General Relativity

[dasmiller]

Why do we "prefer" the simplest model? Might it be closer to the underlying reality?

I'm not sure that there's an "underlying reality" to get closer to.

If we go back to the globe (but not the maps) - we express locations in latitude/longitude, with 0 latitude at the equator and 0 long at some British streetcorner. But we could construct arbitrarily complex systems for doing the same function - suppose the coordinate axes weren't at right angles? Or that we defined 0-0 as the subsolar point, so it wasn't stationary with respect to the Earth's crust? The math might be harder, but would that mean that our coordinate system is "farther" from the underlying reality? I'd argue "no" because there's no underlying reality. The spatial relationships between cities are what they are; the coordinate system (lat/long in this case) is just part of a technique to help us understand the spatial relationships, so the coordinate system itself is entirely made-up.

And so, I suspect, is the universe. These reference frames are completely made-up things; we make them up because when we do certain mathematical operations with them, they let us accurately predict relationships among objects in our universe. But the reference frames themselves are no closer to an underlying reality than the lat/long system.

 

[Tim Thompson]

The Spinning Bucket

It might be worth reviewing Newton's "spinning bucket problem". It is relevant here, and is one of the key features in history that leads, indirectly perhaps, to general relativity ...

http://www.gap-system.org/~history/HistTopics/Newton_bucket.html
http://en.wikipedia.org/wiki/Bucket_argument

 

[Perpetual Student]

Thanks for all the responses above and thanks, Tim Thompson, for those links. Perhaps they will help clarify things
OK, let's look at Newton's two rocks in empty space attached by a rope. If the system rotates the rope is taut if the system does not rotate the rope is slack.
Suppose the rope is slack. Can we make the rope taut by choosing a coordinate system that rotates around the center of gravity of the two rocks?
Or, alternatively, if the rope is taut can we make it slacken by choosing a coordinate system that does not rotate with respect to the rocks?
I don't think we can change reality by the act of choosing a coordinate system. Either the rope/rock system is rotating or not, because either the rope is slack or not.
How do we escape the fact that regardless of coordinate system there is a reality here. Either the system is rotating or not, the reality of which can be readily determined.

 

[schrodingasdawg]

Suppose the rope is slack. Can we make the rope taut by choosing a coordinate system that rotates around the center of gravity of the two rocks? Or, alternatively, if the rope is taut can we make it slacken by choosing a coordinate system that does not rotate with respect to the rocks?

No. Whether the rope is taut or slack is a physical invariant. All coordinate systems will agree on this.

I don't think we can change reality by the act of choosing a coordinate system.

Neither do I. I would suppose that sol et al will also agree with this.

Either the rope/rock system is rotating or not, because either the rope is slack or not.

This is where the disagreement is occurring. If the rope/rock system is slack, a coordinate system can be chosen where the rope is rotating and the centripetal force keeping the rocks in circular motion is due to gravitational forces instead of tension. So while the tautness/slackness of the system is a physical invariant that will be reproduced by all coordinate systems, whether or not the system is rotating is not a physical invariant.

How do we escape the fact that regardless of coordinate system there is a reality here. Either the system is rotating or not, the reality of which can be readily determined.

We don't escape the fact that there is a reality. However, whether the system is rotating is not part of that reality.

 

[Uncayimmy]

No. Whether the rope is taut or slack is a physical invariant. All coordinate systems will agree on this.

Will they agree it is straight and not curved? If it's curved and it's a rope, then it's not slack in "reality." Is that where you're heading, PS?

 

[Reality Check]

Thanks for all the responses above and thanks, Tim Thompson, for those links. Perhaps they will help clarify things
OK, let's look at Newton's two rocks in empty space attached by a rope. If the system rotates the rope is taut if the system does not rotate the rope is slack.
...

In Newton's two rocks in empty space attached by a rope, the rope is stated to be taut since the rock have been set to be rotating. Thus it is never slack. He used this in an empty universe to argue for absolute space.
If the rope is slack then we have a different physical system . A coordinate change does change one physical system into another physical system. It is a way of describing the same physical system differently.

 

[sol invictus]

Suppose the rope is slack. Can we make the rope taut by choosing a coordinate system that rotates around the center of gravity of the two rocks?
Or, alternatively, if the rope is taut can we make it slacken by choosing a coordinate system that does not rotate with respect to the rocks?

