Question about Gravity and curvature of space time

[jkl71]

You may find this useful, but as a warning in some ways this analogy is a bit deceptive, I’ll explain why later.

Imagine two people are at the Earth’s equator, fairly close together and both decide to walk due North (their paths are lines of longitude and are geodesics). They start off on parallel paths and both are traveling on straight lines. However, if they make careful measurements they’ll find that they are approaching each other. From the outside perspective it’s pretty clear what’s happening, the surface of the Earth is curved and parallel lines can converge. If the people walking don’t know about this, they might conclude there is a force causing them to accelerate towards each other.

In general relativity bodies that are not acted upon by any force (in this context by force I mean what Newton might call a non-gravitational force) travel on space-time geodesics. Since space-time can be curved, initially parallel geodesics can converge. So two observers moving on geodesics can seem to accelerate towards each other although neither is accelerating. For example, a body in free fall in the Earth’s gravitational field will seem to accelerate towards the Earth, but a better perspective is that the world lines of this body an the Earth, which are both geodesics, are converging because of space-time curvature.

Why is the analogy a bit deceptive? In the first case it’s just a curved spatial surface and time is taken as an external parameter (basically it’s the usual Newtonian point-of-view). In the case of GR, it’s space-time itself that’s curved and there is no external time.

Also consider a person standing on the surface of the Earth. From the Newtonian perspective one would usually say that person is not accelerating. From the GR perspective it’s more natural to say that the person is accelerating, in other words deviating from a space-time geodesic, because of the force from the ground on their feet (imagine a rocket in deep space traveling at a constant acceleration).

 

[pervect]

Yes I understand that light speed is always constant in a vacuum, slows down in a medium, but I'm afraid I don't get that video.

What part don't you get?

Let's break it down into xxxxx four parts. The first part is understanding that one can represent space-time via a diagram, a space-time-diagram.

The second part is that if you have 1 space and one time dimension, your space-time diagram is a two dimensional diagram.

The third part, is to understand that you can draw a two-dimensional space-time diagram on a curved 2d surface (which we will visulaize as being the 2d surface of some 3d object).

The fourth part is to understand the consequences of drawing this, by interpreting the results of the diagram (which is basically a map) physically. This entails a bit of geometry, and also the ability to go back from the abstract space-time diagram to what it represents.

Which of the above parts do you get, and which do you not get?

 

[Christine88]

You may find this useful, but as a warning in some ways this analogy is a bit deceptive, I’ll explain why later.

Imagine two people are at the Earth’s equator, fairly close together and both decide to walk due North (their paths are lines of longitude and are geodesics). They start off on parallel paths and both are traveling on straight lines. However, if they make careful measurements they’ll find that they are approaching each other. From the outside perspective it’s pretty clear what’s happening, the surface of the Earth is curved and parallel lines can converge. If the people walking don’t know about this, they might conclude there is a force causing them to accelerate towards each other.

In general relativity bodies that are not acted upon by any force (in this context by force I mean what Newton might call a non-gravitational force) travel on space-time geodesics. Since space-time can be curved, initially parallel geodesics can converge. So two observers moving on geodesics can seem to accelerate towards each other although neither is accelerating. For example, a body in free fall in the Earth’s gravitational field will seem to accelerate towards the Earth, but a better perspective is that the world lines of this body an the Earth, which are both geodesics, are converging because of space-time curvature.

Why is the analogy a bit deceptive? In the first case it’s just a curved spatial surface and time is taken as an external parameter (basically it’s the usual Newtonian point-of-view). In the case of GR, it’s space-time itself that’s curved and there is no external time.

Also consider a person standing on the surface of the Earth. From the Newtonian perspective one would usually say that person is not accelerating. From the GR perspective it’s more natural to say that the person is accelerating, in other words deviating from a space-time geodesic, because of the force from the ground on their feet (imagine a rocket in deep space traveling at a constant acceleration).

Ok so does that mean that the acceleration of gravity is basically just an illusion? Or a quirk of geometry?

 

[Nugatory]

Ok so does that mean that the acceleration of gravity is basically just an illusion? Or a quirk of geometry?

No, the acceleration (as we're using the word here) is real. Release an object above the Earth's surface, and it and the the Earth's surface will move closer together. We choose to interpret that as the object falling towards a stationary Earth with some speed. That speed will increase with time - and that's more or less by definition acceleration.

