Comparison of Special Relativity with a Galilean "preferred frame" theory

Well suppose, as in the SI definition, you're using caesium atoms to define your own measure of the second. The problem is, that only works because all Caesium atoms in nature are the same size and the transition between their energy levels is the same. If isotropic scaling were a symmetry in physics then there would be no reason that should be the case: for any given system allowed by the laws of physics, scaling symmetry would imply that a larger and slower evolving, but otherwise identical, version of the same system should also be able to exist in nature. So you wouldn't really be able to define a standard time unit in a unique, reproducible, and non-arbitrary way.

What exactly do you mean by this?

I doubt that approach is going to work. The first postulate asserts a velocity dependent symmetry of all the laws of physics. It's a really hefty postulate and the way it is usually stated (as in your earlier quote) has a lot of implicit baggage associated with it. "Reference frame" notably: reference frames are useful concepts largely because we presuppose a relativity-type symmetry of physics between them. Think back again to the SI definition of the second: when is the last time you actually used Caesium atoms to measure time? Presumably you never have, and a relativity-type symmetry is the main reason you never needed to, because it implies that any other physical system would work just as well as a time standard. Without a relativity-type symmetry, just because you calibrated your Casio watch to measure the same second as defined by Caesium atoms doesn't mean that they'll be equivalent time measures for an observer in motion, which in turn would make the whole concept of a reference frame somewhat arbitrary and ill-defined.

Personally I doubt that you're going to be able to derive relativity's first postulate without implicitly assuming it in some way.

Well you answered your own question:

The first postulate alone doesn't rule out Galilean invariance as a possible relativity theory, so the first postulate (and anything strictly equivalent to it) can never make the second postulate redundant.


To be honest, personally I don't see much point in trying to "derive" relativity from postulates anyway - especially if you're trying to "derive" the first postulate (which is pretty much the definition of a relativity theory). The result always seems to be semi-handwavy and you find a host of implicit assumptions the author is making. Einstein's own original formulation of relativity is no exception in this regard and would probably make a serious mathematician cry. (That said, if you properly define what a "relativity theory" is right from the beginning, I do think it's interesting to be able to show that there are only a few possible candidate relativity theories.)

Relativity is really just a generalisation of the observation that we have rotational symmetry in physics. Why does nobody think it's important to derive a "principle of rotational symmetry" from postulates, like everyone seems to insist on doing with relativity? It's not like you couldn't do it in the same style:

Postulate #1: the laws of physics are the same in reference systems with their spatial axes oriented differently relative to one another.
Postulate #2: the length of any object is independent of the reference system.​

But what's the point? Is that really a clearer way of defining rotational symmetry than just saying it like it is?

 

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Hmm...it's still not immediately obvious to me that isotropic scaling wouldn't be a symmetry in physics. Your "slower evolving" would require a reference which would be unavailable wouldn't it? We would be back to dealing with the fact that time dilation is never locally identifiable.

It's a poorly-worded attempt at saying that an invariant rate of proper time passage would exhibit (or explain, or be equivalent to) consistent laws of physics in inertial frames.

I'm not saying it's a personal goal of mine but it would certainly be a "simpler" postulate for SR if such an equivalency could be established. Are there classical physical laws which are covered by Galilean Relativity without a time parameter?

Actually, (depending on the wording) the second postulate doesn't rule out Galilean invariance either since everyone could agree that c is infinite. I maintain that the second postulate is not needed.

That's interesting but I suspect it's because we're still trying to make intuitive sense of SR, while rotational symmetry is understood by monkeys.

 

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"rate of proper time" is not invariant. One way to see this is to write out in concrete terms exactly what you think this phrase means.

The rate of my proper time to my co-moving coordinate time is 1:1. The rate of my proper time to some inertial standard coordinate time is \(\sqrt{1 - \frac{\vec{v}^2}{c^2}} \, : \, 1\) where \(\vec{v}\) depends on my history of acceleration and the choice of inertial standard. Likewise comparing "my" proper time to "your" proper time, we see that they are not invariant, as in the Twin Paradox.

