Undergrad Principle of relativity for proper accelerating frame of reference

[Dale]

In the context of SR instead, as @Dale pointed out in post #22, the principle of relativity is actually deliberately 'restricted' to inertial frames only.

And as I pointed out in post 2 the principle of relativity is the same in SR and Newtonian physics.

 

[cianfa72]

ok, so even if the two spaceships have the same (fixed) proper acceleration and the same (increasing) velocities w.r.t LAB inertial frame, they do not maintain a constant relative velocity between them, however.

Nevertheless, since the two spaceships reference frames are related by a Lorentz boost, we can apply the principle of relativity in terms of symmetries formulation to get the result that physical laws stay unchanged in both the frames.

Make sense ? Thank you.

Sorry, is my understanding right ?

 

[Dale]

Sorry, is my understanding right ?

Since your understanding appears to be that there is a difference between SR and Newtonian mechanics regarding the principle of relativity then I would say, no, your understanding is not right.

 

[cianfa72]

Since your understanding appears to be that there is a difference between SR and Newtonian mechanics regarding the principle of relativity then I would say, no, your understanding is not right.

Sorry, not sure to understand

We said that the content of the principle of relativity is the same both in Newtonian mechanics and in SR, however in SR its applicability is deliberately 'restricted' only between inertial frames, right ?

In the case of spaceships undergoing the same proper acceleration, are their rest reference frames related by a Lorentz boost ? In that case the principle of relativity in terms of symmetries formulation should be applicable, dont' you ?

 

[vanhees71]

The special (sic!) principle of relativity states that there are global inertial frames, where Newton's 1st Law holds. Together with the assumptions that for inertial observers time is homogeneous and space is Euclidean as well as that the transformations between inertial frames, connected continuously to the identity, and the spacetime manifolds admits a causality structure, you get only two kinds of spacetime, the Galileo-Newtonian one (a fiber bundle), or Minkowski space (a pseudo-Euclidean affine space with a fundamental form of signature (1,3)) underlying special relativity.

 

[cianfa72]

The special (sic!) principle of relativity states that there are global inertial frames, where Newton's 1st Law holds.

This formulation is different from the principle of relativity as stated by Galileo. AFAIK the Galileo formulation makes no assumption about the state of motion of the frames involved (proper accelerated or not).

In other words Galileo principle of relativity applies as well even if the first frame is proper accelerated (i.e. bodies at rest in it have got the same proper acceleration as measured by accelerometers attached to them) and the second frame is moving with constant relative velocity w.r.t the first frame.

The laws of physics in both frames will be the same even if, of course, they are not in the 'simplest' form as in any inertial frame.

Maybe this difference with SR is 'summarized' by the keyword 'special', I guess...

 

[Dale]

We said that the content of the principle of relativity is the same both in Newtonian mechanics and in SR, however in SR its applicability is deliberately 'restricted' only between inertial frames, right ?

I said that I prefer to deliberately restrict it to inertial frames, but that is my preference for both SR and Newtonian physics. By "I prefer" I intended to indicate that there are other opinions and this is my opinion.

 

[cianfa72]

I said that I prefer to deliberately restrict it to inertial frames, but that is my preference for both SR and Newtonian physics.

ok, this way I got your point.

 

[Dale]

ok, this way I got your point.

Actually, my apologies, looking back I never said "I prefer". I had intended it, but not stated it explicitly, so the misunderstanding was entirely my fault and not yours.

 

[cianfa72]

Actually, my apologies, looking back I never said "I prefer". I had intended it, but not stated it explicitly, so the misunderstanding was entirely my fault and not yours.

no problem

Coming back to the following point:

In the case of spaceships undergoing the same proper acceleration, are their rest reference frames related by a Lorentz boost ? In that case the principle of relativity in terms of symmetries formulation should be applicable, dont' you ?

what do you think about ?

 

[vanhees71]

This formulation is different from the principle of relativity as stated by Galileo. AFAIK the Galileo formulation makes no assumption about the state of motion of the frames involved (proper accelerated or not).

In other words Galileo principle of relativity applies as well even if the first frame is proper accelerated (i.e. bodies at rest in it have got the same proper acceleration as measured by accelerometers attached to them) and the second frame is moving with constant relative velocity w.r.t the first frame.

The laws of physics in both frames will be the same even if, of course, they are not in the 'simplest' form as in any inertial frame.

Maybe this difference with SR is 'summarized' by the keyword 'special', I guess...

Well the point is that Newtonian mechanics is formulated in inertial frames, and Lex I states that there exist global reference frames (Newton made even the stronger assumption that there is an absolute global reference frame), but this got immediately (and as we know today rightfully) criticized by his arch enemy, Leibniz, who already then stated that one can establish only "relative motion". The physical laws, once formulated in inertial frames, always are the same, but it's of course mathematically no problem to express them wrt. to non-inertial reference frames. The same holds true for special relativity. It's only that non-inertial frames in SR are much more complicated and usually apply only in parts of the complete Minkowski spacetime, i.e., a single coordinate chart usually covers only a part of the complete Minkowski spacetime, but that's not a real obstacle since you can always introduce complete atlasses to cover the entire manifold.

