B Relativity of Measures: A vs B Frame & Light Source S

[AlMetis]

A frame is a convention for attaching numerical labels (for example, x,y,z and t coordinates) to events. Therefore, using a different frame can only change the labels attached to the events, not the events themselves. Light hitting the mirror is an event.
That’s why.

I agree.
I have included another diagram that is (hopefully) less ambiguous than the word salads I’ve been posting.
The diagram plots the Galilean transformation of the same motion of source, mirror and light predicted to be observed by two inertial frames in motion.
The relativistic (Lorentz) transformations which hold the constancy of c in both frames, removes the (apparent) conflict of “different events” observed by each frame predicted by Galilean relativity. To remove the conflict, the slower time of the moving frame (S from E’s frame) allows it to measure a shorter length S-A of longer light path S-c+3v which holds S’s measure of light speed at c.
In other words, the shorter you measure a longer path, the slower your time must run to arrive at the same ratio of d/t = c.

 

[Sagittarius A-Star]

the (apparent) conflict of “different events” observed by each frame predicted by Galilean relativity.
View attachment 322343

Also with Galilean transformation there must be aberration of a vertically emitted light-particle (which was assumed to exist by James Bradley). The light-particle must strike A in the left and in the right diagrams. In the right diagrams, it must have the speed $\sqrt{c^2+v^2}$, according to the Galilean transformation.

 

[AlMetis]

Yes, the illustration is just showing there is a difference between frames, whether speed, aberration, length, time....something has to give to define the same physical event that is not apparent in the initial conditions of both frames.

Attached is a more legible version.

 

[PeroK]

What's the relevance of the Galilean transformation here?

 

[AlMetis]

What's the relevance of the Galilean transformation here?

It’s to help me understand the line of reasoning that gets us from Galilean to Einsteinian.
I understand Galilean fails to describe the empirical evidence of light, the Lorentz transformations give us the conversion from frame to frame that mathematically satisfies the empirical evidence.
I don’t understand how Lorentz transformations represent reality. My examples are pointing to some of what I think are the most difficult physical aspects to understand when we go from Galilean to Einsteinian.

For a point of reference - Poincaré/Lorentz relativity was mathematically as effective as Einstein’s but required the physically absolute interpretation of the transformations. As this is beyond experimental testing, it was dropped in favour of Einstein’s relativity that uses the principle of relativity in place of absolute. It has been successfully tested in every increasing accuracies and physical convolutions - life of particles etc.
I want to understand how it changes our understanding of physical dimension.

 

[PeroK]

It’s to help me understand the line of reasoning that gets us from Galilean to Einsteinian.

There's no logical way, IMO, to get from Galilean to Einsteinian physics. That was the problem. It's the other way round. Once you have SR, you get Newtonian physics as a low speed approximation. SR is not an extension of classical physics. SR didn't evolve from classical physics: not in the critical differences. SR was something new and unimagined within classical physics. Classical physics is a special case of SR.

 

[Ibix]

I want to understand how it changes our understanding of physical dimension.

Minkowski noticed that the Lorentz transforms were an analogue of Euclidean rotation in a non-Euclidean space now known as Minkowski spacetime, so he proposed that Einstein's maths could be interpreted as implying the existence of a 4d spacetime. That turned out to be useful in understanding gravity, since Newtonian gravity is incompatible with relativity. It was basically a matter of recognising the mathematical theory of (pseudo-)Riemannian manifolds in the maths coming out of physics.

A more abstract approach is to start with the principle of relativity and show that there are only two coordinate transforms consistent with this - Galileo and Lorentz. There's no need to impose the invariance of light speed. Then you simply check by experiment which kind of universe we're in.

 

[Sagittarius A-Star]

I understand Galilean fails to describe the empirical evidence of light, the Lorentz transformations give us the conversion from frame to frame that mathematically satisfies the empirical evidence.

Yes. As @Ibix mentioned: Either there exists a finite invariant speed or not.

• If not, the assumed invariance of causality implies t'=t, that means the Galilei transformation must be valid.
• If yes, the only possible transformation between inertial frames is the Lorentz transformation.