What is invariant in Special Relativity is the integrated proper time along a particular time-like path, \(f(\lambda)\) between two space-time events. \(\Delta \tau = \frac{1}{c} \int_{\lambda_A}^{\lambda_B} \sqrt{c^2 \left( \frac{d f_t}{d \lambda} \right)^2 - \left( \frac{d f_x}{d \lambda} \right)^2 - \left( \frac{d f_y}{d \lambda} \right)^2 - \left( \frac{d f_z}{d \lambda} \right)^2 } \, d\lambda = \frac{1}{c} \int_{\lambda_A}^{\lambda_B} sqrt{c^2 \dot{f}_t^2 - \dot{f}_x^2 - \dot{f}_y^2 - \dot{f}_z^2 } \, d\lambda\) where \(f(\lambda_A) = A = \left( A_t \\ A_x \\ A_y \\ A_z \right)\), \(f(\lambda_B) = B = \left( B_t \\ B_x \\ B_y \\ B_z \right)\), \( \dot{f} = \frac{df}{d\lambda}\) and everywhere \(c \dot{f}_t \gt \sqrt{\dot{f}_x^2 + \dot{f}_y^2 + \dot{f}_z^2}\).

A related invariant, is that for any two events in space-time \(c^2 ( \Delta t )^2 - ( \Delta x )^2 - ( \Delta y )^2 - ( \Delta z )^2\) is invariant.

Examples of this are here (2011/09/27), here (2012/08/20), and here (2013/01/18).

This test model (with K as a parameter) is equivalent to assuming Newton's law first law of motion and then equating inertial motion with every straight line in the 4-dimensional Cartesian coordinates.

You still need to rely on data to distinguish the Galilean from the Lorentzian case. Just assuming the consistency of the laws of electromagnetism doesn't rule out that they are only consistent with respect to a special standard of inertial rest.

i.e. when you set \(K = c^{-2}\) you find this choice is identical to postulating the invariance of the interval or requiring the invariance of the speed of light in vacuum.

 

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Well there's no a priori reason it shouldn't be, and a consequence would be that many of relativity's standard predictions (like length contraction and time dilation) would not be unique. The absence of isotropic scaling symmetry is effectively an additional postulate (though one that's easily justified on observational grounds) required in the derivation of these predictions.

I'm not so sure now, but when I read you talking about the "local invariance of proper time" earlier, I got the impression you were touching on this. If you were saying that any two observers at rest with respect to one another should always agree on how long 1 second is (based on some or any operationally defined time standard), then you are effectively disallowing isotropic scaling as a symmetry.

What do you mean by "invariant rate of proper time passage"?

I don't know if this is what you mean, but in relativity, proper time is invariant in the sense that the interval \((\Delta \tau)^{2} \,=\, (\Delta t)^{2} \,-\, (\Delta x)^{2}\) is the same in all inertial reference frames. But you run into difficulties if you try to use that as a replacement postulate for the first (principle of relativity) postulate. The immediate question is: what are \(\Delta t\) and \(\Delta x\)? Measured distance and time intervals in some given reference frame. Fine. Measured how? If you say "measured with wooden rulers and Casio watches" then you don't have relativity. You just have a theory about wooden rulers and Casio watches. If you're expecting the same result by any measurement standard then you're assuming different measurement conventions are compatible and interchangeable, so you're effectively assuming the principle of relativity anyway.

No, obviously, because a time parameter is explicitly included in the Galilean transformation itself. What are you getting at?

But a finite invariant speed - which is how the second postulate is widely understood, given it refers to an already measured and finite quantity - certainly does rule out Galilean invariance. In any case, Galilean invariance is not the same thing as Lorentz invariance and you will need a postulate (which may be easy to justify on experimental grounds, but will logically be applied as a postulate nonetheless) in order to rule one out in favour of the other.

