How do projectors project the color black?

Projectors do not project the color black. This makes sense since black is really the absence of light, and you can't project something that does not exist. When a projector sends a beam of light on to a wall or a projector screen so that an image is formed on the wall or screen, the parts of the image that look black are really a very dim white color (which we sometimes call gray). The projector sends some light to all parts of the image, including the parts that we perceive as black. Some white light is indeed beamed to the parts of the image that are supposed to be black, but the light is typically dim enough in these regions that they look black to our eyes when surrounded by areas of the image that are receiving much more light and therefore are much brighter.

On the left is an illustration of what a black-and-white checkerboard image would look like projected on the wall if a projector could literally project black. Note that the dark regions in the projected image would be darker than the regions of the wall outside of the projected image. Since black is actually just the absence of light and not something that can literally be projected, the illustration on the right shows what we actually see. Regions of the projected image that are supposed to be black are actually a dim white color, which our brains interpret as black if the image contrast is high enough. Note that the "black" squares in the checkerboard on the right are actually brighter than the regions of the wall outside of the projected image. Public Domain Image, source: Christopher S. Baird.

Our human eyes and brains are designed to evaluate a color based on how it looks relative to the colors of the surrounding objects, rather than based on the absolute spectral content of the color. For instance, look at the image below. Directly surrounding both black dots are patches of blue that are the exact same color. They look different to humans because of the way our brains perceive color based on the color of surrounding areas. You can convince yourself that the areas right around the two black dots are the same color blue by holding a thick piece of paper up to this image with two holes cut out right over the black dots, allowing you to see the colors directly around the dots without seeing the rest of the image.

The blue colors immediately surrounding both black dots are the exact same color. They look like different colors because of the way our brains perceive color based on the color of surrounding areas. Public Domain Image, source: Christopher S. Baird.

The tendency of our brains and eyes to evaluate a color based on its relation to nearby colors is actually beneficial. Consider a standard yellow banana. A ripe banana placed next to an unripe banana is always more yellow and less green than the unripe banana, no matter what type of light is shining on it. The relative color difference between the two bananas does not change, even if the light source does change. In contrast, the absolute color of the bananas does change as the light source changes. For instance, objects that are in shade on a sunny day are more blue than objects in direct sunlight, because they are being illuminated by the whitish blue sky rather than the white sunlight. Therefore, a ripe banana sitting in direct sunlight has an absolute color of yellow, while the same ripe banana has an absolute color of yellowish green when in the shade on a clear sunny day. If humans were only able to evaluate colors according to their absolute spectral content, then we would bizarrely conclude that ripe bananas become unripe every time they go in the shade. By basing our color perception on relative color differences rather than absolutes, our brains are able to link object color to intrinsic properties of the object (such as ripeness), rather than assume that object color only tells us the color content of the light source.

Returning to the case of the image cast on the wall by the projector, in order for our brains to perceive a part of the image as black, it just has to be less bright than all the other parts of the image. In this way, a projector can throw some white light on a wall and convince you that it is black. There are a few ways that you can convince yourself of this concept. First of all, turn off the room lights, turn on the projector, and then send an image from your computer to the projector that is completely black everywhere. Look closely and you will see that the area of the wall on which the projector is projecting is brighter than an area of the same wall that is out of the projector's reach, despite the fact that the projector is supposed to be projecting perfect blackness. Now turn off the projector. You will notice that the wall darkens when the projector turns off, because it has stopped sending the dim white light that is supposed to represent the black colors in the image.

As another way to convince yourself of this, turn the projector on and have it display an image that has black and white regions. Now turn on and off the room lights. You will notice that when the room lights are on, the black regions in the image no longer look very black. They still look black in comparison to the other parts of the image, but they look like a washed-out, muted black color. This is because our color perception depends on relative color differences. When you turn on the room lights, you flood both the black and white regions of the projected image with white light, thereby decreasing the contrast between dark and light in the image. This decreases your eyes' ability to use relative color differences to perceive a dim white color as a black color. For this reason, a projected image is most vivid and convincing when the image has a high contrast between the brightness of its white regions and the darkness of its black regions. High contrast is achieved by building a projector that can send out a lot of light to some regions and send out very little light to other regions. High contrast is also achieved by simply turning off the room lights and shuttering the windows. The fact that we watch projected movies in darkened rooms is direct evidence that projectors can't emit literal black, but instead emit dim white light which is interpreted as black when the contrast is high enough.