User talk:WillowW/Inertial frames

WillowW
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Inertial Frames

Thank you very much for fixing up the photon article. I really liked your edits with your careful phrasing. You really have a way with words! So I was wandering if you'd be willing to help fix some things on the inertial frame of reference article as well. The definition on that page has always bugged me, but unfortunately I'm having trouble deciding a better way to word it myself and due to the fundemental nature of the topic I'd prefer to discuss it constructively with someone knowledgeable and calm headed first before making any edits or having to deal with misunderstanding of my intention on the talk page. Since you seem incredibly well intentioned and calm headed (for example your above post), I'd appreciate your suggestions.

The current definition is actually from your edit here. "An inertial frame is a coordinate system in which Newton's first and second laws of motion are valid." You changed this from just referring to the first law of motion. Many text books (and even wikipedia) just use Newton's second law to define a force. Besides the fact that the second law refers to "net force" (implying that the forces are additive), the second law doesn't seem to add much more than this definition. (As even mentioned in the force article, several prominent physicists have found the lack of a more explicit definition of force to be problematic.) The only point in bringing this up, is that you may be getting more out of Newton's second law than most people take, and if so, you may be able to clarify the definition of inertial frame by stating explicitly what you wish to be obtained by mentioning Newton's second law. Anyway, onto the heart of the matter...

First, I must admit that defining an inertial frame is difficult and physicists for the most part usually "know" what is meant by such a term even without a good explicit definition. However, even in literature, this has sometimes led to confusion where inertial frames are occassionally taken to be nothing more than a coordinate system where Newton's first law holds true (obviously the authors don't pause to define an inertial frame, but it is implicit in how they treat the frames in the paper).

Where I find fault with this definition is that most physicists agree that Newton's first law is not sufficient to define an inertial frame. I have seen people try to add something to the effect that "space should be described isotropically" and "time should be described homogenously". This better describes what people mean, but just shifts the problem to defining what specificly it means to be "described isotropically or homogenously".

Maybe this suggests an inertial frame is best defined by its metric. I don't know. What do you suggest?

Thanks, -- Gregory9 10:25, 16 August 2006 (UTC)

Hi, Gregory9! Thanks for your really kind words — it's really encouraging to know that people like my contributions! But I'm also a little abashed; I feel my own flaws and limitations keenly, and hope you won't be disappointed in me later on.

I see what you mean about the delicacy of defining inertial frame of reference, and I think I need to brood on it, and maybe find some books at the library. I also need to remember what I was thinking when I made that edit! ;) Can I get back to you in a few days? It fits in well with Cleonis' request above, too. Thanks for being patient with me! :) Willow 12:07, 17 August 2006 (UTC)

Hi, this is Cleonis responding.

I visited WillowW's page to see if maybe there was a comment to my post. I saw this entry, and I cannot resist adding a comment.

I am in favor of the following definition of 'inertial frame of reference' (newtonian dynamics) An inertial frame of reference is a frame of reference in which Newton's laws of motion hold good.

Another possibility - mathematically equivalent - is the following definition: An inertial frame of reference is a frame of reference in which the laws of conservation of momentum hold good. (Conservation of linear momentum and conservation of angular momentum)

Newton's third law is equivalent to asserting conservation of momentum when two (or more) objects are exerting a force on each other. At the same time, the law of conservation of momentum (in the case of objects exerting a force on each other) serves as an operational definition of 'inertial frame of reference'. Conservation of momentum holds good if and only if you are mapping the motion in an inertial frame of reference.

I think the following 'what if' scenario is quite interesting, and maybe it is worthwile to include it in the 'inertial frame of reference' article.

What if through the ages astronomers would only have had the planets, and none of the stars, would Newton still have been able to formulate his laws of motion?

Planetary motion has always been mapped with respect to the fixed stars. When planetary motion is mapped with respect to the Sun and the fixed stars, then Kepler's law of areas holds good.

Clearly, without any of the fixed stars providing a reference, it would have been much harder, but not impossible, to discover Kepler's law of areas. With or without seeing the fixed stars, there is only one frame of reference in which Kepler's law of areas holds good.

In order to appreciate the validity of Newton's laws of motion, the significance of the equivalence class of inertial frames of reference must be recognized.

In order to appreciate the valitidy of the equivalence class of inertial frames of reference, the significance of conservation of momentum must be recognized.

Cleonis | Talk 13:20, 17 August 2006 (UTC)

Hi all, this is a very interesting topic, although I'm not sure that I can do it justice. Regarding my earlier edit, I think my thoughts were that the force laws for physical interactions (e.g., Coulomb's law) could be defined a priori, not a posteriori from the Newton's second law. Otherwise, it seems that we couldn't even use Newton's first law to define an inertial frame, since we wouldn't be able to detect empirically the absence of external forces except by a lack of acceleration (geodesic motion). But perhaps I'm not understanding it correctly.