Be aware that there is something else called "proper acceleration" which the falling object is not experiencing. But as I said... That's something different.

 

[Christine88]

What part don't you get?

Let's break it down into xxxxx four parts. The first part is understanding that one can represent space-time via a diagram, a space-time-diagram.

The second part is that if you have 1 space and one time dimension, your space-time diagram is a two dimensional diagram.

The third part, is to understand that you can draw a two-dimensional space-time diagram on a curved 2d surface (which we will visulaize as being the 2d surface of some 3d object).

The fourth part is to understand the consequences of drawing this, by interpreting the results of the diagram (which is basically a map) physically. This entails a bit of geometry, and also the ability to go back from the abstract space-time diagram to what it represents.

Which of the above parts do you get, and which do you not get?

I don't get any of it. I don't know enough about all of this to understand a graph like that.

 

[Dale]

I don't get any of it. I don't know enough about all of this to understand a graph like that.

Do you mind me asking what is your background? Usually by the time someone can ask the question you started with they have seen "position vs time" diagrams, which is pervect's point 1 and 2.

 

[A.T.]

The guy in the video lost me when he retuned the graph back to zero gravity and then proceeded with his explanation as though there was still gravity. How could that be if the graph was returned to zero gravity?

He merely switches between two different interpretations of gravity:

Einsteins interpretation: The free falling worldline is straight (force free), while the space-time coordinates are curvilinear.

Newtons interpretation: The free falling worldline is curved (bend by the force of gravity), while the the space-time coordinates are Cartesian.

Since the little moon is already in motion through space-time it falls toward the large planet without the need of an external force to get it moving because it's already in motion. Does that sound right?

Yes, the space time distortion changes the directions w.r.t the space & time axes, which manifest itself as accelerations in space.

I still don't get the acceleration though.

If you get why a free falling object starts moving in space, do you get the acceleration. To start moving from rest it needs to accelerate.

But note that this is just coordinate acceleration, a geometrical effect. A free falling object doesn't experience proper acceleration. A free falling accelerometer measures zero. In terms of proper acceleration, the apple is not accelerating down, but the branch and surface are accelerating up.

 

[jkl71]

Ok so does that mean that the acceleration of gravity is basically just an illusion? Or a quirk of geometry?

I wouldn't use the words illusion or quirk. In the approximation that you can ignore other forces objects move on geodesics. Space-time can be curved (the source being the stress-energy tensor) and this can make the geodesics "accelerate" (I'm using quotes because I'm using terms pretty roughly, the concepts and terminology can be made more rigorous) towards each other, e.g. they can start off parallel and then converge and intersect.

Consider an object in free fall in the Earth's gravitational field (of course the Earth is also in the gravitational field of this object). From the Newtonian point of view there is a force between the two objects and that causes them to accelerate towards each other. From the GR point of view the object and Earth still accelerate towards each other (again being a bit rough with terms), but there is no force. The acceleration arises from the way the object and Earth space-time geodesics converge.

In a sense there is acceleration of gravity in GR, but it's not the result of a force. It's the result of the way space-time geodesics converge. Typically the term acceleration in GR is used for forces that cause objects to not follow geodesics.

 

[Christine88]

Do you mind me asking what is your background? Usually by the time someone can ask the question you started with they have seen "position vs time" diagrams, which is pervect's point 1 and 2.

I just started Engineering school this year.

 

[Christine88]

He merely switches between two different interpretations of gravity:

Einsteins interpretation: The free falling worldline is straight (force free), while the space-time coordinates are curvilinear.

Newtons interpretation: The free falling worldline is curved (bend by the force of gravity), while the the space-time coordinates are Cartesian.


Yes, the space time distortion changes the directions w.r.t the space & time axes, which manifest itself as accelerations in space.If you get why a free falling object starts moving in space, do you get the acceleration. To start moving from rest it needs to accelerate.

But note that this is just coordinate acceleration, a geometrical effect. A free falling object doesn't experience proper acceleration. A free falling accelerometer measures zero. In terms of proper acceleration, the apple is not accelerating down, but the branch and surface are accelerating up.

