There’s a lot of threads about theoretical physics around at the moment, so please excuse another.
I cannot grasp what spacetime actually is. Explanations I’ve read have described it as being the “fabric of the universe”. Does this actually mean that the universe is made out of spacetime, as well as everything within it? Am I made out of spacetime or is it just a device used for explaining how the universe works?
Secondly, the rubber sheet and bowling ball analogy is commonly used to explain the bending effects of mass on spacetime. Now, this is a dumbed down explanation, but I’m curious as to what is actually causing the bending of spacetime. The bowling ball bends the rubber sheet because gravity attempts to pull the ball down to the ground, but what about spacetime? Also, thinking about the rubber sheet, suppose that I placed two balls on the sheet. Their downward push would create a trough of some sort in which they would both reside. Having read “The Elegant Universe”, the illustrations showing this bending effect seem to imply that spacetime returns to being a perfectly flat surface a short distance away from a mass. Is this a limitation of the illustrations, or is this what actually happens, contrary to what would happen with the rubber sheet?
The fabric of the universe is indeed spacetime, either of four dimensions, according to relativity theory, or eleven dimensions, according to M-theory. Matter exists inside of spacetime. If you use the analogy of a raisin cake instead of the rubber sheet, the cake is spacetime and the raisins represent the matter, which includes you.
Gravity does indeed warp spacetime. Again, the raisin cake analogy works better for understanding than the rubber sheet analogy does. Two raisins obviously distort the cake surrounding them. The distortion depends on the mass of the raisin. A big fat heavy raisin bulges the cake more than a tiny light raisin. Even so, no matter how heavy the raisin, the cake is tremendously huge and the measurable effect of the raisin can only be felt so far. This is why the rubber sheet appears to flatten out after a while. The gravity of any mass can technically be felt everywhere but is so small that real-world affects soon disappear.
The distortion creates other effects. Very loosely (and inaccurately) put, you can think of the cake thinning out between the two raisins. This makes them want to move toward one another, but they can’t do so in perfectly straight lines because of the overall distortion. And neither can any other raisin affected by this distortion. This warpage can be seen in our real universe in the form of light bending in the gravity well of the sun, for example.
The time dimension is harder to visualize, but think of it this way. You can’t have a measurement or an effect until time has passed. The distortion is a representation of an effect moving in time from one raisin to the next at the speed of light. Without that time dimension we can’t even begin to talk of any spatial effects, which is why Einstein realized they are so thoroughly mixed together as to form one larger thing, spacetime.
Nitpick: gravity is spacetime. That is gravity is nothing more than an expression of the fact that spacetime is curved.
Now, curvature is, on one level, a measure of the fact that “parallel” lines move closer together or further apart. If you and I stand a mile apart on the equator and both walk north, then even though we both start off pointing in the same direction, the distance between us will slowly decrease. On the surface of the earth, this seems perfectly natural since we can observe that 2-d curved surface from a 3-d vantage point. In 4-d spacetime, we can’t look at it from some 5-d vantage point, so straight lines moving towards each other seem (to us) to curve and we posit some sort of attractive force between them before we understand that really it’s just that the space we’re moving through that is curved.
My take from reading Sir Herman Bondi’s bood on relativity is that space-time is a system of coordinates that we use to describe the motions of physical objects in the universe.
And, to answer the OP: the concise answer is that stress-energy curves spacetime. This is a measure of all the matter, energy, momentum, pressure, and so on at each point in space. The Einstein equation has on one side a certain expression for the curvature of spacetime and on the other side the stress-energy. Add a bit of matter or energy to the second side and you have to add a bit of curvature to the first to keep them equal.
As for “how” energy curves spacetime and what the spacetime manifold “is”… that’s more a question for the philosophers, imho.
Yes. The range of the gravitational force, is infinite. If you visualize the gravitational force as a bending of spacetime, that bending had also better extend to infinity.
On the other hand, spacetime can be “practically flat” a ways away from the source, and is usually assumed so by physicists. No, it’s not justifiable. See current thread on mathematical vs. physics thinking.
Actually, come to think of it, I don’t see an a priori reason that space couldn’t be truly flat a finite proper distance from a source. There are plenty of functions around, and even smooth (infinitely differentiable) ones, which are identically zero everywhere but on a compact region. In fact, to rigorously justify the basis of GR (which most physicists never bother to do), you need to use a lot of them.
Also, a nitpick: the “force” of gravity is (as I understand it) a convenient shorthand in GR, but not really a force as thought of in Newtonian mechanics. If you were arguing from intuition that our old thought of gravity had infinite range, so the new one should, so curvature can be affected far out, fine. If you meant to invoke some GR concept of force, you’re actually begging the question a bit here. Again, I could probably cook up a toy universe that’s flat outside a compact region.
A simpler and albeit not complete way to understand time as a dimension can be thought of this way…
Any point in 3D space is defined by three coordinates: X, Y, Z. So, you could say lets meet for lunch at a restaurant that is 3 miles north, 2 miles west and on the fifth floor from some known location (say your house…doesn’t matter as long as we both know the starting point). You can think of X, Y and Z as forward/backward, right/left, up/down or north/south, east/west, up/down.
However, while this defines a point in 3D space it is not complete. What time do we meet for lunch? Which day? Time can be considered a 4th coordinate that further defines a point in space/time. You could give coordinates for where the earth is in space but the earth is constantly moving. Without the time coordinate your three coordiantes have a better chance of putting you where the earth was or where it will be rather than put you on earth.
Sure it’s justifiable. For any nonzero curvature, arbitrarily small, I can find a point sufficiently distant from a mass that the curvature would be less than that value. There is some amount of curvature which is so small that it would have no detectable effects (of course, this amount depends on technology, but it exists), so at a distance greater than where the curvature is smaller than this threshhold of detectability, spacetime is practically flat.
And while there are plenty of infinitely-differentiable functions which are identically zero everywhere but on a compact region, it was my understanding that there are no analytic functions with that property. Is this not correct?
I meant, it’s not rigorously justifiable to say that it is zero. The simple fact is that you’re throwing something away, which gets under a mathematician’s skin. Every time I see someone assume at the beginning that spacetime is flat in a given region and then never go back to make sure that the tiny error they introduced doesn’t blow up in their face, I have to take a long, deep breath.
Correct. An analytic function is (by definition) determined by its germ at a given point. Who said that the appropriate functions in GR were analytic, though? In fact, if you insist on using analytic functions it gets a lot harder to show you have any metrics to work with, to have a well-founded notion of integration (sort of important for Lagrangian GR like Ashtekar uses…), and so on.