Expanding the definition of inertial coordiante systems.

[duordi134]

Einstein has a thought experiment with two trains which he uses to prove linear motion without acceleration is inertial. Inertial means there is no physical test which will prove which train is moving and which is stationary, no coordinate system is preferred and that coordinate system are relative.

Consider a space station which is similar to a spoke wheel with identical sleeping quarters around the rim. In each sleeping chamber there is a camera, a viewing screen and some very accurate physics experimental equipment. The space station is rotating around a hub which is not accelerating linearly. The hub rotation is not accelerating (increasing or decreasing the rotation rate).
The passengers in the sleeping quarters may communicate through the camera and view screen. The view screen will always show another chamber upright but the angle between the chambers is give at the bottom of the screen. The passengers may preform any experiment they wish. Regardless of the experiment chosen the results will be the same regardless which chamber the specific test is preformed in. There is no physical experiment which all the passengers can preform which will identify a specific chamber as the preferred coordinate system.

This indicates all of the chambers are inertial.

Based on this thought experiment I can make the following statement.
All objects with identical acceleration A are inertial with one another.

Special relativity was based on the premise that all coordinate systems which have zero acceleration are inertial which would agree with my definition of inertial systems.

General relatively is based on the premise that all coordinate systems are in a free fall condition following a straight line with zero acceleration. A curved motion is attributed to a property of curved space time. Because General Relativity defines all coordinate system to have zero acceleration General relativity also would agree with my definition of inertial systems.

Please correct any misunderstanding I have.

Thank you in advance.

Duordi

 

[Nugatory]

This indicates all of the chambers are inertial.

None of the chambers are inertial - coriolis and centrifugal forces will be observed in all of them.

The passengers in the sleeping quarters may communicate through the camera and view screen. The view screen will always show another chamber upright but the angle between the chambers is give at the bottom of the screen.

With this restriction, you have excluded an entire class of experiments, namely any in which they both observe the same event. Obviously with this class of experiments excluded they have no way of distinguishing the rooms, but this has nothing to do with inertialness or the equivalence of their coordinate systems. You've just arbitrarily prevented them from making the measurements that would allow them to construct the relationships between compartments.

As an aside, you seem to be assuming that the communication between camera and viewing equipment is instantaneous. Of course it's not really instantaneous, so the denizens of your rotating space station can study the communication pathways between the rooms by doing experiments analogous to the measurement of the two-way speed of light. I clap my hands; when you see me clapping my hands in your screen you clap yours; I then see you clapping your hands in my screen and note the time between that and when I originally clapped my hands. If you haven't been careful to obfuscate things by adding delay loops and the like, the experimenters will eventually (confined humans with unlimited time on their hands have been surprising their captors for millenia) be able to map out the entire space station; if you have obfuscated the communication pathways, it just reinforces the point that you're arbitrarily restricting the experiments that they can perform.

 

[mfb]

This indicates all of the chambers are inertial.

No. They are identical, but none of the systems is an inertial system. They can all perform the same experiment to show this: try to let something float. It will hit the floor soon. Alternatively, they can observe the coriolis force. They also can use more complex setups and see the Sagnac effect for a direct verification that they are rotating around something.

Based on this thought experiment I can make the following statement.
All objects with identical acceleration A are inertial with one another.

You can write down that statement, but it is wrong.

General relatively is based on the premise that all coordinate systems are in a free fall condition following a straight line with zero acceleration.

It is not. You can use other coordinate systems if you like.

 

[Fredrik]

General relatively is based on the premise that all coordinate systems are in a free fall condition following a straight line with zero acceleration.

As mfb already said, this isn't accurate. But you could say that GR is based on the premise that all motion is described by curves in spacetime, and specifically that free-fall motion is described by "straight lines" (the technical term is "geodesics"). Acceleration can then be defined as a measure of how much the motion deviates from free-fall motion.

A curved motion is attributed to a property of curved space time.

The metric determines both the curvature and which lines are to be considered "straight". Deviation from free-fall motion is always caused by something other than spacetime geometry. For example, right now the floor below you is what's causing your "curved motion" (deviation from free-fall motion).

 

[Nugatory]

General relativity is based on the premise that all coordinate systems are in a free fall condition following a straight line with zero acceleration.

