User talk:Ems57fcva/sandbox/Equivalence Principle

Newtonian inheritence

Hi EMS, I think the aritcle is fine. I think you have chosen the level of difficulty well. To be curious at all about the principle of equivalence you must allready possess some physics knowledge, and what you write suits that knowledge an that curiosity, I think.

There is an ambiguity in your text that I think needs to be adressed.
From Newtonian dynamics we have the inheritence that most people are accustomed to categorizing gravity as a force.

You write: 'the local presence of a force that acts of all objects'. Since most people are accustomed to thinking of gravity as a force acting downwards, they may assume that you are talking about detecting the presence of a downward force inside an accelerating spaceship as the essence of the principle of equivalence. I know what you have in mind, but a lot of people will not know, and they may make a wrong assumption.

When I tell people about GR, I often use the following example to illustrate what I mean by force:
When I make a stack of 10 weighing scales each weighing one kg, then the bottom scale will read 9 kg (because that's what he's carrying) the next one will read 8 kg etc.

I call the interaction that is transmitted by mechanical contact a force. The bottom weighing scale has the job of transmitting the amount of force that is necessery for accelerating all the scales above him, and so on. Then I contrast that with gravity and inertia. There is no discernable transmitter of gravity and inertia. Each particle that the object consists of is individually affected. So the mediator of this interaction must be a field.

Most people are very accustomed to thinking of gravity as 'just another force', making it hard to explain the principle of equivalence. Often I don't write 'force', but 'mechanical force', or 'force like the thrust of rocket engines' to stress the difference with a pervasive field. --Cleon Teunissen | Talk 13:01, 30 Mar 2005 (UTC)

Coordinate systems are mental constructs

You are using the expression 'an accelerated frame of reference'.

I think you should state explicitly with respect to what this acceleration is.

Definition
If a test mass is released to move without any constraint, it will move inertially. A frame of reference that is co-moving with an inertially moving test mass is an inertial frame of reference.

When a mechanical force such as the thrust of a rocket engine is being exerted on an object, the object will be deviated from the trajectory that an inertially moving test mass would follow. This deviation is the kind of acceleration that is caused by a force being exerted. A frame of reference that is co-accelerating with an object that is being accelerated by a mechanical force is an accelerating frame of reference.

In the context of general relativity, this is the only way that inertial motion and acceleration due to a mechanical force can be defined: by observing how a test mass moves in the given situation.

A frame of reference is a mental construct. A frame of reference is an abstraction, derived from observations of aspects of physical motion. I think it is somewhat circular to describe an observer as 'being in a accelerated frame of reference', because the only way to define the concept of 'accelerated frame of reference' is to note that it's co-accelerating with an object that is being accelerated by a mechanical force.

Besides acceleration due to a mechanical force being exerted, general relativity recognizes another form of acceleration. Acceleration due to gravitational interaction. A free falling, free moving test mass is moving inertially with respect to the local inertial frame (this statement is circular, here it is an assertion that the definition has real physical meaning.) However, seen from a sufficiently larger perspective, it may be recognized that the test mass is moving in curved space-time. As seen from the larger perspective, the motion may be recognized as accelerated motion due to moving through curved space-time.

In space there is only one way to accelerate: by accelerating some other matter in the opposite directon. An astronaut on a spacewalk who switches off his magnetic boots and jumps "up", accelerates mostly himself but also the space station: away from the common center of mass. When he goes back by pulling on the "umbilical cord", he moves mostly himself but also the space station - towards the common center of mass. In all this, the common center of mass remains moving inertially. Another example of this is a rocket engine, that throws mass backwards with as much velocity as possible --Cleon Teunissen | Talk 20:31, 30 Mar 2005 (UTC)

Definition of force

There is an ambiguity in your text that I think needs to be adressed. From Newtonian dynamics we have the inheritence that most people are accustomed to categorizing gravity as a force.

True

You write: 'the local presence of a force that acts of all objects'. Since most people are accustomed to thinking of gravity as a force acting downwards, they may assume that you are talking about detecting the presence of a downward force inside an accelerating spaceship as the essence of the principle of equivalence.

That actually is what I have in mind, and is what Einstein was thinking of.

I know what you have in mind, but a lot of people will not know, and they may make a wrong assumption.

When I tell people about GR, I often use the following example to illustrate what I mean by force: When I make a stack of 10 weighing scales each weighing one kg, then the bottom scale will read 9 kg (because that's what he's carrying) the next one will read 8 kg etc.

I call the interaction that is transmitted by mechanical contact a force.

