Help Historical evolution of understanding of SR

[JustinLevy]

I realize that the understanding of physics and the perferred description or mathematical representation of our understanding evolves over time. So I have a feeling that the answer to my question is strongly a mix of history and physics. Please, when at all possible add some of the historical context into your answers to help flesh out my understanding. -Thank you.


My questions involves:
It seems to me that what special relativity "actually says and doesn't say" has changed over the years, and this can cause confusion to some new students. How did what physicists consider "Special Relativity" evolve over the years? In particular, in what way has what they feel it predicts/covers change? Is there a definitive "current" source that every phycist would agree on, or once people get nit picky will the number of opinions approximate the number of people asked?

(And yes, that last part will probably be inevitably played out on this board. And please no crackpots, there is no place for that in this discussion. People like Herbert Dingle and the opinions of non-mainstream scientists do NOT count. I'm asking about the evolution of (and current) mainstream understanding of SR.)



Some people may wonder why I think anything evolved at all. So let me give my understanding on this. Let me start from the end and then see the path from 1905 to the current now.

Current:
When I speak to theoretical phycists (mostly grads and profs), they seem to view SR as merely a requirement of "global lorentz symmetry" for lagrangians describing a system in an inertial coordinate systems. And what survives in GR is "local lorentz symmetry" for lagrangians describing a system.

This view automatically gives an upper speed limit in an inertial coordinate system which all such systems agree upon (the fact that it happens to be the speed of light in this case is merely a coincidence based on light being massless, it seems in this view of relativity as just a symmetry requirement, that it doesn't require light to be this speed. But experiment shows that it is.)

I'm sure some people will dislike this wording, but it is undoubtably a common view (the distinctions this causes will become more clear in a bit).



The beginning 1905 -
(English translation, for I can't read german.)http://www.fourmilab.ch/etexts/einstein/specrel/www/

Two postulates
1) ``Principle of Relativity'' - the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good
2) light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body


Inertial frames seem to be implied for #2 (maybe my bias is causing history revisionism, can someone comment on this), so I will assume he originally meant it as such.

It should become instantly clear what I mean by "evolving understanding" just by looking at his original phrasing here. Einstein showed that Newton's laws of mechanics needed revising ... so what frames does he mean by "frames of reference for which the equations of mechanics hold"? This is the first example in many where every one who reads it "knows what he means", and fills in what he feels is the essence of that phrases understanding.

In fact, any term which has survived many centuries we treat like this. What is a Force? What is an inertial frame? They are very hard to define precisely without making the use you wish for becoming tautological. This is because when we see these phrases we fill in with the "essence" of what we understand these phrases to mean. I remember reading an article by Nobel laureate Frank Wilczek complaining about the notion of "Forces" for a similar reason.


1907 - The Geometric view
Hermann Minkowski introduced a way of viewing special relativity as a 4-dimensional space-time with lorentzian symmetry. For example, the four-vector relating to energy-momentum is a geometrical object - it is invarient. It is only our coordinate representation of it which changes depending on our choice of coordinate labels. This led eventually to a coordinate free way of looking at things.


1915/1916 - General relativity
Now the old theory of relativity is just a special case, and that 'principle of relativity' finally becomes "special relativity" (not sure when this phrase was first used). Now any coordinate system is acceptable, and at least in current times, the geometical view becomes dominant (which also shapes how SR is introduced in many cases).


Modern "common phrasing" of SR in introductory books (that don't take the symmetry approach I mentioned at the begginning which theorists seem to consider the "modern view") is usually along the line of:
First postulate - Special principle of relativity - The laws of physics are the same in all inertial frames of reference.
Second postulate - Invariance of c - The speed of light in a vacuum according to an inertial frame of reference is a universal constant, c.




So, my large gaps in the history asside, to help show some of the evolution of ideas, let me point out some questions that throughout this history I am not sure everyone would have agreed on depending on their "version" of SR:

In the beginning notice that Einstein doesn't say "all laws of physics" will be the same for all inertial frames. He specifically mentions the laws of electrodynamics. So if they had data on the life times of energetic muons appearing longer to us (due to the weak force - not handled by electrodynamics), would physicists of that time considered that support of relativity? Or would they consider that a trivial extension of relativity?

In the beginning, Einstein seems to be restricting himself to "all frames of reference for which the equations of mechanics hold good". SR shows the second and third laws of Newton need some adjusting, so is he suggesting we define an inertial frame by Newton's first law alone (frames in which unperturbed objects move in straight lines with constant velocity)? Clearly not (although I have seen people use this definition of inertial frame), for this does not restrict how we synchronize our clocks, so I can easily get frames where the speed of light is not a universal constant.

Those last two are just warm ups to show that our understanding of several terms have evolved some over the years. But not much physical content is there. Let's look at something that people probably would really have varied their opinions on over the years.

Let's say we have an inertial coordinate system (unprimed) and I define a new coordinate system (primed) as follows:
t' = t + A
x' = x
y' = y
z' = z

Where A is a constant. I sure everyone would agree that this is still an inertial coordinate system. So the "common phrasing" above seems to require the laws of physics be invarient to this transformation. But time translational symmetry gives energy conservation.

Would you say special relativity requires energy conservation?

Similarly, I can do this with spatial translation. Would you say special relativity requires momentum conservation?

Now consider this one:
t' = t
x' = -x
y' = -y
z' = -z

I very much believe phycists of 1905 would consider this an inertial coordinate system. Would they believe special relativity requires parity symmetry in the laws of physics?

While it was after Einstein's time, in 1957 the laws of physics were experimentally shown to exhibit parity violation. No one currently worries if this violates special relativity (I don't know if anyone ever did, does anyone have some historical info?). Considering special relativity the way the theorists worded it above, there is no worry as special relativity is merely interested in Lorentz symmetry.

But why? And how? did this come to be?
How do people decide what the "essence" of the theory is, so that it evolves without anyone considering it changing?

What really amazes me, is that with the current reduction to just requiring Lorentz symmetry, this understanding of SR could have been true even if light did use a medium. Just as sound propagation in a medium fits fine in SR as the lagrangian still has lorentz symmetry. Although if light did use a medium it would have taken us a lot longer to discover SR, so we should be grateful it doesn't.


In my opinion, it seems the theory has evolved quite a bit. It makes it more difficult when people seem to not acknowledge this and there is no precise and definitive current phrasing.


I am very interested in hearing your views on this (and especially corrections and additions to the historical context here).

 

[JesseM]

The beginning 1905 -
(English translation, for I can't read german.)http://www.fourmilab.ch/etexts/einstein/specrel/www/

Two postulates
1) ``Principle of Relativity'' - the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good
2) light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting bodyInertial frames seem to be implied for #2 (maybe my bias is causing history revisionism, can someone comment on this), so I will assume he originally meant it as such.

...

In the beginning notice that Einstein doesn't say "all laws of physics" will be the same for all inertial frames. He specifically mentions the laws of electrodynamics. So if they had data on the life times of energetic muons appearing longer to us (due to the weak force - not handled by electrodynamics), would physicists of that time considered that support of relativity? Or would they consider that a trivial extension of relativity?

Interesting post, but just as a note on the comments quoted above, in the 1905 paper you linked to, in section 2 Einstein defines the two postulates as:

1. The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of co-ordinates in uniform translatory motion.

2. Any ray of light moves in the "stationary" system of co-ordinates with the determined velocity c, whether the ray be emitted by a stationary or by a moving body. Hence velocity = light path/time interval where time interval is to be taken in the sense of the definition in § 1.