No, it is based on the premise that objects in free fall follow geodesics. This has absolutely nothing to do with coordinates, and indeed the Einstein field equations $G=8\pi{T}$ contain no coordinates.

Coordinates only come into the picture when we want to describe the motion of a body relative to some point in spac time, and then we choose whatever coordinate system makes the calculation easiest; there is no reason why the origin of that coordinate system should be following a free fall path. Probably the best known and most generaly useful non-trivial coordinates are Schwarzschild coordinates, and an object at rest in those coordinates is most assuredly not in free fall.

 

[WannabeNewton]

An inertial coordinate system, both in SR and in GR (with the latter being localized by the curvature scale), is simply one that at each point has zero acceleration as measured by an accelerometer there and zero rotation as measured by a set of mutually orthogonal gyroscopes (compass of inertia) there.

In SR both the local rotation (with respect to the compass of inertia) and the global rotation (with respect to the distant stars) agree so you cannot have a system rotating in any sense whilst remaining inertial; as others have noted one can easily test for rotation by observing the presence of centrifugal/Coriolis forces or the Sagnac effect. In GR, when in a rotating space-time (e.g. Kerr space-time), the local and global definitions of rotation will fail to agree and so you can (and do) have inertial systems non-rotating locally that nonetheless rotate relative to the distant stars.

 

[duordi134]

Space station responce.

Some very good responses.

First let me correct an error I found on my own.
All objects with an identical acceleration magnitude are inertial with one another.

Now..

1. It is true that the Coriolis acceleration will appear but it will appear in each chamber identically. No chamber is unique. All coordinate systems (chambers) are relative.

2. Centrifugal acceleration will occur but it will be identical in each chamber. No chamber is unique.

3. The communication travels first to the center hub and than to any other occupant so the distance a message travels between any two occupants is identical. The communication will not be instantaneous but it will be identical in time of transfer. The clocks of all of the occupants will move at an identical rate.

If you really want to determine what order the chambers are in the view screen gives a differential angle between occupants so you are free to construct a map showing where everyone is with respect to each other. You can argue which of you is up and which is down but there is no way to prove anyone is correct or incorrect.

4. In Einsteins train thought experiment the occupants of the train were able to see through the windows from one train to another but no where else. I am following his lead.

5. I can use the term Geodesics instead of straight lines if you prefer.

6. My definition of coordinate systems is a set of properties of motion for an object at a specific time. Some coordinate systems (objects) are inertial and some are not. I use the term in this manner because Einstein did.

7. Inertial systems have come to be understood as coordinate systems (objects) in free fall (GR) or coordinate system (objects) with zero acceleration (SR) because they are the only examples we use.

The true definition of inertial coordinate systems are coordinate systems (objects) each with a set of properties of motion if measured by an observer will follow the predictions of a set of physical laws. Two observer may not agree on the measurements and/or predictions but each will agree that an identical set of physical laws hold true.

I think I have covered everything.
If not give me a second chance.

Duordi

 

[DrGreg]

The true definition of inertial coordinate systems are coordinate systems (objects) each with a set of properties of motion if measured by an observer will follow the predictions of a set of physical laws.

No, the true definition is the one that the consensus of scientists agree upon, and it's not yours. (WannabeNewton has given one way of characterising them; i.e., where there is no proper acceleration and no rotation.)

 

[duordi134]

Science is not Democratic.

Dr Greg.

You are incorrect.
The true definition is the one Einstein created, not you, me, or the consensus of scientists.
You can disagree Einstein if you wish but you can not change his theory by having a majority vote.

No proper acceleration and no rotation was Einstein's basis for SR which is one example of a group of inertial coordinate systems.

This did not prevent Einstein from creating GR which allows acceleration.

Any set of coordinate systems which comply with the relativity principle will be inertial.

Duordi

 

[PeterDonis]

No proper acceleration and no rotation was Einstein's basis for SR which is one example of a group of inertial coordinate systems.

Yes, but your rotating space station doesn't meet this requirement, since it has both proper acceleration and rotation.

This did not prevent Einstein from creating GR which allows acceleration.

SR can deal with acceleration (and rotation) perfectly well. What it can't deal with is tidal gravity; that's why Einstein created GR. But your example does not have any tidal gravity, so SR can handle it perfectly well.

Any set of coordinate systems which comply with the relativity principle will be inertial.

