General Relativity

[Uncayimmy]

You seem to have jumped from saying "both descriptions can't be valid at the same time" to saying "therefore one or the other is real".

There's no "seem" about it. That's exactly what I did. It's a perfectly natural reaction for which I feel no shame.

Try that on the map analogy. Two maps, two projections (the function that tells you how "stretched" the map is, and therefore how to convert map distances into real distances). The projections are different, so they can't both be valid for both maps. You should pick one map and one projection and use that for everything, or carefully switch from one to the other. But obviously neither is more real than the other.

On a personal note, Sol, I've noticed lately that your patience seems to be growing thin and your attention to detail lacking. That's understandable. Dealing with cranks can wear on you, and dealing with people like myself, perfect examples of "a little knowledge is a dangerous thing," is no picnic.

I said already that I am not seeking a preferred frame of reference. I said that I would use whatever reference was most expedient for solving my problem. What seems to be the issue here is how these frames of reference correlate to reality, which is loosely defined as the physical world.

If I win a free space shuttle trip and take a look at the earth from space, I'm gonna say that the reality is that it's a big ball. So while various map projections are excellent for particular tasks, a globe probably represents most closely what I see. Of course, the globe itself is not reality - it's a representation of reality. Then again, on my drive from Phoenix to Cape Canaveral, I'm going to think that a flat map most closely represents that so-called reality. None of this bothers me in the least.

But let's instead go to a race track and watch two dragsters go at it. As I understand it I could describe the race as the earth moving under the vehicles, both of which use their engines to stay in place. If I plug in all the numbers, I'll get the same results as if the ground was stationary and the vehicles moved. I've no problems with that. I can accept that I really can't perform a test to know which way it happened in the physical world.

Where I think the confusion exists is this human desire to relate the description to our experiences. Why would pressing the accelerator cause the earth to move under the vehicle? Since both vehicles follow essentially the same route in the physical world, I would think that it not possible for the earth to be stationary and moving at the same time. So physically I think there must be one thing that "actually" happened, which has no bearing on the validity of my descriptions.

Intuitively I think that if I make enough observations I can arrive at understanding what "really" happened. I have not done this, so I don't know. I'm not particularly bothered by this, but I think it irritates the hell out of Perpetual Student, assuming I am describing things properly from his perspective.

 

[Perpetual Student]

Intuitively I think that if I make enough observations I can arrive at understanding what "really" happened. I have not done this, so I don't know. I'm not particularly bothered by this, but I think it irritates the hell out of Perpetual Student, assuming I am describing things properly from his perspective.

Yes, but I would substitute the word "perplexes" for "irritates."

 

[schrodingasdawg]

Suppose in your highway example I look at the car next to me. It would seem that your descriptions would have to apply to both of us. It would seem that it can't be the earth moving relative to my stationary car and the other car moving on a stationary earth at the same time. It would seem it's one or the other. So the natural inclination in my puny mind is to think that one of them is "real" while the other is merely a description. It would also seem that given enough blood flow to the brain I could figure out a way to determine which one of many is "real."

Right. When you're describing the two cars on the highway, say one going at 80 mph and the other at 120 mph, you have to choose a particular set of coordinates to describe the situation. You can choose one where the road is stationary, and the cars do have those speeds; you can choose one where you are stationary, so that the road is going at 80 mph southwards (assuming you're both heading north in the road-stationary frame) and the other driver is going at 40 mph north; you can choose one where the other car is stationary, the road is going 120 mph south, and you're going 40 mph south; or you can choose any other reference frame where perhaps none of these three things is stationary.

Now, you can describe each of the cars separately using different frames of reference if you want. i.e. you can use the frame where you're stationary to describe your car and the frame where the speed-demon is stationary to describe his car. If you want to make any direct comparison between both you and the speeder, you need to keep in mind that you've described both in different coordinate systems and you need to apply a change of coordinates to make the comparison.