I think the best way to make intuitive sense of SR is to think of it as a meta theory asserting a space-time symmetry of physics analogous to rotational symmetry. Because that's really what SR is.

(I say "meta theory" because SR isn't really a theory in the usual sense of the word. Regular physics theories make predictions about the dynamics and behaviour of matter, fields, quantum states, and generally whatever "stuff" you think exists in nature and are applying your theory to. SR, by contrast, primarily makes a prediction about other theories. If you are given all the laws of physics, and they happen to be Lorentz invariant, then asserting relativity as a theory in itself is technically redundant. If, for the sake of argument, you think all of physics is governed by relativistic mechanics and electromagnetism for example, then there is nothing predicted by relativity about nature that isn't already predicted by these theories.)

 

Yet, you are STILL posting absolute nonsense.

At that time you were just piling up errors. In fact, you are STILL posting fringe stuff. Once the errors were pointed out to you, you changed your "song". It took you 5 days to get things right, practically you had to reverse each one of your erroneous statements.

 

This is of course what I meant; the rate of proper time of the observer. In other words, functioning clocks held in each hand will exhibit the same rate of "clocking". Tidal forces could break this symmetry, which is why it applies to SR but not GR.

That's interesting, thx!

 

LOL, this is getting more and more amusing by the post.

You finally got this one, took quite long to learn this.

Has nothing to do with "tidal forces". It has to do with another effect , what would that effect be?

 

Ah, this is interesting. What you're saying is that if observers A and B are co-moving and inertial, they would mutually agree that the rate of time passage is the same, however this would not be the case if one of the observer's spatial dimensions were scaled up by some value? I would agree on this point, but both observers would continue to claim that their respective rate of proper time passage had not changed. I'm still confused if you are claiming that isotropic scaling symmetry does or does not exist in SR and how that relates to what I'm discussing.

Please read my reply to rpenner, perhaps that will help.

OK I'll need to think about this after work.

What I'm after is, can SR's first postulate (regarding constancy of physics) be completely explained by an invariant rate of local proper time passage? Are there other parameters which, if they were not invariant, would provide a counter-example to the first postulate (and which themselves do not ultimately rely upon t)?

I'm just pointing out that Galilean invariance is just a special case of Lorentz invariance, and they both do technically fall under SR if you use the following wording for the second postulate:

YES! I came to the same conclusion. Not my best work but you might get the gist of it here

 

You are still making up kooky terms that reflect your ignorance of the subject


1. You can't "explain" a postulate, much less "by an invariant rate of local proper time passage"
2. The first postulate is not about "constancy of physics".
3. There is no such thing as "an invariant rate of local proper time passage". The "rate" is variant, not invariant. The name of the invariant is "proper time" and this was explained to you several times already by at least 3 posters, last time by rpenner in post 243.
4. Parameters cannot "provide a counter-example to the first postulate".
5. A simple counter-example falsifies your statement: Maxwell equations are Lorentz invariant but aren't Galilei invariant.

Your whole post is a word salad.

 

1. You can reformulate and simplify a postulate.
2. Of course it is.
3. I can't help your lack of ability for abstract thought. See all those words that rpenner and przyk are writing? That means that "something" is being communicated and explored.
4. I don't think they can either; if they could then we could not reformulate the first postulate into one only referring to time. Also, see #3.
5. Maxwell's equations are not a postulate in SR; they could have a different form if c had a value other than 0 < c < infinity.

 

Despite your back-pedaling, what you claimed is that you could "explain a postulate". "Reformulate and simplify" is not the same thing as "explain". Postulates are axioms, they cannot be "explained".

Nope, you are persisting in your hilarious error.

No, it means that you are being corrected. They are writing the same thing I am writing, that you don't even know your basic terms, yet you are pretending to "discuss" physics with us.

Then why are you posting nonsense? The "reformulation" uses either:

a. the first postulate only (the one that you are unable to post correctly)
b. the invariance of the metric \(ds^2\), not of "time" as you keep claiming.