Which "relativity principle" are you referring to? There are two that are pertinent for this discussion: the special principle and the general principle. Since you are talking about "inertial" coordinate systems, you appear to be using the special principle, which only applies to inertial systems, which, as you noted, must have zero proper acceleration and zero rotation. (Actually, zero proper acceleration is sufficient, because any rotating system will have nonzero proper acceleration anywhere that is not on the axis of rotation.) So that principle doesn't even apply to your space station example.

The general principle says that *all* coordinate systems are equivalent, including ones with proper acceleration (and rotation), but it also says that "fictitious forces", such as centrifugal force or Coriolis force, may appear in non-inertial systems, which allows you to distinguish inertial from non-inertial systems even in GR. The general principle certainly does *not* say that all states of motion are equivalent.

 

[PeterDonis]

None of the chambers are inertial - coriolis and centrifugal forces will be observed in all of them.

More generally, an object at rest with respect to any of the chambers will have a nonzero proper acceleration. That's the most general way of specifying what it means to be non-inertial.

 

[PeterDonis]

All objects with an identical acceleration magnitude are inertial with one another.

This doesn't even make sense unless the acceleration magnitude is zero. Any object with nonzero proper acceleration is non-inertial. There's no such thing as "inertial with one another"; objects with identical nonzero proper acceleration might be at rest with respect to one another, but that doesn't make them inertial. Whether or not a given object is inertial has nothing to do with how it's moving relative to some other object; you just measure the object's own proper acceleration.

1. It is true that the Coriolis acceleration will appear but it will appear in each chamber identically. No chamber is unique. All coordinate systems (chambers) are relative.

2. Centrifugal acceleration will occur but it will be identical in each chamber. No chamber is unique.

3. The communication travels first to the center hub and than to any other occupant so the distance a message travels between any two occupants is identical. The communication will not be instantaneous but it will be identical in time of transfer. The clocks of all of the occupants will move at an identical rate.

None of which has anything to do with whether the chambers (more precisely, objects at rest in anyone of the chambers) are inertial. None of them are, because an object at rest in any chamber has nonzero proper acceleration.

You can argue which of you is up and which is down but there is no way to prove anyone is correct or incorrect.

Assuming all these observers are at the same distance from the hub, yes. But if they are at different distances from the hub, and they are each at rest in their respective chambers, they will have different proper accelerations, and can distinguish themselves from each other by this means.

7. Inertial systems have come to be understood as coordinate systems (objects) in free fall (GR) or coordinate system (objects) with zero acceleration (SR)

Yes; and as I've already noted, the chambers in your space station don't meet this criterion because objects at rest in them have nonzero proper acceleration.

 

[duordi134]

Inertial coordiante systems do not require fictional forces.

My rotating space station does not comply with SR.

SR does not allow acceleration.
If you are using SR with acceleration and rotation you are not using the SR Einstein developed.

What is relativity?

If in the case of Einsteins train thought experiment two trains are passing and the passengers on both trains can see each other through the windows. If one train accelerated all of the train
passengers in both trains would agree which train was accelerating. The train which has
passengers tumbling toward one end of the train is the accelerating train. The coordinate systems are not relative because we can tell the difference between them.

So a coordinate system which is accelerating is not inertial with a coordinate system which
is not accelerating unless you add fictitious forces to keep everything in place.

If on the other hand you have two trains with equal acceleration the passengers will not be
able to determine a difference between the trains as both train will have passengers falling toward one end.

So the trains with equal acceleration are inertial with each other and the trains which have zero
acceleration are inertial with each other but the trains which are accelerating are not inertial with the trains which are not accelerating.

Once you have defined a set of inertial coordinate systems then the laws which govern the systems may be defined. This is not so easily done as there is a lot more to SR than just the train thought experiment. The equations for motion and energy are not intuitive.

No fictitious forces are necessary if the coordinate systems are inertial because both coordinate systems will give identical results. Fictitious forces are required to force equality between non-inertial coordinate systems.

This does not mean the resulting physical laws governing a set of inertial coordinate systems will be convenient.

By convenience I mean unless you are intending to solve a problem which has all accelerations velocities and displacements limited to a single line the calculations become very difficult.

Fortunately many problems can be solved within these limitations.

