Why no privlidged reference frame?

[JDługosz]

I'm not a physicist and not a mathematician. I just want to understand things, and have read "advanced layman" books.

It's often touted that Noether's theorem shows that the uniformity of space (the same everywhere; no preferred origin) implies conservation of momentum. More properly, the fact that the laws of motion don't indicate any absolute position will give conservation of momentum.

However, it appears that reversing it, conservation of momentum implies that space is uniform, does not by itself imply that velocity is also not absolute.

Given that space "exists", I'm trying to crystallize the idea that the lack of special rules is what gives us the rules of motion that actually exist. The lack of special rules means there is no origin, and only certain types of natural motion is possible ("natural") without adding more to it. So, is there any reason why lack of a preferred velocity is just as bare-bones as lack of a preferred position?

Care to discuss philosophical reasons and beauty?

--John

 

[{~}]

I'm not sure what you mean by space existing. Space is defined by the things in it (particles, fields ect.) There is really no such thing as empty space. That would just be nothing.

Now if you ignore relativistic effects your speed of travel would seem arbitrary. It is possible to travel great distances in a small amount of time for you but when you consider relativity the distance traveled is shorter from your point of and the trip took longer from an outside point of view.

The is a motion vector that is not arbitrary but has the same magnitude for everything in the universe. This is the vector sum of relative velocity and relative rate of flow of time between any two inertial (non accelerating) frames.

 

[Antiphon]

The theorem says that if you make a transformation and the laws of physics are the same in the transformed experiment, then there is a quantity connected to that transformation that is a conserved quantity. If you do an experiment facing north and do it again facing west you should get the same result (forget about the Earth's magnetism and rotation for a moment.). The consevef quantity is angular momentum and the transformation is rotation. For linear momentum, the transformation is translation. It gets
more involved but the idea is the same for other conserved quantities.

 

[MaxwellsDemon]

The relativity of velocity is a really old idea. What this is saying is that if you have two isolated observers moving "way out there" in space, then there is no experiment you can perform to determine who is "really" moving...the best they can do is talk about the relative speed between the two. Even Newton makes this relativity of motion assertion in his description of mechanics...it was kind of abandoned in electromagnetic theory for a while, but Einstein elevated it to universal status and tacked on his finite speed of causality. I think its preferred for two reasons, Occam's Razor and experimental predictive power, i.e. there is no experimental result that requires an assumption of absolute motion for its description.

 

[JDługosz]

Puzzle me this: I postulate a toy universe (e.g. a computer program) where "natural motion" is to accelerate, increasing the magnitude of the velocity by universal growth factor. This requires a preferred reference frame (absolute velocity) to formulate. But, the behavior is the same no matter where you start from. It seems to have translational symmetry. But it does not conserve momentum.

Does "translational symmetry" not quite mean what the popular explanation states (no absolute labeling of position), but also requires there to be no absolute measure of velocity? What am I missing?

 

[Dale]

Puzzle me this: I postulate a toy universe (e.g. a computer program) where "natural motion" is to accelerate, increasing the magnitude of the velocity by universal growth factor. This requires a preferred reference frame (absolute velocity) to formulate. But, the behavior is the same no matter where you start from. It seems to have translational symmetry. But it does not conserve momentum.

You are a little confused here about how to apply Noether's theorem. For such a toy universe the Lagrangian would be of the form:

$$L=\frac{1}{2}mv^2+mgx$$
where the acceleration is of magnitude g in the x direction.

This Lagrangian is translation-symmetric in y and z, but not x. So according to Noether's theorem momentum would be conserved in y and z but not x, as you could easily demonstrate by solving the equations of motion.

 

[JDługosz]

You are a little confused here about how to apply Noether's theorem. For such a toy universe the Lagrangian would be of the form:

$$L=\frac{1}{2}mv^2+mgx$$
where the acceleration is of magnitude g in the x direction.

This Lagrangian is translation-symmetric in y and z, but not x. So according to Noether's theorem momentum would be conserved in y and z but not x, as you could easily demonstrate by solving the equations of motion.

So, even though the rule of natural motion as stated doesn't involve x, the Lagrangian does, owing I think to the need for the derivatives. I was thinking that would be the case, but don't know enough to set it up from scratch. I roughly understand that the Lagrangian is the sum of kinetic and potential energy, but I don't understand what "action" is.

 

[Dale]

Yes, that is correct. The action is the integral of the Lagrangian along the trajectory.

 

[IRobot]

I would ask 'Why a a privileged frame?"; "Why should Nature have wanted that physical laws would change between Galilean frame and others?".

 

[JDługosz]

Yes, that is correct. The action is the integral of the Lagrangian along the trajectory.

Thanks. What would it look like if (as I was thinking) the x-direction is not singled out in the law of motion, but rather the particle accelerates along whatever direction it is moving in before? That is, a particle at rest (in "the" reference frame) remains at rest. If set in motion, it runs away faster and faster. Its vector's length increases by a constant factor in equal time units, like compound interest.

