Kind of newbie question about gravity

[Tiago]

I understand what you're saying about the rolling balls on rubber sheet, but have you seen the Elegant Universe by Brian Greene? He actually explains the GR, not quite like rolling balls on a rubber sheet but he does say that the distortion of spacetime caused by the presence of a large mass (such as the sun) makes it so that other planets don't move in straight lines, but rather follow the natural path of the warped / distorted spacetime. This is, at least, the explanation that I've accepted as correct in my limited knowledge on the matter. I find this is much easier to understand at a planetary level than on a local level. Because on a planetary level we're just talking about warped spacetime and masses following those distorted paths, and on a local level the rules are different. What I'm probably confused about are two concepts such as feeling gravity (acceleration of the ground underneath our feet) and knowing it exists even though I'm in free fall. If I fall of a building I can't go on a straight line because the Earth's presence is distorting the spacetime or because it accelerated to "get me"? Does this even make sense?

Thanks

 

[A.T.]

I find this is much easier to understand at a planetary level than on a local level. Because on a planetary level we're just talking about warped space-time and masses following those distorted paths, and on a local level the rules are different.

The rules are the same. You probably just misunderstood a (potentially misleading) pop-sci explanation of the plantary level, which confuses paths in space with paths in space-time. A correct GR explanation of orbits (or anything with more than 1 spatial dimension) cannot be easily visualized in one diagram.

 

[PeterDonis]

the distortion of spacetime caused by the presence of a large mass (such as the sun) makes it so that other planets don't move in straight lines, but rather follow the natural path of the warped / distorted spacetime

Brian Greene has a way of explaining physics that, while it isn't exactly wrong, tends to easily generate misconceptions (and those often lead to threads here on PF...). In this case, the misconception is stated explicitly: "planets don't move in straight lines" is wrong. The paths of planets are straight lines--in curved spacetime. More precisely, they are geodesics, which are the equivalent of straight lines in a curved manifold. (The "straight lines" Greene was probably referring to are straight lines in a hypothetical flat "background", but that background doesn't exist and there's no way to actually physically measure these hypothetical "straight lines", so thinking about them only causes confusion.)

If I fall of a building I can't go on a straight line because the Earth's presence is distorting the spacetime or because it accelerated to "get me"?

If you fall off a building, you are moving in a straight line--more precisely, as above, a geodesic in the curved spacetime around the Earth. The Earth's surface accelerates upward to "get" you.

The reason this view is preferable is that it gives an easy way of telling what is moving in a "straight line" and what isn't: test for weightlessness. When you fall off the building, you are (ignoring air resistance) weightless. The Earth's surface is not. So you are the one moving in a straight line.

 

[Tiago]

Brian Greene has a way of explaining physics that, while it isn't exactly wrong, tends to easily generate misconceptions (and those often lead to threads here on PF...). In this case, the misconception is stated explicitly: "planets don't move in straight lines" is wrong. The paths of planets are straight lines--in curved spacetime. More precisely, they are geodesics, which are the equivalent of straight lines in a curved manifold. (The "straight lines" Greene was probably referring to are straight lines in a hypothetical flat "background", but that background doesn't exist and there's no way to actually physically measure these hypothetical "straight lines", so thinking about them only causes confusion.)
If you fall off a building, you are moving in a straight line--more precisely, as above, a geodesic in the curved spacetime around the Earth. The Earth's surface accelerates upward to "get" you.

The reason this view is preferable is that it gives an easy way of telling what is moving in a "straight line" and what isn't: test for weightlessness. When you fall off the building, you are (ignoring air resistance) weightless. The Earth's surface is not. So you are the one moving in a straight line.

Of course I probably understand the idea behind what Brian Greene explained, but not the complexity of its concepts. And it's good that I don't, cause I'm not a physicist, I'm merely someone that admires physics, live fascinated by it and largely because of enthusiasts like Brian Greene that make these subjects easy to follow for someone not linked to the field. Of course, like you said, that probably causes a lot of misconceptions and you guys have to deal with newbies with dumb questions :)

Anyway, I'm going to take advantage of my newbie position and insist on this stupid question. I understand that the Earth's surface accelerates to get whoever is in freefall with no proper acceleration (did I get this right?), but this is due to the Earth's own rotating movement (or acceleration, this I didn't get). But if everyone on the planet suddenly jumped of a building, how can the planet accelerate in every direction to get everyone? This is confusing and I'm asking this, probably without the basic knowledge to understand your explanation, but I'll risk it anyway :)

Thanks!

 

[A.T.]

