I Principle of relativity for proper accelerating frame of reference

[cianfa72]

TL;DR Summary

Application of principle of relativity for proper accelerating frames of reference in the context of classic Newton theory and SR

Hi,

I've a doubt about the application of the principle of relativity as follows.

Assume as principle of relativity the following statement: It is impossible by any experiment performed inside a "closed" laboratory to say whether we are moving at constant velocity or staying at rest.

Consider the following scenario in the context of Newton classic theory: the analysis of a physical process in two different reference frame sharing the same proper acceleration (i.e. bodies at rest in each of the two frames respectively have the same proper acceleration as measured by an accelerometer attached to them).

The principle of relativity says the equations describing the given physical process in the two reference have to be the same (of course we need to take in account the pseudo forces appearing to act on masses due to non-inertial accelerating reference frames used).

What about in the context of SR ? Does the principle of relativity has to say something only for inertial frames of reference having constant relative velocity (i.e. only for frames in which bodies at rest have got zero proper acceleration) ?

Thanks.

 

[Dale]

The principle of relativity is unchanged between SR and Newtonian mechanics.

 

[cianfa72]

The principle of relativity is unchanged between SR and Newtonian mechanics.

ok, so even in SR do exist accelerated frames of reference (same proper acceleration for bodies at rest in each of them as measured by accelerometers) having constant relative velocity ?

In those frames the laws of physics should stay unchanged. Is that the case ?

 

[pervect]

The idea that accelerating frame is like a non-accelerated frame with the addition of "fictitious forces" works for Newtonian physics, but it needs to be modified for special relativity. Effects happen in accelrating frames that don't fit into this mold. The most common and important example is "gravitational time dilation". As an example of what I mean by this, the bow and stern of a rigidly accelerating spaceship have different proper accelerations, and clocks ticking at the bow and stern of the spaceship do not remain synchronized.

Thomas precession is another example of a SR effect that occurs in accelerating frames that doesn't occur in Newtonian physics. THomas precession however doesn't affect stationary gyroscopes in an accelerated frame, only moving ones.

This does not contradict Dale's remarks, because the notion that the only difference between accelerated and unaccelerated frames is the addition of fictitious forces is not really a good statement of the principle of relativity.

 

[cianfa72]

As an example of what I mean by this, the bow and stern of a rigidly accelerating spaceship have different proper accelerations, and clocks ticking at the bow and stern of the spaceship do not remain synchronized.

ok, focusing on this example: if we take two rigidly accelerating spaceships (with the same profile of proper acceleration along each of them) can we apply the principle of relativity to them ?

 

[Dale]

ok, focusing on this example: if we take two rigidly accelerating spaceships (with the same profile of proper acceleration along each of them) can we apply the principle of relativity to them ?

Yes. Although, some statements of the principle of relativity are better than others.

 

[PeroK]

ok, focusing on this example: if we take two rigidly accelerating spaceships (with the same profile of proper acceleration along each of them) can we apply the principle of relativity to them ?

Why do you need two spaceships?

 

[cianfa72]

Why do you need two spaceships?

My point is the following: Galileo principle of relativity applies not only to inertial frames but even to not-inertial constant (proper) accelerated frames having constant relative velocity (it definitely makes sense in the context of Newtonian mechanics).

Then what about in the context of SR ? I was trying to single out two (proper) accelerated frames (spaceships) having constant relative velocity to ask if we can continue to apply the principle of relativity even to them.

 

[martinbn]

I don't think this has anything to do with the relativity principle. You are saying that if you have two reference frames that have the same proper acceleration then the physics in them will be the same. Well, of course, under the same circumstances the same thing will happen, what else!

 

[cianfa72]

Yes. Although, some statements of the principle of relativity are better than others.

That does mean the two frames (rigidly accelerating spaceships with same profile of proper acceleration along them) have constant relative velocity ?

 

[cianfa72]

I don't think this has anything to do with the relativity principle. You are saying that if you have two reference frames that have the same proper acceleration then the physics in them will be the same. Well, of course, under the same circumstances the same thing will happen, what else!