Here is where we diverge. I do not limit forces to those things transmitted by physical contact. For example, a magnet can act of a piece of iron without being in contact with it.

The bottom weighing scale has the job of transmitting the amount of force that is necessery for accelerating all the scales above him, and so on. Then I contrast that with gravity and inertia. There is no discernable transmitter of gravity and inertia. Each particle that the object consists of is individually affected. So the mediator of this interaction must be a field.

That is fine, but then what is the nature of the field? In a rocket, the gravitational field exists due to the floor of the rocket being accelerated from travelling on a timelike geodesic. On the earth, it is also due to the floor of your house being accelerated from travelling on a timelike geodesic. In both cases, the observer detects the field based on the effect that it has on how objects move relative to themself. (This is the same way that one usually detects an electromagnetic field, btw).

Most people are very accustomed to thinking of gravity as 'just another force', making it hard to explain the principle of equivalence. Often I don't write 'force', but 'mechanical force', or 'force like the thrust of rocket engines' to stress the difference with a pervasive field.

I realize that gravity is not a "real" force. Perhaps I need to explain that better. I will think on that. However, the basic definition I give will stand. It is an important point of the Equivalence Principle that gravitation really is perceived to be due to a force until the observer realizes that its characteristics mean that he or she is the one being accelerated.

--EMS 22:07, 30 Mar 2005 (UTC)

Cleon Teunissen | Talk 03:05, 31 Mar 2005 (UTC) Gravity is a force, that's not the point, but the way it is mediated is unlike that of any other force. Magnetism is also mediated by a field, but that field only interacts with a part of the matter, not with all matter. The inclusiveness of gravitational interaction is uncanny.

My criterium for calling something a force is that it is an interaction in which momentum is transferred. Space probes are sometimes send on trajectories through the solar system that include flyby's, to get a sling-shot effect. The voyager missions relied on that too: Jupiter deflected their trajectory, and "catapulted" them, to Saturn. (This technique is also known as 'gravitational assist')

When a spacecraft is accelerating, the floor of the space-craft is exerting a force on anything resting on the floor of the spacecraft, accelerating it. If the astronaut is holding a ball in his hand, his grip makes that ball co-accelerate with him. When he releases his grip, the ball ceases to accelerate, as no force is being exerted on it, it just cruises on with the velocity it had when it was released, and it is quickly overtaken by the floor of the spacecraft.

I use the above description rather than any other because the above discription enables me to maintain a consistency in my definitions.

Definition: when I am not being accelerated with respect to the local inertial frame then I am weightless. When I feel weight I know that I am being accelerated with respect to the local inertial frame.

Definition: when two forces are opposite and equal, there will be no acceleration.

I prefer to interpret the situation inside the accelerating spacecraft as a situation in which there is one mechanical force: the accelerating force. When objects are being accelerated, inertia manifests itself. Inertia opposes change of velocity, but it is unable to prevent the change of velocity. There is only manifestation of inertia as long as a mechanical force is being exerted, accelerating the object (with respect to the local inertial frame). As soon as the accelerating force ceases, manifestion of inertia ceases, and the object continues in inertial motion.

Gravitational interaction is mediated by curvature of space-time geometry. When a space-craft is very far away from any other mass, so it is in space that is very close to zero-curvature space-time geometry, and the space-craft is accelerating, then it is interacting with zero-curvature space-time geometry. Zero-curvature space-time geometry is not the mediator of any force.

Matter interacts with space-time geometry whenever it is being accelerated with respect to the local inertial frame. This is valid both in curved space-time geometry and in zero-curvature space-time geometry. When matter is being accelerated with respect to the local inertial frame inertia manifests itself, opposing (but not preventing) the acceleration.

That is how I understand the principle of equivalence. The description of the physics of gravity and the physics of manifestation of inertia are to be unified in a single description. Inertia is the result of interaction of matter with a universal inertial field. In zero-curvature space-time the following is valid: when an object has a constant velocity in a constant direction there is no interaction with the universal inertial field. There is interaction with the universal inertial field only when an object is being accelerated with respect to the inertial frame. In curved space-time inertia manifests itself when an object is being deviated from inertial motion. To move along a geodesic is inertial motion.

A magnet has a field of its own, the magnetic field is an "occupant" of space.

Gravitational interaction does not "deploy" a field of its own, gravitational interaction is mediated by adapting something that is already there: the universal inertial field. Matter induces a curvature of space-time geometry and the curvature of space-time geometry is the mediator of graviational interaction. --Cleon Teunissen | Talk 03:05, 31 Mar 2005 (UTC)

Cleon -

I think that you are doing avery good job of making something simple complicated. Your "universal inertial field" is nothing more than the geodesics of spacetime. The modern Principle of Inertia is:

An object will follow a given timelike geodesic unless it is acted on by a force.