A better answer may be derivable from Einstein's theory of general relativity. If I recall correctly, the Bianchi identities and Einstein's field equations ensure that the divergence of the stress-energy tensor is zero; thus, the conservation of energy-momentum is an intrinsic, coordinate-independent property of space-time. I also seem to recall another Einstein paper (from ~1940) with Bruno Hoffmann and Leo Infeld, in which they showed that the law of geodesic motion on space-time (equivalent to Newton's first law, I think) can be derived from only the continuity conditions on the curvature of space-time. I have to brood more on the whole question, though, since it doesn't seem transparent to me yet. Hoping for better news to follow, Willow 19:48, 17 August 2006 (UTC)

If you reason in terms of the general theory of relativity, the answer will be different in some respects, but fundamentally the same. To obtain a solution to the field equations at all, it is necessary to assume what the metric will be at spatial infinity. The only assumption that leads to physically realistic solutions is that further and further away from space that contains matter, space-time converges asymptotically to Minkowski space-time. Mathematically, Minkowski space-time is the equivalence class of inertial frames of reference. So you end up with the same mutual dependency that you have in newtonian dynamics. Recognition of the equivalence class of inertial frames of reference makes it possible to formulate laws of motion at all; the laws of motion inherently single out the equivalence class of inertial frames of reference. I think it is impossible to get a away from that mutual dependency, nor is there any need to get away from that mutual dependency.

None of the laws of physics can be formulated a priory, I think. All axioms of a particular theoretical framework are dependent on the way they are embedded in the theory as a whole in order to carry any meaning. --Cleonis | Talk 20:52, 17 August 2006 (UTC)

Willow, of course, feel free to take all the time you need. As is obvious from my post, I need time to brood on this as well. I would caution however, that considering "the force laws for physical interactions (e.g., Coulomb's law) could be defined a priori, not a posteriori from the Newton's second law." and then defining an inertial frame as one in which these forces obey Newton's second law, you would be turning special relativity into a tautology. (ie. The laws of physics are the same in all inertial frames of reference because an inertial frame is defined as such.)

Defining an inertial frame is indeed tricky. Take your time to think about it, there's no rush. I'll keep mulling it over, and I look forward to whatever insight you can offer. -- Gregory9 21:26, 17 August 2006 (UTC)

Hey all, just a quick clarification of what I was thinking when I said a priori (although it may still be muddled thinking!) It seems like one could define a standard force between two identifiably identical objects under identical conditions, e.g., the repulsive force between two electrons separated by one Angstrom. You could tell if you were in an accelerated frame if a measurement of that force (say, with a tiny spring ;) did not yield its standard value. So, strictly speaking, you need two standard forces to set in equilibrium, here, the electrostatic and the spring forces.

Perhaps more simply, you might be able to define a standard trajectory. In an inertial frame, the two electrons should fly apart in a standard way, rectilinearly if I'm not mistaken. You could then tell whether you were in an accelerated frame by examining the trajectories of the standard electrons upon releasing them and looking for "non-standard" accelerations. Perhaps the radioactive decay of an α-particle might be a more practical "standard" for this kind of experiment. Anyway, just a few random thoughts, Willow 22:08, 17 August 2006 (UTC)

Inertial frames: The question when a definition is tautological

Gregory9 wants to avoid a definition that he regards as a tautology. I think that the definition that I favor is not tautological. If the definitions I gave would be mere tautologies, then they would not encompass a theory with such rich physical content.

I wrote the wikipedia article about Inertial space to focus on the physical content. The concept of inertial space is pervasive in physics, both in newtonian physics and in relativistic physics. Due to its never being absent, inertial space is often overlooked; many textbook writers do not mention inertial space at all. Many textbook writers use the expression "inertial frame of reference" as a substitute for the expression inertial space, making it hard for the reader to obtain a clear view. As long as the expression 'inertial frame of reference' is used ambiguously, defining it will be problematic.

isotropy

Let a spaceship be accelerating with respect to inertial space (as measured by accelerometers). Let two objects be ejected, one forwards and one backwards, with equal force. Then it is seen that the object that is ejected backwards moves away from the spaceship faster than the object that was ejected forwards. This can be expressed in the following way: when a spaceship is accelerating with respect to inertial space, then measurements indicate that there is an anisotropy.

Let a number of spaceships be in inertial motion (as measured by accelerometers). The shaceships have a velocity with respect to each other. Then for all spaceships applies: two objects that are ejecte in opposite directions with equal force move away from the ship at an equal rate (As measured by the ejecting ship) Whenever a spaceship is in inertial motion then measurements indicate that space is isotropic.

isotropy of space as a criterium

This criterium of isotropy of space for ejection/emission of objects/particles is given a more central place in special relativity. Special relativity accomodates that photons carry momentum. If an object in inertial motion emits two photons of identical energy in opposite directions, then the momentum of the emitting object is conserved. Phrased in another way: when an object is in inertial motion, then photons that are emitted in opposite directions move away from the ship at an equal rate.

According to special relativity, particles and electromagnetic radiation are equally subject to the principle of inertia. The shift from newtonian dynamics to relativistic dynamics is that the principle of inertia is given a more central place.

Principle of inertia

The principle of inertia is embodied in the law of conservation of momentum: if there is an interaction between two objects (of if particles are created., etc) then the momentum of the common center of mass of the set of all involved objects/particles will be conserved. This law holds good if the motion is mapped with respect to an inertial frame of reference. This serves as an unambiguous definition of the concept 'inertial frame of reference'.

