Question from I. KINEMATICAL PART § 1. Definition of Simultaneity

[arrell]

Suppose we have two frames (frame 1 and frame 2) which are passing each other infinitesimally close at a relative velocity v with respect to x axes in both frames..
We have two points, a and b, in frame 1 with separation L along the x axis.
We have two other points, c and d also in frame 1 where point c is infinitesimally above (y axis) point a and point d is infinitesimally above point b (so c and d are also with separation L along the x axis).

We fire lasers simultaneously from a, b, c, and d (all in frame 1) orthogonally to vector v towards frame 2 .
In frame two, there is a fully reflective surface opposite to point a and b
and a laser detecting sensitive surface opposite to c and d.

Let g and h be the points where the laser beams (which were fired from c and d) strike and are detected in frame 2.

Let e and f be the points in frame 1 where the laser beams (which were fired from a and b) strike in frame 1 after being reflected from frame 2.

Questions:
What is the separation in frame 1 of e and f in terms of L?
What is the separation in frame 2 of points g and h in terms of L (with respect to a coordinate system in frame 2 that is congruent to that of frame 1 when the two frames are at rest)?

 

[JesseM]

We fire lasers simultaneously from a, b, c, and d (all in frame 1) orthogonally to vector v towards frame 2 .

When you say "towards frame 2", what does that mean exactly? Frames don't have a particular location in space, they're more like coordinate systems that assign coordinates to every point in space. Do you mean "towards the origin of frame 2"? If so, where are the origins of frame 1 and 2 located at the moment the lasers are fired?

 

[arrell]

For purposes of this question, we are considering a plane surface that is in frame 2 (ie moving at the same v as frame 2) and is parallel to the axis of the points in frame 1 and infinitesimally close to the axis of the points in frame 1. Also at the time of the lasers firing, the plane surface extends in
the +x and -x directions sufficiently that the laser pulses all strike somewhere on the plane. I should have also made clear that for purposes of the question, the lasers are emitting an infinitesimally small pulse (ie a photon).

 

[JesseM]

For purposes of this question, we are considering a plane surface that is in frame 2 (ie moving at the same v as frame 2) and is parallel to the axis of the points in frame 1 and infinitesimally close to the axis of the points in frame 1.

If the surface is infinitesimally close to the point where the photons from the lasers are emitted, then they will be detected and reflected infinitesimally close to the point where they were emitted.

 

[arrell]

Thanks for the answer

Do you mean that when the distance from e to f in frame 1 is measured by a frame 1 measuring stick that the separation = L ?

Do you mean that when the distance from g to h in frame 2 is measure by a frame 2 measuring stick that the separation = L ?

 

[JesseM]

Thanks for the answer

Do you mean that when the distance from e to f in frame 1 is measured by a frame 1 measuring stick that the separation = L ?

Do you mean that when the distance from g to h in frame 2 is measure by a frame 2 measuring stick that the separation = L ?

If the distance from g to h is L in frame 1 where they are at rest, it must be [tex]\sqrt{1 - v^2/c^2}L[/tex] in frame 2 where they are moving with velocity v.

 

[reilly]

arrel -- You have posited the basic train expt. of Einstein, but in a somewhat more complex arrangement. So, what's to worry? The solution to your problem is pretty much explained in most basic books on relativity. Read, and you will find the answers to your problem. It's all about the LT.
Regards,
Reilly Atkinson

 

[Physicsguru]

arrel -- You have posited the basic train expt. of Einstein, but in a somewhat more complex arrangement. So, what's to worry? The solution to your problem is pretty much explained in most basic books on relativity. Read, and you will find the answers to your problem. It's all about the LT.
Regards,
Reilly Atkinson

Reilly, why do you place so much faith in the Lorentz transformations. Are you positive that simultaneity is relative?

Regards,

Guru

PS: One other thing, what is the speed of a charge density wave relative to the center of inertia of that which supports it? Greater or less than c.