What more needs to be said? An object released in an accelerating spaceship will follow a timelike geodesic to the floor of the vehicle just the same as an oject released above the suface of the Earth will follow a timelike geodesic down to it. All that the Principle of Equivalence is is a rule for determining if one is in an accelerated frame of reference. Remember that all views are valid: It is perfectly fine to see inertially moving objects as being accelerated with respect to you even though you are really the one being accelerated. That is really how you see it. All that is needed is the ability to reconcile your view with the of the free-falling observer or even that of someone orbitting the Earth in a spaceship to their own.

In short, just let yourself "see" events from any point of view. After all, no viewpoint is preferred in relativity.

--EMS 05:04, 31 Mar 2005 (UTC)

Coordinate systems are mental constructs

A frame of reference is a mental construct.

Frames of reference are not mental constructs. Item: When you are driving south of a street, your are getting closer to places south of you. When you are driving north, they are getting farther away. Those are two different frames of reference. So frames of reference are quite real. Just standing up and going into the next room is enough to change your frame of reference.

As for coordinate systems: Here you are on a better footing. The coordinate system is a mathematical construct used for describing events in a frame of reference. Reality is not marked off (unless we have done it for our own benefit, like with a football field). However, all coordinate systems are valid. Indeed, that is one of the founding principles of GR, and why the Einstein Field Equations are such a bear to work with: To solve them you not only need a stress-energy distribution, but also a coordinate system against which to describe it.

Ah, that turns out to be a case of babylonian confusion. As far as I knew 'frame of reference' and 'coordinate system' were two expressions for one and the same thing, I am accustomed to using them interchangebly. For me, 'coordinate system' can be any reference grid, cartesian, polar coordinates. What is for you the difference? --Cleon Teunissen | Talk 09:56, 31 Mar 2005 (UTC)

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More edits

I have a revision ready, which I think addresses some of your earlier comments. I think that the result is a stronger page.

I will move it over in the next day or two.

Cleon - Be advised that I will take "ownership" of this page. On the Sagnac page I am willing to play a supporting role. On this page however I think I need to shephard it closely. Even so, comments are still appreciated. --EMS 05:43, 31 Mar 2005 (UTC)

I have chosen a very specific area to be the "expert" in: the physics of rotation. So for me four articles are 'close to my heart': centrifugal force, coriolis force, fictitious force, and Sagnac effect. --Cleon Teunissen | Talk 09:39, 31 Mar 2005 (UTC)

Good. I am not interested in stepping on your toes, but for this page I am willing to do so. You actually seem to do a fairly good job in those areas of yours. So this is probably a good "division of labor".

What is the important characteristic of the Principle of Relativity?

It is not clear whether we are applying the same Principle of Relativity. I will write some about what for me the important characteristic of the Principle of Relativity is.

In the theory of special relativity, the Principle of Relativity is satisfied when the physics is indistinguishable from one velocity to another. Even if a space-ship has accelerated to a velocity very close to the speed of light, once the engines have stopped and the space-ship is moving inertially the crew on board will experience normal space-time, everything behaves the way the crew is used to. Also, if a second ship is co-moving with the first ship the two ships will not experience any problem in radio-communication.

I do not use a criterium in the form of trying to see whether there is a preferred frame of reference, the criterium I use is whether it is possible to detect a difference in the local physics.

Of course it is agreed by everyone that acceleration with respect to the local inertial frame of reference is always measurable with a local measurement. There are very smal piezo-electric accelerometers that use kinematics to measure the acceleration, and it is also possible to build a linear version of a Sagnac interferometer. When the linear laser accelerometer has two frequencies of light being generated rather than one then it is being accelerated.

So the class of accelerating frames of reference is distinguishable from the class of inertial frames of reference. The theory of special relativity was logically based on the observations that no difference could be observed. In the case of being accelerated with respect to the local inertial frame: that is measurable, so the type of relativity of the theory of special relativity does not extend to accelerated motion.

Next is the question whether there is a separate Principle of Relativity for being accelerated either in zero-curvature space-time geometry or in curved space-time geometry. Are those indistinguishable? The weak principle of relativity acknowledges that curved space-time has gradients, leading to tidal effects that are measurable with sufficiently sensitive equipment.

