'd have to ask certain questions, like, what's the proposed mechanism here? You mentioned spectra. Spectra of what? If light is bouncing off of it (making a shadow, as in DexX's example), then what is the light bouncing off of, and why shouldn't we therefore be able to see it, and how could a dead person make it happen? If it makes a sound, then let's speculate on what is causing compression and rarefaction of the elastic medium we call air, and-- again-- how this is linked to being a dead person.
One question I would have to ask is, if a disembodied human soul can effect changes on the physical environment, why does it ever need a body, and why can't we seem to do ghost tricks while living? Seems like it'd come in handy, right? You could send messages without a cell phone, spy on people invisibly, and all kinds of neat stuff.
A more basic question is why our minds go straight to supernatural explanations when confronted with weird events. It's one thing to say, "the cat sees something I don't see," and quite another to say "the cat sees the ghost of Mrs. Dinglethwaite." And I'm speaking not as a detached, cool, scientific observer, but as a person who has had hair-raising-- literally hair-raising-- experiences of the weird and strange and hard-to-explain kind. I've seen ghosts (or faries or aliens) and I'm scared of them. But there is always the possibility that I hallucinated, or that my perceptions tricked me.
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