 

[JesseM]

Reilly, why do you place so much faith in the Lorentz transformations. Are you positive that simultaneity is relative?

What does that question mean? In a philosophical sense, it's possible there could be some single "true" definition of simultaneity, relativity just says there's no physical experiment we can do that will pick out a preferred frame--this is equivalent to saying that all the fundamental laws of physics will turn out to be Lorentz-invariant. And besides Maxwell's laws, all the new laws of physics which have been discovered in the twentieth century have either been Lorentz-invariant, or are understood to be approximations to a more fundamantal Lorentz-invariant theory (like how nonrelativistic QM is understood as an approximation to relativistic quantum theories).

 

[Physicsguru]

What does that question mean? In a philosophical sense, it's possible there could be some single "true" definition of simultaneity, relativity just says there's no physical experiment we can do that will pick out a preferred frame--this is equivalent to saying that all the fundamental laws of physics will turn out to be Lorentz-invariant. And besides Maxwell's laws, all the new laws of physics which have been discovered in the twentieth century have either been Lorentz-invariant, or are understood to be approximations to a more fundamantal Lorentz-invariant theory (like how nonrelativistic QM is understood as an approximation to relativistic quantum theories).

The meaning of simultaneity is physical, not philosophical. Maybe the meaning of the word 'simultaneity' is philosophical, but the concept at issue here is physical.

Next, the laws of physics must be formulated so as to be invariant under a Galilean transformation, if simultaneity is absolute.

Take Newton's second law for example:

[tex] \vec F = m \vec a [/tex]

Let S denote an inertial reference frame. Let the statement [tex] \vec F = m \vec a [/tex] be necessarily true in this frame.

Now, let the origin of frame S` be moving in reference frame S. We wish to inquire as to whether or not the second law of Newton is also true in frame S`.

Suppose that the position vector of a particle in frame S is given by:

[tex] \vec r = x \hat i+y \hat j+z \hat k [/tex]

and that the position vector of the particle in frame S` is given by:

[tex] \vec r^\prime [/tex]

Let the position vector of the origin of S` in frame S be given by:

[tex] \vec R [/tex]

The following statement about the vector triangle in S is true at moment in time t in frame S:

[tex] \vec R + \vec r^\prime = \vec r [/tex]

For a Galilean transformation we have to have t=t`, therefore dt=dt` hence:

[tex] \vec V + \vec v^\prime = \vec v [/tex]

Where V is the velocity of the origin of frame S` in frame S, and v is the velocity of the particle in frame S, and v` is the velocity of the particle in frame S`.

Now, suppose that at moment in time t=0, force F is exerted on the particle of inertia m, so that in frame S we have:

[tex] \vec F = m \frac{dv}{dt}[/tex]

The external force acting upon the particle is not a frame dependent quantity. So let F` denote the force acting upon the particle in the primed frame. The condition for S` to be an inertial reference frame is:

F-F`=0

So that we have:

[tex] \vec F - \vec F^\prime = m \frac{dv}{dt} - m \frac{dv^\prime}{dt}

[/tex]

Which leads to:

[tex] \vec F - \vec F^\prime = m \frac{d}{dt}( \vec v - \vec v^\prime)

[/tex]

Therefore:

[tex] \vec F - \vec F^\prime = m \frac{d\vec V}{dt}

[/tex]

So the condition for S` to be inertial is for that V above be constant as measured in frame S.

Regards

Guru

 

[JesseM]

Next, the laws of physics must be formulated so as to be invariant under a Galilean transformation, if simultaneity is absolute.

OK, so do you agree that all of the most fundamental laws discovered in the twentieth century were invariant under the Lorentz transformation but not under the Galilei transformation? So if the real fundamental laws were Galilei-invariant that would require a pretty radical revamping of all of modern physics, yet the new laws would have to reproduce all the predictions made by the old laws which have been verified experimentally to a high degree of accuracy. The unlikeliness of this is why most physicists "place so much faith in the Lorentz transformations".