The role of the 2nd postulate of SR - which is not really needed - is to separate between these two possibilities. Experiments showed, that light moves in vacuum with this invariant speed.

Einsteins new idea was to distinguish between $t$ and $t'$ in combination with defining time as that, what a standardized clock shows. He dropped Newtons assumption, that an absolute time exists.

 

[AlMetis]

There's no logical way, IMO, to get from Galilean to Einsteinian physics. That was the problem. It's the other way round. Once you have SR, you get Newtonian physics as a low speed approximation. SR is not an extension of classical physics. SR didn't evolve from classical physics: not in the critical differences. SR was something new and unimagined within classical physics. Classical physics is a special case of SR.

Yes, I didn’t mean to suggest there is a one to one evolutionary correspondence from Galilean to Einsteinian relativity.If I take SR at face value, of course it works perfectly. But I’m not a “shut up and calculate” person.

My diagrams show a distinction that is not made clear in most explanations of SR I’ve read.
When we are given the information on the left of my diagram, we have no reason to expect, or predict an asymmetry in the observations via Galilean or Einsteinian relativity.
When the same motion is observed relative to the light,(right side of diagram) Galilean relativity fails. It predicts what is not consistent with empirical evidence.
When we apply the Lorentz transformation from the left to the right, Einstein’s relativity works on the right (pulse strikes A) predicted observation equals evidence once more.
But we don’t regain the symmetry.
The symmetry is not exclusive to Galilean relativity, it is also predicted by Einsteinian relativity on the left and lost on the right.

Why did it disappear?

 

[PeterDonis]

I don’t understand how Lorentz transformations represent reality.

They give you the proper way to transform your view of reality in one inertial frame, to the correct view of reality in another inertial frame.

But we don’t regain the symmetry.

What symmetry are you talking about?

 

[Dale]

A more abstract approach is to start with the principle of relativity and show that there are only two coordinate transforms consistent with this - Galileo and Lorentz. There's no need to impose the invariance of light speed. Then you simply check by experiment which kind of universe we're in

Well, you do still need more than the first postulate. You can either use the second postulate or you can use experiment. But you need something more.

I do like postulating the invariance of the spacetime interval. That is one single postulate that actually does get you all the way. Of course, it isn’t so intuitive, but then again this is relativity we are talking about.

 

[Dale]

But we don’t regain the symmetry.
The symmetry is not exclusive to Galilean relativity, it is also predicted by Einsteinian relativity on the left and lost on the right.

Why did it disappear?

What symmetry? You must be misunderstanding something because all of the predicted symmetries of SR have been experimentally confirmed. So either you think some symmetry is predicted and it isn’t, or you are not actually looking at the symmetry that you think you are.

 

[AlMetis]

They give you the proper way to transform your view of reality in one inertial frame, to the correct view of reality in another inertial frame.

Yes, they do this by translating numerical values of a measure in one frame, to the numerical values of the same measure in another frame. Why do our measures of dimension change with our motion relative to another frame? Einstein explained “what” happens, time slows, length contracts, mass increases and when all of these are calculated via the Lorentz transformation they accurately predict the “amounts” that each change. The Lorentz transformation explains how to get there, it does not explain why we have to go there. What aspect of reality is defined in the need for the Lorentz equations? If they didn’t work and light was still a constant, reality would be different a very different thing.

 

[AlMetis]

What symmetry are you talking about?

I've marked the symmetry break in this copy.

 

[PeterDonis]

they do this by translating numerical values of a measure in one frame, to the numerical values of the same measure in another frame.

Lorentz transformations don't transform "measures", they transform coordinates.

What aspect of reality is defined in the need for the Lorentz equations?

The geometry of spacetime.

If they didn’t work and light was still a constant, reality would be different a very different thing.

You have no basis for this claim unless you have some alternate theory in which the transformations are different from the Lorentz transformations but the speed of light is still the same in all frames. And we already now such a theory is impossible if we accept the principle of relativity, because the POR only allows two possibilities, Galilean transformations (infinite invariant speed) and Lorentz transformations (finite invariant speed). In the former case, the speed of light is not the same in all frames. So your claim is assuming an impossibility.