So I don't see why you say that he was only talking about electrodynamics in the first postulate, he makes a general statement about all "laws by which the states of physical systems undergo change". Likewise, I don't see any ambiguity in the fact that the second postulate is specifically referring to inertial frames, since in section 1 he defines the "stationary" system as "a system of co-ordinates in which the equations of Newtonian mechanics hold good".

 

[JustinLevy]

So I don't see why you say that he was only talking about electrodynamics in the first postulate, he makes a general statement about all "laws by which the states of physical systems undergo change".

Good point. I forgot he reworded the principles again later in the paper. (I quoted those from the beginning of the paper.)

Interesting that in the first time he states it, the first postulate seems to relate all inertial frames, whereas the second time he states it, it only relates frames moving relative to each other (supporting better the concept of SR dealing "only" with lorentz symmetry as suggested by the phrasing by the theorists).

Thanks for the comments.

 

[paw]

I would venture that the evolution in SR you refer to is more a change, or improvement if you prefer, in mathematical formalism than an actual change in the theory. That and an evolution in language.

 

[meopemuk]

Let's say we have an inertial coordinate system (unprimed) and I define a new coordinate system (primed) as follows:
t' = t + A
x' = x
y' = y
z' = z

Where A is a constant. I sure everyone would agree that this is still an inertial coordinate system. So the "common phrasing" above seems to require the laws of physics be invarient to this transformation. But time translational symmetry gives energy conservation.

Would you say special relativity requires energy conservation?

Yes. The invariance of physical laws with respect to time translations implies the conservation of the total energy of an isolated system.

Similarly, I can do this with spatial translation. Would you say special relativity requires momentum conservation?

Yes. The invariance of physical laws with respect to space translations implies the conservation of the total momentum

Now consider this one:
t' = t
x' = -x
y' = -y
z' = -z

I very much believe phycists of 1905 would consider this an inertial coordinate system. Would they believe special relativity requires parity symmetry in the laws of physics?

I am not sure what physicists in 1905 would say. In my view, this transformation cannot be realized in practice: you cannot physically prepare a mirror image of a reference frame. Therefore, inversion is not an inertial transformation, strictly speaking. Therefore, the principle of relativity cannot guarantee the invariance of all physical laws with respect to inversion, and parity conservation cannot be rigorously proven. This is in agreement with experiment.

Eugene.

 

[rbj]

This view automatically gives an upper speed limit in an inertial coordinate system which all such systems agree upon (the fact that it happens to be the speed of light in this case is merely a coincidence based on light being massless,

the principles of SR do not require a notion of the particle nature of waves (such as E&M waves of which visible light is a subset) at all. the fact that photons have zero rest mass comes as a consequence that they, as particles, are believed to move at a speed of c for any reference frame. "the fact that it happens to be the speed of light" is because light happens to be EM radiation and this upper speed limit applies to all things ostensibly instantaneous. as dealt with in GR, it also applies to gravity.

if you're holding a charge (or mass) and I'm holding another charge (or another mass) and i give my charge a big jerk, you, holding your charge, are going to feel it. the Coulomb law (or Newton's law of gravitation) implies that such a disturbance would be felt instantaneously, as if the speed of propagation of the EM interaction was infinite. but it's not. no interaction propagates at infinite speed or any speed exceeding c. the effect of the big jerk cannot be felt by you sooner than distance/c units of time (as observed by a third party that is equi-distant from us both). i presume the same could be said if the action was gravity or any other interaction that has effect that spans empty space.

it seems in this view of relativity as just a symmetry requirement, that it doesn't require light to be this speed. But experiment shows that it is.)

no, light happens to be that speed because light is an EM wave and the speed of the effect of the EM interaction (as well as gravity, etc) is c. what experiments told us, back in the days when the meter was the distance between two scratches on a platinum-iridium bar, was that the speed of this propagation (when the interaction happened to be E&M at visible light frequencies) is somewhere around 300,000 km/s. but that speed is, itself, actually part of the ruler of nature. it is what it is.

 

[Anonym]

I realize that the understanding of physics and the perferred description or mathematical representation of our understanding evolves over time. So I have a feeling that the answer to my question is strongly a mix of history and physics. Please, when at all possible add some of the historical context into your answers to help flesh out my understanding.

The answer to your questions is a book and strongly a mix of history, physics and personal interpretations and knowledge. Fortunately, the understanding is the collective result of all that. I will try to discuss only one particular example:”Principle of Relativity”.

Let us start from interpretations. You read English translation, for you can't read German. I read Russian translation, for the same reason. You wrote:

``Principle of Relativity'' - the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good.

“The equations of mechanics hold good.” – A. Einstein is not able to express himself that way. In addition, my speller requires “well”. The Russian translation is completely different and fit the later definitions. Now, you enter your interpretation when you call it ``Principle of Relativity''. It is only introduction into what is going on. When the math-ph said: “These two principles we define as follows”, the completely different level of wording will follow as JesseM explained to you.

Now let us consider physics. The physical principles are the top axiomatical level and the evolution of the principle of relativity occurred through the following line: N. Copernicus, G. Galileo, A. Einstein, E.P. Wigner and C.N. Yang. Originally the principle of relativity was introduced by N. Copernicus in form very close to the A. Einstein general relativity. The concrete mathematical realization was done by G. Galileo. The evolution went from the Newtonian mechanics (classical analysis and 3D+1T) to the Maxwell ED (vector analysis and Minkowski 4D) to the A. Einstein GR (tensor analysis and curved space-time). QM story is in progress (functional analysis, group theory, “blurred” space-time and the relative phases).

Regards, Dany.

 

[JustinLevy]

Yes. The invariance of physical laws with respect to time translations implies the conservation of the total energy of an isolated system.

...

Yes. The invariance of physical laws with respect to space translations implies the conservation of the total momentum.

You missed the point of those lines (I appologize if my writing isn't very clear). Yes of course those symmetries lead to those conservation laws, that was not the question.

Since one would consider those inertial coordinate systems as well, would you then say that special relativity requires momentum and energy conservation?

[regarding the parity transformation]
In my view, this transformation cannot be realized in practice: you cannot physically prepare a mirror image of a reference frame. Therefore, inversion is not an inertial transformation, strictly speaking.

"cannot physically prepare a mirror image of a reference frame"? That phrase doesn't seem to make sense. A coordinate system is not a physical entity.

A coordinate system is merely a way of labelling spacetime events. So I made a new labelling based on a previous labelling. You cannot deny this labelling exists.

The question is then two-fold. Would you consider this new labelling an inertial coordinate system? If not, I'm very much interested in your definition of an inertial coordinate system for I don't see what you are using to distinguish one as an inertial frame and one as not.

Second, if you agree that new labelling is an inertial coordinate system, then would you say SR requires the laws of physics to look the same in this frame (ie. be invarient to a parity transformation)?

Do you think your answer is being affected by the knowledge that there are parity violations (which those back in the early 1900's didn't know)?

------------------------------

rbj:
I think you may have misunderstood what I was asking and stating (or at least the context in which I was stating that).

I understand relativity limits the speed of propagation of information (interactions or otherwise). The point is that the modern view seems to be that special relativity is merely stating the physical laws have lorentz symmetry. Stated this way, if light didn't travel at the "speed limit c", but E+M was still described with a lorentz symmetry (which can be done if light has mass), that would be fine with this interpretation of relativity.