Or, more simply, you can use the same coordinate system to describe both cars at the same time. You're right that multiple different coordinate systems cannot be applicable at the same time, but all are equally valid: the you-stationary, speeder-stationary, or road-stationary frames of reference can all be used to describe the three objects.

If you could take that ball and run with it, I would be appreciative. I'm not disputing anything you've said. I'm just trying to wrap my head around it.

Nah, it's fine. The difference between a crackpot and someone curious about a subject is usually pretty obvious, and I think it's clear that you fall into the latter group.

 

[Perpetual Student]

That doesn't make any more sense than what you said before. What's the "function" you claim the "real earth" "has"?

The minute you write down any such function, you're no longer talking about the "real earth", you're talking about a map (or metric) of it. But then we're back where we started.

I see that and it is quite troublesome as I attempt to get to a "real" perspective of the universe. After all, our models do only model!

That doesn't make sense for the same reason as above.

As UncaYimmy pointed out a globe would be the best representation: it would have the simplest function, one dealing with non-varying scale.
Isn't that fact (the simpler function) at the root of the reason why the globe as a closer representation to the actual earth? If the globe were as close as we could make it to the actual size of the earth, it would be even better since no scale factor would be needed. Similarly, I believe that the simplest functions should give us the closest representation of the actual universe.

How?

Well, OK, then you do argee that (schrodingasdawg) "Using your rockets to change the angular velocity of your spacecraft doesn't set the universe into motion any more than turning around rotates the universe. Regardless of whether you use a comoving or revolving frame, the angular velocity of your ship is changed by the rockets: it's your ship that was set into motion, not the universe."
Sorry, I'm confused.

 

[Uncayimmy]

Yes, but I would substitute the word "perplexes" for "irritates."

No offense was intended by my choice of words. When I'm perplexed enough to pursue a solution and don't find one readily, I get irritated. I guess I was projecting.

 

[Kwalish Kid]

I see that and it is quite troublesome as I attempt to get to a "real" perspective of the universe. After all, our models do only model!

But what else could you ask for?

Similarly, I believe that the simplest functions should give us the closest representation of the actual universe.

Given the incredible complexity of the universe, why should this be the case?

Well, OK, then you do argee that (schrodingasdawg) "Using your rockets to change the angular velocity of your spacecraft doesn't set the universe into motion any more than turning around rotates the universe. Regardless of whether you use a comoving or revolving frame, the angular velocity of your ship is changed by the rockets: it's your ship that was set into motion, not the universe."

But here's the question: are you moving relative to some universal standard or is the universe rotating and you accidentally happened to match that rotation with your rockets?

 

[sol invictus]

As UncaYimmy pointed out a globe would be the best representation: it would have the simplest function, one dealing with non-varying scale.

No - that completely misses the point. Spacetime is curved (and in a very complex way). That's why I picked a curved space - the 2D surface of the earth - as an example. Replacing it with a globe destroys the analogy.

Well, OK, then you do argee that (schrodingasdawg) "Using your rockets to change the angular velocity of your spacecraft doesn't set the universe into motion any more than turning around rotates the universe. Regardless of whether you use a comoving or revolving frame, the angular velocity of your ship is changed by the rockets: it's your ship that was set into motion, not the universe."

The rockets only affect your ship, not the rest of the universe, yes. Whether its your ship or the universe that ends up in motion depends on frame, though - your rockets might just rotate the ship, or they might resist the force that would otherwise rotate it and stop it from rotating, while the rest of the universe ends up rotating.

 

[Perpetual Student]

No - that completely misses the point. Spacetime is curved (and in a very complex way). That's why I picked a curved space - the 2D surface of the earth - as an example. Replacing it with a globe destroys the analogy.

Yes, that does miss the point! OK, I've got it.

The rockets only affect your ship, not the rest of the universe, yes. Whether its your ship or the universe that ends up in motion depends on frame, though - your rockets might just rotate the ship, or they might resist the force that would otherwise rotate it and stop it from rotating, while the rest of the universe ends up rotating.