Of course not. A force meter in the middle of the rope reads something. That's a physical experiment, so it must be a coordinate invariant.

I don't think we can change reality by the act of choosing a coordinate system.

As I've said I don't know how many times so far, we cannot affect the results of physical experiments by our choice of coordinates. So if "the results of physical experiments" are what you mean by "reality", then yes.

And if that's the case, you'd better think pretty hard about your questions regarding whether rotation is "real".

Either the rope/rock system is rotating or not, because either the rope is slack or not.

You've just contradicted yourself.

The rope is either slack or not. In both cases we can choose coordinates in which the rocks are rotating, or not.

How do we escape the fact that regardless of coordinate system there is a reality here. Either the system is rotating or not, the reality of which can be readily determined.

What - have you now decided that "tense rope" = "rotation" or something? What will you do when there's no rope, or when some ropes are slack and some aren't?

Did you read my response to Roboramma?

 

[Thabiguy]

ETA: I posted this before I noticed sol's reply, so apologies if parts of it kinda repeat what he's already written.

Thanks for all the responses above and thanks, Tim Thompson, for those links. Perhaps they will help clarify things
OK, let's look at Newton's two rocks in empty space attached by a rope. If the system rotates the rope is taut if the system does not rotate the rope is slack.

You mean, if it rotates in an inertial frame.

Suppose the rope is slack. Can we make the rope taut by choosing a coordinate system that rotates around the center of gravity of the two rocks?
Or, alternatively, if the rope is taut can we make it slacken by choosing a coordinate system that does not rotate with respect to the rocks?

Of course not.

I don't think we can change reality by the act of choosing a coordinate system.

If by reality you mean measurable quantities, then you're of course right.

Either the rope/rock system is rotating or not, because either the rope is slack or not.

The correct conclusion should have been: Either the rope is slack or not, and whether the rope/rock system is rotating or not is a different issue.

Again, rotation simply means moving in circles. When the rope is taut and you choose a rope-centered, rope-fixed coordinate system, then no part of your system is moving in circles; it's trivial to verify. The rocks are provably stationary in that coordinate system, so any insistence that they are going in circles in that coordinate system, because the rope is taut, does not make sense.

Perhaps you want to say that - say, when the rope is slack - that any system in which the rocks are rotating is "inferior" to the one in which they aren't. Well, this is another way of saying that there is a special class of frames - the inertial frames - in which the laws of physics are particularly convenient for some purposes. That's what special relativity tells us.

But look what you've done: you've chosen a scenario in flat spacetime and without gravity. Yes, it could be argued that when you restrict yourself to that, general relativity is an overkill, because special relativity handles those cases just as well and is simpler.

But general relativity is not restricted to flat spacetime, it's a theory of gravity and it's meant to deal with much more complicated setups than the one you propose. In more general situations, your equivalence rotation=taut just breaks down. For example, when the two rocks orbit each other, then the rope will be slack, and yet the rocks will rotate with respect to distant stars. Or, if the rocks are falling towards a massive body, the rope may be taut (due to tidal forces), even if the rocks don't rotate with respect to distant stars.

These were just simple examples; the point is that when you've got sufficiently curved spacetime, inertial frames go out the window, so one way or another, any coordinate system you may choose will be "inferior" according to your standard. Thus the choice of coordinate system ultimately becomes a matter of convenience.

 

[schrodingasdawg]

Will they agree it is straight and not curved? If it's curved and it's a rope, then it's not slack in "reality." Is that where you're heading, PS?

I don't know what you mean. Can you elaborate?

 

[Perpetual Student]

Thanks, again to everyone. I would very much like to get a grasp on this.
Here goes: So, out in empty space, I have a system of two rocks tied to a rope and the system is rotating so that a force meter in the middle of the rope confirms the forces associated with that rotation. I can choose a reference frame under GR that has the universe rotating around the rock/rope system, but we would know that the system is really rotating because of the meter. So, (choice 1) there is a real description of events here, the choice of frame is irrelevant.
Now, I thought I understood that we could conjure up a reference frame that has the system not rotating, using fictitious gravity (whatever that means) to account for the force seen on the meter and a rotating universe to explain the same situation (choice two).
So, which is it?