 

[JDługosz]

I would ask 'Why a a privileged frame?"; "Why should Nature have wanted that physical laws would change between Galilean frame and others?".

Well, rotating frames and accelerating frames are "different" in our real world. What's inherently special about what we call inertial frames? Specifically, is this implied by isotropism? It's not obvious that having no preferred labeling of coordinates is sufficient in itself to demand that velocities must be relative as well.

My larger thoughts on the matter are: Newton's laws are "compelled" by having a minimal number of ingredients and not having any other special cases added in. Natural motion is what's allowed without adding in more stuff.

 

[Dale]

Thanks. What would it look like if (as I was thinking) the x-direction is not singled out in the law of motion, but rather the particle accelerates along whatever direction it is moving in before? That is, a particle at rest (in "the" reference frame) remains at rest. If set in motion, it runs away faster and faster. Its vector's length increases by a constant factor in equal time units, like compound interest.

Noether's theorem only applies to systems that can be described by a Lagrangian. I can't think of one that would do what you describe. That doesn't mean that there isn't one, just that I am not clever enough to figure it out so I can't answer the question.

 

[Cleonis]

I'm not a physicist and not a mathematician. I just want to understand things, and have read "advanced layman" books.
[...]
Given that space "exists", I'm trying to crystallize the idea that the lack of special rules is what gives us the rules of motion that actually exist. The lack of special rules means there is no origin, and only certain types of natural motion is possible ("natural") without adding more to it. So, is there any reason why lack of a preferred velocity is just as bare-bones as lack of a preferred position?

The way I understand it, you are enquiring whether there is some plausibility argument showing that galilean relativity arises necessarily.

I don't think such a plausibility argument exists. We have to assume galilean relativity in order to frame a theory of motion at all, but we can't reduce it to something more fundamental. In other words: as is, galilean relativity is not reducible.


We also have the counter-intuitiveness of galilean relativity.
In classical physics, just as in its successor relativistic physics, physical properties are attributed to space. (In general relativity this takes the form of attributing to spacetime the possibility of being curved.)

The laws of motion are properties of space.
Let me write some things about contrast between matter and space.
Matter has physical properties, and it has constituent parts that can be tracked through time. Space on the other hand, while having physical properties, does not have constituent parts that can be tracked through time.

That is counter-intuitive in itself. How is that possible: to have physical properties, yet have no constituent parts that can be tracked through time?


For the homogeneity of space we may be tempted to use something tangible as a metaphor. Let's say a ship is afloat on an a featureless body of water. Being featureless there is no reference for location; you can take any point as the origin of a coordinate system. But of course the crew can always determine the ship's velocity relative to the water. The crew can use dead reckoning to make a journey and return to the very same spot. (Two ships can move out of sight of each other, and use dead reckoning to return to each other at a pre-arranged point in space and time.)

As we know, this metaphor breaks down: Galilean relativity asserts that the homogeneity carries over from position to the first time-derivative of position: velocity. It is as if everything is moved over a notch. A single notch: the homogeneity does not extend to the second time-derivative of position: acceleration.

Spaceships floating in space can use something analogous to dead reckoning: when thrusters are fired the onboard accelerometers give the change of velocity, and direction of the change of velocity, and integrating those data you can plot the ship's course. Two spaceships can separate, and relying only on their accelerometry based dead reckoning they can rejoin. No mechanical model can explain why velocity-space is relative, and acceleration-space is absolute.


I think all that is profoundly mysterious. At present we have no prospect of reducing this to something more fundamental. In order to formulate laws of motion at all the relativity of inertial motion must be assumed, and we take it from there.

 

[JDługosz]

Cleonis, that is food for thought. First, your analogy of water and space having no "parts" suggests that you might not be able to tell your velocity by looking at space going by. Why just one notch? Perhaps it's not possible to formulate a law that treats acceleration or higher as non-absolute, so it basically goes up as high as it's able to?

Physical properties of space? Think about programming a classic video arcade game. The ships, missiles, asteroids, etc. are objects with properties including positions. The rules of the simulation update the position values over time. Other rules proscribe interaction between objects whose properties match certain criteria; to wit when their position attributes are nearly matching. There is no "space" data structure. It is an emergent concept from position of objects and that position matters.

The same thing could be implemented as a cellular automaton with cells connected such that motion moves objects to adjacent cells. The position of an object is not a stored number anywhere, but is implicit based on which cell it is stored in. Position only has meaning in that other objects might be moved to the same cell and thus interact.

Either way, the result is the same. In the latter case, the automa cells "have properties" in storing objects and communicating with neighbors. In the former case, there is no problem with property-less space because there is no such data structure.

Interestingly, the former would be a better choice to implement "entanglement", paired objects at different positions that still interact. It's just a different selection criteria for deciding which objects interact.