To repeat, it seems that (with the Lorentz symmetry only interpretation) "it doesn't require light to be this speed. But experiment shows that it is."

------------------------------

I would venture that the evolution in SR you refer to is more a change, or improvement if you prefer, in mathematical formalism than an actual change in the theory. That and an evolution in language.

So if you feel it has just been made more rigorous, then what would you venture is the modern statement of SR? And given that, what is your interpretation on what SR says (or doesn't say) about the symmetries relating inertial coordinate systems other than lorentz transformations?

-------------------------------

Anonym:
I may be misunderstanding you. Are you saying the translation should be more along the lines of "valid for all frames of reference for which the equations of mechanics hold well"?

Regardless of the wording in the original document, how would you currently word it... What do you feel is the modern statement of the principles of SR?


Thank you everyone for your contributions so far.

 

[pervect]

My $.02.

A lot of this is really ancient history. Newton's (1643-1728) formulation of physics is still taught in high school, but at the graduate level the Lagrangian approach (1788, according to the Wikipedia) and the Hamiltonian approach (1833, ibid) has more or less superseded Newton's original approach. Personally don't know exactly when the Lagrangian approach gained favor, though.

If you're comfortable with the Lagrangian formulation being exactly equivalent to Newton's formulation (with some particular Lagrangian), you shouldn't mind much if Einstein's original work has been reinterpreted in a Lagrangian formulation (with a different Lagrangian than Newton's).

I would say that special relativity plus a statement that physics can be described by an action principle does imply that special relativity has a conserved momentum and energy, via Noether's theorem. I think for clarity is best to include an explicit mention of both relativity and an action principle though, rather than to say that "relativity" must have a conserved energy. Technically, an action principle is more closely related to the Hamiltonian formulation than the Lagrangian formulation. I haven't really thought much about attempting to keep relativity while throwing out the action principle enough to know if it's even possible, if that's what you're asking about. I don't see that as being very likely to happen.

 

[JesseM]

"cannot physically prepare a mirror image of a reference frame"? That phrase doesn't seem to make sense. A coordinate system is not a physical entity.

A coordinate system is merely a way of labelling spacetime events. So I made a new labelling based on a previous labelling. You cannot deny this labelling exists.

The question is then two-fold. Would you consider this new labelling an inertial coordinate system? If not, I'm very much interested in your definition of an inertial coordinate system for I don't see what you are using to distinguish one as an inertial frame and one as not.

I'd say it definitely is an inertial coordinate system--you can certainly define it physically in terms of the measurements on a system of inertial rulers and synchronized clocks, just as Einstein did. If you have an existing inertial coordinate system defined this way, you just need to rotate each ruler about the origin by 180 degrees.

Second, if you agree that new labelling is an inertial coordinate system, then would you say SR requires the laws of physics to look the same in this frame (ie. be invarient to a parity transformation)?

Do you think your answer is being affected by the knowledge that there are parity violations (which those back in the early 1900's didn't know)?

This is an interesting point. Does the most general form of the Lorentz transformation (ie one which doesn't assume the axes of different coordinate systems are parallel to one another) allow for such a transformation between parity-reversed coordinate systems, or does it only allow one set of axes to be a rotated version of the other? If the former, why don't parity violations imply that the laws of physics are not actually Lorentz-symmetric? Anyone know the answer?

 

[rbj]

rbj:
I think you may have misunderstood what I was asking and stating (or at least the context in which I was stating that).

I understand relativity limits the speed of propagation of information (interactions or otherwise). The point is that the modern view seems to be that special relativity is merely stating the physical laws have lorentz symmetry. Stated this way, if light didn't travel at the "speed limit c", but E+M was still described with a lorentz symmetry (which can be done if light has mass), that would be fine with this interpretation of relativity.

To repeat, it seems that (with the Lorentz symmetry only interpretation) "it doesn't require light to be this speed. But experiment shows that it is."

what experiment are you referring to?

 

[Anonym]

My $.02.

Cheap.

A lot of this is really ancient history.

I don’t agree with you. The ancient period usually related to Greeks and the modern physics started with N. Copernicus.

I would say that special relativity plus a statement that physics can be described by an action principle does imply that special relativity has a conserved momentum and energy, via Noether's theorem.

I would say that the statement that there exists set of laws that the physical system obey imply the conservation of energy, momentum and angular momentum via Noether's theorem. The special relativity compare with Galileo introduce causality.

Regards, Dany.

 

[meopemuk]

I would say that special relativity plus a statement that physics can be described by an action principle does imply that special relativity has a conserved momentum and energy, via Noether's theorem. I think for clarity is best to include an explicit mention of both relativity and an action principle though, rather than to say that "relativity" must have a conserved energy.

This is one way to think about conserved quantities. But there is another way which does not involve the assumption about the validity of the action principle. In Wigner's approach to relativistic quantum mechanics, the principle of relativity is realized by constructing an unitary representation of the Poincare group in the Hilbert space of the system. Then operators of total observables (total energy H, total momentum P, and total angular momentum J) are represented by Hermitian generators of this representation, and conservation of H, P, and J follows simply from zero commutators of these operators with H in the Poincare Lie algebra.

"cannot physically prepare a mirror image of a reference frame"? That phrase doesn't seem to make sense. A coordinate system is not a physical entity.

A coordinate system is merely a way of labelling spacetime events. So I made a new labelling based on a previous labelling. You cannot deny this labelling exists.

In my understanding a reference frame is a physical inertial observer with her measuring devices. The goal of physics is to describe measurements performed by different observers and their connections to each other. From this point of view, preparing a "mirror image of a reference frame" is a non-trivial thing.

Simple relabeling of spacetime events cannot change anything in physics. Physical predictions cannot depend on the choice of the coordinate system, e.g., cartesian vs. spherical.

Eugene.

 

[JesseM]

In my understanding a reference frame is a physical inertial observer with her measuring devices. The goal of physics is to describe measurements performed by different observers and their connections to each other. From this point of view, preparing a "mirror image of a reference frame" is a non-trivial thing.

But what is so difficult about simply rotating all your rulers 180 degrees around the origin, so that the 1 meter mark on a given ruler is now where the -1 meter mark was before the rotation and so forth?

Simple relabeling of spacetime events cannot change anything in physics. Physical predictions cannot depend on the choice of the coordinate system, e.g., cartesian vs. spherical.

Relabeling coordinate systems cannot change physical predictions, but it can change the equations to express the same laws of physics in the new coordinate system. Many symmetries in physics involve statements about the equations being unchanged in different coordinate systems related by a certain transformation, like translation invariance or lorentz invariance. So, it's interesting to ask whether Lorentz symmetry implies parity symmetry, or whether the most general form of the Lorentz transformation does not allow you to flip each individual ruler in this way (perhaps only rigid rotations of all three rulers at once are allowed).

 

[meopemuk]

But what is so difficult about simply rotating all your rulers 180 degrees around the origin, so that the 1 meter mark on a given ruler is now where the -1 meter mark was before the rotation and so forth?

I think we need to distinguish two different kinds of transformations that can be applied to inertial observers or reference frames or laboratories. I will call them inertial transformations and reparameterizations.

Inertial transformations Suppose that we have one laboratory. This laboratory has everything necessary for perfoming physical measurements: It has an origin and three mutually perpendicular axes x, y, and z erected from the origin. Let us agree that the axes form a right-handed system. The laboratory also has a standard yardstick, so observer in the laboratory can measure cartesian coordinates of points in space by projecting these points on the axes. He also have a standard (e.g., atomic) clock for measuring time, and all kinds of devices for measuring particle momenta, energies, spins, etc.