Again I am not interested in assigning any frame of reference a status of being preferred. I only look at whether it is distinguishable from other frames by means of local measurement. --Cleon Teunissen | Talk 09:33, 31 Mar 2005 (UTC)

I think that the principles of Relativity state what is important at the very start. "The laws of physics are the same ...". The really is it in a nutshell. For the Special Principle, it is completed with "... in all intertial frames of reference". Add in the constancy of the speed of light with respect to an observer, and you have the fundamental postulates of Special Relativity.

For the general theory, the General Principle drops the word "inertial". Then Einstein goes on and also adds in the principle of general covariance. This is another relativity principle but it ends with "... in all coordinate systems". So whereas SR was good (as stated in 1905) only in inertial frames of reference mapped with a Cartesian coordinate system, GR had to be good no matter how the spacetime was mapped. That is why $G=T$ is at once so profound and also so inaccessible: It conforms to those principles, but in the process ends up being highly, highly abstract. --EMS 15:56, 31 Mar 2005 (UTC)

Re: What is the important characteristic of the Principle of Relativity?

I gather you have dropped the criterium of whether frames of reference are distinguishable. For example, the light in a ring laser interferometer behaves differently, depending on whether the interferometer is in a stationary frame of reference or in a rotating frame of reference. But if the laws of physics are the same in both frames then the laser light should behave in the same way in both situations.

So how do do deal with that? It looks like an inconsistency to me.

What inconsistency? The light does behave in the same way in each frame of reference. You get the same phase shift in both frames of reference!

The only way out, it appears, is to have a meta-law of physics. A meta-law that subsequently determines which laws are "active" in which frames.

The same laws are active in both frames. What changes is how their action is perceived. In one case, the speed of light in uniform and what is differs are the lengths of the paths when the ring interferometer is rotated. In the other case, the lengths of the paths are the same and what varies is the (coordinate) speed of the light. Note that the light is travelling along null gedesics between the mirrors in both frames of reference, and what has changed is the characterstics of those geodesics.

The metric you gave in the Sagnac effect seems to serve as a meta-law. It is, it seems, an algorithm to make laws of physics. It has $\omega$ in it. So for every rotating frame of reference, co-rotating with a particular rotation rate, you get an appropriate law of physics for that particular rotating frame. --Cleon Teunissen | Talk 18:54, 31 Mar 2005 (UTC)

That is one way of looking at it I guess. A "meta-law" implies a higher level of abstraction, and indeed that is what you have in GR. To me, the law of interia is that objects will follow timelike geodesics unless they are acted on by a force. Similarly, light follows light-like (or null) geodesics (where $ds=0$). The metric simply describes what those geodesics are. As you go from one view to another, the coordinate system changes, and often so does the metric. (In SR for example, an acceleration from one inertial frame to another leaves the metric unaffected but changes your view of the spacetime none-the-less. Of course in the Sagnac case, the metric does change.)

Personally, I think that you need to get used to the very different mindset of GR. It is truly general. --EMS 05:27, 1 Apr 2005 (UTC)

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The same laws are active in both frames. What changes is how their action is perceived.--EMS 05:27, 1 Apr 2005 (UTC)

OK, that parallels for example the space-time interval, that is invariant under Lorentz transformation, and individual observers, moving at different velocities, who both choose to view from their own, co-moving frame of reference, will not observe the same proportion of space and time (separating the two events at each end of that space-time interval). Knowing special relativity, they can mutually agree on the correctnses of each set of observations.

Here is how I currently understand general relativity:
The mathematics of general relativity uses invariants, that are mapped to a chosen frame of reference of an observer. Depending on the choice of frame of reference, all kinds of things are present in frame-specific proportions.

I have been thinking about an accelerating observer. Plotted on a Minkowski time-line diagram, his world line is a curved line.

It's actually a hyperbola. Constant acceleration in Relativity is actually called "hyperbolic motion".

A geometrical mapping can be performed that renders his worldline on the diagram into a straight line (and all other lines into curved lines.) I assume that the physics that is occurring in the space-ship of the accelerating observer is a physics that is consistent with the physics that occurs in space-time that is curved by gravitational space-time curvature.

That is an excellent rendition of the Equivalence Principle, especially with the use of "consistent" instead of "identical".

So if an observer in an accelerating space-ship performs this mapping, a force appears in the calculation.

Only if the observer assumes that his motion is inertial.  :-)

In the case of special relativity, the mapping of one space to another uses the relative velocity.
In the metric that you gave in the Sagnac effect article omega is used; the rotation rate with respect to the inertial frame of reference. In the case of linear acceleration: if you map from one accelerating frame to another you get how much more force there is from one frame compared to the other. If you map from the inertial frame of reference, then you get how much force there is, since the inertial frame of reference is the one frame without manifestation of inertia.