Duordi

 

[pervect]

If on the other hand you have two trains with equal acceleration the passengers will not be
able to determine a difference between the trains as both train will have passengers falling toward one end.

So the trains with equal acceleration are inertial with each other

Sorry, as was already explained this simply isn't what the term "inertial" means. Rotating frames of reference are never inertial. Also, you can say whether or not a frame is inertial or not without comparing it to another frame.

See for instance "Gravitation" by Misner Thorne and Wheeler, pg 18, figure 1.7. A physical device involvin spring projected balls and sheets with holes in them is discussed as a way to test whether or not a frame is inertial. They also mention that Harold Waage actually built such a device.

If you can't find Gravitation online (through google books, for instance) you can see a similar discussion in "Light and Matter", a free online SR book http://www.lightandmatter.com/html_books/genrel/ch01/ch01.html under the heading "Operational definition of a Lorentz frame".

There are various ways of excluding rotation, besides the device that Harold Waage built. What's important for this discussion is that rotating frames are explicitly ruled out as being inertial by modern textbooks, and it is an error to think that rotating frames are inertial.

 

[duordi134]

Pervect it has been a while, I hope you are doing well.

You are defining inertial as if SR (zero acceleration and zero rotation) is the only set of coordinate systems which are relativistic and you may be right.

All I am asking for is a physical experiment which could be preformed in any of the chambers (my first post) which would indicate a different result between the identically accelerating chambers. This will indicate the non-inertial, non-relativistic condition between two equally accelerating coordinate systems.

If I could have found one myself I would not have posted but every test I can think of gives the same results in all chambers which means they must be relativistic and inertial with respect to each other.

You can use my space station as a start and modify it if you need to or give your own constraints.

I already agree that a zero acceleration coordinate system and an accelerating coordinate system are not inertial so there would be little point in this proof.

In the mean time I will be reading your references.

Thanks

Duordi

PS
I checked out the reference material which proves an accelerating and a non-accelerating frame is not inertial and relative.

As stated above I agree with this.
My question was are two equally accelerating/ rotating frames of reference inertial and relative.

The apparatus described would give the same results in any of the chambers because the balls would follow identical paths and strike the plate in the same location.

 

[PeterDonis]

SR does not allow acceleration.

You are going to have to either back up this claim with a mainstream reference or drop it. And I strongly doubt that you can find a mainstream reference to support it, since at least one mainstream reference already given in this thread, Misner, Thorne, & Wheeler (referenced by pervect) has a whole chapter (Ch. 6) explaining how to handle acceleration in SR.

If you are using SR with acceleration and rotation you are not using the SR Einstein developed.

So what? The SR we use today has been refined and improved over more than a century since Einstein published his original paper, and includes contributions from a lot of people besides Einstein. In fact, the most comprehensive model of SR we have today, 4-dimensional flat spacetime, was discovered by Minkowski, not Einstein.

If one train accelerated all of the train passengers in both trains would agree which train was accelerating.

Because accelerometers on one train would read nonzero, and on the other would read zero, yes.

The coordinate systems are not relative because we can tell the difference between them.

Telling the difference isn't a matter of coordinate systems; it's a matter of a direct physical observable, the accelerometer reading, being different. That fact is independent of coordinates.

So a coordinate system which is accelerating is not inertial with a coordinate system which is not accelerating

Again, this makes no sense; coordinate systems (or observers) aren't "inertial with" one another. "Inertial" is a one-place predicate, not a two-place predicate; a given observer or coordinate system is either inertial, or not, purely due to measurements made by that observer, without reference to any other. You really need to fix this repeated error in your terminology.

If on the other hand you have two trains with equal acceleration the passengers will not be able to determine a difference between the trains as both train will have passengers falling toward one end.

Sure, but that doesn't make either one inertial. As I've already pointed out several times.

 

[Fredrik]

Dr Greg.

You are incorrect.

This is almost never a good way to start a post.

The true definition is the one Einstein created, not you, me, or the consensus of scientists.
You can disagree Einstein if you wish but you can not change his theory by having a majority vote.

Definitions and conventions change all the time. The original definition of "topological space" associated that term with what we now call a Hausdorff topological space. The original definition of "Hilbert space" associated that term with what we now call a separable Hilbert space.