We haven't defined yet where in the Universe this laboratory is located, or in what century, or what are the orientations of its coordinate axes, or what is the speed of the laboratory. So, we can imagine identical copies of the laboratory situated in different places, at different times, moving with respect to each other, or having different orientations. We can also say that these laboratories are connected to each other by inertial transformations - space and time translations, rotations, and boosts.

The most fundamental law of physics - the principle of relativity - says that identical experiments performed in each of these laboratories would yield identical results. This is a deep and non-trivial physical statement which has numerous important consequences for all physics.

You can notice that I didn't include inversion in the list of inertial transformations. It is easy to see that "inverted" laboratory is not exactly equivalent to the set of laboratories listed above. Its coordinate axes form a "left-handed" system, not the "right-handed" one. Inversion is quite different from translations, rotations, and boosts described above. Inversion would require making a mirror image of all measuring devices in the laboratory, all atoms and nuclei from which these devices are made. This is rather non-trivial transformation, and the principle of relativity doesn't guarantee that all laws of physics will be exactly the same in the "inverted" laboratory. And we know that mirror symmetry is not an exact symmetry of nature.

Reparameterizations Instead of applying real physical inertial transformations to laboratories we can decide to do something more trivial. For example, in a given laboratory we can redefine the unit of length. Or we can decide to use spherical coordinates instead of cartesian ones. Or we can use a clock with different rate. By doing this, we haven't actually done any physical transformation. We simply relabeled or reparameterized space-time labels of events. This would change equations by which we describe physical phenomena. For example, in the spherical coordinate system the angle $\theta$ of a free-moving particle will have a non-linear dependence on time. However, this reparameterization has no effect on physics. This is a purely mathematical change of description. It has nothing to do with the principle of relativity.

One example of such a reparameterization is the change of signs of all measured coordinates, which you proposed. I hope I made it clear that this sign change (or, as you suggested rotation of all axes by 180 degrees) is much more trivial that physical "reflection" of the laboratory.

Relabeling coordinate systems cannot change physical predictions, but it can change the equations to express the same laws of physics in the new coordinate system. Many symmetries in physics involve statements about the equations being unchanged in different coordinate systems related by a certain transformation, like translation invariance or lorentz invariance.

Translation invariance and Lorentz invariance are not invariances with respect to trivial relabelings of coordinates. They are invariances with respect to real physical "inertial transformations" of laboratories - shifts or boosts.

So, it's interesting to ask whether Lorentz symmetry implies parity symmetry, or whether the most general form of the Lorentz transformation does not allow you to flip each individual ruler in this way (perhaps only rigid rotations of all three rulers at once are allowed).

As I tried to explain above, flipping individual rulers is an example of a trivial reparameterization. It would change the mathematical form of equations, but it wouldn't have any physical significance. Creating a physical "mirror image" of the laboratory (which would also flip the rulers as a byproduct) is a non-trivial transformation, which has not been done in practice ever. The principle of relativity does not apply in this case. However, this principle does apply in the case of inertial transformation called "rotation", i.e., when all three rulers are rotated at once.

Eugene.

 

[JesseM]

You can notice that I didn't include inversion in the list of inertial transformations. It is easy to see that "inverted" laboratory is not exactly equivalent to the set of laboratories listed above. Its coordinate axes form a "left-handed" system, not the "right-handed" one. Inversion is quite different from translations, rotations, and boosts described above. Inversion would require making a mirror image of all measuring devices in the laboratory, all atoms and nuclei from which these devices are made.

Why would it require that, though? It is not really a requirement that two laboratories moving inertially relative to one another be exact duplicates down to the last atom, only that they use the same general type of equipment to run the same general types of experiments.

Reparameterizations Instead of applying real physical inertial transformations to laboratories we can decide to do something more trivial. For example, in a given laboratory we can redefine the unit of length. Or we can decide to use spherical coordinates instead of cartesian ones. Or we can use a clock with different rate. By doing this, we haven't actually done any physical transformation. We simply relabeled or reparameterized space-time labels of events. This would change equations by which we describe physical phenomena. For example, in the spherical coordinate system the angle $\theta$ of a free-moving particle will have a non-linear dependence on time. However, this reparameterization has no effect on physics. This is a purely mathematical change of description. It has nothing to do with the principle of relativity.

I disagree it has nothing to do with the principle of relativity--after all, the fact that the equations of the laws of physics remain unchanged when you reparametrize your cartesian coordinate system using the Lorentz transformation logically implies the notion that the the outcomes of experiments in two laboratories moving inertially relative to one another will be the same (whether it implies that the outcomes will be the same in a mirror-reversed laboratory depends on whether the Lorentz transformation relates a given inertial coordinate system to its mirror-reversed image--I imagine it doesn't, though I'm not sure). And in terms of mathematical physics, it's easier to express precisely what it means to say the laws of physics are unchanged by a certain coordinate transformation than it is to give the conceptual explanation of an "inertial transformation" like yours--I think if you look in any physics textbook that rigorously defines a symmetry like Lorentz symmetry or translation invariance, they would define it in this coordinate-based way, although you're welcome to show I'm wrong about that if you know of any counter-examples.

Translation invariance and Lorentz invariance are not invariances with respect to trivial relabelings of coordinates. They are invariances with respect to real physical "inertial transformations" of laboratories - shifts or boosts.

I disagree that there is anything "trivial" about this, since the physical consequences of such a symmetry are exactly the ones you label as "invariances with respect to real physical inertial transformations" (do you think it would be possible to imagine a universe where the equations of the laws of physics were invariant under the Lorentz transformation but the outcomes of experiments in different labs moving inertially with respect to each other would not be identical?) And like I said, I think textbooks would define these invariances in coordinate-based ways since there is no need for verbal/conceptual arguments in this case.

As I tried to explain above, flipping individual rulers is an example of a trivial reparameterization.

Why is the physical act of flipping a ruler any more trivial than the physical act of rotating all the rulers simultaneously (rotation invariance) or moving them to a different location (translation invariance) or changing their velocity (Lorentz invariance in relativity)?

Creating a physical "mirror image" of the laboratory (which would also flip the rulers as a byproduct) is a non-trivial transformation, which has not been done in practice ever.

Again this argument doesn't make sense to me, since creating an exact duplicate of a laboratory at the atomic scale but with a different velocity also has not been done in practice ever, but that does not stop us from talking about Lorentz invariance. Also, if the laws of physics did obey parity-flipping invariance, are you suggesting it would be impossible to actually verify this experimentally without creating such a perfect mirror image of a laboratory? The fact that we could write down the equations of the laws of physics as determined experimentally in one lab, and see that mathematically they would not change under a parity-reversing coordinate transformation, would not be a sufficient demonstration?

 

[Anonym]

Again this argument doesn't make sense to me, since creating an exact duplicate of a laboratory at the atomic scale but with a different velocity also has not been done in practice ever, but that does not stop us from talking about Lorentz invariance.

Suppose that we have one laboratory. This laboratory has everything necessary for perfoming physical measurements: It has an origin and three mutually perpendicular axes x, y, and z erected from the origin…

In Wigner's approach to relativistic quantum mechanics, the principle of relativity is realized by constructing an unitary representation of the Poincare group in the Hilbert space of the system. Then operators of total observables (total energy H, total momentum P, and total angular momentum J) are represented by Hermitian generators of this representation, and conservation of H, P, and J follows simply from zero commutators of these operators with H in the Poincare Lie algebra.