OK, I've also got that. Both explanations are equally valid. I see that this is a profound concept and I would imagine it leads to important consequences that this layman can only imagine.
However, we know that I decided to rotate my ship at a particular moment in time and that it is not possible that I just happened to choose the moment that the universe started to rotate in the opposite direction and, consequently, that I prevented my ship from rotating with the universe. How can we ignore the decision that I made and not conclude that there is a preferred frame, at least in this narrow context? GR cannot stand alone with no further context, excluding common sense.
Sol, do you really believe that we cannot make any distinction between these two possible frames in order to conclude that only one of them actually reflects reality?

 

[Mashuna]

However, we know that I decided to rotate my ship at a particular moment in time and that it is not possible that I just happened to choose the moment that the universe started to rotate in the opposite direction and, consequently, that I prevented my ship from rotating with the universe. How can we ignore the decision that I made and not conclude that there is a preferred frame, at least in this narrow context? GR cannot stand alone with no further context, excluding common sense.
Sol, do you really believe that we cannot make any distinction between these two possible frames in order to conclude that only one of them actually reflects reality?

From my (entirely layman's) understanding, in the example above in can be argued that your ship was previously rotating with the universe, then you applied a counter force to stop that rotation. It's not the case that the universe suddenly started rotating in the opposite direction when you applied thrust.

 

[Uncayimmy]

However, we know that I decided to rotate my ship at a particular moment in time and that it is not possible that I just happened to choose the moment that the universe started to rotate in the opposite direction and, consequently, that I prevented my ship from rotating with the universe.

What Mashuna said. It seems to me you started with a preferred reference frame, which was a stationary universe and rocket. How do you know it wasn't already moving and you just slowed down?

 

[Roboramma]

Personally I look at it all like this: there are certain invariant properties that don't depend on which set of coordinates you happen to be using. Before Einstien, for instance, the distance between two points was thought to be one such invariant. The time between two events taking place was another. It makes sense to us to see the world that way: that no matter how you look at it, these two things are the same distance from each other.
But when we measure that distance, we can choose any arbitrary coordinate system to make our measurement. I could say, for instance, Jim is five feet to the left of Sally. Making Sally, I suppose, the origin of my coordinate system.
Then again, I could say, "Sally is five feet in front of the door, Jim is five feet further still." Which puts the door at the origin and the line which crosses through Sally and the door the x-axis. 10-5 still equals 5. And if I tilted the axis 45 degrees, I'd still get the same answer.
On the other hand, I could figure out their exact latitude and longitude, and then measure the distance that way. Here, too, the distance works out to be the same.

You might ask the question, "Yes, but which of these is the real coordinate system, which is of these describes the universe that we live in?" Well, they all do. "How far is Sally from the real origin" doesn't have a valid answer.

We can also pick a coordinate system which is moving relative to Jim and Sally. One in which, say, the train that's passing by outside would measure as stationary. What's interesting is that according to Galilean relativity, the distance between Jim and Sally remains the same. In fact, everything is the same in this system, except that the velocities of objects measure differently. That's not an invariant, but the velocity of this object relative to that one is.

Now when Einstein came along we found that with a moving frame, the distance and time actually do vary, and thinking about it so too do relative velocities. But there are still invariant properties. I think the spacetime intervals between events is one such thing. So to ask questions like, "Yes, but how far apart are Jim and Sally, really?" No longer has a valid answer. The answer is different in different coordinate systems, and there is no way to distinguish the 'right' one from the 'wrong' one. But the invariant properties don't differ.

Just like you can't say how far Sally is from the "real" origin, you can't say if she's "really" moving or not. And, based on Sol's posts, it seems to me that the same argument applies to rotation as well. But it doesn't matter, what's "real" are those invariant properties. At least, that's my layman's view.

Happy to have the experts correct my misunderstandings.