I understand that the Earth's surface accelerates

Yes proper acceleration of 1g upwards, as an accelerometer placed on a table will confirm.

to get

It doesn't accelerate to get anyone. It accelerates just to stay in place, at a constant distance from the center. Remember that this is proper acceleration, not a change in velocity (coordinate acceleration).

whoever is in freefall with no proper acceleration

Yes, free fall = zero proper acceleration.

but this is due to the Earth's own rotating movement

No. The Earths rotation has nothing to do with it. A non-rotating planet would have gravity as well.

how can the planet accelerate in every direction to get everyone?

Proper acceleration away from the center doesn’t imply movement away from the center. Even in classical mechanics you can accelerate towards a center without getting closer to it (uniform circular motion). In curved space-time you can also accelerate away from a point without getting further away from it (without any rotation). Again, remember that the relevant acceleration is proper acceleration, not coordinate acceleration.

Do you understand the apple animation posted above (post #35)? When the green apple hangs on the branch, it has only an upwards force on it, so it has upwards proper acceleration. Proper acceleration upwards just says it has a curved (non-geodesic) path in space time. It says nothing about moving upwards.

To see what happens with an apple on the opposite side of the Earth, at the same time, have a look at this link, which shows both sides and the inside:
http://www.adamtoons.de/physics/gravitation.swf

 

[PeterDonis]

I understand that the Earth's surface accelerates to get whoever is in freefall with no proper acceleration (did I get this right?),

Locally, yes, you can view it this way: with respect to a local inertial frame in which the free-falling apple is at rest, the Earth accelerates upward (where here we mean "accelerates" in the Newtonian sense of the second derivative of position with respect to time) and hits the apple.

But this doesn't work globally, because spacetime around the Earth is curved, not flat. The curvature means the local inertial frames at different points on the Earth do not "line up" the way they would in flat spacetime (i.e., far out in empty space away from all gravitating bodies). So there is no single global inertial frame in which an apple falling in England and an apple falling in Australia are both at rest. So we can't view the Earth surface as globally accelerating (in the above Newtonian sense) in all directions at once, because that view of "acceleration" only works in an inertial frame.

GR solves this problem by not requiring global inertial frames; if we want to view things globally, we can use a system of coordinates centered on the Earth just fine. But these coordinates will not be an inertial frame--obviously, because free-falling objects, like falling apples and orbiting spacecraft , are not either at rest or moving in a straight line at a constant speed in this frame. GR also uses a different definition of acceleration, called "proper acceleration", which, as A. T. said, is just the reading on an accelerometer attached to the object. The advantage of this definition is that it is frame-independent; the reading on a given accelerometer attached to a given object is the same no matter what coordinates we use. But, as A. T. pointed out, this definition breaks the link between acceleration and motion: the Earth's surface can perfectly well be accelerating in all directions at once in this sense.

 

[Tiago]

I think what makes it confusing (at least for me) is understanding the principle of equivalence. Of course it's easy to understand why a person would feel an artificial gravitational field if traveling upwards on a rocket in empty space. I would feel the force of the acceleration of the rocket pushing me down and that would create an artificial sensation of gravity (assuming it's accelerating at 9.8m/s2). This is easy to understand because it's intuitive. But how do you transfer this principle to a circular surface such as a planet? If I was in a circular rocket accelerating in space, we all would feel the push downwards just the same. If we could imagine the rocket to be circular and have several floors, wouldn't we all feel the push the same direction? If it would work like the Earth, we would be pushed to the center of the rocket, right?

 

[A.T.]

But how do you transfer this principle to a circular surface such as a planet?

You have gravity in the rocket, because it follows a curved path in space time. The same is true for every part of the surface.

If it would work like the Earth

It's not like the Earth because the rocket in in flat space-time, so you cannot have outwards proper acceleration at each part.

 

[Tiago]

Is this video accurate on ilustrating GR?

 

[A.T.]

Is this video accurate on ilustrating GR?

It's not clear which aspect of GR is tries to illustrate. The distorted 3D grids might be an attempt to illustrate the purely spatial distortion (without the time dimension), but that cannot explain gravity (you need the time dimension for that).

And even for the spatial part it is wrong, because the distorted 3D grid still encompasses the same total volume an undisturbed grid would (within the same boundary). But the key feature of curved space is that there is more volume inside a boundary that you would expect based on Euclidean geometry. You cannot visualize this by shifting nodes around in an 3D grid. What you can do is:

- Reduce 3D space to a 2D surface with a dent, which has more surface area than a flat plane would, which corresponds to the extra volume the 3D space actually has.

- Use sector models: http://www.spacetimetravel.org/sectormodels1/sectormodels1_en_w.pdf

 

[jbriggs444]

The distorted 3d grid has more proportionally more coordinate volume assigned to the Earth than would otherwise be the case. That is, if the Earth occupies more volume, it covers more grid lines in the chart, so the grid lines have to be bent toward the earth. That part makes a certain amount of sense.