A frame of reference is just that, and to me the view in and of itself lacks any "manifestation of inertia", (or at least in independent of it). Inertia manifests itself in the movement of objects along spacelike geodesic paths. Once you realize that, you come up with the same force acting on the observer in the spaceship due to the spaceship's acceleration whether you are on the spaceship, watching it wiz by in free space, or at looking at it through a telescope on a planet.

Perhaps one way of looking at it is that the view affects the metric, and the metric determines the geodesics. Do remember that spacetime is and the events which occur within it are. Relativity has no requirement that all observers view the same events in the same way, only that those views be reconcilable according to certain ascertainable rules. After all, in SR, even the order of events are mutable (at least for spacelike separated events).

--EMS 17:00, 1 Apr 2005 (UTC)

--Cleon Teunissen | Talk 08:21, 1 Apr 2005 (UTC)

The Physics of time dilation

I assume that the physics that is occurring in the space-ship of the accelerating observer is a physics that is consistent with the physics that occurs in space-time that is curved by gravitational space-time curvature. --Cleon Teunissen | Talk 08:21, 1 Apr 2005 (UTC)

That is an excellent rendition of the Equivalence Principle, especially with the use of "consistent" instead of "identical". --EMS 17:00, 1 Apr 2005 (UTC)

As is well documented, in the years between 1907 and 1915 Einstein used that line of thinking to explore ways to find the equations of general relativity. I try to follow that line of thought. An observer is maintaining his position on a rotating disk; further away from the hub there is stronger relativistic time dilation compared to close to the hub, indicating that when there is an "incline" in time dilation over a volume of space, then a force is required to maintain status quo.

Here is what I am really curious about: what did Einstein think about the physics of an accelerating space-ship? Did he assume that inside the space-ship, a force is acting towards the floor? Or did he work with a more abstract conception, that any fluctuation in the mechanical force, accelerating the ship, travels from the tail of the ship to the nose with the speed of light. The force affects the space-ship's tail first and the nose nanoseconds later, so that the tail of the accelerating ship is in a different Lorentz frame than the nose, an "incline" in time dilation.

So if an observer in an accelerating space-ship performs this mapping, a force appears in the calculation.

Only if the observer assumes that his motion is inertial.  :-)

I need to correct myself here, I was inconsistent on a point where I usually am very cautious. What appears in the calculation is a manifestation of inertia. In response to the force being exerted, inertia manifests itself. I suppose some observers may choose to interpret that manifestation of inertia (appearing in the calculation) as a force.

If an observer has set his mind squarily on the assumption that his motion is inertial, then he will gladly interpret the manifestation of inertia as a force that counteracts the mechanical force. (If the observer is aware that he is being accelerated, then he may interpret the force that has appeared in the calculation as representing the force that is accelerating him.)

The inertial frame of reference is the one frame without manifestation of inertia.

A frame of reference is just that, and to me the view in and of itself lacks any "manifestation of inertia",

Yes, I should correct that. A world line that is at all times parallel to the curvature of space-time is an inertial world line. A frame of reference that follows exactly the path of that world line is an inertial frame of reference. An object at rest with respect to that inertial frame of reference will be free of manifestation of inertia.

When an object is moving inertially with respect to the inertial frame of reference, and space is mapped to an accelerating frame of reference, then the calculation will indicate that a force is accelerating the object with respect to the current frame of reference. Of course that force isn't actually there, it is an artifact of the inappropriate choice of mapping the space. In general, mapping space to an accelerating frame of reference is certain to produce unwanted artifacts, only mapping to a frame that is co-accelerating with an accelerated object is sometimes useful.

Mapping to an inertial frame of reference has the property that no artifacts will show up. --Cleon Teunissen | Talk 19:24, 1 Apr 2005 (UTC)

Re: The Physics of time dilation

Cleon - You wrote:

When an object is moving inertially with respect to the inertial frame of reference, and space is mapped to an accelerating frame of reference, then the calculation will indicate that a force is accelerating the object with respect to the current frame of reference.

That is a good way of looking at it.

Of course that force isn't actually there, it is an artifact of the inappropriate choice of mapping the space.

Here is I will disagree somewhat. I see no problem with an acceletated mapping of spacetime. The "force" is not due to the mapping, but instead to an inappropriate assumption regarding the form of the geodesics of spacetime in that frame of reference. Using the Equivalence Principle, you can infer that you are in fact in a accelerated frame of reference. That after all is what Einstein did for being on the surface of the Earth.