How we define the laboratory (inertial ref frame) at the atomic scale? I mean we need an origin, three mutually perpendicular axes x, y, and z erected from the origin and the relative velocity. QM doesn’t allow doing that.

Regards, Dany.

 

[JesseM]

How we define the laboratory (inertial ref frame) at the atomic scale?

I'm not saying we do, I was responding to meopemuk's statement "Inversion would require making a mirror image of all measuring devices in the laboratory, all atoms and nuclei from which these devices are made."

 

[meopemuk]

Why would it require that, though? It is not really a requirement that two laboratories moving inertially relative to one another be exact duplicates down to the last atom, only that they use the same general type of equipment to run the same general types of experiments.

Two laboratories moving relative to one another (e.g, on Earth and on a spaceship) can be made sufficiently identical, if not down to the last atom, then to the precision acceptable for experiments. It has been demonstrated empirically that for all practical purposes these laboratories are, indeed, equivalent. They have the same laws of physics.

Nobody has ever prepared a mirror image of a laboratory. So, it is not obvious that the principle of relativity can be applied in this case. Most likely, it is not applicable, because we know that the parity is not conserved in weak interactions. Therefore, inversion should not be a member of the group of inertial transformations. Of course, if one is not interested in the physics of weak interactions, then she can approximately enlarge the Poincare group by space and time inversions. But this would be an approximation.

I disagree it has nothing to do with the principle of relativity--after all, the fact that the equations of the laws of physics remain unchanged when you reparametrize your cartesian coordinate system using the Lorentz transformation logically implies the notion that the the outcomes of experiments in two laboratories moving inertially relative to one another will be the same (whether it implies that the outcomes will be the same in a mirror-reversed laboratory depends on whether the Lorentz transformation relates a given inertial coordinate system to its mirror-reversed image--I imagine it doesn't, though I'm not sure). And in terms of mathematical physics, it's easier to express precisely what it means to say the laws of physics are unchanged by a certain coordinate transformation than it is to give the conceptual explanation of an "inertial transformation" like yours--I think if you look in any physics textbook that rigorously defines a symmetry like Lorentz symmetry or translation invariance, they would define it in this coordinate-based way, although you're welcome to show I'm wrong about that if you know of any counter-examples.

I don't think that in the general case boost transfromations of observables - such as space-time coordinates of particles - can be reduced to simple reparameterization by means of Minkowski diagrams.

Let us consider the following example. Suppose that we know the state of the particle in a reference frame O. We know particle position $\mathbf{r}$ , velocity $\mathbf{v}$, momentum, spin orientation, etc. as measured by this observer. What would be results of measurements of the same observables by observer O' displaced in space with respect to O? The answer is simple

$$\mathbf{r}' = \mathbf{r} - \mathbf{a}$$

where $\mathbf{a}$ is the amount of displacement. Other observables remain exactly the same as in O. It is easy to find particle observables in the rotated reference frame as well.

Now, let's ask a less trivial question. What are the values of observables in the reference frame displaced in time with respect to O? If the particle is free than the answer is simple: velocity, momentum, and spin orientation will be the same (they are conserved quantities), and particle position can be found by rather universal formula

$$\mathbf{r}' = \mathbf{r} + \mathbf{v} t$$

But what if this particle is a part of an interacting multiparticle system? Then the result of time translation cannot be expressed by any universal formula. We need to know positions and velocities of other particles in the system and interactions between them, if we want to know how observables of our particle transform with respect to time translations. This example shows that transformations between inertial reference frames can be rather non-trivial.

The next question: what can we say about boosts? Are boost transformations of particle observables simple, universal, and interaction-independent (like space translations and rotations), or they are non-trivial and interaction-dependent (similar to time translations)? What do you think?

Why is the physical act of flipping a ruler any more trivial than the physical act of rotating all the rulers simultaneously (rotation invariance) or moving them to a different location (translation invariance) or changing their velocity (Lorentz invariance in relativity)? Again this argument doesn't make sense to me, since creating an exact duplicate of a laboratory at the atomic scale but with a different velocity also has not been done in practice ever, but that does not stop us from talking about Lorentz invariance.

The laboratories obtained from each other by inertial transformations are identical. If you move the observer from one laboratory to the other he wouldn't notice the change. However, he would immediately notice the switch if he was placed in the laboratory with flipped rulers, because the rulers no longer formed the right-handed system.

Also, if the laws of physics did obey parity-flipping invariance, are you suggesting it would be impossible to actually verify this experimentally without creating such a perfect mirror image of a laboratory? The fact that we could write down the equations of the laws of physics as determined experimentally in one lab, and see that mathematically they would not change under a parity-reversing coordinate transformation, would not be a sufficient demonstration?

The absence of the inversion invariance has been proven experimentally. In order to do that, physicists assumed that laws of physics wouldn't change in the mirror-image laboratory. From this assumption they derived certain predictions about nuclear decays. These predictions were found to be violated in experiments. So, it was concluded that there is no exact inversion invariance, even though nobody has ever built a mirror-image laboratory.

Eugene.

 

[Anonym]

I'm not saying we do, I was responding to meopemuk's statement "Inversion would require making a mirror image of all measuring devices in the laboratory, all atoms and nuclei from which these devices are made."

I follow your discussion with Meopemuk and I understand your arguments. But I ask you both the additional question: why and if it is necessary the laboratory to obey laws of Classical Physics? Do you have an idea how the def of the inertial frame may be extended using Wigner's approach?

Regards, Dany.

 

[JesseM]

Two laboratories moving relative to one another (e.g, on Earth and on a spaceship) can be made sufficiently identical, if not down to the last atom, then to the precision acceptable for experiments. It has been demonstrated empirically that for all practical purposes these laboratories are, indeed, equivalent. They have the same laws of physics.

But this is all rather vague and subjective...what does "sufficiently identical" mean? I'd think, for example, that if one lab was using rulers made of wood to establish a coordinate system while another was using rulers made of metal, this would be an irrelevant difference...but if the material constitution of the rulers is irrelevant, then it shouldn't matter whether or not the atoms of rulers used to establish a parity-reversed coordinate system are themselves parity-reversed.

Nobody has ever prepared a mirror image of a laboratory. So, it is not obvious that the principle of relativity can be applied in this case.

I'd say that the "principle of relativity" is not sufficiently precise when defined in verbal terms, this is why one needs a more precise mathematical definition like the statement that the laws of physics are invariant under a poincare transform. But you seem to disagree that coordinate-based definitions are acceptable. If that's still the case, it would be helpful if you'd answer my earlier question--do you agree that the physical notion that the results of experiments are the same in sealed labs moving inertially relative to one another follows as a logical consequence of the statement that if you determine the equations for the laws of physics in a coordinate system defined by measurements on rulers and clocks of an inertial observer, then these equations must be unchanged by a Lorentz or Poincare transformation?

I don't think that in the general case boost transfromations of observables - such as space-time coordinates of particles - can be reduced to simple reparameterization by means of Minkowski diagrams.

