Can a physical lens filter make a digital camera shoot true black-and-white?

Pardon me while I get a little metaphysical for a bit. "Color" as we understand it isn't a real property of anything in the universe. It's something created by our vision system — a complicated interaction in our eyes and brains. It's useful for things like "don't eat the poison berries", "look at that tiger over in the grass", and, more recently, "stopping our vehicles at intersections".

This sense is based on something which is a real property of objects in the universe: different materials scatter, reflect, and absorb different wavelengths of light in different ways. Our eyes have receptors which are differently-sensitive to different wavelengths of light, and the vision system translates that to what we call color.

Color itself can be thought of in a lot of different ways. One way which is helpful in this circumstance is to break it down into chromaticity and luminance — luminance is basically "brightness", and chromaticity is... the other color stuff — hue (red, orange, yellow, green, blue...) and saturation or colorfulness. Dividing the concept of color in this way works nicely with our mental model — but isn't actually immediately translatable back to the physical universe.

A filter which resulted in black and white would need to filter out the chromaticity and pass through only the luminance, because that's what a "black and white" photo basically is — just a record of the brightnesses, without all the other "color stuff".

But, I don't know of any way to do that. It certainly isn't possible with something similar to the kind of filters we normally use. Those just block either certain wavelengths (in the case of color filters or UV or infrared filters) or generally all wavelengths to a small degree (in the case of neutral-density filters). A "filter" which converted to black and white would have to actually transform the wavelength in some way (since light with no wavelength is ... darkness), rather than filtering it. This would probably involve some kind of nonlinear metamaterial and nothing I can explain with my high-school-level knowledge of physics. And it would have to convert all different wavelengths to the same wavelength, or else scatter them randomly so the result would be white light; this seems like it's probably preposterous. I feel safe in saying that even if it were possible, the result wouldn't be something you can attach to a camera and carry around.

On the other hand, one can certainly record just the brightness. That's what black and white film does, and it's actually what digital photosites do, too. They're inherently just measures of brightness, but today's digital cameras use filters to record brightness only in certain wavelengths, measuring blue, green, and red separately. (This roughly matches how human vision works, so we can combine that back to make a full-color image.) If you have one of the few cameras made without these filters (like the Leica M Monochrom), you just get a black and white image.

Of course, another approach is to filter out everything except one specific wavelength. You can see this in Jerry Coffin's answer here, or in this other question involving an almost-monochromatic sodium-vapor light. That's black-and-some-single-color rather than black-and-white, but may be close to you want. Of course, that cuts out quite a lot of light, and the other downside is that it also cuts out brightness levels from other colors — so you'll just see variance in green (or whatever selected color) and nuances of shade in the other colors won't register at all.