Why is the concept of proper acceleration important in General Relativity?

[earamsey]

Hi,
If gravity is result of curved spacetime;
1.why are there eliptical orbits around spherical bodies?
2.Why don't I stop accellerating when I reach the lowest point of the curvature, ie, on surface of Earth I am at lowest point of curvature I should stop accellerating (maybe fall back up) ?

 

[mathman]

For questions like these, Newtonian mechanics works just fine.
1. The shape really doesn't matter. Gravity of spherically symmetric bodies is the same as for a point with the same mass at the center. The possible orbits are conic sections.
2. You stop moving when you hit the ground because the ground is in the way. The acceleration is still there. That's indicated by the fact that you weigh something when you step on a scale. Your weight is a measure of force.

 

[robinson]

If you have a non rotating body, let's say, a long stick, (to start off with), and this body is heading towards a large gravity source, like, the earth. It is a long stick, but not that massive, so our theoretical stick will be captured, and fall into an orbit around the earth.

It's a thought question, so as to how that can happen isn't the important part, it is about curved spacetime and gravity. We can add thrusters to slow it down, whatever. But only to get to the point where gravity will cause an orbit. We want no thrust at the point where things start to curve.

What I'm wondering about, is will our stick, (which starts out passing by with the long axis pointing past the earth), will gravity (or curved spacetime) actually cause the stick to continue it's course, but curve around the earth?

Does gravity, in relativity, does it actually cause the path of objects to curve, and to them, they are still going "straight"? Does an observer on the stick (OK we could call it a long thin spaceship I guess) feel like nothing has happened, assuming there is no acceleration, to get into orbit?

Or if the stickship simply passes by, and is curved into a new trajectory, does an observer on the stickship feel and change? Like when a car goes around a corner? Or is it all free fall to them?

I'm trying to imagine what happens to an object when it is curving. Like light, they say that when it "bends" due to a massive gravity field, the photons don't actually bend, the light thinks it is still going straight, spacetime bends. Curves.

Is this true for objects as well?

 

[robinson]

Wow. That sounds terrible. Let me try and clean that up.

1) Does an object in orbit travel through curved spacetime?

2} Does the object experience any acceleration as it orbits? The kind that an observer onboard can detect?

3) Is it all free fall? Or can you feel a change when a gravity source changes your trajectory?

4) Will a long rod, whose long axis is at right angles to a gravity source, will it maintain a right angle as it curves around the planet? Or will it point in the same direction? In other words, if the rod is "pointed" towards a star, as it starts to orbit, does it rotate away from the star?

5) Will pictures help explain this?

 

[A.T.]

1) Does an object in orbit travel through curved spacetime?

Yes

2} Does the object experience any acceleration as it orbits? The kind that an observer onboard can detect?

No

3) Is it all free fall?

Yes

4) Will a long rod, whose long axis is at right angles to a gravity source, will it maintain a right angle as it curves around the planet?

That depends. But let's say both ends of the rod have the same distance to the source of gravity, and the oribit is a circle. In that case the rod will maintain a right angle to the line between its center an the source of gravity

 

[robinson]

Thanks. I thought that was the case, but wasn't sure.

So an object in "free fall" is actually falling in curved spacetime? Is that correct?

 

[A.T.]

Thanks. I thought that was the case, but wasn't sure.

So an object in "free fall" is actually falling in curved spacetime? Is that correct?

I don't know what "falling in spacetime" would mean. I would rather say:
An object in "free fall" is advancing on a straight path in spacetime:
http://www.relativitet.se/spacetime1.html

 

[Oscar Wilde]

Forgive me if I'm wrong, but don't objects in orbit experience centripetal acceleration? The observer wouldn't necessarily "feel" it, but that doesn't mean it would not exist.

 

[A.T.]

Forgive me if I'm wrong, but don't objects in orbit experience centripetal acceleration? The observer wouldn't necessarily "feel" it, but that doesn't mean it would not exist.

Depends on the definitinon of acceleration. In GR acceleration means the proper acceleration that you can "feel" (measure with a accelerometer). Unlike the coordinate acceleration (velocity derivate) proper acceleration is absolute (independent of the frame of reference).

 

[robinson]

Here's the thing then. If an object gets captured by another object, so that it goes into orbit around the much larger object, it isn't experiencing acceleration, like an object that is falling towards an object. Instead it is moving at a constant velocity (hence the free fall or weightlessness experience, like the ISS), in a straight line, which seems curved because of the warped space time.

Is that even close?

 

[A.T.]

Here's the thing then. If an object gets captured by another object, so that it goes into orbit around the much larger object, it isn't experiencing acceleration, like an object that is falling towards an object. Instead it is moving at a constant velocity (hence the free fall or weightlessness experience, like the ISS), in a straight line, which seems curved because of the warped space time.

Is that even close?

Almost. The key is: straight line in space-time = no acceleration. But it does not mean a constant velocity in space. Unlike in flat space-time, a straight line in curved space-time doesn't imply a constant velocity in space.

The velocity in space we observe is changing, so in a flat space-time the path of the object would be curved, which means acceleration. The idea is to warp space-time in a way that makes the path of every free faller straight, so they aren't accelerated anymore.

 

[robinson]

I'm confused.

