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Gravity from general relativity

000ike
Posts: 11,196
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7/22/2013 6:05:43 PM
Posted: 3 years ago
I need a few clarifications.

1) The stardard illustration of the bowling ball on the rubber sheet seems to imply that space curves around the massive object. Does space curve around the object or toward the object?

2) I don't understand how an object can accelerate without a force acting on it. How is that possible? If gravity is not a force, but just a distortion of the spacetime fabric, then what is it that causes an object to accelerate toward the massive object if it was not initially in motion?

3) I get that objects orbit around massive objects because they have linear velocities in curved spacetime just on the precipice of the disturbance caused by the massive object. But what is it hat would cause someone to plummet to the ground if he's in the sky, if there's no force to initiate his acceleration?
"A stupid despot may constrain his slaves with iron chains; but a true politician binds them even more strongly with the chain of their own ideas" - Michel Foucault
Poetaster
Posts: 587
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7/22/2013 6:17:19 PM
Posted: 3 years ago
At 7/22/2013 6:05:43 PM, 000ike wrote:
I need a few clarifications.

1) The stardard illustration of the bowling ball on the rubber sheet seems to imply that space curves around the massive object. Does space curve around the object or toward the object?

Spacetime is distorted by the object's mass and is described using a vector field directed toward the object; the sheet illustration exploits dimensional suppression to cater to the human imagination.

2) I don't understand how an object can accelerate without a force acting on it. How is that possible? If gravity is not a force, but just a distortion of the spacetime fabric, then what is it that causes an object to accelerate toward the massive object if it was not initially in motion?

The actual curvature of spacetime is the cause; the cause of the curvature, is in turn, the massive objects themselves. In this way, the relation is reciprocal.

3) I get that objects orbit around massive objects because they have linear velocities in curved spacetime just on the precipice of the disturbance caused by the massive object. But what is it hat would cause someone to plummet to the ground if he's in the sky, if there's no force to initiate his acceleration?

This is also caused by the curvature of spacetime; all of the arrows of the vector field are pointing toward the source of distortion, with their magnitudes at each point governed by a scalar potential.
"The book you are looking for hasn't been written yet. What you are looking for you are going to have to find yourself, it's not going to be in a book..." -Sidewalker
the_croftmeister
Posts: 678
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7/22/2013 6:20:21 PM
Posted: 3 years ago
At 7/22/2013 6:17:19 PM, Poetaster wrote:
At 7/22/2013 6:05:43 PM, 000ike wrote:
I need a few clarifications.

1) The stardard illustration of the bowling ball on the rubber sheet seems to imply that space curves around the massive object. Does space curve around the object or toward the object?

Spacetime is distorted by the object's mass and is described using a vector field directed toward the object; the sheet illustration exploits dimensional suppression to cater to the human imagination.

2) I don't understand how an object can accelerate without a force acting on it. How is that possible? If gravity is not a force, but just a distortion of the spacetime fabric, then what is it that causes an object to accelerate toward the massive object if it was not initially in motion?

The actual curvature of spacetime is the cause; the cause of the curvature, is in turn, the massive objects themselves. In this way, the relation is reciprocal.

3) I get that objects orbit around massive objects because they have linear velocities in curved spacetime just on the precipice of the disturbance caused by the massive object. But what is it hat would cause someone to plummet to the ground if he's in the sky, if there's no force to initiate his acceleration?

This is also caused by the curvature of spacetime; all of the arrows of the vector field are pointing toward the source of distortion, with their magnitudes at each point governed by a scalar potential.

I would add here that it is important to remember that there is no acceleration. In a gravity well, the inertial frame actually moves towards the earth (giving the appearance of acceleration).
Poetaster
Posts: 587
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7/22/2013 6:24:52 PM
Posted: 3 years ago
At 7/22/2013 6:20:21 PM, the_croftmeister wrote:
At 7/22/2013 6:17:19 PM, Poetaster wrote:
At 7/22/2013 6:05:43 PM, 000ike wrote:
I need a few clarifications.