Let us consider the following example. Suppose that we know the state of the particle in a reference frame O. We know particle position $\mathbf{r}$ , velocity $\mathbf{v}$, momentum, spin orientation, etc. as measured by this observer. What would be results of measurements of the same observables by observer O' displaced in space with respect to O? The answer is simple

$$\mathbf{r}' = \mathbf{r} - \mathbf{a}$$

where $\mathbf{a}$ is the amount of displacement. Other observables remain exactly the same as in O. It is easy to find particle observables in the rotated reference frame as well.

Now, let's ask a less trivial question. What are the values of observables in the reference frame displaced in time with respect to O? If the particle is free than the answer is simple: velocity, momentum, and spin orientation will be the same (they are conserved quantities), and particle position can be found by rather universal formula

$$\mathbf{r}' = \mathbf{r} + \mathbf{v} t$$

But what if this particle is a part of an interacting multiparticle system? Then the result of time translation cannot be expressed by any universal formula. We need to know positions and velocities of other particles in the system and interactions between them, if we want to know how observables of our particle transform with respect to time translations. This example shows that transformations between inertial reference frames can be rather non-trivial.

I don't understand your point--which specific point of mine are you disagreeing with here? Are you saying we can't determine whether a given set of laws are invariant under a coordinate system using mathematics alone? Why do you see it as problematic that you need the complete initial conditions for a system in order to determine its future behavior? For any dynamical system, to calculate its behavior you need two things--a complete set of initial conditions, and a general set of laws of physics which can generate the dynamics for an arbitrary set of initial conditions. Once we have found the general laws in this coordinate system by experiments with lots of different possible initial conditions, then to check whether the laws of physics are invariant under a coordinate transformation we just apply the coordinate transformation to the equations of the general laws themselves (a purely mathematical test), you don't even need to worry about any particular set of initial conditions. Of course if the equations are unchanged by the transformation, then if you transform the initial conditions of a particular physical system into the new coordinate system, you're guaranteed to get the same physical predictions about its future behavior when you apply the same general laws to these new initial conditions.

The next question: what can we say about boosts? Are boost transformations of particle observables simple, universal, and interaction-independent (like space translations and rotations), or they are non-trivial and interaction-dependent (similar to time translations)? What do you think?

I'm totally lost. I don't understand what this issue of "interaction-dependence" is supposed to relate to, and I also don't understand what it means for a coordinate transformation to be interaction-dependent or interaction-independent--surely it is the laws of physics themselves which are one or the other? And when you seemed to say earlier in your post that spatial translations are just a coordinate matter but time translations are not, I also don't get this--does this somehow relate to your point about interaction-dependence? Are you saying that if we know the position and time of an event in one coordinate system, somehow we wouldn't automatically know the position and time of the same event in a time-translated coordinate system?

The absence of the inversion invariance has been proven experimentally. In order to do that, physicists assumed that laws of physics wouldn't change in the mirror-image laboratory. From this assumption they derived certain predictions about nuclear decays.

But how can you explain how they "derived" these predictions, if according to you the only way to test predictions about parity symmetry would be to actually build a laboratory out of parity-flipped particles? Since I think that testing parity symmetry is simply a matter of seeing what equations particles obey, then seeing if the equations are unchanged by a purely mathematical transformation, of course it doesn't surprise me that they could find particles behaving in a way that would not be unchanged under this operation and thus show that parity symmetry is violated, but since you emphasize the need for a lab composed of parity-reversed matter I don't see how you can account for this.

 

[meopemuk]

I follow your discussion with Meopemuk and I understand your arguments. But I ask you both the additional question: why and if it is necessary the laboratory to obey laws of Classical Physics? Do you have an idea how the def of the inertial frame may be extended using Wigner's approach?

Regards, Dany.

This is a good question, but I am afraid I don't have a good answer. In the Poincare group approach it is assumed that the parameters of space-time translations, rotations and boosts of inertial laboratories can be determined simultaneously and precisely. So, it is tacitly assumed that inertial reference frames are classical objects.
This doesn't look consistent with fundamental quantum uncertainties.

Perhaps, one can try to avoid the inconsistency by noticing that the Poincare group only requires the translation, rotation, and boost vectors between frames to be exactly determined. This doesn't exactly mean that positions, velocities, and orientations of frames themselves are simultaneously measurable. So, maybe there is a loophole?

I've seen a couple of papers where people tried to do physics with "quantum frames". But these approaches didn't look convincing to me. I have no idea how one can do relativistic quantum physics without Poincare group.

Eugene.

 

[Anonym]

The absence of the inversion invariance has been proven experimentally. In order to do that, physicists assumed that laws of physics wouldn't change in the mirror-image laboratory. From this assumption they derived certain predictions about nuclear decays. These predictions were found to be violated in experiments. So, it was concluded that there is no exact inversion invariance, even though nobody has ever built a mirror-image laboratory.

To the best of my knowledge and memory, the absence of the inversion invariance has been observed experimentally using standard measurement apparatus (CI). You just count the number of electrons emitted in the beta decay and see that left is not equal to right. The quantitatively correct theoretical explanation was obtained assuming that the weak int. is described by V-A (M.Gell-Mann and R.P.Feynman).

Since I think that testing parity symmetry is simply a matter of seeing what equations particles obey.

The equations particles obey were written 11 years later by S.Weinberg.

Regards, Dany.

 

[JesseM]

The equations particles obey were written 11 years later by S.Weinberg.

The full laws of the electroweak force maybe, but I was thinking along the lines of specific "laws" for the specific situation where parity was violated, like a simple law telling you the probability that beta particles will be emitted in a particular direction by a cobalt nucleus with a given spin.

 

[meopemuk]

I'd say that the "principle of relativity" is not sufficiently precise when defined in verbal terms, this is why one needs a more precise mathematical definition like the statement that the laws of physics are invariant under a poincare transform. But you seem to disagree that coordinate-based definitions are acceptable. If that's still the case, it would be helpful if you'd answer my earlier question--do you agree that the physical notion that the results of experiments are the same in sealed labs moving inertially relative to one another follows as a logical consequence of the statement that if you determine the equations for the laws of physics in a coordinate system defined by measurements on rulers and clocks of an inertial observer, then these equations must be unchanged by a Lorentz or Poincare transformation?

My verbal description of the principle of relativity can be cast in a precise mathematical form. This has been done in two famous papers which form a foundation of relativistic quantum theory

E.P. Wigner "On unitary representations of the inhomogeneous Lorentz group", Ann. Math. 40 (1939), 149

P. A. M. Dirac, "Forms of relativistic dynamics", Rev. Mod. Phys. 21 (1949), 392.

The idea is that the principle of relativity + quantum mechanics requires that there is a unitary representation $U_g$ of the Poincare group in the Hilbert space of each physical system. If we want to find how certain observable $F$ transforms between two reference frames (related by the group element g), we need to calculate the following product

$$F' = U_g F U_g^{-1}$$...(1)

In some cases this transformation law coincides with the transformation law for 4-tensors (or 4-vectors, or 4-scalars). In other cases there are differences. I haven't seen any convincing independent derivation of Lorentz transformation formulas. All derivations that I know contain some unjustified assumptions. So, I tend to believe that transformation formulas obtained from (1) are more rigorous.

I'm totally lost. I don't understand what this issue of "interaction-dependence" is supposed to relate to, and I also don't understand what it means for a coordinate transformation to be interaction-dependent or interaction-independent--surely it is the laws of physics themselves which are one or the other? And when you seemed to say earlier in your post that spatial translations are just a coordinate matter but time translations are not, I also don't get this--does this somehow relate to your point about interaction-dependence? Are you saying that if we know the position and time of an event in one coordinate system, somehow we wouldn't automatically know the position and time of the same event in a time-translated coordinate system?