I've marked the symmetry break in this copy.

Your diagram on the right is not correct for the SR case, so all you are showing is that Galilean relativity makes a wrong prediction.

 

[Dale]

I've marked the symmetry break in this copy.

I don’t get it. I don’t know what you are calling asymmetry there

 

[strangerep]

What aspect of reality is defined in the need for the Lorentz equations?

The principle that all inertial (i.e., non-accelerating) observers perceive the same laws of physics.

That, plus a couple of technical points about Lie groups implies a group of transformations among inertial reference frames that, in general, involve 2 universal constants: one has dimensions of inverse velocity squared (usually identified with $1/c^2$) and another with either dimensions of inverse time or else inverse length squared (the latter corresponding to the de Sitter group). The 2nd constant is usually ignored, leaving us with the Poincare group, of which the Lorentz transformations are a subgroup.

The value of the $1/c^2$ must then be determined from experiment.

 

[Ibix]

Why do our measures of dimension change with our motion relative to another frame?

On a Euclidean plane, why do the x and y separations between things change when you rotate coordinate axes? Because you've changed which directions in space you call the x axis and the y axis.

On a Minkowski plane, why do the x and t separations between things change when you boost coordinate axes? Because you've changed which directions in spacetime you call the x axis and the t axis.

That's really all that's going on here. Relativity allows you some freedom to choose which direction in spacetime to call "only moving in time" and "only moving in space". That choice affects how you choose to describe things in much the same way a Euclidean rotation might change a long narrow thing into a short wide one.

 

[AlMetis]

Lorentz transformations don't transform "measures", they transform coordinates.

How far apart are these coordinates?

 

[AlMetis]

The geometry of spacetime.

That’s true, I should have said the geometry of spacetime represented by the need to employ the Lorentz transformations between inertial frames expresses a continuum of fundamental dimension we don’t, but I would like to, understand.

 

[AlMetis]

You have no basis for this claim

The basis I have is they “do” work , light “is” a constant and reality “is” what it is.

 

[AlMetis]

Your diagram on the right is not correct for the SR case, so all you are showing is that Galilean relativity makes a wrong prediction.

Both sides are Galilean representations. The red line labeled ‘c_3v” on right side at t=1 represents the time dilated position of A relative to E (in principle, not to scale). If this is not correct, please explain why.

 

[jbriggs444]

How far apart are these coordinates?

What are you talking about?

There is a formula that we can use to determine "how far apart" two coordinate four-tuples are from one another. Let us identify two events $E_1$ and $E_2$ with coordinates $(x_1, y_1, z_1, t_1)$ and $(x_2, y_2, z_2, t_2)$ respectively. The "how far apart", $s$ between the two events is given by:$$s^2 = (x_1 - x_2)^2 + (y_1 - y_2)^2 + (z_1 - z_2)^2 - (t_1 - t_2)^2$$If we shift to an a new reference frame where the same two events $E_1$ and $E_2$ are identified with coordinates $({x'}_1, {y'}_1, {z'}_1, {t'}_1)$ and $({x'}_2, {y'}_2, {z'}_2, {t'}_2)$ then the same formula applies:$${s'}^2 = ({x'}_1 - {x'}_2)^2 + ({y'}_1 - {y'}_2)^2 + ({z'}_1 - {z'}_2)^2 - ({t'}_1 - {t'}_2)^2$$Further, it will turn out to be the case that $s^2 = s'^2$.

So "how far apart" is an invariant measure. It looks like it is a function of coordinates, but that is just because we specified our events with coordinates. It is actually a function that takes two events as input and spits out a [squared] separation as the output.

This "how far apart" formula is the "metric" for the flat space-time of special relativity. When you get to the curved space-times that general relativity can contemplate, the metric will not have this exact form. Though in the limit of increasingly nearby events, it will approximate this form ever more closely.

If you are evaluating the separation between the two ends of a physical rod, then you are asking for the separation between the world-line corresponding to the one end and the world-line corresponding to the other end. That is not the separation between two events -- it is the separation between two trajectories.