Misner, Thorne and Wheeler

I quote from Gravitation, by Misner, Thorne and Wheeler (paragraph 12.3):

Point of principle: how can one write down the laws of gravity and properties of spacetime first (paragraph 12.1) and only afterward (here) come to grip with the nature of the coordinate system and its nonuniqueness? Answer: (a quotation from paragraph 3.2, slightly modified) "Here and elsewhere in science, as emphasized not least by Henri Poincaré, that view is out of date which used to say: 'define your terms before you proceed'. All the laws and theories of physics, including Newton's laws of gravity, have this deep and subtle character that they both define the concepts they use (here Galilean coordinates) and make statements about these concepts"

--Cleonis | Talk 07:01, 19 August 2006 (UTC)

As I see it, you are putting the emphasis of conservation laws in the incorrect place. Noether's theorem shows us that symmetries in the physical laws yield conserved quantities. You are trying to subsume the form of physical laws into the defintion of coordinate system that is chosen.

Besides, it is very common to discuss a system which does not conserve momentum in the context of an inertial frame. For example the humble particle in a harmonic well. Or any example including dissipation such as friction.

-- Gregory9 09:15, 21 August 2006 (UTC)

It is not clear to me what makes you use the expression 'subsume'. What I can say is that in my view there can be no such thing as the laws of physics being subordinate to the definitions of the concepts that are used. In that sense I am positive that there is no such thing as 'subsuming the form of physical law into the definition of coordinate system that is chosen'. --Cleonis | Talk 15:06, 21 August 2006 (UTC)

I recognize that I am not a proficient writer. So I appologize if the wording of that phrase was not clear. Let me try again: You are attempting to incorporate the form of physical laws into the definition of a coordinate system.

Coordinate systems are used to label events. What events actually happen, what Lagrangians are used to model a system, etc. are something above and beyond how we decide to label events. This is why, when at all possible, the modern approach is to describe the physical laws in a coordinate independant manner. Therefore the physical laws are a concept independent of the coordinate system and I feel it is completely inappropriate to try to incorporate this separate idea into the definition of the labelling scheme itself.

Also, you ignored my point that it is very common to discuss a system which does not conserve momentum in the context of an inertial frame.

-- Gregory9 21:31, 21 August 2006 (UTC)

The metric singles out inertial frames?

The more I think about it, the more I lean towards using the metric to define an inertial frame. It is what separates the inertial frame from all the other ways to label events. In essence this is using SR's second postulate (along with maybe a given "standard" of length or time) to define an inertial frame. This does not make SR a tautology because this does not require that all physical laws have Lorentz symmetry.

I still feel a bit uncomfortable with this, though I cannot specifically point out what causes my hesitation. So I need some time to think all this over as well. I look forward to your comments. -- Gregory9 21:39, 21 August 2006 (UTC)

Removed non-contributing thread

I have removed the thread 'Oscillation of single particle in harmonic potential well' because it is a distraction from the intended subject. Gregory wrote: "you ignored my point that [...]" , which I took as a request to elaborate on the point that Gregory had raised. I agree with Gregory that subsequently our comments on each other have been going around in circles. Since the thread 'Oscillation' makes no useful contribution I have removed it. (The complete thread remains available in the page history; version-ID 71436396) I will not watch this talk page for a while. I'll check back in a couple of weeks or so. --Cleonis | Talk 12:15, 24 August 2006 (UTC)

The Stanford Encyclopedia of Philosophy article

I copy and paste from above:

First, I must admit that defining an inertial frame is difficult and physicists for the most part usually "know" what is meant by such a term even without a good explicit definition. [...] -- Gregory9 10:25, 16 August 2006 (UTC)

I copy and paste from User_talk:WillowW#Inertial_frames

But before considering such things, of course we still need to figure out the best way to define an inertial frame in the first place :) -- Gregory9 20:21, 2 September 2006 (UTC)

There is something puzzling here.
I think we agree that students of physics learn effortlessly what is meant by the expression 'inertial frame of reference' through a process of assimilation. The expression 'inertial frame of reference' is naturally and unambiguously embedded in the body of mechanics knowledge. Textbook writers feel no need to provide an explicit definition of what they mean by 'inertial frame of reference'. Students of physics just pick up on it.

Whenever a concept is learned so effortlessly and naturally, it is inconceivable that this concept should be resistent to explicit definition. So what is the obstacle here?

I recommend the following article from the Stanford Encyclopedia of Philosophy.
Space and Time: inertial frames
The SEP article is very much indepth, it is a rich source of information and insights (including a lot of historical information).
The Stanford Encyclopedia of Philosophy articles that deal with physics subjects are written by accomplished physicists with a keen interest in the philosophy of Physics. Contributing authors are of course expected not to give their personal view, but to reflect the overall views that are current in the scientific community. The SEP is a very ambitious encyclopedic project: aiming for the highest academic standards.
about the SEP
--Cleonis | Talk 23:32, 2 September 2006 (UTC)