One major task of physics is to connect descriptions of the same physical system seen from different inertial reference frames. One example is dynamics: we know a full description of the system in one reference frame (by this I mean an "instantaneous" frame characterized by certain time position, orientation and velocity, i.e., all 10 parameters of the Poincare group) and we are trying to find the description (values of observables) in the time-translated reference frame. If we know positions, velocities, etc. of all particles in the system today, what will be the values of these observables tomorrow? You would probably agree with me that this is not a trivial task. We cannot directly invoke the time-translation-invariance to solve this problem. The time-translation invariance simply tells us that the relationship between particle observables today and tomorrow is exactly the same as for any days N and N+1 in the past or in the future. But to find this relationship we need to solve complicated equations of motion of interacting particles. So, describing time translations is not a trivial business.

Things are much easier with space translations and rotations. If we know particle observables in one (instantaneous) reference frame, it is easy to find the values of the same observables in the frame displaced in space or rotated. Not a big deal.

The important question is about boosts. Suppose we have a full set of observables for the physical system seen from one (instantaneous) frame. Can we calculate the observables seen by another moving reference frame? Can we simply apply universal Lorentz transformation formulas, which depend only on the relative velocity of the frames? If this is true, then we can regard boosts as universal ("geometrical") transformations, similar to space translations and rotations.

Or we need to solve complicated interaction-dependent equations, just as we did in the case of time translations? Then Lorentz transformation formulas will be just approximations. I believe that the latter answer is correct.

But how can you explain how they "derived" these predictions, if according to you the only way to test predictions about parity symmetry would be to actually build a laboratory out of parity-flipped particles? Since I think that testing parity symmetry is simply a matter of seeing what equations particles obey, then seeing if the equations are unchanged by a purely mathematical transformation, of course it doesn't surprise me that they could find particles behaving in a way that would not be unchanged under this operation and thus show that parity symmetry is violated, but since you emphasize the need for a lab composed of parity-reversed matter I don't see how you can account for this.

The preferable and direct way of the experimental check of the inversion invariance would be preparation of the mirror-imaged laboratory. We both know that this is impossible. However, there are other indirect ways to check the invariance, without actually leaving our university lab. This requires assuming the invariance and calculating consequences of this assumption for processes seen in the usual laboratory.

There is a full analogy with boost transformations. The direct way of checking the invariance of physical laws in moving reference frames is to actually have a fast-moving frame and to perform all kinds of experiments there. However, we don't have such fast moving frames in Earth conditions. Actually, most tests of special relativity were performed in usual stationary laboratories. Then why are we sure that the invariance with respect to boost is a valid symmetry? Because, by assuming this symmetry we derived all kinds of predictions which can be verified in stationary laboratories, e.g., in accelerators. And all these predictions were confirmed by experiment.

Eugene.

 

[Anonym]

The full laws of the electroweak force maybe, but I was thinking along the lines of specific "laws" for the specific situation where parity was violated, like a simple law telling you the probability that beta particles will be emitted in a particular direction by a cobalt nucleus with a given spin.

Our session is about the evolution of knowledge. All “laws” starting from C.N.Yang and T.D. Lee via M.Gell-Mann and R.P.Feynman to S.Weinberg were “cooked” to demonstrate the parity violation in weak interaction which is the experimental fact.

Regards, Dany.

 

[JesseM]

My verbal description of the principle of relativity can be cast in a precise mathematical form. This has been done in two famous papers which form a foundation of relativistic quantum theory

E.P. Wigner "On unitary representations of the inhomogeneous Lorentz group", Ann. Math. 40 (1939), 149

P. A. M. Dirac, "Forms of relativistic dynamics", Rev. Mod. Phys. 21 (1949), 392.

The idea is that the principle of relativity + quantum mechanics requires that there is a unitary representation $U_g$ of the Poincare group in the Hilbert space of each physical system. If we want to find how certain observable $F$ transforms between two reference frames (related by the group element g), we need to calculate the following product

$$F' = U_g F U_g^{-1}$$...(1)

In some cases this transformation law coincides with the transformation law for 4-tensors (or 4-vectors, or 4-scalars). In other cases there are differences. I haven't seen any convincing independent derivation of Lorentz transformation formulas. All derivations that I know contain some unjustified assumptions. So, I tend to believe that transformation formulas obtained from (1) are more rigorous.

While I don't follow the details, this looks like a purely mathematical requirement--to check whether the laws of physics satisfied this requirement, would it be necessary to actually experimentally determine the laws of physics in more than one frame?

One major task of physics is to connect descriptions of the same physical system seen from different inertial reference frames. One example is dynamics: we know a full description of the system in one reference frame (by this I mean an "instantaneous" frame characterized by certain time position, orientation and velocity, i.e., all 10 parameters of the Poincare group) and we are trying to find the description (values of observables) in the time-translated reference frame. If we know positions, velocities, etc. of all particles in the system today, what will be the values of these observables tomorrow? You would probably agree with me that this is not a trivial task. We cannot directly invoke the time-translation-invariance to solve this problem.

But time-translation, just like spatial translation, is just about relabeling events, it makes no sense to say that time translation somehow requires dynamical predictions while spatial translation doesn't. If you're only interested in the initial state of a system rather than its behavior over time, than a time translation would just involve changing the time coordinate of the set of events representing the initial state, while a spatial translation would involve changing the spatial coordinates of these events. On the other hand, if you're interested in finding the coordinates of events throughout spacetime in your coordinate system, rather than just the coordinates of the events representing the initial conditions, then you have just as much need to calculate the dynamical behavior from the initial state in the spatially translated coordinate system as you do in the time translated coordinate system.

The time-translation invariance simply tells us that the relationship between particle observables today and tomorrow is exactly the same as for any days N and N+1 in the past or in the future. But to find this relationship we need to solve complicated equations of motion of interacting particles. So, describing time translations is not a trivial business.

Things are much easier with space translations and rotations. If we know particle observables in one (instantaneous) reference frame, it is easy to find the values of the same observables in the frame displaced in space or rotated. Not a big deal.

This argument makes no sense to me, for the reasons above. Whether you need to calculate dynamics depends on whether you want to find the coordinates of events over an extended period of time or just the coordinates of the events corresponding to initial conditions, it has nothing at all to do with whether you're using a time-translated coordinate system or a spatially translated one.

There is a full analogy with boost transformations. The direct way of checking the invariance of physical laws in moving reference frames is to actually have a fast-moving frame and to perform all kinds of experiments there. However, we don't have such fast moving frames in Earth conditions. Actually, most tests of special relativity were performed in usual stationary laboratories. Then why are we sure that the invariance with respect to boost is a valid symmetry? Because, by assuming this symmetry we derived all kinds of predictions which can be verified in stationary laboratories, e.g., in accelerators. And all these predictions were confirmed by experiment.

But this "assuming symmetry" you talk about is just the notion of wanting to make sure that the laws of physics in our frame would remain the same under a coordinate transformation, which is what I've been saying all along. There is no need to actually have measuring-systems corresponding to each coordinate system, you can just do experiments in a single system and then checking for any type of symmetry like Lorentz invariance or translation invariance is a purely mathematical exercise.

 

[meopemuk]

But time-translation, just like spatial translation, is just about relabeling events, it makes no sense to say that time translation somehow requires dynamical predictions while spatial translation doesn't.

