Some questions about general relativity

PeterDonis

But this "accelerates each particle equally" is frame-dependent; if you are freely falling yourself, you won't see other freely falling objects "accelerate"; they will just float next to you. We are so familiar now with video from the Space Shuttle and the International Space Station, that shows the astronauts and objects around them all floating weightless, that it may be hard to imagine how much of an insight it was to realize in 1907, as Einstein did, that "if a person falls freely, he will not feel his own weight". (Einstein later called this "the happiest thought of my life".)

The fact that we can make the "acceleration due to gravity" disappear by falling freely ourselves is why Einstein wanted a different definition of acceleration, one that wouldn't be dependent on adopting a particular point of view. We see freely falling objects "accelerate" because we're at rest on the surface of the Earth; but our theory of physics shouldn't be dependent on being at rest on the surface of the Earth. Defining acceleration as proper acceleration--i.e., as feeling weight--is general: it works everywhere, and doesn't depend on adopting a particular point of view.

PeterDonis

Because "moving in the time axis" is just "existing"; it's just moving from now towards tomorrow.

Warp

If you think about the visualization I described in my previous post, throwing a ball upwards would be (in that visualization) having an initial speed of the ball disc away from the Earth disc. As both discs move in the time axis (which is being curved by the Earth), the ball will follow a parabolic path due to its own inertia (because that's the shortest path in this curved coordinate system.) In our three-dimensional "slice" (which would be the plane in the visualization) it looks to us like the ball just goes up (away from the Earth) and then comes down, when in reality it's following a parabolic spacetime curve.

That doesn't really explain anything... :/

PeterDonis

What would count as an explanation? Do you think you can somehow avoid "moving in time" from now to tomorrow? That's what the "time axis" represents: one point on that axis is your "now", another one is your "tomorrow", and since you can't avoid moving from now to tomorrow, you can't avoid moving along the time axis from one point to another.

Warp

Something that helps me understand, or even get a faint idea.

There's nothing that forces something to move in the three spatial axes, so why is the time axis different, even though for the calculation of movement due to gravity it's basically handled as just a fourth geometry axis?

The three spatial axes are free to be traversed in any direction at any speed, but the fourth time axis is not. You cannot travel it backwards, you cannot stop. "Something" forces you to travel it in one direction, and your speed is determined by something else than your acceleration (basically mass determines it, rather than anything else, although I may be really wrong here.)

So it's both handled as "just another dimensional axis", but it's also quite different from the other three. This is, AFAIK, a fundamental aspect of GR, but I just cannot grasp it.

I know, but I don't understand why. What is it that makes everything traverse the time axis?

PeterDonis

Because you can't stay at the same point of time the way you can stay at the same point of space. That's just a physical fact. If you want to know why that physical fact is a fact, well, nobody knows. So perhaps there is no "explanation" that will meet your requirements in this particular case.

Actually every object's "speed through time" is the same (technical point: this is true for every object with nonzero rest mass; things like light that have zero rest mass work a bit differently). Mass, or more precisely energy, is like "momentum through time", not "speed through time"; your "speed through time" is your "momentum through time" (i.e., your energy), divided by your mass (i.e., your energy), so it ends up being the same for everything.

There may not be any answer to this beyond "that's just the physical fact". See above.

Warp

I'm not sure I'm satisfied with that answer because it just sounds like "yeah, in principle you could travel in the time axis at will, but for some reason that nobody knows, you are forced to travel in one direction and you can't travel the other way" even though, if I have understood correctly, that's not what GR postulates. AFAIK being able to "travel" freely in the time axis would cause all kinds of paradoxes and would break GR. I think that GR handles time differently from the spatial axes (but I know nothing about the GR equations, so it all goes well above my head.)

PeterDonis

GR doesn't really postulate anything as far as whether or not you can travel along the time axis at will like you can along the spatial axes. It models spacetime as a 4-dimensional thing that just "is", not as something "moving" through the time axis. Interpreting the time dimension as something that objects "move" through is a way of linking up the spacetime model with our everyday experience, but you can do all the math and make all the predictions in GR without ever using it.

Mathematically, the reason the time axis is different is that it has an opposite sign in the metric. This is true in SR as well; take a look at the Minkowski line element:

[tex]d\tau^2 = dt^2 - dx^2 - dy^2 - dz^2[/tex]

The [itex]dt^2[/itex] term has opposite sign from the other terms; that means the interval [itex]d\tau^2[/itex] is not positive definite. That is, the interval between two distinct points can be positive, zero, or negative. That's not possible in ordinary Euclidean geometry: there, the distance between two points can only be zero if the points are identical, and it can never be negative.

What all this means is that, in spacetime, there is something fundamentally different about a timelike interval with [itex]d\tau^2 > 0[/itex], vs. a spacelike interval with [itex]d\tau^2 < 0[/itex] or a null interval with [itex]d\tau^2 = 0[/itex]. They are three physically distinct kinds of intervals. The same is true in GR; the only difference there is that the line element can look different than the formula above, due to spacetime curvature.

But you'll notice that nowhere in any of this did I talk about anything "moving" along a curve, or through an interval. If two points are separated by a timelike interval, that means some timelike curve connects them, so some object's worldline can pass through both points. But that's just a fact about the geometry of spacetime, in the same way that the statement "the Earth's equator passes through Quito, Ecuador and Nairobi, Kenya" is a fact about the geometry of the Earth's surface. (I don't know that that's exactly a fact, btw; those cities are close to the equator but probably not exactly on it. But it illustrates what I'm getting at.) We can describe the geometry of spacetime without talking about anything "moving" in it, just as we can describe the geometry of the Earth without talking about any objects moving on it.