1) The stardard illustration of the bowling ball on the rubber sheet seems to imply that space curves around the massive object. Does space curve around the object or toward the object?

Spacetime is distorted by the object's mass and is described using a vector field directed toward the object; the sheet illustration exploits dimensional suppression to cater to the human imagination.

2) I don't understand how an object can accelerate without a force acting on it. How is that possible? If gravity is not a force, but just a distortion of the spacetime fabric, then what is it that causes an object to accelerate toward the massive object if it was not initially in motion?

The actual curvature of spacetime is the cause; the cause of the curvature, is in turn, the massive objects themselves. In this way, the relation is reciprocal.

3) I get that objects orbit around massive objects because they have linear velocities in curved spacetime just on the precipice of the disturbance caused by the massive object. But what is it hat would cause someone to plummet to the ground if he's in the sky, if there's no force to initiate his acceleration?

This is also caused by the curvature of spacetime; all of the arrows of the vector field are pointing toward the source of distortion, with their magnitudes at each point governed by a scalar potential.

I would add here that it is important to remember that there is no acceleration. In a gravity well, the inertial frame actually moves towards the earth (giving the appearance of acceleration).

Yes, this is important: Galilean relativity implies that absolute velocity does not exist (i.e. it's frame-dependent), but its equations imply that acceleration is absolute (frame-independent).

In general relativity, acceleration is finally made frame-dependent (you can get it to disappear by choosing the right coordinate system).
"The book you are looking for hasn't been written yet. What you are looking for you are going to have to find yourself, it's not going to be in a book..." -Sidewalker
000ike
Posts: 11,196
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7/22/2013 6:27:19 PM
Posted: 3 years ago
At 7/22/2013 6:17:19 PM, Poetaster wrote:
At 7/22/2013 6:05:43 PM, 000ike wrote:
I need a few clarifications.

1) The stardard illustration of the bowling ball on the rubber sheet seems to imply that space curves around the massive object. Does space curve around the object or toward the object?

Spacetime is distorted by the object's mass and is described using a vector field directed toward the object; the sheet illustration exploits dimensional suppression to cater to the human imagination.

2) I don't understand how an object can accelerate without a force acting on it. How is that possible? If gravity is not a force, but just a distortion of the spacetime fabric, then what is it that causes an object to accelerate toward the massive object if it was not initially in motion?

The actual curvature of spacetime is the cause; the cause of the curvature, is in turn, the massive objects themselves. In this way, the relation is reciprocal.

3) I get that objects orbit around massive objects because they have linear velocities in curved spacetime just on the precipice of the disturbance caused by the massive object. But what is it hat would cause someone to plummet to the ground if he's in the sky, if there's no force to initiate his acceleration?

This is also caused by the curvature of spacetime; all of the arrows of the vector field are pointing toward the source of distortion, with their magnitudes at each point governed by a scalar potential.

but that's not very specific. I'm told that the curvature causes the acceleration, I guess what I'm asking is how (in simple terms). The way the orbit of the Earth around the sun is explained, the Earth is already in straight motion and then enters a depression in the fabric and then perpetually revolves around it. That makes enough sense. What is it that causes a stationary object (stationary relative to the the object propagating the gravitational field) to fall down? Why would a sky diver, who's suspended high in Earths atmosphere plummet to the ground if he is not already in motion? You mentioned a vector field, but how can there be a field without a force?
"A stupid despot may constrain his slaves with iron chains; but a true politician binds them even more strongly with the chain of their own ideas" - Michel Foucault
the_croftmeister
Posts: 678
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7/22/2013 6:29:38 PM
Posted: 3 years ago
At 7/22/2013 6:27:19 PM, 000ike wrote:
At 7/22/2013 6:17:19 PM, Poetaster wrote:
At 7/22/2013 6:05:43 PM, 000ike wrote:
I need a few clarifications.