 

[sol invictus]

However, we know that I decided to rotate my ship at a particular moment in time and that it is not possible that I just happened to choose the moment that the universe started to rotate in the opposite direction

It's "not possible"? Why?

The frame in which your ship never moves is an extremely special one, but it's certainly "possible". As Mashuna and UncaYimmy point out, another possibility is to choose a frame where your ship is initially rotating and later comes to rest. Of course there are an infinite number of other possibilities in between and not in between.

consequently, that I prevented my ship from rotating with the universe. How can we ignore the decision that I made and not conclude that there is a preferred frame, at least in this narrow context?

We're back to where we started - in that case there's a simple model for the specific set of phenomena we're discussing (that your ship was at rest in a locally flat space and then started to rotate under the influence of its rockets) and a whole bunch of less convenient models. Generally in science we prefer the simplest model, I'll grant you that. But that's all we can say (and I'll refer you back to the capacitance example and let you decide whether spherical or cylindrical or some other coordinates are "real").

Sol, do you really believe that we cannot make any distinction between these two possible frames in order to conclude that only one of them actually reflects reality?

If "only one of them reflects reality" then GR isn't just incomplete - it's wrong. GR's predictions for the results of physical experiments are identical in those two frames, so you could never under any circumstances falsify one and not the other (any more than you could decide between two frames in relative motion in SR).

And to be honest I have trouble imagining how it could be otherwise, how you could have a correct and complete theory that doesn't allow you to use whatever coordinates you choose.

 

[sol invictus]

But it doesn't matter, what's "real" are those invariant properties.

That's correct - the results of experiments cannot possibly depend on what coordinates you use; therefore, all physical results depend only on coordinate invariants.

Just like you can't say how far Sally is from the "real" origin, you can't say if she's "really" moving or not. And, based on Sol's posts, it seems to me that the same argument applies to rotation as well.

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]

From my (entirely layman's) understanding, in the example above in can be argued that your ship was previously rotating with the universe, then you applied a counter force to stop that rotation. It's not the case that the universe suddenly started rotating in the opposite direction when you applied thrust.

Why would the universe be rotating around me?

 

[Thabiguy]

Why would the universe be rotating around me?

Perhaps the problem is that you consider rotation to be a physical phenomenon, something that happens "for real". But rotation is really only a description, a tag that we put on the underlying physical phenomena when we want to explain them. Whether something is rotating or not depends entirely on your choice of coordinate system. Your house, most people would say, is not rotating, because in the reference system they use it really is not. In another coordinate system, it is rotating, once every 23 hours and 56 minutes. Both statements are true in the respective systems.

"Rotation" is just a name for a particular way that certain numbers change in your system of describing things. If you are concerned with what's real, then look at things that can be measured. For example, when you're standing in a ship that you would call rotating, there will be a measurable force between your feet and the floor, you will see that the path of a thrown ball is curved. Those are real phenomena that can be actually measured.

And the point is that everyone will agree on those. They may just have different explanations for the same effects, but surely there's no problem with there being multiple ways to explain something. For example, one observer will explain the effects with the rotation of your ship and inertia, in another observer's coordinate system the observed effects are due to centrifugal and Coriolis forces - but those are just words, saying the same thing differently. Both will agree what weight the scales will show when you step on them, both will agree what number on the target you hit with your ball. And they will all agree that when you fire the rockets, you will feel and observe the effects, and the rest of the universe won't feel anything. Isn't that the reality and certainty you want?

When you accept that 'rotation' is just a short word for some particular way that numbers dance in your way of explaining stuff (which is fine to use if it's clear what you mean), you may see that the question "is it really rotating or not?" makes about as much sense as the question "is your rounded height really even or odd?"

Did that help?

 

[Perpetual Student]

Perhaps the problem is that you consider rotation to be a physical phenomenon, something that happens "for real". But rotation is really only a description, a tag that we put on the underlying physical phenomena when we want to explain them. Whether something is rotating or not depends entirely on your choice of coordinate system. Your house, most people would say, is not rotating, because in the reference system they use it really is not. In another coordinate system, it is rotating, once every 23 hours and 56 minutes. Both statements are true in the respective systems.