Even in GR the inertial frames play still a very dominant role, but as a truly local concept, i.e., at each spacteime point you can introduce an inertial frame (through the Fermi-Walker transported tetrades of any free-falling observer, which then of course are parallel transported along the corresponding time-like geodesic world line). The great difference is that there are no more global but only local inertial frames and that the theory is generally covariant in the sense of a gauge symmetry. The observables are defined as local observables and as such operationally defined wrt. a corresponding local inertial frame.

 

[Dale]

no problem

Coming back to the following point:

what do you think about ?

Let me show the issue explicitly in Newtonian physics so that you can see why I don't like the unrestricted formulation of the principle of relativity and also see that it applies for Newtonian physics.

Consider an standard 2D Newtonian inertial frame $(x,y)$ with universal "absolute" time $t$. Now, consider a rotating frame defined by $$X=x \cos (t \omega )-y \sin (t \omega )$$ $$
Y=x \sin (t \omega )+y \cos (t \omega ) $$ Consider a second (primed) inertial frame $$x'=v t+x$$ $$y' = y$$ and a corresponding (primed) rotating frame $$
X'=x' \cos (t \omega )-y' \sin (t \omega ) $$ $$ Y'=x'
\sin (t \omega )+y' \cos (t \omega ) $$

Now, the laws of physics in the non-inertial frames, $X$ and $X'$, are the same, including the same inertial forces. But what is the relationship between the two non-inertial frames? They are moving apart from each other at a constant relative velocity $v$ in the inertial frames. But after some algebra we find that in the non-inertial frames the relationship is: $$X = X'-t v \cos (t \omega ) $$ $$ Y = Y'-t v \sin (t \omega ) $$ meaning that in the non-inertial frames the separation is not at a constant relative velocity.

This is what I meant earlier when I said:

This one is a little poor when applied to accelerating reference frames because the concept of constant relative velocity becomes more difficult to define. It can still be made to work, but it is not straightforward.

So the reason that I don't like the "generic frame" formulation is that it is not actually as generic as it seems. The "constant relative velocity" must be measured from an inertial frame, and is not necessarily a constant relative velocity in the non-inertial frame. Since you must specifically introduce an inertial frame anyway, I prefer to have the whole formulation in terms of the necessary inertial frame. Then the relationship between two non-inertial frames can be derived as a corollary to the principle of relativity which is defined in terms of the inertial frames.

 

[cianfa72]

ok got it. Always in Newtonian mechanics what about the scenario in which the two non-inertial frames are linearly accelerated w.r.t. the inertial frames with the same (proper) acceleration ?

In that case they are moving apart with a constant relative velocity even in the two non-inertial frames, right ?

 

[Dale]

In that case they are moving apart with a constant relative velocity even in the two non-inertial frames, right ?

Yes, that is correct and can be derived similarly to how I did above for rotating frames.

 

[PeterDonis]

Galilean principle of relativity makes no assumption about the state of motion of frames involved.

I don't know where you are getting this from. The Galilean principle of relativity specifically talks about frames with zero acceleration. In Newtonian mechanics, these are inertial frames as Newtonian mechanics defines them. In SR, these are inertial frames as SR defines them; GR keeps the same definition but clarifies that in the presence of gravity inertial frames can only be defined locally. The difference between the Newtonian and relativistic definitions only becomes relevant in the presence of gravity, since Newtonian mechanics considers gravity to be a force and relativity does not.

 

[PeterDonis]

This formulation is different from the principle of relativity as stated by Galileo. AFAIK the Galileo formulation makes no assumption about the state of motion of the frames involved (proper accelerated or not).

Please give a reference for this claim.

 

[PeterDonis]

the unrestricted formulation of the principle of relativity

I'm not sure there is any such thing in Galilean relativity, Newtonian mechanics, or SR. Every formulation of the principle of relativity that I'm aware of prior to General Relativity explicitly restricted it to inertial (unaccelerated) states of motion.

 

[Dale]

The difference between the Newtonian and relativistic definitions only becomes relevant in the presence of gravity, since Newtonian mechanics considers gravity to be a force and relativity does not.

Yes, this is a good point which I often forget to point out. It is also possible to geometrize Newtonian gravity so that even with gravity the relevant concepts are the same. This gives a curved Newtonian space + time, but it is definitely a niche topic.

 

[Dale]

I'm not sure there is any such thing in Galilean relativity, Newtonian mechanics, or SR. Every formulation of the principle of relativity that I'm aware of prior to General Relativity explicitly restricted it to inertial (unaccelerated) states of motion.

Well, in Einstein's 1905 paper he stated it as "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." That could certainly be read as @cianfa72 described. It does not explicitly mention inertial frames or restrict the systems of coordinates.

I don't think that was Einstein's intention, but nevertheless I can see where @cianfa72 is coming from in his understanding.