1) The stardard illustration of the bowling ball on the rubber sheet seems to imply that space curves around the massive object. Does space curve around the object or toward the object?

Spacetime is distorted by the object's mass and is described using a vector field directed toward the object; the sheet illustration exploits dimensional suppression to cater to the human imagination.

2) I don't understand how an object can accelerate without a force acting on it. How is that possible? If gravity is not a force, but just a distortion of the spacetime fabric, then what is it that causes an object to accelerate toward the massive object if it was not initially in motion?

The actual curvature of spacetime is the cause; the cause of the curvature, is in turn, the massive objects themselves. In this way, the relation is reciprocal.

3) I get that objects orbit around massive objects because they have linear velocities in curved spacetime just on the precipice of the disturbance caused by the massive object. But what is it hat would cause someone to plummet to the ground if he's in the sky, if there's no force to initiate his acceleration?

This is also caused by the curvature of spacetime; all of the arrows of the vector field are pointing toward the source of distortion, with their magnitudes at each point governed by a scalar potential.

but that's not very specific. I'm told that the curvature causes the acceleration, I guess what I'm asking is how (in simple terms). The way the orbit of the Earth around the sun is explained, the Earth is already in straight motion and then enters a depression in the fabric and then perpetually revolves around it. That makes enough sense. What is it that causes a stationary object (stationary relative to the the object propagating the gravitational field) to fall down? Why would a sky diver, who's suspended high in Earths atmosphere plummet to the ground if he is not already in motion? You mentioned a vector field, but how can there be a field without a force?

You were told wrong. The curvature does not cause acceleration, the object still follows a geodesic (like a straight line but in curved space) and thus experiences no acceleration.
the_croftmeister
Posts: 678
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7/22/2013 6:31:10 PM
Posted: 3 years ago
Perhaps a better question is the following. If an object moving perpendicular in a gravitational field curves inwards, why wouldn't a stationary object curve inwards (just without any tangential velocity) which effectively amounts to a straight line back to earth.
000ike
Posts: 11,196
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7/22/2013 6:33:03 PM
Posted: 3 years ago
At 7/22/2013 6:29:38 PM, the_croftmeister wrote:

You were told wrong. The curvature does not cause acceleration, the object still follows a geodesic (like a straight line but in curved space) and thus experiences no acceleration.

I hope this isn't a stupid or redundant question, but then what causes the acceleration? We observe that things fall down and accelerate as they do so. What's the explanation for that?
"A stupid despot may constrain his slaves with iron chains; but a true politician binds them even more strongly with the chain of their own ideas" - Michel Foucault
Wnope
Posts: 6,924
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7/22/2013 6:33:51 PM
Posted: 3 years ago
At 7/22/2013 6:05:43 PM, 000ike wrote:
I need a few clarifications.

1) The stardard illustration of the bowling ball on the rubber sheet seems to imply that space curves around the massive object. Does space curve around the object or toward the object?

2) I don't understand how an object can accelerate without a force acting on it. How is that possible? If gravity is not a force, but just a distortion of the spacetime fabric, then what is it that causes an object to accelerate toward the massive object if it was not initially in motion?

3) I get that objects orbit around massive objects because they have linear velocities in curved spacetime just on the precipice of the disturbance caused by the massive object. But what is it hat would cause someone to plummet to the ground if he's in the sky, if there's no force to initiate his acceleration?

At 7/22/2013 6:05:43 PM, 000ike wrote:
I need a few clarifications.

1) The stardard illustration of the bowling ball on the rubber sheet seems to imply that space curves around the massive object. Does space curve around the object or toward the object?

2) I don't understand how an object can accelerate without a force acting on it. How is that possible? If gravity is not a force, but just a distortion of the spacetime fabric, then what is it that causes an object to accelerate toward the massive object if it was not initially in motion?

3) I get that objects orbit around massive objects because they have linear velocities in curved spacetime just on the precipice of the disturbance caused by the massive object. But what is it hat would cause someone to plummet to the ground if he's in the sky, if there's no force to initiate his acceleration?