 

[PeterDonis]

You are defining inertial as if SR (zero acceleration and zero rotation) is the only set of coordinate systems which are relativistic

No, he's saying that only coordinate systems with zero acceleration (and zero rotation) are inertial. He's *not* saying they're the only ones allowed in SR.

All I am asking for is a physical experiment which could be preformed in any of the chambers (my first post) which would indicate a different result between the identically accelerating chambers.

If you restrict things to measurements purely within each chamber, there isn't one. So what?

This will indicate the non-inertial, non-relativistic condition between two equally accelerating coordinate systems.

You appear to be very confused about the terms you are using.

* None of the chambers are inertial; they are all accelerating.

* All of the chambers are "relativistic"; we can use SR to describe physics in all of them perfectly well.

* With respect to measurements made purely within each chamber, all the chambers are indistinguishable (which is what I think you really mean by "relativistic"); measurements purely within each chamber can't tell you which one you're in (since you've stipulated that the acceleration of all of them is the same, so they're all at the same radius from the hub of the station and therefore have the same centrifugal and Coriolis forces within them).

* However, the fact that the chambers are indistinguishable by purely local (i.e., restricted to within each chamber) measurements does *not* make them inertial, much less "inertial with each other" (which, as I've pointed out repeatedly, doesn't even make sense--"inertial" is a one-place predicate), and does not mean you can't use SR to describe them.

I already agree that a zero acceleration coordinate system and an accelerating coordinate system are not inertial

No, you agree that they are not indistinguishable. Please fix your terminology.

 

[duordi134]

For PeterDonis

"If you restrict things to measurements purely within each chamber, there isn't one. So what?"

I gave the same freedoms Einstein gave in the train though experiment.

So what freedoms are you suggesting you should have?

And Yes, I agree they are indistinguishable if you prefer.
How would you differentiate between inertial, relativistic and indistinguishable?

Perhaps that is my area of misunderstanding.

Einsteins point was you could not tell which train was moving and which was not moving.
This made the two coordinate systems relativistic and indistinguishable.

I have always thought that an "inertial frame of reference" was equivalent to indistinguishable and relativistic because I defined Inertial fame of reference as a set of coordinate systems governed by identical physical laws.

This is not exactly the case.
Although inertial frames of reference are governed by a common set of physical laws and the coordinate systems which have identical acceleration also are governed by a common set of physical laws this in itself is not enough to define accelerating systems as inertial.

Thanks for the help

Duordi

 

[PeterDonis]

So what freedoms are you suggesting you should have?

I'm not suggesting anything. I have no problem with the fact that the chambers are indistinguishable using local measurements.

How would you differentiate between inertial, relativistic and indistinguishable?

I thought my previous post already did that, but here's a quick recap:

"Inertial" means "zero proper acceleration, zero rotation". (Actually, as I pointed out in a previous post, "zero proper acceleration" is sufficient, because in any rotating frame there will be nonzero proper acceleration off-axis. Note, however, that two frames with the same nonzero proper acceleration might still have different rotation, i.e., different Coriolis forces.) An inertial frame is one in which an object at rest (i.e., with constant space coordinates) is inertial.

"Indistinguishable" means just what it says: all local measurements give the same results.

"Relativistic" can have a variety of meanings, and I'm not sure what you have been using it to mean other than as another (badly chosen) term for "indistinguishable". I don't think any of its standard meanings are actually relevant to this discussion.

Einsteins point was you could not tell which train was moving and which was not moving.

This made the two coordinate systems relativistic and indistinguishable.

No, it just made them indistinguishable. There is no standard meaning of the term "relativistic" that means what you are saying here.

I have always thought that an "inertial frame of reference" was equivalent to indistinguishable and relativistic because I defined Inertial fame of reference as a set of coordinate systems governed by identical physical laws.

Well, that's the wrong definition of "inertial frame of reference", so it's not surprising that you have been getting confusing results when you use it. For the correct definition, see above.

I'm not sure there's a standard term for a set of coordinate systems governed by identical physical laws; the best term would probably be "equivalent", by analogy with the equivalence principle. Then we could say that in SR, all inertial frames are equivalent, and all frames with the same nonzero proper acceleration (and rotation) are equivalent, but the latter are not equivalent to the former. In GR, all coordinate systems are equivalent, since the physical law in GR is the Einstein Field Equation, which applies to all coordinate systems.