Maybe it would help to go back to the Sagnac frame. In that view, there is a "cetrifugal force" that increases in strength as one moves away from the axis of the rotation. However, the metric is of a form such that an object initially at spatial rest in the coordinate system is on a timelike geodesic of motion that indeed accelerates outward with respect to the spatial coordinates of the rotatating system.

BTW - The metric for an accelerated observer in free space is $ds^{2}=(1+az)^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}$ where $z$ is the direction of acceleration (with respect to an inertial frame of reference) and $a$ is the proper (or self-perceived) acceleration of the observer. This metric has geodesics of spacetime that accelerate "downward" with respect to the coordinate system.

GR is a geometrical theory. Just let the geometry drive things, and you can map a spacetime from any viewpoint or frame of reference.

--EMS 20:28, 1 Apr 2005 (UTC)

Rotating mercury mirror world and geodesics

In the coriolis effect article, I use a rotating mercury mirror (as is employed in astronomy), to model a world that is governed by a centripetal force field. The force directed towards the center of rotating is proportional to the distance to the center (possibly you have read the coriolis effect article already.)

In geometrodynamics, the laws of motion are in a form that is uncommitted to any frame of reference. When mapped to a particular frame of reference, the geometry describes geodesics. The geodesics of rotating mercury mirror world are elliptical orbits, (and circular orbits, which are symmetric ellipses).

To my knowledge, when there is an actually rotating mercury mirror, and the the laws of motion are mapped to a frame of reference that is co-rotating with the mercury mirror, then that mapping is highly informative. The mapping not only shows the geodesics of that world, it also shows exactly how an object will be deviated from inertial motion if a force is exerted on that object.

However, if you have a swimming pool that is stationary with respect to the local inertial frame of reference, and you perform the same mapping to a rotating frame of reference then the geodesics you get do not match the actual geodesics of the space of that swimming pool.

Oh yes they do! Doing that mapping means that you (or at least your frame of reference) is now rotataing. When you are co-rotataing with the mirror, those are now the geodesics for an observer at rest with respect to the mirror. When the mirror is not rotataing, they are still valid, but not for the observer at rest with respect to the mirror.

One area of concern with repect to my response is how you are handling the deformation of the mirror. If the sides being higher means that your hovercraft is now being affected by the "force" of gravity (which is appropriate for modeling the interactions on the 2D manifold of the surface of the pool), then my statement above stands. If the hovercraft is moving geodesically with respect to the pool (which is doable but more difficult), then a more complicated explanation is needed.

--EMS 04:10, 3 Apr 2005 (UTC)

Whether an object is accelerating or not is measurable by local measurement, by kinematic or interferometric measurement. An observer who measures that he is not accelerating,(or just a modest acceleration) and who has only access to local measurement has the following option: to infer that is not (or modestly) acccelerating with respect to the local inertial frame of reference. If he is highly imaginative, he may consider the possibility that he is currently inside a tiny space-ship inside a humungus space-ship that is accelerating hard. He maps space to the frame of reference that is co-accelerating with the humungus space-ship, and he imagines himself as moving along a geodesic of humungus-space-ship-space.

You get my drift, I hope. Mapping to the local inertial frame is garanteed to be informative, both for an inertially moving observer and an observer who is being accelerated by a mechanical force. But mapping to an accelerating frame of reference is informative only if it is the co-accelerating frame. If it is not the co-accelerating frame, then you would need supporting hypotheses to explain the mismatch. --Cleon Teunissen | Talk 06:56, 2 Apr 2005 (UTC)

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One area of concern with repect to my response is how you are handling the deformation of the mirror. --EMS 04:10, 3 Apr 2005 (UTC)

As a first stage I think about the rotating mercury mirror as the actual device. The "saucer" is preformed; the inclination of the rotating mercury (proportional to the distance to the center), employs the Earth's gravity to obtain a centripetal force "field". The physics of the actual device is happening in 3 dimensions of space, but for the sake of simplicity I disregard the motion in the up-down direction.

The next stage is to remove everything but the centripetal force field, and to define it as a world with 2 dimensions of space and one of time. This 2D world is somewhat like a solid rotating disk. But this solid rotating disk can only represent the pervasive centripetal force. I need test masses to move without friction, and yet be subject to the centripetal force all the time. So the paraboloid, the "saucer" of the rotating mercury mirror, is the best approximation, I think. --Cleon Teunissen | Talk 10:56, 4 Apr 2005 (UTC)

Geodesics for an observer

I will try and focus the difference of opinion between you an me a little more. I use a concept of geodesics where the geodesics are related to physical objects. I throw out a bunch of test masses, and the trajectories I subsequently observe are the geodesics of that space. I think gravitational time dilation is real physics, and I think the gravitational time dilation (and space deformation) is independent of the choice of frame of reference of the observer.