 

[sol invictus]

Thanks, again to everyone. I would very much like to get a grasp on this.
Here goes: So, out in empty space, I have a system of two rocks tied to a rope and the system is rotating so that a force meter in the middle of the rope confirms the forces associated with that rotation. I can choose a reference frame under GR that has the universe rotating around the rock/rope system, but we would know that the system is really rotating because of the meter. So, (choice 1) there is a real description of events here, the choice of frame is irrelevant.
Now, I thought I understood that we could conjure up a reference frame that has the system not rotating, using fictitious gravity (whatever that means) to account for the force seen on the meter and a rotating universe to explain the same situation (choice two).
So, which is it?

I don't understand your question - it's not either/or. The statements labeled "choice 1" and "choice 2" are both true according to GR (apart from a few minor semantic quibbles that I don't think are essential, like "fictitious gravity" and precisely what you mean by "irrelevant").

 

[Uncayimmy]

I don't know what you mean. Can you elaborate?

I was thinking about moving trains and dropping balls. If I'm on a moving train and drop a ball, it follows a straight path to the floor of the train car. For an observer on the ground it follows a curve on the way down. If I attach a string to the ball before I drop it and let it stretch out behind the ball, is that string straight? According to me, yes. What about the ground based observer?

I could be to totally blowing this, so think simple mistake rather than profound inference.

 

[Hellbound]

I was thinking about moving trains and dropping balls. If I'm on a moving train and drop a ball, it follows a straight path to the floor of the train car. For an observer on the ground it follows a curve on the way down. If I attach a string to the ball before I drop it and let it stretch out behind the ball, is that string straight? According to me, yes. What about the ground based observer?

I could be to totally blowing this, so think simple mistake rather than profound inference.

The ground observer would see it as straight, too...it just moves sideways at the same rate as the ball.

 

[Uncayimmy]

The ground observer would see it as straight, too...it just moves sideways at the same rate as the ball.

I disagree.

http://www.phys.unsw.edu.au/einsteinlight/jw/module1_Galileo_and_Newton.htm

 

[Perpetual Student]

I don't understand your question - it's not either/or. The statements labeled "choice 1" and "choice 2" are both true according to GR (apart from a few minor semantic quibbles that I don't think are essential, like "fictitious gravity" and precisely what you mean by "irrelevant").

OK, so you did understand my poorly worded question and the answer is both. I am not the one here who first used the term "fictitious forces" and "fictitious gravity." I'm merely trying to understand what is meant by those terms.
So, both?
Let me try this: I'm cruising along in intergalactic space and I come across two rocks, string and meter as previously described, which is a system, that from my perspective is rotating. So, I want to know what is really happening. I put myself in synchronous rotation with the rocks for further analysis, so I now have the universe revolving around me. Can I conclude that the rocks are really rotating because the meter and string shows it or not?

 

[Hellbound]

I disagree.

http://www.phys.unsw.edu.au/einsteinlight/jw/module1_Galileo_and_Newton.htm

Nothing there contradicts me, that I can see.

The ball would follow an arc, but as the person holding the string is moving at the same rate as the ball, the string would be straight and moving sideways. I've attached a photo that should explain it (please don't criticize the lack of artistic talent ).


The blue represents the path the ball fills over the time, the straight black lines the string. The black circles are the positions of the ball at the various time intervals.

Just to make clear, it's the appearance of the string my earlier post was concerned with, not the ball path. Hope this clears up any confusion.

 

[sol invictus]

OK, so you did understand my poorly worded question and the answer is both. I am not the one here who first used the term "fictitious forces" and "fictitious gravity." I'm merely trying to understand what is meant by those terms.
So, both?
Let me try this: I'm cruising along in intergalactic space and I come across two rocks, string and meter as previously described, which is a system, that from my perspective is rotating. So, I want to know what is really happening. I put myself in synchronous rotation with the rocks for further analysis, so I now have the universe revolving around me. Can I conclude that the rocks are really rotating because the meter and string shows it or not?

You've lost me again.