Acceleration as I understand it is constantly changing velocity. Velocity is distance over time.

 

[Christine88]

I wouldn't use the words illusion or quirk. In the approximation that you can ignore other forces objects move on geodesics. Space-time can be curved (the source being the stress-energy tensor) and this can make the geodesics "accelerate" (I'm using quotes because I'm using terms pretty roughly, the concepts and terminology can be made more rigorous) towards each other, e.g. they can start off parallel and then converge and intersect.

Consider an object in free fall in the Earth's gravitational field (of course the Earth is also in the gravitational field of this object). From the Newtonian point of view there is a force between the two objects and that causes them to accelerate towards each other. From the GR point of view the object and Earth still accelerate towards each other (again being a bit rough with terms), but there is no force. The acceleration arises from the way the object and Earth space-time geodesics converge.

In a sense there is acceleration of gravity in GR, but it's not the result of a force. It's the result of the way space-time geodesics converge. Typically the term acceleration in GR is used for forces that cause objects to not follow geodesics.

So acceleration of gravity is a phenomenon that no one really understands and can only be explained by mathematical representation?

 

[Christine88]

He merely switches between two different interpretations of gravity:

Einsteins interpretation: The free falling worldline is straight (force free), while the space-time coordinates are curvilinear.

Newtons interpretation: The free falling worldline is curved (bend by the force of gravity), while the the space-time coordinates are Cartesian.


Yes, the space time distortion changes the directions w.r.t the space & time axes, which manifest itself as accelerations in space.If you get why a free falling object starts moving in space, do you get the acceleration. To start moving from rest it needs to accelerate.

But note that this is just coordinate acceleration, a geometrical effect. A free falling object doesn't experience proper acceleration. A free falling accelerometer measures zero. In terms of proper acceleration, the apple is not accelerating down, but the branch and surface are accelerating up.

Coordinate acceleration. Yes I think that is where I'm running into problems. I'm thinking in terms of actual acceleration as opposed to theoretical acceleration.

 

[Dale]

I just started Engineering school this year.

You may want to hold off on your questions for a few months. In that time your classes should teach you about position vs time graphs. Then pervects comments should be clear.

 

[Dale]

So acceleration of gravity is a phenomenon that no one really understands and can only be explained by mathematical representation?

No. For many people the mathematical representation leads directly to understanding. Just because you don't understand it after a day or two of study doesn't mean that nobody does, nor even that you won't be able to eventually.

 

[A.T.]

Acceleration as I understand it is constantly changing velocity.

That is "coordinate acceleration" while "proper acceleration" is what an accelerometer measures.

I'm thinking in terms of actual acceleration as opposed to theoretical acceleration.

No idea what you mean by "theoretical acceleration". I would suggest you learn the commonly used terms above, instead of inventing your own. And "actual acceleration" usually refers to "proper acceleration", not to the "acceleration as you understand it".

 

[PeterDonis]

I just started Engineering school this year.

How familiar are you with Special Relativity? If you haven't spent any time on that, General Relativity will, I think, be quite a stretch.

 

[Christine88]

How familiar are you with Special Relativity? If you haven't spent any time on that, General Relativity will, I think, be quite a stretch.

I'm working on it.

 

[berkeman]

Thread closed briefly for Moderation...

Thread re-opened after some cleanup.

 

[PeterDonis]

Coordinate acceleration. Yes I think that is where I'm running into problems. I'm thinking in terms of actual acceleration as opposed to theoretical acceleration.

The terminology that is used here in SR and GR is probably different than what you are used to. Coordinate acceleration is the rate of change of coordinate velocity with coordinate time (and I think this is more or less what you were describing as your understanding of the term "acceleration"). In Newtonian physics, this is what the term "acceleration" without qualification usually means. But the "coordinate" qualifier is important: by changing coordinates, you can make it look as if one and the same object has very different accelerations. For example, if I drop a rock, the rock accelerates downwards with respect to coordinates in which the Earth is fixed; but I could also choose coordinates in which the rock is fixed and the Earth and I are accelerating upwards. As far as coordinate acceleration goes, both of these coordinates are equally valid; neither one is more "real" than the other.