Why ? It is in force of the principle of relativity that we can say that.

 

[martinbn]

Why ? It is in force of the principle of relativity that we can say that.

Why? You have to spaceships with the same acceleration. The people inside perform experiments. Why do you need the principle to say that they will have the same results?

 

[cianfa72]

Why? You have to spaceships with the same acceleration. The people inside perform experiments. Why do you need the principle to say that they will have the same results?

Sorry...but if not involved in this or similar cases, which is the content/purpose of the principle or relativity ?

That's more or less the same scenario (with no proper acceleration) of the Galileo description of experiments performed in two different 'closed' cars having constant relative velocity.

 

[robphy]

Suppose a ball sitting on a frictionless table is held in place in an accelerating rocket ship. If that ball is released by the holder so that the frictionless table is now the only object interacting with the ball, does the ball remain in place in that ship?

If not, measure the position after 1 sec.

Suppose another accelerated ship performs the same experiment, does the ship obtain the same displacement?

Do the accelerations matter?
By saying “same velocity”, in what frame?

 

[cianfa72]

Suppose a ball sitting on a frictionless table is held in place in an accelerating rocket ship. If that ball is released by the holder so that the frictionless table is now the only object interacting with the ball, does the ball remain in place in that ship?

No, I would say no. Nevertheless I didn't get your point.

 

[robphy]

I updated my post

 

[Dale]

That does mean the two frames (rigidly accelerating spaceships with same profile of proper acceleration along them) have constant relative velocity ?

I don't think that is implied. What is implied is the converse: if they have constant relative velocity then they will have the same profile of proper acceleration. However, as I said before, some formulations of the principle of relativity are better than others. 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.

 

[cianfa72]

I don't think that is implied. What is implied is the converse: if they have constant relative velocity then they will have the same profile of proper acceleration. However, as I said before, some formulations of the principle of relativity are better than others. 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, which could be a better formulation of the principle of relativity in this case ?

 

[Dale]

For special relativity I prefer: the laws of physics are the same in all inertial reference frames. And for general relativity I prefer: the laws of physics are the same in all reference frames.

Another formulation that I like is an explicit statement of the symmetries: (SR) the laws of physics are unchanged for rotations, translations, and boosts. (GR) the laws of physics are unchanged for any change of coordinates. The last means that the laws can be written as tensors, which in some sense is an almost trivial statement but was not obvious for many years so I think it is worth stating explicitly.

 

[vanhees71]

This is a somewhat dangerous formulation of the meaning of "general covariance" for GR. You can in fact make any theory generally covariant. Also in SR you work with tensor (and sometimes also various kinds of spinor) fields and you can use any non-Lorentzian spacetime coordinates you like (it's as in Euclidean affine space, where you can use any "curvilinear coordinates" you like; you are not restricted to Cartesian coordinates).

The physical symmetry principles are global Poincare invariance for special relativity with the proper orthochronous Poincare group as the spacetime symmetry and making this global Poincare invariance local to formulate GR. The physical principle of relativity is then the statement that there's the class of global inertial reference frames in SR and for GR there's at any point in spacetime a class of (local) inertial frames. Note that using this point of view you get standard GR with a pseudo-Riemannian (Lorentzian) manifold as the spacetime model as long as you restrict yourself to the macroscopic classical physics (i.e., (continuum) mechanics + the em. field), while you get Einstein-Cartan theory, i.e., a differentiable pseudo-metrical Lorentzian manifold with torsion, when including also spinor fields.

From this point of view the general covariance of GR (or Einstein-Cartan theory) is rather a gauge symmetry, i.e., a local symmetry. It is rather an redundancy in the description of the physical observables, which are necessarily gauge invariant to make physical sense.

 

[cianfa72]

For special relativity I prefer: the laws of physics are the same in all inertial reference frames. And for general relativity I prefer: the laws of physics are the same in all reference frames.

Surely, this way we're basically limit the scope of the principle of relativity only to inertial frames (actually excluding proper accelerated reference frames).

