One thing has always really bothered me about general relativity (probably due to gaps in my understanding). As I understand it, any object with mass distorts spacetime around it – a common analogy is to imagine an object on an elastic sheet: the sheet bends under the weight of the object (a nice graphic of this is at the top of this page).
Here’s the difficulty I’m having: in order to bend or distort something, you have to apply a force to it. So if the Sun (or other massive object) distorts nearby spacetime, that means the Sun must somehow apply a force to spacetime itself. I have a hard time wrapping my head around that – how does having mass allow an object exert a force on space itself?
The sheet thing is not a great analogy to start with. I think it’s better to think about the warping of spacetime as a property of spacetime itself, rather than something “outside” of spacetime which pushes on it.
It is better to understand the analogy as “spacetime has the geometric properties that a curved manifold has” than “spacetime is actually bent in some direction”. The former are properties we can determine while remaining inside the universe. The latter we’d have to leave the universe to verify.
As an example of one of those properties, imagine a very long, very heatproof measuring tape. We wrap it around the Sun and measure its circumference. To get the radius of the Sun, you might think you could just take the circumference and divide by 2*pi, but if you instead dip your measuring tape in the sun to its midpoint, you get a distance that is about 2km larger than that.
This is exactly what would happen if space was kind of “bowed out” into another dimension a little bit, so on that basis we can say that space is curved or bent or warped or what have you. But all you’re really measuring is distances inside the universe – how that relates to any outside-the-universe behaviour is an extrapolation.
A lense distorts the light that passes through it but doesn’t exert a “force” per se. It’s sort of like the presence of matter does, in fact, distort spacetime itself. And it very much acts like a lense.