No one claimed that "Maxwell equations are a postulate" , this is a non-sequitur. The point was that you do not understand the notion of invariance and how it applies differently in special vs. Galilei relativity.

Huh? The form of the equations does not depend on the value of c. In the integral formulation they don't even depend on c, period. In the differential formulation some depend on the c symbol. It is fun to watch you digging yourself deeper and deeper. You really need to take an introductory class in SR and one on electromagnetism as well. Even worse, you are starting to delve into GR with even more ridiculous claims.

LOL, this one is the most hilarious of the lot. Can the absolute value of a scalar be anywhere else than in the interval 0 to \(\infty\)?

 

Go play, Tach. The adults are having a conversation.

 

No one is "conversing" with you. A lot of us are trying in vain to correct your crank misconceptions. But we can't fix delusional. With every post, you become closer to Farsight.

 

I am saying that if scaling were a symmetry in physics, there would be no "absolute" notion of distance or time interval, just like there is no "absolute" notion of velocity in Galilean or Einsteinian relativity. In reality we do observe "absolute" distance and time scales in the universe: all hydrogen atoms are the same size, the duration of the second as defined using caesium atoms is unique, a block consisting of Avogadro's number of iron atoms at a given temperature is always the same volume, and so on. You may also be familiar with the fact that ants can carry 50 times their own body weight while we can't, or that if King Kong existed he'd probably overheat or crumple under his own body weight. These are examples of a lack of scaling symmetry in physics.

What does that mean, again? The problem is that you have to be careful to make sure you're not saying something useless and trivial here. For example, if you define proper time as measured by some clock, then the clock will always measure the same time as itself and is "invariant" in that sense. Or if you have two clocks that are properly synchronised, then yes, they will measure the same time (pretty much because you defined them to) and proper time is also "invariant" in that sense. But these are rather useless statements.

Neither really, though it could depend on how you define SR. Isotropic scaling symmetry can coexist with Lorentz symmetry - Lorentz symmetry neither requires it nor prohibits it. But many predictions usually associated with relativity, like length contraction and time dilation, implicitly assume scaling is not a symmetry.

Lorentz symmetry implies that, for any physical system that is allowed by the laws of physics, a moving, length contracted, and time-dilated version of that system (i.e. Lorentz-transformed but otherwise identical arrangement of atoms, etc.) is also allowed. But if scaling is also a symmetry then e.g. a scaled up and even slower evolving version is also allowed, so the relativistic time dilation and length contraction factors would not be unique. They would be possible length contraction and time dilation factors, but not the length contraction and time dilation factors associated with moving objects.

See my own response above.

How about we reverse the question: suppose proper time were not invariant in the sense you mean. What would that look like? Can you give an example of a hypothetical observation we might make or experimental result we might find that we don't in reality?

Well, again, what do you mean by "invariant"? We often talk about quantities that are "invariant" in relativity, but saying that is only meaningful largely because we're implicitly assuming the first postulate of relativity is true, like I explained in my previous post with proper time. If you're not assuming the first postulate is true, you have the problem of having to explicitly define a lot of things that are normally taken for granted when talking about relativity.

Not really, because it is always intended and understood that c refers to the experimentally measured and finite speed of light, of around 3x10[sup]8[/sup] m/s. If that were not clear it would be considered a defect in the definition, and not an inclusion of the Galilean transformation as part of Einsteinian relativity.

It's true that the Galilean transformation is sometimes described as a limiting case of the Lorentz transformation with \(c \rightarrow \infty\). But that only works in a limited sense. For starters, c is a dimensionful quantity and we can (and often do) pick units such that c = 1, so in that respect taking the limit to infinity is a bit ill-defined. Also as far as I know it isn't possible (or at least not trivial) to apply some notions of Minkowski geometry or adapt General Relativity to the Galilean transformation. It also makes a difference for the rest of physics whether physics is Galilean or Lorentz invariant (special relativity combined with quantum physics has a lot more implications than Galilean relativity with quantum physics, for example). In many ways they really are qualitatively different transformations.