For special relativity I prefer: the laws of physics are the same in all inertial reference frames. And for general relativity I prefer: the laws of physics are the same in all reference frames.

Another formulation that I like is an explicit statement of the symmetries: (SR) the laws of physics are unchanged for rotations, translations, and boosts.

Are the two spaceship's rest frame of reference with the same profile of proper acceleration related by a boost ?

 

[Dale]

Surely, this way we're basically limit the scope of the principle of relativity only to inertial frames (actually excluding proper accelerated reference frames).

Yes, that is correct and intended. SR in this way can still handle non-inertial frames, but the application of the principle of relativity is deliberately restricted. This avoids the difficulties I mentioned above with your approach.

Are the two spaceship's rest frame of reference with the same profile of proper acceleration related by a boost?

If two spaceships are related by a boost and one is undergoing proper acceleration then the other will undergo the same proper acceleration.

 

[Dale]

You can in fact make any theory generally covariant.

Yes, this is specifically what I was referencing by "almost trivial", but still worth stating.

 

[cianfa72]

If two spaceships are related by a boost and one is undergoing proper acceleration then the other will undergo the same proper acceleration.

ok, but as you said that does not imply they have a constant relative velocity, though. Nevertheless we can continue to apply the principle of relativity in terms of symmetries formulation (boost in this case).

Hence, since related by a boost, the laws of physics are the same in both the spaceships undergoing the same profile of proper acceleration (basically the equations of the physics laws are the same in the two reference frame).

 

[cianfa72]

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.

Do you mean employing the concept of MCIRF (Momentarily Comoving Inertial Reference Frame) ?

 

[vanhees71]

ok, but as you said that does not imply they have a constant relative velocity, though. Nevertheless we can continue to apply the principle of relativity in terms of symmetries formulation (boost in this case).

Hence, since related by a boost, the laws of physics are the same in both the spaceships undergoing the same profile of proper acceleration (basically the equations of the physics laws are the same in the two reference frame).

Yes, it's indeed a bit mind-boggling if it comes to non-inertial frames in relativistic physics. The most simple example coming to mind is Bell's spaceship paradox, which nevertheless illustrates well the obstacles with accelerated (in this case even non-rotating) reference frames in SR. As a side effect it also illustrates the pretty strange properties of Born-rigid bodies. Rumor has it that Bell kept his colleagues at CERN busy for some time discussing his "spaceship paradox"...

For an elementary treatment, see

https://itp.uni-frankfurt.de/~hees/pf-faq/srt.pdf

 

[robphy]

That does mean the two frames (rigidly accelerating spaceships with same profile of proper acceleration along them) have constant relative velocity ?

This might be helpful
https://www.geogebra.org/m/ETgBGFSX
(My Bell spaceship/rocket visualization.)

In my earlier questions above,
I had this visualization in mind.

 

[cianfa72]

This might be helpful
https://www.geogebra.org/m/ETgBGFSX
(My Bell spaceship/rocket visualization.)

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.

 

[robphy]

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.

I don't think the statement is specific enough.

There are certainly laws of physics, independent of any observer.
The principle of relativity suggests that the equations take the same form among
inertial observers.

Are you trying to extend this to arbitrary observers?
I would think the principle fails in relating a noninertial uniformly-accelerating observer and an inertial one.

Or are you trying to extend this to equal-proper-acceleration observers?
If their worldlines are distinct, no boost transformation will map one worldline onto the other.
A translation is needed.

Maybe the analysis of a simple thought-experiment would be more specific.

 

[cianfa72]

There are certainly laws of physics, independent of any observer.
The principle of relativity suggests that the equations take the same form among
inertial observers.

Are you trying to extend this to arbitrary observers?
I would think the principle fails in relating a noninertial uniformly-accelerating observer and an inertial one.

My point is quite straightforward.

Galilean principle of relativity makes no assumption about the state of motion of frames involved. Its statement involves just the constant relative velocity between the two frames (that's fine in Newtonian mechanics). Then for frames having constant relative velocity it follows that if one frame is inertial then even the other one is inertial too.

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.

 

[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).