 

[cianfa72]

I don't think that was Einstein's intention, but nevertheless I can see where @cianfa72 is coming from in his understanding.

Yes, I get the same understanding from the book I used for the undergraduate in Engineering (in Italian).

 

[cianfa72]

Coming back to the example of @Dale in post #42 I think the logic to conclude that physics laws are the same in the two rotating frames is as follows:

The two inertial frames involved (with coordinates $x,y,t$ and $x',y',t$ respectively) are moving apart with constant relative velocity. By virtue of the principle of relativity restricted to inertial frames the laws of physics (i.e. their set of equations) are the same in the two inertial frames. Then we perform exactly the same transformation starting from each of them (what changes are just the names of coordinates used for the inertial and the rotating frame involved). That does mean the set of equations (i.e. physical laws) have to change accordingly in the same way in both the non-inertial rotating frames (with coordinates $X,Y,t$ and $X',Y',t$ respectively).

Is that correct ? Thanks.

 

[Dale]

Yes, that is correct. Note how even though the principle of relativity in this case was stated in terms of inertial frames, it did allow you to derive the equivalence of the two non-inertial frames.

 

[cianfa72]

Yes, that is correct. Note how even though the principle of relativity in this case was stated in terms of inertial frames, it did allow you to derive the equivalence of the two non-inertial frames.

Yes, got it thanks. Thus the case of post #43 (linearly accelerated frames with constant proper acceleration) is really a 'special' case in which the two non-inertial frames move apart with constant relative velocity.

 

[Dale]

Yes, got it thanks. Thus the case of post #43 (linearly accelerated frames with constant proper acceleration) is really a 'special' case in which the two non-inertial frames move apart with constant relative velocity.

Yes, and I don’t know if there are any other non-inertial frames that are also in that same “special case” category.

 

[cianfa72]

ok, let's go back to the analysis in SR of the two accelerated spaceships with the same proper acceleration (i.e. accelerometers at rest in each of them measure the same constant proper acceleration). Just to take it simple we neglect their lenghts.

From the point of view of an SR inertial frame they undergo a motion with the same coordinate acceleration. Since the transformations from the inertial frame to end up in each of the two accelerated frames (coordinate systems) - in which each spaceship is at rest respectively - are the same then using the same argument as in post #51 we can conclude that physics laws have to be the same in both spaceships (i.e. same set of equations).

 

[PeterDonis]

Well, in Einstein's 1905 paper he stated it as "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." That could certainly be read as @cianfa72 described.

I don't think so. Einstein is not saying that the relative motion of the two systems of coordinates is uniform and translatory, which is what @cianfa72 has been describing (and which I agree can be the case for two systems of non-inertial coordinates). He is saying that the motion of each system of coordinates, in itself, is uniform and translatory. That is only true of inertial coordinates (basically it's the same as saying there are no fictitious forces in the given system of coordinates).

 

[Dale]

I don't think so. Einstein is not saying that the relative motion of the two systems of coordinates is uniform and translatory, which is what @cianfa72 has been describing (and which I agree can be the case for two systems of non-inertial coordinates). He is saying that the motion of each system of coordinates, in itself, is uniform and translatory. That is only true of inertial coordinates (basically it's the same as saying there are no fictitious forces in the given system of coordinates).

While you can read it that way too, I still see that one could read it as @cianfa72 did. I do think your reading is correct, but his is possible.

 

[PeterDonis]

I still see that one could read it as @cianfa72 did.

Perhaps that one sentence taken in isolation, but it is not in isolation. The rest of the paper makes clear that only inertial coordinates are intended. For example, the clock synchronization procedure Einstein describes does not work for non-inertial coordinates. Also, the second postulate, that rays of light move at $c$, requires inertial coordinates.

 

[PAllen]

While you can read it that way too, I still see that one could read it as @cianfa72 did. I do think your reading is correct, but his is possible.

The first paragraph of the first numbered section indicates that only inertial coordinate systems are considered. The phrase “such that the laws of Newton hold good” was at the time a well known way of saying an inertial frame, specifically that no fictitious forces need be introduced. The whole rest of the 1905 paper assumes this definition.

 

[Dale]

Perhaps that one sentence taken in isolation, but it is not in isolation.

Agreed, but nonetheless often people take such statements out of context or otherwise misunderstand it. The first time I read that statement I understood it as did @cianfa72. I later understood it in context, but his reading is not unreasonable and I sympathize with his confusion.

Furthermore, a version of the principle of relativity can be applied somewhat as he indicated. The laws of physics, including the inertial forces, are indeed the same for two non-inertial frames each having the same transformation from two different inertial frames. It is not as simple as he posed originally, but there is a core of truth in his thought. That coupled with poor wording of the principle of relativity makes for a reasonable misunderstanding.

There is no need to further justify Einstein’s intent. I recognize that intent and that is not the issue. The issue is simply that the misunderstanding is a reasonable one and deserves to be addressed directly.