 

[Physicsguru]

OK, so do you agree that all of the most fundamental laws discovered in the twentieth century were invariant under the Lorentz transformation but not under the Galilei transformation? So if the real fundamental laws were Galilei-invariant that would require a pretty radical revamping of all of modern physics, yet the new laws would have to reproduce all the predictions made by the old laws which have been verified experimentally to a high degree of accuracy. The unlikeliness of this is why most physicists "place so much faith in the Lorentz transformations".

Welp they're wrong about simultaneity, hence the fundamental laws of physics should not be made to be invariant under a Lorentz transformation, so yeah some re-vamping needs to be done. I don't think the laws of physics care whether or not we know them.

Regards,

Guru

 

[JesseM]

Welp they're wrong about simultaneity, hence the fundamental laws of physics should not be made to be invariant under a Lorentz transformation

We don't "make" the laws of physics, we discover them. Physicists may use Lorentz-invariance as a criterion when postulating new equations, but it's experiment that decides which postulated equations describe nature and which don't. And Lorentz-invariant theories are the ones that have consistenly been favored by experiment in the twentieth century--do you deny this?

 

[arrell]

arrel -- You have posited the basic train expt. of Einstein, but in a somewhat more complex arrangement. So, what's to worry? The solution to your problem is pretty much explained in most basic books on relativity. Read, and you will find the answers to your problem. It's all about the LT.
Regards,
Reilly Atkinson

Just a simple answer to my questions would be of real help to me and I would be very appreciative.

I'm just simply trying to follow Einstein's paper line by line. My 'posit' was to try to find out if what I thought had to be a basis for one step in his derivation is what I think it is.

If the length in both cases (e to f and g to h) is not L, I'm in a whole lot of trouble early on at almost the start of the derivation he's making in
I. KINEMATICAL PART § 2. On the Relativity of Lengths and Times

In terms of the physical world, e to f in frame 1 seems like it has to be L.

If the distance from g to h is L in frame 1 where they are at rest, it must be [tex]\sqrt{1 - v^2/c^2}L[/tex] in frame 2 where they are moving with velocity v.

If g to h in frame 2 were to be:
[tex]\sqrt{1 - v^2/c^2}L[/tex]

then when the photons are reflected from frame 2 they are now originating in frame 2 and symmetry would seem to indicate that e to f in frame 1 would be:
[tex]\sqrt{1 - v^2/c^2}(\sqrt{1 - v^2/c^2}L)[/tex] which isn't exactly the same as L. As I say, I get into a lot of trouble about there if g to h is not equal to L.

You really do have to carefully follow the experiment and the points exactly as I layed it out to see what I am trying to figure out.

 

[JesseM]

If g to h in frame 2 were to be:
[tex]\sqrt{1 - v^2/c^2}L[/tex]

then when the photons are reflected from frame 2 they are now originating in frame 2 and symmetry would seem to indicate that e to f in frame 1 would be:
[tex]\sqrt{1 - v^2/c^2}(\sqrt{1 - v^2/c^2}L)[/tex] which isn't exactly the same as L. As I say, I get into a lot of trouble about there if g to h is not equal to L.

Why would symmetry indicate that? If an object at rest in one frame has length X in that frame, its length in another frame moving at v relative to the first one will be [tex]X\sqrt{1 - v^2/c^2}[/tex]. But if an object moving in one frame has length X in that frame, its length in the second frame will not be [tex]X\sqrt{1 - v^2/c^2}[/tex].

You really do have to carefully follow the experiment and the points exactly as I layed it out to see what I am trying to figure out.

Well, as I said before, your language is confusing, since photons don't "originate" in one frame vs. another, I guess what you're saying is that the source of the photons is at rest in a given frame. You also have a profusion of points which are infinitesimally close to each other--if a and c and g and e are all infinitesimally close to one another, what is the point of giving them different labels, for example? Both frames will assigning a and c and g and e the same coordinates anyway.