Beyond that, I won't try to defend the depiction.

 

[A.T.]

The distorted 3d grid has more proportionally more coordinate volume assigned to the Earth than would otherwise be the case. That is, if the Earth occupies more volume, it covers more grid lines in the chart, so the grid lines have to be bent toward the earth. That part makes a certain amount of sense.

Distorting the 3d grid merely shifts volume around, between the coordinate cells. It doesn't add to total volume as the actual spatial distortion does.

 

[Tiago]

I'm going to risk bringing a different subject that I think relates to everything we've been talking about. Of course, before Einstein, gravity according to Newton was much easier to understand. And in some websites, people will still say that gravity is a force that pulls to the center of the Earth. And that gravity is the attraction between bodies (which, of course, can't be true, since Einstein proved light is also bent by gravity). I think, besides Einstein's equations were more accurate than Newton's, his model to understand gravity is not as intuitive.

What I understand: the presence of mass bends spacetime, making planets and asteroids and whatever move through straight lines that are curved by the presence of bigger masses. This is an easy idea to understand, though it needs simplification to illustrate. At a local level (on Earth), if someone jumps of a building, the body will follow a straight line the same as if it were in space. But the straight line is bent towards the Earth, so it follows that path.

What I don't understand: how is the above compatible with "the ground meeting the body and not the body meeting the ground"? Is the fact that the planet is moving what creates our own feeling of weight?

thanks

 

[bhobba]

What I don't understand: how is the above compatible with "the ground meeting the body and not the body meeting the ground"? Is the fact that the planet is moving what creates our own feeling of weight?

In GR the gravitational force is simply an artefact of your coordinate system. The most natural coordinate system is a small freely falling one. In that frame there is no gravitational field, it is locally inertial, and the Earth comes towards you.

Their are a number of routes to GR. One is simply to take the above observation to its logical conclusion. When you 'stitch' all these small internal frames together you get curved space-time.

Thanks
Bill

 

[Nugatory]

What I don't understand: how is the above compatible with "the ground meeting the body and not the body meeting the ground"? Is the fact that the planet is moving what creates our own feeling of weight?

Suppose that you and I are both standing at the equator, one kilometer apart. We both start walking due north. We smack into one another at the north pole. Did you meet me or did I meet you? We can think about it either way, and they're both right - the actual physical phenomenon we're observing is that the distance between us shrank from one kilometer to zero, and that is described equally well by the "I met you" picture and the "you met me" picture.

 

[A.T.]

What I don't understand: how is the above compatible with "the ground meeting the body and not the body meeting the ground"?

As I already told you, the distinction between "ground meeting the body" and "body meeting the ground" doesn't have any physical significance. The only absolute (coordinate independent) fact is that they do meet. Their movement is just a question of which coordinates you choose to describe the meeting.

 

[Tiago]

Ok, it makes sense. It's basically an adaptation of Einstein's Special Theory of Relativity, right? It doesn't really matter if the ground hits the person or if the person hits the ground, it all depends on reference frame. Still, if a person jumps of a building and an external observer is on a spaceship in outter space watching both of them (the Earth and the falling person), what would he see? He would see them both moving towards each other, right? Just like he would see those two guys walking north and he would say they both met each other.

 

[A.T.]

Still, if a person jumps of a building and an external observer is on a spaceship in outter space watching both of them (the Earth and the falling person), what would he see?

Depends on how the spaceship moves relative to them. Also note that there is a difference between what an physical observer at some position sees, and what happens at distant positions according to some coordinates where he is at rest.

 

[Nugatory]

Ok, it makes sense. It's basically an adaptation of Einstein's Special Theory of Relativity, right?

You've got it. (Strictly speaking, it's not an adaptation of SR, it's a different application of the basic insight behind SR, but that's a quibble here).

 

[Tiago]

Knowing that it's off subject here, but there is something else that's been mindblowing me. Of course we know that gravity affects time and the presence of big masses such as planets will cause big distortions in spacetime which slows things down (if I'm floating in space, won't I be distorting spacetime for.. say.. an ant? Won't the ant gravitate towards me?).

Anyway. we know that velocity also affects time, and if I'm traveling very close to the speed of light, time would go slower for me than people on Earth. But only the speed of light is constant, everything else is relative to another frame of reference. So if I say the Earth is traveling at N km/h, that would have to be relative to something (perhaps the Sun). So, in order for me to be traveling at 99% of the speed of light, what reference frame am I using? Earth? I need to be going at the speed of the Earth, plus the 99% of the speed of light? And which velocity of Earth? Relative to what?

 

[Nugatory]

That's two questions. Best to start two new threads, as the question that started this one has been answered.