"Rotation" is just a name for a particular way that certain numbers change in your system of describing things. If you are concerned with what's real, then look at things that can be measured. For example, when you're standing in a ship that you would call rotating, there will be a measurable force between your feet and the floor, you will see that the path of a thrown ball is curved. Those are real phenomena that can be actually measured.

And the point is that everyone will agree on those. They may just have different explanations for the same effects, but surely there's no problem with there being multiple ways to explain something. For example, one observer will explain the effects with the rotation of your ship and inertia, in another observer's coordinate system the observed effects are due to centrifugal and Coriolis forces - but those are just words, saying the same thing differently. Both will agree what weight the scales will show when you step on them, both will agree what number on the target you hit with your ball. And they will all agree that when you fire the rockets, you will feel and observe the effects, and the rest of the universe won't feel anything. Isn't that the reality and certainty you want?

When you accept that 'rotation' is just a short word for some particular way that numbers dance in your way of explaining stuff (which is fine to use if it's clear what you mean), you may see that the question "is it really rotating or not?" makes about as much sense as the question "is your rounded height really even or odd?"

Did that help?

Nonsense! Rotation is accompanied by forces that can be measured. For example, the earth's rotation causes a decrease in the weight of objects at the equator compared to their weight at the poles. I understand that an alternative view that the whole universe is rotating around the earth can also work under GR with the application of fictitious forces, etc. Which do you think is really happening?

 

[Perpetual Student]

It's "not possible"? Why?

The frame in which your ship never moves is an extremely special one, but it's certainly "possible". As Mashuna and UncaYimmy point out, another possibility is to choose a frame where your ship is initially rotating and later comes to rest. Of course there are an infinite number of other possibilities in between and not in between.

My point was that the two events would have to occur precisely at the same time. It has the same probability of randomly choosing a given point on the real line. What would you say that is?

... Generally in science we prefer the simplest model, I'll grant you that. But that's all we can say (and I'll refer you back to the capacitance example and let you decide whether spherical or cylindrical or some other coordinates are "real").

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

If "only one of them reflects reality" then GR isn't just incomplete - it's wrong. GR's predictions for the results of physical experiments are identical in those two frames, so you could never under any circumstances falsify one and not the other (any more than you could decide between two frames in relative motion in SR).

I don't get that. Why can't GR be correct but incomplete? Can't you just see it as a powerful tool, an excellent model, but other input is needed to complete our grasp of reality?

Consider an electrical device that is simply a black box with two wires coming out of it. We have instruments with which we can measure the capacitance, the resistance, the inductance, etc. Our instruments are "correct" in that we can measure all the properties mentioned above. But, those instruments are limited in that the actual internal geometry and materials of the device remain a mystery to us. That does not make the instruments and our understanding of those properties incorrect; it simply makes them incomplete. We would need something else, like an x-ray machine to learn more.

 

[schrodingasdawg]

My point was that the two events would have to occur precisely at the same time. It has the same probability of randomly choosing a given point on the real line. What would you say that is?

The coordinate system would be chosen so that the two events occur simultaneously.

Why do we "prefer" the simplest model?

Because it's simpler.

I don't get that. Why can't GR be correct but incomplete? Can't you just see it as a powerful tool, an excellent model, but other input is needed to complete our grasp of reality?

The issue here is that GR is inherently coordinate-independent. Physics interprets the statement "some coordinate systems are special" as "the laws of physics take a special form in these coordinate systems," which is in direct contradiction with GR. If by "some coordinate systems are special," you mean "some coordinate systems are special in some metaphysical way, but the (empirically detectable) laws of physics are the same in all possible coordinate systems," that's in violation of Occam's razor.

If "some coordinate systems are special" means that "the laws of physics aren't any different in these coordinate systems, but the math is easier to work out," there isn't any disagreement there, except with what's meant by "special."