Could you explain what it is you're trying to figure out, exactly?

 

[arrell]

Why would symmetry indicate that?

My understanding is that
frame 2 is moving at v with respect to frame 1 (and points stationary in frame 1)
and symmetrically
frame 1 is moving at v with respect to frame 2 (and points stationary in frame 2)

Well, as I said before, your language is confusing, since photons don't "originate" in one frame vs. another, I guess what you're saying is that the source of the photons is at rest in a given frame.

Photons have to emit from somewhere. In this case, I chose that photons emit originally from points a, b, c, and d which are stationary in frame 1.

You also have a profusion of points which are infinitesimally close to each other--if a and c and g and e are all infinitesimally close to one another, what is the point of giving them different labels, for example? Both frames will assigning a and c and g and e the same coordinates anyway.

Only
a and c
b and d
are infinitesimally close.
While
a and b
c and d
are separated by significant distance L.


When the photons originally from a and b are reflected from frame 2, the points at which they are reflected from frame 2 are stationary in frame 2
so now
the photons travel back towards frame 1 which is moving at v with respect to the reflection points and strike at points e and f in frame 1.

 

[JesseM]

Why would symmetry indicate that?

My understanding is that
frame 2 is moving at v with respect to frame 1 (and points stationary in frame 1)
and symmetrically
frame 1 is moving at v with respect to frame 2 (and points stationary in frame 2)

OK, I think I see the problem. It's true that if the two points on the source which shoot the photons have a separation of L in frame 1, then they will have a separation of [tex]\sqrt{1 - v^2/c^2}L[/tex] in frame 2. But what you have to realize is that this does not mean frame 2 will see events c and d (or g and h) happen at a distance of [tex]\sqrt{1 - v^2/c^2}L[/tex] apart; the reason is that if events c and d (or g and h) happen at the same time in frame 1, then they happen at different times in frame 2, so in the time between them the source will have moved somewhat in frame 2. If g happens at coordinates x=0, t=0 in frame 1, and h happens at coordinates x=L, t=0 in frame 1, then using the Lorentz transform we can show that in frame 2, g happens at x'=0, t'=0 while h happens at x'=[tex]\frac{L}{\sqrt{1 - v^2/c^2}}[/tex], t'=[tex]\frac{-vL}{c^2 \sqrt{1 - v^2/c^2}}[/tex].

Note that in frame 2, at time t'=0 the first laser gun was at position x'=0 and the second laser gun was at position x'=[tex]\sqrt{1 - v^2/c^2}L[/tex], so since both guns move a distance vt' in any time interval t', at the earlier time t'=[tex]\frac{-vL}{c^2 \sqrt{1 - v^2/c^2}}[/tex] both laser guns must have been a distance [tex]\frac{v^2 L}{c^2 \sqrt{1 - v^2/c^2}}[/tex] to the right, meaning the first laser gun was at position x'=[tex]\frac{v^2 L}{c^2 \sqrt{1 - v^2/c^2}}[/tex] and the second was at position x'=[tex]\sqrt{1 - v^2/c^2}L \, + \, \frac{v^2 L}{c^2 \sqrt{1 - v^2/c^2}} = L(\frac{1 - v^2/c^2}{\sqrt{1 - v^2/c^2}} \, + \, \frac{v^2/c^2}{\sqrt{1 - v^2/c^2}}) = \frac{L}{\sqrt{1 - v^2/c^2}}[/tex], just where h must have happened in frame 2 according to the Lorentz transformation.

Well, as I said before, your language is confusing, since photons don't "originate" in one frame vs. another, I guess what you're saying is that the source of the photons is at rest in a given frame.

Photons have to emit from somewhere. In this case, I chose that photons emit originally from points a, b, c, and d which are stationary in frame 1.