For example: the solar system: the Sun, the Planets, the Moons of the planets, Comets, all the rotations of these celestial bodies. The planets are the test masses of the solar system, the shapes of the geodesics of the solar system are mostly due to the gravitational space-time curvature of the Sun. (But check out Oterma. External link: trajectory of Oterma) The moons that are orbiting planets are following geodesics that are mostly due to the space-time curvature of their planet. So the local inertial frame of reference in the neighbourhood of Jupiter is co-orbiting the Sun together with Jupiter, etc etc, for all the planets.

All the local inertial frames of the solar system have something in common: they do not rotate perceptibly with respect to each other. The planetary orbits are elliptical, and when perturbations by other planets are taken into account (and the relativistic shift in the orbit of mercury is taken into account) it is seen that the lines connecting aphelion and perihelion of each of the planets do not rotate with respect to each other.

In the case of the rotating mercury mirror: when the mercury mirror is rotating then test masses that are released (in this example the 'tiny hovercrafts') will move along elliptical trajectories, I call these trajectories geodesics. When I stop the mercury from rotating, a physical act, then the surface becomes flat and the test masses will from that moment on move along straight lines, the geodesics are straight now. That is how I use the concept geodesics. If a physical change can make the geodesics change, then they are real, physical geodesics.

I think that you can stop right here! Geodesics are a very definite mathematical concept, arising out of a metric equation in a very well defined way. Technically, the hovercrafts are NOT moving along geodesics. Instead their movement is a combination of coriolis "force" which arises as you documented (due to a geodesic deviation on a 2D manifold), and a centerwards "force" that is in fact due to the pull of gravity of your hovercraft and the mirror being distorted due to the rotation.

Now, if you want to create an appropriate 2+1 metric, you can have those shovercraft moving along geodesics. (I hold my nose on this since that metric and those geodesics are a fiction -- The hovercraft are not on geodesics of (the real 3+1) spacetime since they are not in free-fall. However, pretending that this metric and its geodesics actually exist is about to be quite useful.) In this case, one now has to be careful about what is being rotated. Rotating the coordinate system (as we did to explain the Sagnac Effect and as you do to explain the Coriolis Effect) is not the same as rotating the mirror due to the deformation that arises and which creates a cetripetal "force" that the 2+1 metric must account for. In GR, things are similar: Rotataing a mass results in its surrounding spacetime being described by the Kerr Solution, which rotataing the coordinate system aroung an otherwise stationary massive object results in the observed spacetime being described by the rotataing Schwarzschild Solution.

So my points are:

• Be careful about what you call a geodesic,
• An observed path of motion for an object in not necessarily along a geodesic. Only if it is not under this influence of any force (such as being pushed upwards by mercury) is that the case, and
• Be aware that rotating your view of a spacetime is often not the same thing as rotating a mass within the spacetime.

By the way, you probably should remove the hovercraft business from your Coriolis Effect page since that is due to multiple causes. At the least, calling those paths "geodesics" is highly misleading and will cause confusion about what geodesics really are. Here is a rule: If you cannot produce a metric equation and show that certain simple observed motions arise from its geodesic equations, you have no business citing the term in your work. (For the coriolis effect itself you can do this, although your animation is the best way of showing people what is going on.)

--EMS 16:03, 4 Apr 2005 (UTC)

In the concept of geodesics that I use the viewpoint of the observer is not a factor. The observer is supposed to be able to switch between viewpoints readily. If in a calculation an observer maps from one space to another then the geodesics valid in the new frame are, of course, different from the geodesics valid in the previously calculated frame. I am only interested in those calculations that yield geodesics that coincide with the physical geodesics of the system that I am thinking about. --Cleon Teunissen | Talk 13:22, 4 Apr 2005 (UTC)

Cleon Teunissen | Talk 16:16, 4 Apr 2005 (UTC) On the coriolis effect page I do not call the trajectories geodesics. I call them interchangebly 'elliptical trajectories' and 'elliptical orbits'. Please doublecheck that sort of thing before you accuse.

The coriolis effect page is aimed at explaining the way the coriolis effect affects atmospheric motions. The animation is not by me, and it is wrong. It is oversimplified. I only inserted it to accomodate people with very little understanding of physics.