In any case, I don't think many physicists would recognise Galilean invariance as Einsteinian relativity, and if you show a derivation of Einsteinian relativity that also allows the Galilean transformation, you can expect a lot of eyes to roll.

Er, I in case it's not clear I meant that the Lorentz transformation is similar to a rotation in some respects. For example it has an invariant (pseudo) norm and scalar product, and an angle-like parameter (called the "rapidity"). As a result, many notions from Euclidean geometry carry over with only minor modification to relativity.

If you think I meant that you can rearrange \((\Delta \tau)^{2} \,=\, (\Delta t)^{2} \,-\, (\Delta x)^{2}\) to the more Euclidean-like \((\Delta t)^{2} \,=\, (\Delta \tau)^{2} \,+\, (\Delta x)^{2}\) and from there just treat relativity as Euclidean geometry, then it doesn't work that way and I think you'll find that trick only works in the most trivial of examples.

 

OK, we make two identical clocks, and both appear to be functioning normally. They reside next to each other in the lab, yet they do not stay synchronized; the rate they appear to be "clocking" at differs by location. In this world, would we also be unable to accept SR's first postulate? Of course, because observations would change based on where they were made. Is the fact that both clocks do stay synchronized sufficient to predict SR's first postulate? I think yes but I'd like a counter-example.

No I wasn't making any proclamations about SR being Euclidean, it was just a connection-through-analogy that I made in my mind. Reread the post I linked to. If we consider 4-momentum to be analogous to length and relative velocity to be analogous to rotation away from the perpendicular perspective of that length, then SR length contraction becomes analogous to foreshortening. It was just an observation with no real consequences. An obvious and more proper connection between SR and rotation is of course Penrose-Terrell effects.

 

Err, maybe in the "RJBeery relativity". In reality, you are dead wrong.

No, it doesn't. You need to stop posting kooky stuff , the rate is the same.

 

Wow...

 

You need to take an introductory class to learn the basics. You are posting pure nonsense.

 

So when you talk about the clock rates being invariant, you mean invariant with respect to position? I.e. if I test the rate of a clock placed in different positions with respect to a reference clock, I won't find that the relative rate of the clocks depends on where I put the first one? If so it sounds like you are describing a special case of translation symmetry in physics.

Not necessarily, though obviously yes if you read SR's first postulate as implicitly assuming translation symmetry (which along with rotational symmetry, SR as it is usually understood does assume). But there's no a priori reason you couldn't have Lorentz invariance (as opposed to Poincaré invariance) without translation symmetry.

Which would let you identify a preferred location in space, but not necessarily a preferred state of rest or absolute motion with respect to it. Like I said, it depends on how you prefer to formulate and understand relativity (i.e. do you consider translation symmetry part of the definition of relativity, or simply a separate symmetry?). Whether relativity makes sense without translation symmetry is an opinion I'll let you decide for yourself.

Well I'd say no because translation symmetry doesn't imply Lorentz symmetry (or Galilean symmetry for that matter). For a toy example, consider a universe in which the only things that exist are clocks that tick at a rate that depends only on their (absolute) velocity. For an actual historical example, just take the state of physics near the end of the nineteenth century, which consisted of Newtonian mechanics and electrodynamics. Newtonian mechanics is Galilean invariant but not Lorentz invariant, while electrodynamics is Lorentz invariant but not Galilean invariant. Together, the combination is neither Galilean nor Lorentz invariant, but it is translation invariant in the "rest frame" (the one frame in which both Newtonian mechanics and electrodynamics are valid in their canonical textbook form).

Actually if you'd stopped at length contraction, that would be fine. I'd see Terrell rotation as a simply visual effect, a bit like the way objects look smaller when they're further away from you.

 

Terrell rotation is all about the signal delay of a finite speed of light. As przyk says, it is a visual effect.