It may help to think of a magnetic.

If two magnetics are close to each other, you don't need an outside "force" to get them to pull towards each other.

The magnetic force is inherent in the magnets. Bigger magnet means bigger magnetic force.

If a magnet is floating in space and comes close to another magnet, it may just make a slight adjust due to attraction.

But if a small magnet flies past a planet-sized magnet, the small magnet will appear to "fall from the sky" as though being driven by a force.
Poetaster
Posts: 587
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7/22/2013 6:46:39 PM
Posted: 3 years ago
At 7/22/2013 6:27:19 PM, 000ike wrote:
At 7/22/2013 6:17:19 PM, Poetaster wrote:
At 7/22/2013 6:05:43 PM, 000ike wrote:
I need a few clarifications.

1) The stardard illustration of the bowling ball on the rubber sheet seems to imply that space curves around the massive object. Does space curve around the object or toward the object?

Spacetime is distorted by the object's mass and is described using a vector field directed toward the object; the sheet illustration exploits dimensional suppression to cater to the human imagination.

2) I don't understand how an object can accelerate without a force acting on it. How is that possible? If gravity is not a force, but just a distortion of the spacetime fabric, then what is it that causes an object to accelerate toward the massive object if it was not initially in motion?

The actual curvature of spacetime is the cause; the cause of the curvature, is in turn, the massive objects themselves. In this way, the relation is reciprocal.

3) I get that objects orbit around massive objects because they have linear velocities in curved spacetime just on the precipice of the disturbance caused by the massive object. But what is it hat would cause someone to plummet to the ground if he's in the sky, if there's no force to initiate his acceleration?

This is also caused by the curvature of spacetime; all of the arrows of the vector field are pointing toward the source of distortion, with their magnitudes at each point governed by a scalar potential.

but that's not very specific. I'm told that the curvature causes the acceleration, I guess what I'm asking is how (in simple terms). The way the orbit of the Earth around the sun is explained, the Earth is already in straight motion and then enters a depression in the fabric and then perpetually revolves around it. That makes enough sense. What is it that causes a stationary object (stationary relative to the the object propagating the gravitational field) to fall down? Why would a sky diver, who's suspended high in Earths atmosphere plummet to the ground if he is not already in motion? You mentioned a vector field, but how can there be a field without a force?

Good questions. But why do you think that that "earth-falling" case is materially different from the "sun-falling" case? They're each governed by the same equations according to general relativity.

To answer your question of why motion begins at all, the "scalar potential" which I mentioned comes in handy (by that I mean it's necessary). The "potential" here is potential energy; the potential energy tells us how much motion, basically, the field can give some object at any point. That is how the motion begins: the potential energy starts doing work on the object; the object then is measured to accelerate.

The potential energy itself is explained by the fact that spacetime obeys certain conservation laws; it "tends to minimize" its 4-D "surface area", just like the tendency of a water droplet to form a sphere. It's a minimization principle of energy; it happens because energy is conserved.
"The book you are looking for hasn't been written yet. What you are looking for you are going to have to find yourself, it's not going to be in a book..." -Sidewalker
the_croftmeister
Posts: 678
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7/22/2013 6:51:17 PM
Posted: 3 years ago
At 7/22/2013 6:33:03 PM, 000ike wrote:
At 7/22/2013 6:29:38 PM, the_croftmeister wrote:

You were told wrong. The curvature does not cause acceleration, the object still follows a geodesic (like a straight line but in curved space) and thus experiences no acceleration.