To explain the coriolis effect I do not use rotation of the coordinate system. My explanation of the coriolis effect takes place in the context of the inertial frame of reference. Occasionally I point out how something looks as seen from the perspective of the rotating system, but that is only about how it looks not about the physics that is going on. In my opinion it is inappropriate to refer to changing to a rotating coordinate system in explaining the physics of the coriolis effect.

Inertia is inertia. It will not introduce unnecessary complexity by suggesting there are three different phenomena, where there is actually a single phenomenon: manifestation of inertia. It makes no sense to suggest the existence of separate phenomena: linear manifestation of inertia, coriolis manifestation of inertia and centrifugal manifestation of inertia. That differentiation is an artifact of burdening a theory with unnecessary complexity. --Cleon Teunissen | Talk 16:16, 4 Apr 2005 (UTC)

You are right that I should have referred to your Coriolis Effect page before saving my thoughts. My appologies for that sloppiness.

--EMS 18:14, 4 Apr 2005 (UTC)

Interpreting space-time geometry as real physics

In trying to find out the content of geometrodynamics I encountered many contradicting interpretations. I follow the interpretation that to me seems the most convincing one.

As I wrote before, I think that gravitational space-time curvature is real physics. I think there is a one-on-one correspondence between the space-time curvature and the pattern of geodesics of space.

I think that in the case of a space-ship being accelerated by a mechanical force and the same space-ship standing on the surface of the Earth the same physics is going on. In both situations the nose-end of the space-ship is not in the same Lorentz frame as the tail-end of the space-ship, and that "incline", that unlevelness in time dilation over the length of the ship is always associated with manifestation of inertia. In the case of the accelerating space-ship the accelerating force is first in the causal chain and the unlevelness of time-dilation is a consequence of that. In the case of space-time curvature the unlevelness of time dilation is a pervasive property of the space-time surrounding the gravitating mass.

All atomic clocks on Earth that are at sealevel run in synchrony. Atomic clocks at the equator are further away from the center of the Earth, and also they move at a higher velocity than stations closer to the axis of the Earth. The two influences on the amount of time dilation cancel. That is not a coincidence, of course. The sealevel surface of the Earth must be an iso-chrone, a surface of level time-dilation. It it wouldn't be level, mass would redistribute itself until it is level. --Cleon Teunissen | Talk 09:14, 5 Apr 2005 (UTC)

the causal chain

I feel I need to explain one more thing.

In the case of the space-ship being accelerated by a mechanical force I need to identify what I call 'the causal chain'. In the interpretation of geometrodynamics that I follow it is mandatory to specify with mathematical precision what is part of the accelerating frame and what is not. This is especially important in the case of rotation. What is the reach, the extent of the frame of reference that is co-rotating with a merry-go-round? If the co-rotating frame would extent to infinity, it would imply an infinite force, and I agree with the opinion that whenever a theory implies an infinite force somewhere that is a sign of the theory breaking down. So a mathematical formula must be provided that specifies the extent of the rotating frame. In the interpretation of geometrodynamics that I follow only those objects that are part of the causal chain are considered to be part of the accelerating frame.

The co-rotating frame of reference for a merry-go-round does extend to infinity, and it is perfectly valid to note that long before you get there (like at $r=c/\omega$) you will need an infinite force to get something to co-rotate with you (placing it at rest in your frame of reference). However, do note that your rotataing does not make the rest of the universe counter-rotate against you. What has changed is your view of the universe, not the universe. So no force is used, as a coordiante system is a construct used to mark reality and make it accessible to mathematical manipulation/modeling. Such constructs can be changed at will. [OTOH, Changing the universe is a somewhat different matter.  :-) ]

--EMS 14:33, 6 Apr 2005 (UTC)

In the case of the space-ship: any object that is attached to the space-ship in such a way that it is being co-accelerated is subject to the causal chain. When there is no force, no momentum transferring interaction between the ship and an object inside the ship, then that object will move along a geodesic of the local inertial frame.

When light is travelling inside a cavity in the accelerating space-ship then that light will move along a lightlike geodesic of the local inertial frame of reference. --Cleon Teunissen | Talk 12:59, 6 Apr 2005 (UTC)

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By the way, you may consider moving the content of this current User_talk:Ems57fcva/sandbox/Equivalence_Principle page, in order to "clean" this discussion page. I feel I have flooded this discussion page. If you select the move tab you can move everything to, say:
sandbox/Equivalence Principle/archive1
The current sandbox/Equivalence Principle page will then contain a redirect to that archive page, you can change that redirect into a link to the archive page. This method of moving the content of a discussion page preserves internal wikipedia linkages. --Cleon Teunissen | Talk 12:59, 6 Apr 2005 (UTC)