1) Is an object in orbit accelerating?

If so ..

1.a) In what reference frame?
1.b) Can the acceleration be measured?

2) Is an object on the surface of the Earth accelerating?

3) If you say no, what about a tall tower on the earth, say, 22,500 miles high, is the top of that accelerating?

4) Is a geosynch satellite at the same height accelerating?

 

[robinson]

And what relationship to warped spacetime do any of those objects have?

 

[A.T.]

1) Is an object in orbit accelerating?

Proper accerleration: no
Coordinate acceleration: depends on the reference frame

1.a) In what reference frame?

Proper accerleration: is the same in every reference frame (absolute)
Coordinate acceleration: see above

1.b) Can the acceleration be measured?

Proper accerleration: yes, with an accelerometer
Coordinate acceleration: yes, as the derivate of velocity

2) Is an object on the surface of the Earth accelerating?

Proper accerleration: yes
Coordinate acceleration: depends on the reference frame.

3) If you say no, what about a tall tower on the earth, say, 22,500 miles high, is the top of that accelerating?
4) Is a geosynch satellite at the same height accelerating?

If you omit the mass of the rest of the tower, both cases are the same.
Proper accerleration: no
Coordinate acceleration: depends on the reference frame.

And what relationship to warped spacetime do any of those objects have?

If proper accerleration = 0 they advance straight ahead in warped spacetime.

 

[jnorman]

inre:
"2} Does the object experience any acceleration as it orbits? The kind that an observer onboard can detect?

No"

AT, sorry, but you are incorrect here. the observer would surely be able to detect that the direction his ship is traveling is constantly changing. acceleration is a change in velocity - since velocity is a vector, this can mean a change in speed, or a change in direction.

 

[A.T.]

inre:
"2} Does the object experience any acceleration as it orbits? The kind that an observer onboard can detect?

No"

AT, sorry, but you are incorrect here. the observer would surely be able to detect that the direction his ship is traveling is constantly changing. acceleration is a change in velocity - since velocity is a vector, this can mean a change in speed, or a change in direction.

I assumed that "experience any acceleration" meant proper acceleration. The object doesn't experience proper acceleration.

You are referring to coordinate acceleration, which is frame dependent (as is velocity). In the objects own frame (where the object is a rest) coordinate acceleration is zero as well. Coordinate acceleration would be not zero in a frame attached to the Earth's center for example. But that is not what I understood by "what the object experiences".

 

[Mentz114]

inre:
"2) Does the object experience any acceleration as it orbits? The kind that an observer onboard can detect?

No"

AT, sorry, but you are incorrect here. the observer would surely be able to detect that the direction his ship is traveling is constantly changing. acceleration is a change in velocity - since velocity is a vector, this can mean a change in speed, or a change in direction.

Have you seen film of people in the space stations ? They are in free fall (in orbit) and do not experience any proper acceleration.

 

[jnorman]

that question did not specify proper acceleration. my answer is correct.

 

[A.T.]

that question did not specify proper acceleration.

Nor did it specify coordinate acceleration, which you assume

my answer is correct.

Not in general. Your answer assumes a certain frame in which the velocity of the object is changing (coordinate acceleration.). But the onboard observer could just as well pick a frame, where he has no coordinate acceleration.

 

[robinson]

Thanks for all the input.

 

[jnorman]

AT - you are being pedantic to defend your incorrect answer. you are the one who arbitrarily assumed a specific category of acceleration. the "general" definition of acceleration is exactly as i stated, and includes changes in direction.

 

[robinson]

The issue of the reference frame is something of great interest at the moment. Is there another topic, or can we expand on it here?

 

[Dale]

As is the case with most things in relativity there are two "flavors" of acceleration, one is coordinate acceleration and the other is proper acceleration. Just like with coordinate time and proper time all reference frames agree on proper acceleration and different reference frames disagree on coordinate acceleration.

Traditonally the unspacified word "acceleration" is understood to refer to "proper acceleration" by default. This is why people might say "velocity is relative but acceleration is absolute". They are referring to proper acceleration. It is, of course, always best to be explicit in order to avoid this sort of miscommunication.

 

[jnorman]

dale - thanks for a balanced comment.

 

[JesseM]

One other point is that it's pretty meaningless to talk about coordinate acceleration in general relativity without specifying a choice of coordinate system, since all coordinate systems are equally valid in GR, and every object will have nonzero coordinate acceleration in some coordinate systems.

 

[A.T.]

AT - you are being pedantic to defend your incorrect answer

No, you are defending your meaningless answer. As JesseM says:

it's pretty meaningless to talk about coordinate acceleration in general relativity without specifying a choice of coordinate system

you are the one who arbitrarily assumed a specific category of acceleration.

Well you assumed two things arbitrarily:
1) specific category of acceleration : coordiante acceleration
2) specific frame of reference where the coordiante acceleration is not zero

the "general" definition of acceleration is exactly as i stated, and includes changes in direction.

Not in General Relativity. As DaleSpam already stated:

Traditionally the unspecified word "acceleration" is understood to refer to "proper acceleration" by default.

And the original question was even specifically about the acceleration experienced by the object. This term usually refers to absolute proper quantities like proper time, which is the time experienced by the object.