I hope this isn't a stupid or redundant question, but then what causes the acceleration? We observe that things fall down and accelerate as they do so. What's the explanation for that?
We observe things falling down, they do not accelerate. This is an important distinction. In curved space, if we do not experience a force, we all follow a geodesic. Imagine objects on the surface of the Earth. If you follow the geodesics (lines of latitude) from the equator in parallel (initially) then if we are travelling at the same speed we both eventually reach the North pole (collide). Parallel paths at one instant (the paths of the skydiver and the earth) do not have to remain parallel in curved space and this is why the path of the skydiver appears to involve acceleration (but in actual fact is 'straight' which guarantees no acceleration). In reality there is none and the only force we actually experience is the upward force of the atmosphere as it is held up by the pressure of all the gas pressing on the Earth.

If you can understand why the Earth can orbit the sun without 'accelerating' based on the bowling ball model, imagine what would happen if you placed a stationary object on the slope. Where would it go?
Poetaster
Posts: 587
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7/22/2013 6:52:26 PM
Posted: 3 years ago
At 7/22/2013 6:29:38 PM, the_croftmeister wrote:
At 7/22/2013 6:27:19 PM, 000ike wrote:
At 7/22/2013 6:17:19 PM, Poetaster wrote:
At 7/22/2013 6:05:43 PM, 000ike wrote:
I need a few clarifications.

1) The stardard illustration of the bowling ball on the rubber sheet seems to imply that space curves around the massive object. Does space curve around the object or toward the object?

Spacetime is distorted by the object's mass and is described using a vector field directed toward the object; the sheet illustration exploits dimensional suppression to cater to the human imagination.

2) I don't understand how an object can accelerate without a force acting on it. How is that possible? If gravity is not a force, but just a distortion of the spacetime fabric, then what is it that causes an object to accelerate toward the massive object if it was not initially in motion?

The actual curvature of spacetime is the cause; the cause of the curvature, is in turn, the massive objects themselves. In this way, the relation is reciprocal.

3) I get that objects orbit around massive objects because they have linear velocities in curved spacetime just on the precipice of the disturbance caused by the massive object. But what is it hat would cause someone to plummet to the ground if he's in the sky, if there's no force to initiate his acceleration?

This is also caused by the curvature of spacetime; all of the arrows of the vector field are pointing toward the source of distortion, with their magnitudes at each point governed by a scalar potential.

but that's not very specific. I'm told that the curvature causes the acceleration, I guess what I'm asking is how (in simple terms). The way the orbit of the Earth around the sun is explained, the Earth is already in straight motion and then enters a depression in the fabric and then perpetually revolves around it. That makes enough sense. What is it that causes a stationary object (stationary relative to the the object propagating the gravitational field) to fall down? Why would a sky diver, who's suspended high in Earths atmosphere plummet to the ground if he is not already in motion? You mentioned a vector field, but how can there be a field without a force?

You were told wrong. The curvature does not cause acceleration, the object still follows a geodesic (like a straight line but in curved space) and thus experiences no acceleration.

This is the minimization principle I was alluding to. However, just because the object itself doesn't experience acceleration doesn't mean spacetime curvature doesn't cause acceleration; an observer in free-fall, as you've described, wouldn't observe their own reference frame as having any curvature precisely because they are in free-fall. Where spacetime curvature is measured, so too will there be acceleration measured (I think).
"The book you are looking for hasn't been written yet. What you are looking for you are going to have to find yourself, it's not going to be in a book..." -Sidewalker
the_croftmeister
Posts: 678
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7/22/2013 7:01:30 PM
Posted: 3 years ago
At 7/22/2013 6:52:26 PM, Poetaster wrote:
This is the minimization principle I was alluding to. However, just because the object itself doesn't experience acceleration doesn't mean spacetime curvature doesn't cause acceleration; an observer in free-fall, as you've described, wouldn't observe their own reference frame as having any curvature precisely because they are in free-fall. Where spacetime curvature is measured, so too will there be acceleration measured (I think).