In relativity, however, the key is to focus on invariants--things that don't depend on what coordinates you choose--because those are what contain the actual physics. In the case of the rock, the key invariants are that the rock feels no acceleration--it is weightless (we are ignoring air resistance here--if it helps, imagine we're dropping the rock on the Moon instead of the Earth)--but I, standing on the surface, do feel acceleration (weight). And that is true regardless of whether we choose coordinates in which I am at rest, or coordinates in which the rock is at rest.

In relativity, this latter kind of acceleration--the kind that is directly felt--is called "proper acceleration", and when the term "acceleration" is used without qualification in relativity, it is more likely than not to mean proper acceleration rather than coordinate acceleration (although many sources unfortunately are sloppy about this). The key reason why we focus on proper acceleration in relativity is that it is the kind of acceleration that requires a force--i.e., it requires something to be done to the object. An object that feels zero proper acceleration will continue in the same state of motion, feeling zero proper acceleration, indefinitely, without anything having to be done to it. The role that curvature of spacetime plays is to determine what states of motion are states of zero proper acceleration (and in GR this is linked to the presence of gravitating masses via the Einstein Field Equation).

So the answer to the question, what starts the rock accelerating downwards? is that nothing has to--the rock's state of motion once I drop it is the "natural" state of motion in that part of spacetime, the one that any object will have if nothing is being done to it. It is I, and the rock before I drop it, who must have something continually being done to us (the Earth's surface pushing up on me, making me feel weight, and my hand pushing on the rock) to maintain our state of motion.

Much of what I've said here has already been said in this thread, but I thought it might be helpful to try to pull it all together.

I'm working on it.

Ok, good. One key thing that I think it will be helpful to keep in mind: in GR, physics in a small patch of spacetime (i.e., a small region of space over a small interval of time) works just like physics in SR. So everything you learn about SR can be carried over to GR as far as local physics is concerned. The difference in GR (i.e., when gravity is present) is that the small local patches of spacetime "fit together" globally in a different way than they do in SR (because in the presence of gravity, spacetime is curved and not flat).

An analogy that is often used is that, in a small enough region of the Earth's surface, you can ignore its curvature and treat it, locally, as if it were a flat plane. But globally, the little flat planes you use in each local area "fit together" differently, because of the curvature of the Earth's surface, than they would if the Earth was globally flat.

 

[jartsa]

So acceleration of gravity is a phenomenon that no one really understands and can only be explained by mathematical representation?

Case1:

Let's say two massive balls are placed some distance apart in empty space. As we have learned, the balls will not accelerate, until they collide. Ball 1 is not accelerating in the space-time that surrounds it, ball 2 is not accelerating in different kind of space-time, this non-acceleration in different space-times causes there to be an increasing relative velocity between the two balls.

Case2:

Same as Case1, except that a pole keeps the balls apart. Now balls are still being in different space-times, but they are also accelerating to opposite directions. These two things cause the distance and the relative velocity of the balls to stay constant.

If you wonder why the balls are in different space-times: One ball has a space-time curving ball on the right side of it, but the other ball has a space-time curving ball on the left side of it.

 

[PeterDonis]

If you wonder why the balls are in different space-times: One ball has a space-time curving ball on the right side of it, but the other ball has a space-time curving ball on the left side of it.

"Different spacetimes" is not the correct way of describing this situation. There is only one spacetime in each case; its geometry is different in each case, but it's still just one spacetime in each case.

In the first case, the one spacetime has two balls and no pole, with the stress-energy of the balls curving the geometry, and the spacetime geometry is dynamic--the balls fall towards each other until they collide, so the geometry changes with time (actually even that's not quite right, since time is one of the dimensions of the geometry--I can unpack further if needed). In this case, the balls are under zero stress (except for the stress due to their self-gravity) and feel zero acceleration.

In the second case, the one spacetime has two balls and a pole, with the stress-energy of all three curving the geometry, and the geometry is static--it doesn't change with time (same caveats here as above). In this case, the balls and the pole are under nonzero stress, and the balls feel nonzero acceleration.

 

[pervect]

I just started Engineering school this year.

I'd hope as an Engineering student, drawing a graph of $x = \frac{1}{2} \, a \, t^2$ would be familiar? A graph of that well-known formula would be a graph of position versus time for a falling object, also called a space-time graph. It's called a space-time graph because the t-axis is time, and the x-axis is space.