What does "really rotating" mean? How do you know that "I now have the universe revolving around me"? What does "the string shows it" mean? If it means the string is under tension, what was the point of going into synchronous rotation with it? You could have just looked at the string to see if it was tense or not, and you can obviously do that even if you're rotating relative to it.

Anyway whatever the answers to my questions are, the answer to yours (to the extent I can guess what you're asking) is no.

For example (to steal from Thabiguy) the rocks might be falling into the gravitational field of something so that the string is taut because of tidal forces (that have nothing obvious to do with rotation). Or perhaps there's a combination of both tidal forces and rotation at work. How can you distinguish? I don't know any way that works in general - which is why I advocate the position that there is no true distinction.

 

[Perpetual Student]

You've lost me again.

Sorry, I am really trying.

What does "really rotating" mean? How do you know that "I now have the universe revolving around me"? What does "the string shows it" mean? If it means the string is under tension, what was the point of going into synchronous rotation with it? You could have just looked at the string to see if it was tense or not, and you can obviously do that even if you're rotating relative to it.

OK, I my rotation is irrelevant. Scratch that part.

Anyway whatever the answers to my questions are, the answer to yours (to the extent I can guess what you're asking) is no.

OK, no.

For example (to steal from Thabiguy) the rocks might be falling into the gravitational field of something so that the string is taut because of tidal forces (that have nothing obvious to do with rotation). Or perhaps there's a combination of both tidal forces and rotation at work. How can you distinguish? I don't know any way that works in general - which is why I advocate the position that there is no true distinction.

Well, let me modify the experiment by putting it in intergalactic space, so there are no gravitational, electric, thermodynamic, etc. forces that can have any measurable influence. But we do see distant galaxies. Now what? Can we conclude that the system is really rotating or not?

 

[sol invictus]

Well, let me modify the experiment by putting it in intergalactic space, so there are no gravitational, electric, thermodynamic, etc. forces that can have any measurable influence. But we do see distant galaxies. Now what? Can we conclude that the system is really rotating or not.

Why do you need the distant galaxies? If they can't have any effect, they could be either "really rotating" or "really not rotating", so how can they help?

Regardless, if there are no other forces or any sources of energy or mass, the spacetime is flat. In that case, I refer you to my previous post. Since you still refuse to explain what you mean by "really rotating", that will have to do.

Well... it's a little more subtle than that. If you're in flat empty space there is a special set of frames - the inertial frames of SR. Those can be characterized mathematically in various ways (they're coordinate systems in which the metric takes Minkowski form, the Christoffel connection is zero, etc.).

What we can say - to be very precise - is that you will never be able to do an experiment that distinguishes an inertial frame in which your apparatus is at the origin from one in which it's not. Similarly you cannot distinguish between inertial frames in which it's at constant velocity with respect to the origin. But you can distinguish inertial frames in which it's accelerating (including rotating around, say, the origin) from those in which it's not (Newton's bucket is a famous example). So in that sense, acceleration and rotation are absolute in flat spacetime.

However everything gets far more complex if the spacetime is curved. There are no longer any inertial frames - but the existence of inertial frames was essential for the statements I just made. If the spacetime is "asymptotically flat" (i.e. gets closer and closer to flat the farther you go from some finite collection of localized sources of energy) you can use that to make some definite statements (including about rotation). But in general, in curved space it's very hard to say anything like the above paragraph that applies outside a small region. And of course in either flat or curved spacetime you can always use either of two coordinate systems that are rotating with respect to each other, and get predictions from each that are perfectly consistent.

 

[Perpetual Student]

Why do you need the distant galaxies? If they can't have any effect, they could be either "really rotating" or "really not rotating", so how can they help?

Well, I thought we might need some refererence to determine whether I am rotating relative to the rest of the universe. But maybe we don't need it.

Regardless, if there are no other forces or any sources of energy or mass, the spacetime is flat. In that case, I refer you to my previous post. Since you still refuse to explain what you mean by "really rotating", that will have to do.

From your previous post, you said, "But you can distinguish inertial frames in which it's accelerating (including rotating around, say, the origin) from those in which it's not (Newton's bucket is a famous example)."
So, that tells me that if the meter shows the forces associated with rotation, it is unambiguous that the rock/rope system is rotating and the universe is not revolving around the system. OK?