Depends what you mean by measurement of acceleration. It also depends on the kind of curvature. A helical path on the surface of a cylinder would not experience any acceleration (I don't think) unless it was rotating (relative to the cylindrical coordinates) because there is no Gaussian curvature (I think that's what it's called, my differential geometry is a little rusty). You can only measure an 'acceleration' if the parts of the object follow geodesics that do not preserve the overall shape of the entity. You can measure the fact that other objects appear to be accelerating towards or away from you, but this is not real acceleration. I was talking about acceleration that alters the geodesic you are following as opposed to visible acceleration (which results from the fact that all our observations are effectively a distorting projection onto 3D space).

This is analogous to the statement 'there is no centrifugal force' because it is fictional (doesn't appear in inertial reference frames). Of course you can talk about fictional accelerations if you want (a bit like talking about non-proper time or length in Spec Rel).
Poetaster
Posts: 587
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7/22/2013 7:16:05 PM
Posted: 3 years ago
At 7/22/2013 7:01:30 PM, the_croftmeister wrote:
At 7/22/2013 6:52:26 PM, Poetaster wrote:
This is the minimization principle I was alluding to. However, just because the object itself doesn't experience acceleration doesn't mean spacetime curvature doesn't cause acceleration; an observer in free-fall, as you've described, wouldn't observe their own reference frame as having any curvature precisely because they are in free-fall. Where spacetime curvature is measured, so too will there be acceleration measured (I think).

Depends what you mean by measurement of acceleration. It also depends on the kind of curvature. A helical path on the surface of a cylinder would not experience any acceleration (I don't think) unless it was rotating (relative to the cylindrical coordinates) because there is no Gaussian curvature (I think that's what it's called, my differential geometry is a little rusty).

If the path is imbedded into the surface of the cylinder (the cylinder is the path's "native space"), then you're right to say that it isn't accelerative. However, if the path is "riding" on top of the surface, with the cylinder being imbedded in ordinary 3-space or something, then the path would need to be accelerative (it would require centripetal acceleration).

If we just imagine a helical path traced out in 3-space, then it also is accelerative.

You can only measure an 'acceleration' if the parts of the object follow geodesics that do not preserve the overall shape of the entity. You can measure the fact that other objects appear to be accelerating towards or away from you, but this is not real acceleration. I was talking about acceleration that alters the geodesic you are following as opposed to visible acceleration (which results from the fact that all our observations are effectively a distorting projection onto 3D space).

This is analogous to the statement 'there is no centrifugal force' because it is fictional (doesn't appear in inertial reference frames). Of course you can talk about fictional accelerations if you want (a bit like talking about non-proper time or length in Spec Rel).

Yes, nothing disagreeable here; gravity is a fictional force in general relativity. I was just trying to phenomenologically rationalize the classical observations which we call "acceleration" in relativistic terms.
"The book you are looking for hasn't been written yet. What you are looking for you are going to have to find yourself, it's not going to be in a book..." -Sidewalker
the_croftmeister
Posts: 678
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7/22/2013 7:20:44 PM
Posted: 3 years ago
At 7/22/2013 7:16:05 PM, Poetaster wrote:
Yes, nothing disagreeable here; gravity is a fictional force in general relativity. I was just trying to phenomenologically rationalize the classical observations which we call "acceleration" in relativistic terms.
Ah, fair enough. I was just worried it would confuse the issue talking about the two different kinds of acceleration without making the distinction clear. Yes, classical accelerations appear, the important point being that they don't require causation because they are fictional.
Poetaster
Posts: 587
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7/22/2013 7:26:10 PM
Posted: 3 years ago
At 7/22/2013 6:33:03 PM, 000ike wrote:
At 7/22/2013 6:29:38 PM, the_croftmeister wrote:

You were told wrong. The curvature does not cause acceleration, the object still follows a geodesic (like a straight line but in curved space) and thus experiences no acceleration.

I hope this isn't a stupid or redundant question, but then what causes the acceleration? We observe that things fall down and accelerate as they do so. What's the explanation for that?

The explanation is that inertial paths in one reference frame may appear to be non-inertial in another reference frame.
"The book you are looking for hasn't been written yet. What you are looking for you are going to have to find yourself, it's not going to be in a book..." -Sidewalker