'Course, you're not actually using the word "special": you're using "correct" and "real," so I'm assuming you mean that in the sense of "metaphysically special," which is a statement in violation of Occam's razor since the laws of physics are the same in the "correct" coordinate system as they are in all other coordinate systems.

 

[Michael C]

My point was that the two events would have to occur precisely at the same time. It has the same probability of randomly choosing a given point on the real line. What would you say that is?

Let's imagine that your spaceship is floating free in space, with no rocket engine working. At a specific moment, you start up the rockets that make the ship rotate. From an infinity of possible frames of reference, here are a few that might be used to look at this event:

- One where your spaceship is initially at rest and then starts rotating.
- One where your spaceship is initially rotating and then stops rotating.
- One where you spaceship is always at rest.

In the third frame, the universe will initially be at rest but will start rotating at the moment you turn the rocket engines on. In this frame, new forces will also appear at the moment the universe starts rotating. All this happens because you defined the frame in the way you did. This doesn't mean that some huge change in the universe has actually occurred just the moment you decided to start rotating your spaceship: it simply means that you are using a complicated frame of reference.

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

The "simplest model" will change depending on what part of the universe we want to observe. For instance, for most events that occur on Earth we are probably best served by a frame of reference where the Earth is stationary. In order to analyse the motion of the Solar system, we'll probably choose a frame of reference where the Sun is always at the centre of the frame. If we're studying Saturn's rings, we'll centre our frame on Saturn. In each case, we are best served by a frame of reference where the mathematics come out simplest: then we can most easily calculate things in order to analyse what is happening and make predictions. What the principle of relativity tells us is that in each situation we are at liberty to choose whichever frame makes our sums easier, since all frames are equally valid.

I don't get that. Why can't GR be correct but incomplete? Can't you just see it as a powerful tool, an excellent model, but other input is needed to complete our grasp of reality?

I'm sure that GR is both correct and incomplete. However, if it were to be found out that there was a "real" coordinate system for the universe that was inherently different to all other systems, GR wouldn't be correct at all: it would be totally negated. The very basis of GR is that all frames of reference are equally valid.

 

[Thabiguy]

Nonsense! Rotation is accompanied by forces that can be measured. For example, the earth's rotation causes a decrease in the weight of objects at the equator compared to their weight at the poles. I understand that an alternative view that the whole universe is rotating around the earth can also work under GR with the application of fictitious forces, etc. Which do you think is really happening?

I actually tried to address that in the very post you've replied to.

What really is happening are the forces that can be measured. Whether something is rotating or not depends on the coordinate system.

Maybe the problem lies in the inconsistent interpretation of the word "rotating".

You seem to prefer to use "rotating" to refer to a planet like the Earth, where objects at the equator weigh less than objects at the poles. But all observers agree that objects at the Earth's equator weigh less than objects at the poles (because after all the scales can only show a single reading), so if you insist on this interpretation of "rotating", then the Earth is "rotating" in all coordinate systems. Your objections against coordinate systems in which the Earth is not "rotating" are then moot, because under this interpretation of "rotating", there are no such systems.

On the other hand, when one says that in one coordinate system the Earth is rotating and in another one it is stationary, then the word "rotating" refers to a circular motion (in this case of the surface of the Earth), i.e. that coordinates change in certain regular patterns. Whether this happens, of course depends on the choice of your coordinate system. And the forces felt by objects at the equator and the poles don't depend on whether the Earth is rotating or not.

You can use any coordinate system to predict that objects will weigh less at the equator; in some the Earth will be rotating, in some it won't. That's what I meant when I said that "rotation" is a short for a piece of math in your model and not much more.

In fact, your apparent tendency to equate "rotation" with "bulging at the equator" already comes from presuming a certain kind of coordinate system (inertial frame in flat spacetime). It is in this kind of system that the two are related; but in others, they may not be.

Did that make it clearer or muddier?