If you throw a ball to someone, you'll see that the path it follows is curved. This is true even though the path itself doesn't have any mass. It seems like we can say that the path is curved without having to say that the path is a real physical object.... I mean, you can hold the ball, but you can't hold its path.
I don't think I can give a full explanation, but if you want to read more about it, write back and I'll try to recommend a good book.
The term that's used is actually that "spacetime" is curved.... not just space. But what does this mean anyway?
We have to back up a little to understand all this. First, why are the three space dimensions always treated together? Well, one reason is that the length of an object is the same no matter how you rotate it in the three spatial dimensions. Just think of taking a ruler; no matter what angle you hold it or turn it, it always has the same length. It doesn't matter which way you define as "up-down" or "left-right" or "forward-back," all three spatial dimensions contribute in exactly the right way so that the distance between the ends of the ruler stays the same.
OK, but then why do physicists combine time with space to make "spacetime?" One answer is that what I said in the last paragraph about the ruler isn't exactly true in all cases. As strange as it sounds, the spatial distance between any two points depends on how fast you're moving. The difference is way too small to ever notice it in everyday life (even going on the fastest airplane). But for things that travel near the speed of light, it makes a big difference. So, for example, imagine that two firecrackers flash at different points in space and at different times. The regular 3-dimensional spatial distance between the two flashes depends very slightly on how fast the person measuring them is moving. Now, according to special relativity (and also some very precise experiments) there is a quantity that always stays the same whether you're moving or not. And that quantity comes from a formula that includes both the spatial distance between the two flashes and also includes the time difference between the two flashes. So, when you get near the speed of light, the quantity that stays the same no matter how the observer is moving involves the three spatial distances difference between them. It's all four things (three space plus one time) that go together, not just the three spatial dimensions!
If there's no mass anywhere in our imagined situation with the firecrackers, then that formula for finding the quantity that will always be the same is fairly simple (and it looks a lot like the formula from geometry for just the spatial distance between the flashes but with an extra piece that depends on the time difference between them). We call that "flat spacetime." If you learn about special relativity, you're really learning about how things work in flat spacetime... and it's very cool!
Now I think I can finally get back to your original question.
This formula for finding the quantity that stays the same even for things moving near the speed of light gets more complicated if there is any mass nearby. It turns out that the mass makes the formula different for different points around the masses. So, we started with a simple 3-dimensional distance formula. But by including the effects of things that travel near the speed of light and also including mass, we wound up with a 4-dimensional (3space + 1time) formula that's different at different points in space. However, even though the new formula isn't like the plain-old "distance" that we're used to, we can still imagine that it's sort of like a distance, and just say that the space itself isn't nice and flat like simple 3-dimensional space. I guess you could say itís an analogy because itís nice to visualize the formula as still being the formula for some sort of distance.
Hope this makes sense.
(republished on 07/19/06)