I think we are using different terminology, and this doesn't permit us to understand each other. Let me clarify my terminology a bit.


In physics we are interested in results of experiments. Each experiment consists of two important stages: 1. preparation of the system, 2. measurement. So, the experimental setup must have a preparation device and a measuring apparatus. For example, the preparation of the system may be: putting a raw egg in the boiling water. The measurement is taking the shell off the egg and seeing if it has been boiled.

We can associate inertial frames of reference with both the preparation device (which places the egg in the pot of water) and with the measuring apparatus (which takes the egg out of water and examines it). Normally these two frames of reference are simply time-shifted with respect to each other. The time difference of about 10 min. is required for the measuring apparatus to find a hard-boiled egg.

Now there is this important "principle of relativity" which says that if we apply an inertial transformation to the full laboratory (i.e., the preparation device + the measuring apparatus) then the result of our experiment will not change. The "boiling egg" experiment will produce exactly the same result if performed today or tomorrow. I believe that when you say that "time translation ... is just about relabeling events" you assume that both preparation device and measuring apparatus are time-translated. This case is covered by the principle of relativity. Nothing more to say.

The things become more interesting when you keep the "preparation frame" unchanged and apply time translations to the "measuring frame" only. For example, in the example above, the measurement occurred 10 min. after the preparation. What if we shift the measuring apparatus 5 min. back in time (this means that the egg is extracted from the pot and examined 5 min. after it was placed there)? What if we shift the measurement frame forward in time by 10 hours? Apparently, it IS possible to connect results of measurements in the 10-min-frame with results of measurements in the 5-min or 10-hour-frames. However, establishing this connection would require solving complicated dynamical equations for the egg's time evolution. This is the kind of non-trivial time evolution I was talking about. Sometimes it is said that time translations are dynamical inertial transformations.

By space displacements and rotations I meant displacements and rotations of the measuring frame only. (I am not interested in what happens if the entire laboratory (preparation device + measuring apparatus) is translated or rotated. I already know from the principle of relativity that these transformations will not have any effect.) It is known from experience that observers looking at the boiling egg from different points in space will find the egg's condition exactly the same. This is why one can say that space translations and rotations are kinematical transformation.

What about boosts of the measuring apparatus? Are they kinematical (like space translations and rotations) or dynamical (like time translations)?

Eugene.

 

[Anonym]

I may be misunderstanding you. Are you saying the translation should be more along the lines of "valid for all frames of reference for which the equations of mechanics hold well"?

No. I said that the statement “the equations of mechanics hold good” as well as “the equations of mechanics hold well” is poetry and not a mathematical physics. A.Einstein also is not W.Shakespeare. The Russian translation was edited by I.E.Tamm and Y.A. Smorodinski, the professionals in the mathematical physics.

What do you feel is the modern statement of the principles of SR?

I suggest reading two pages paper by F.J. Dyson “Feynman’s proof of the Maxwell equations”, Am. J. Phys., 58, 209 (1990).

Regards, Dany.

 

[Ich]

I don't know if you might gain any deeper insight in whatever you're debating, but maybe I can contribute.
Einstein definitely wrote "hold good" in the sense of "are valid", unless I miss some fine points in the English meaning.
However, the whole statement is only a subordinate clause in the German original.

Examples of this sort, together with the unsuccessful attempts to discover any motion of the Earth relatively to the ``light medium,'' suggest that the phenomena of electrodynamics as well as of mechanics possesses no properties corresponding to the idea of absolute rest. They suggest rather that, as has already been shown to the first order of small quantities, the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good.

is in fact only one sentence. Unfortunately, it is not easy to translate, partly because you dropped the dative case some thousand years ago, as I have learned from wikipedia. There is a slight difference in the first part, which I try to point out in a poor, but more literal translation:

Examples of this sort, together with the unsuccessful attempts to discover any motion of the Earth relatively to the ``light medium'', lead to the assumption that the notion of absolute rest not only in mechanics, but also in electrodynamics, does not correspond to any properties of the phenomena, but that rather for all frames of reference for which the equations of mechanics hold good, also the same laws of electrodynamics and optics hold good, as has already been shown for the quantities of first order.

The difference that is "Lost in Translation" is that he obviously refers to the Galilean Principle of Relativity of Newtonian mechanics, which he extends to electrodynamics. The whole thing about laws that hold good here or there is merely a restatement of that, not more. It obviously did not occur to (or did not bother) Einstein, that the reinvents said laws a few pages later. Really "you know what I mean" on his side, as I perceive it.

 

[bernhard.rothenstein]

special relativity history

I would venture that the evolution in SR you refer to is more a change, or improvement if you prefer, in mathematical formalism than an actual change in the theory. That and an evolution in language.

I think that the history of special relativity is related with the physical meaning of the speed of light: infinite, finite, finite and invariant.

 

[Anonym]

he obviously refers to the Galilean Principle of Relativity of Newtonian mechanics, which he extends to electrodynamics..

I think that you give precise explanation of the role and meaning of the par. 1. That is what I tried to explain to JustinLevy. I understand JesseM (post #2) said the same. I am not expert in linguistic but the translation without “good” or “well” seems to me better (simply: … will be valid for all frames of reference for which the equations of mechanics hold.).

I don't know if you might gain any deeper insight in whatever you're debating.

I don’t know either yet. By the way, we didn’t discuss the formulation in the par.2. I prefer the one given by L.D. Landau and E.M. Lifsheetz (“Field Theory”): There exists the upper bound of the velocity of the propagation of the interactions. The rest is deduced (Minkowski interval).

However, formulation in the par.1 I still consider poetry. I think that it is impossible to give the independent formulation of the Principle of Relativity outside the language of the theory of continuous groups and the differential geometry (math definition of the reference frames). To enforce my statement, it is the fundamental principle of the Theory of Measurements and has nothing to do with dynamical laws. We know that the axiomatic introduction of the internal degrees of freedom allow rediscovering the connection and thus the external and internal degrees of freedom appear democratically. All that was done by H.Weyl, E.P.Wigner and C.N.Yang. I guess that the proper extension of the ref frames into the quantum domain exists.

Regards, Dany.

P.S. “you dropped the dative case some thousand years ago”. What you have in mind?

 

[Ich]

I am not expert in linguistic but the translation without “good” or “well” seems to me better (simply: … will be valid for all frames of reference for which the equations of mechanics hold.).

I think "hold good" means exactly the same as "is valid"; in any case, that's what Einstein wrote.

I think that it is impossible to give the independent formulation of the Principle of Relativity outside the language of the theory of continuous groups and the differential geometry (math definition of the reference frames).

I've read several operational definitiions, and even though I never checked thoroughly I'd expect such a definition to be possible.

P.S. “you dropped the dative case some thousand years ago”. What you have in mind?

That was an irrelevant and offtopic remark. I just found it impossible to translate the sentence without changing its grammar, because there is no dative in English. Old English still had it.

 

[Anonym]

I've read several operational definitiions, and even though I never checked thoroughly I'd expect such a definition to be possible.

Please, give me a refs on that definitions. I use N.A.Doughty “Lagrangian Interaction”, Addison-Wesley, 1990. See par. 9.5-9.7 (p.197-201) especially.

Regards, Dany.

 

[Ich]

Please, give me a refs on that definitions.

There was sort of a misunderstanding on my side. I meant definitions of an inertial system.

Given an inertial system, it is easy to state that in every single one of them, every internal experiment gives the same results. Is that enough?