I think if you relax a little bit, you might realize that what we are asking you to do is not all that complicated, just rather abstract. Once you realize what a space-time graph is, we can maybe proceed further.

 

[Christine88]

I'd hope as an Engineering student, drawing a graph of $x = \frac{1}{2} \, a \, t^2$ would be familiar? A graph of that well-known formula would be a graph of position versus time for a falling object, also called a space-time graph. It's called a space-time graph because the t-axis is time, and the x-axis is space.

I think if you relax a little bit, you might realize that what we are asking you to do is not all that complicated, just rather abstract. Once you realize what a space-time graph is, we can maybe proceed further.

The math is not a problem. I Can do the math.

 

[TrickyDicky]

As long as you clearly distinguish gravitational, proper and coordinate accelerations as different concepts(even if in certain circumstances some of them may coincide) you'll be fine.

 

[PeterDonis]

As long as you clearly distinguish gravitational, proper and coordinate accelerations as different concepts

Actually, just the last two are enough; "gravitational" acceleration is just a particular case of one of the other two. (And there could be confusion over which one: in the case where I drop a rock, is "gravitational acceleration" the coordinate acceleration of the rock in my rest frame, or the proper acceleration I feel that the rock doesn't?)

 

[A.T.]

in the case where I drop a rock, is "gravitational acceleration" the coordinate acceleration of the rock in my rest frame, or the proper acceleration I feel that the rock doesn't?)

Your proper acceleration is due to electromagnetic repulsion. Gravitational acceleration is just coordinate acceleration in General Relativity.

 

[PeterDonis]

Gravitational acceleration is just coordinate acceleration in General Relativity.

I agree this is the most natural interpretation (and it still makes the term "gravitational acceleration" superfluous), but I don't think the GR literature is consistent on this point. For example, I've seen the term "gravitational acceleration" used to mean the proper acceleration necessary to hold an object stationary (with respect to the timelike Killing vector field of the spacetime).

 

[TrickyDicky]

I don't think that use is standard.
I know wikipedia is not always right but this excerpt looks right to me:"The "acceleration of gravity" ("force of gravity") never contributes to proper acceleration in any circumstances, and thus the proper acceleration felt by observers standing on the ground is due to the mechanical force from the ground, not due to the "force" or "acceleration" of gravity. If the ground is removed and the observer allowed to free-fall, the observer will experience coordinate acceleration, but no proper acceleration, and thus no g-force."

 

[A.T.]

I don't think the GR literature is consistent on this point.

If you talk just about the magnitude g as "gravitational acceleration", it obviously applies to both:
- downwards coordinate acceleration of free faller relative to hover
- upwards proper acceleration of hover

I agree with you that "gravitational acceleration" is not a third type of acceleration, just the name for the magnitude that can apply to both types above.

 

[TrickyDicky]

If you talk just about the magnitude g as "gravitational acceleration", it obviously applies to both:
- downwards coordinate acceleration of free faller relative to hover
- upwards proper acceleration of hover

I agree with you that "gravitational acceleration" is not a third type of acceleration, just the name for the magnitude that can apply to both types above.

You mean the three are actually the same up to sign?

 

[A.T.]

You mean the three are actually the same up to sign?

I mean that two listed vectors have the same magnitude, which is sometimes called "gravitational acceleration" or short "g".

 

[Dale]

You mean the three are actually the same up to sign?

For certain objects in certain coordinate systems in certain spacetimes, I would say "yes", but not in general.

 

[Christine88]

Actually, just the last two are enough; "gravitational" acceleration is just a particular case of one of the other two. (And there could be confusion over which one: in the case where I drop a rock, is "gravitational acceleration" the coordinate acceleration of the rock in my rest frame, or the proper acceleration I feel that the rock doesn't?)

That's where I get confused. Gravitational and coordinate acceleration. Or is it proper and coordinate acceleration? Or proper and gravitational acceleration? I guess I understand proper acceleration.

 

[A.T.]

Or is it proper and coordinate acceleration?

That's the two types of acceleration there are in general. Gravitational acceleration can refer to either of the two types in specific cases.

 

[Dale]

Or is it proper and coordinate acceleration?

Yes. Proper and coordinate acceleration are the two correct technical terms for the two different kinds of acceleration.