In order to apply the "how far apart" formula, you need to identify two events. We conventionally pick out two events that are "at the same time" according to some agreed-upon simultaneity convention.

Each frame has its own simultaneity convention. So each frame will be comparing different pairs of events when they measure the separation between two trajectories.

The two frames are not measuring the same thing. They are measuring different frame-specific hings. This is the same way that the measured width of a strip of paper depends on the angle at which you hold the ruler.

 

[AlMetis]

I don’t get it. I don’t know what you are calling asymmetry there

You and a number of people have said I have unknowingly switched frames of reference resulting in me mistaking the aberration observed in a “different” frame as a change in the observations in the same frame.
From everything posted, this seems to be the most consistent advice.
I need to understand at what point in time, I have switched frames of reference in context of what I think is being observed, in order to see/understand what would be observed.
So please indulge me one step at a time.

In the rest frame of the light source S and a mirror M at rest with S a fixed distance on +y, the path of the light pulse is observed perpendicular to x between emission at t=0 and reflection t=1, while S is in motion along x in the +x direction relative to the frame E.
Then
1. E will see the aberration of the light pulse from t=0 to t=1 as a path at an angle from x toward -x.
(when increasing x is in the right hand direction relative to you, the reader)Yes, no?

 

[Dale]

1. E will see the aberration of the light pulse from t=0 to t=1 as a path at an angle from x toward -x.
(when increasing x is in the right hand direction relative to you, the reader)Yes, no?

Yes. The worldline of the light in $S$ is $$r^\mu=(t,x,y,z)=(t,0,t,0)$$ and in $E$ is $$r^{\mu'}=(t',x',y',z')=(t',-t' v, t' \sqrt{1-v^2},0)$$ using units where $c=1$.

 

[PeroK]

In the rest frame of the light source S and a mirror M at rest with S a fixed distance on +y, the path of the light pulse is observed perpendicular to x between emission at t=0 and reflection t=1, while S is in motion along x in the +x direction relative to the frame E.
Then
1. E will see the aberration of the light pulse from t=0 to t=1 as a path at an angle from x toward -x.
(when increasing x is in the right hand direction relative to you, the reader)Yes, no?

Probably yes. I would have said simply:

In a frame where the source and mirror are at rest, the light path is along the positive y axis.

In a frame, E, where the source and mirror are moving to the right (along the positive x axis), then the light path is at an angle to the x axis.

In terms of aberration, this means that when the light is observed in frame E, the source is further to the right than it appears when the light is received.

 

[AlMetis]

Yes.

Thank you.
@ PeroK, yes, I will likely say more than necessary. It's a symptom of doubt.

2. The principle of relativity requires the laws do not discern one "inertial frame" from another when the Lorentz transformations hold the laws valid in all.Yes, no?

 

[Dale]

2. The principle of relativity requires the laws do not discern one "inertial frame" from another when the Lorentz transformations hold the laws valid in all.Yes, no?

Yes. Another way of stating this is that the laws of physics, both the differential equations and any associated constants, are the same in all inertial frames.

 

[PeterDonis]

How far apart are these coordinates?

The question makes no sense. You can ask what the spacetime interval is between two events (points in spacetime), but that's all.

I should have said the geometry of spacetime represented by the need to employ the Lorentz transformations between inertial frames expresses a continuum of fundamental dimension we don’t, but I would like to, understand.

Why do you think we don't understand it? You should not generalize from your own experience, because you evidently don't have a good grasp of SR, but that doesn't mean nobody does.

The basis I have is they “do” work , light “is” a constant and reality “is” what it is.

None of this justifies the claim I responded to. I gave you reasons why your claim was not justified. You haven't responded to those reasons at all.

Both sides are Galilean representations.

Which means, as I said, that they tell us nothing about SR. But you said your claim about some symmetry not being there after switching frames applies to SR. What are you basing that on?

 

[AlMetis]

Yes.

Thank you.
3. Then by virtue of 2, the motion of S is relative to the rest frame of E and the motion of E is relative to the rest frame of S, in that neither hold the equations 'More" valid in terms of the laws.

Yes, no?