I understand that, I'm just saying it initially confused me when you said that the photons "originate from frame 1" when what you really meant was that they originate from sources which are stationary in frame 1; this is not a typical shorthand used by physicists.

You also have a profusion of points which are infinitesimally close to each other--if a and c and g and e are all infinitesimally close to one another, what is the point of giving them different labels, for example? Both frames will assigning a and c and g and e the same coordinates anyway.

Only
a and c
b and d
are infinitesimally close.
While
a and b
c and d
are separated by significant distance L.

Yes, I understand that--but I am correct that a, c, g, and e are all infinitesimally close, right? Likewise, aren't b, d, f, and h infinitesimally close? If so, you have eight different labels but only two distinct points in spacetime. What do all these points add to the thought-experiment? Why not just say something like "one photon is emitted at a, another is emitted at b at the same time in their rest frame, the distance between a and b is L in their rest frame, what is the distance in another frame moving at v relative to their rest frame?"

When the photons originally from a and b are reflected from frame 2, the points at which they are reflected from frame 2 are stationary in frame 2
so now
the photons travel back towards frame 1 which is moving at v with respect to the reflection points and strike at points e and f in frame 1.

Again, you're talking about "frames" like they're things which can emit and reflect photons, and which occupy some particular position in space, while the standard meaning of a "frame" is just a coordinate system which fills all of space. I think what you really mean is something like "When the photons originally from a and b are reflected from a mirror at rest in frame 2, the points on the mirror where they are reflected are stationary in frame 2 so now the photons travel back towards the source which is at rest in frame 1 and which is moving at v with respect to the reflection points and strike at points e and f on the source which is at rest in frame 1". But again, I don't see what the mirror-reflection adds to the thought-experiment. For one thing, since you're placing the mirror infinitesimally close to the source, then each photon will be emitted, reflected and reabsorbed by the source at the same spacetime coordinates, regardless of which frame you're using. Second of all, even if the mirror wasn't infinitesimally close so these events happened at different locations, the question of how the mirror is moving is irrelevant--the photons will only make contact with the mirror for a single moment, so it doesn't make any difference whether the mirror is at rest in frame 1, at rest in frame 2, or moving in both frames; this will have no effect whatsoever on the coordinates of the three events as seen in either frame.

 

[arrell]

It's true that if the two points on the source which shoot the photons have a separation of L in frame 1, then they will have a separation of [tex]\sqrt{1 - v^2/c^2}L[/tex] in frame 2. But what you have to realize is that this does not mean frame 2 will see events c and d (or g and h) happen at a distance of [tex]\sqrt{1 - v^2/c^2}L[/tex] apart; the reason is that if events c and d (or g and h) happen at the same time in frame 1, then they happen at different times in frame 2, so in the time between them the source will have moved somewhat in frame 2. If g happens at coordinates x=0, t=0 in frame 1, and h happens at coordinates x=L, t=0 in frame 1, then using the Lorentz transform we can show that in frame 2, g happens at x'=0, t'=0 while h happens at x'=[tex]\frac{L}{\sqrt{1 - v^2/c^2}}[/tex], t'=[tex]\frac{-vL}{c^2 \sqrt{1 - v^2/c^2}}[/tex].

The distance from the plane of frame 1 to frame 2 (orthogonal to the direction of v) is constant. No matter where a, b, c, and d are located on the x-axis of frame 1, when the photons are fired simulaneously towards the plane of frame 2, the photons will travel the same distance and at the same velocity c. If they leave frame 1 simultaneously, there is no way in a physical world where they can arrive at the plane of frame 2 other than simultaneously or the reflected photons arrive back in frame 1 other than simultaneously. The photons are traveling in the y direction in the framework of a Lorentz transform. There is no preference between them. There are no observers or relational photons involved in the x direction during the travel of the photons in this situation. There are just after the fact 'measurers' who measure the separation of e and f, g and h after those points have all been created.

 

[JesseM]

The distance from the plane of frame 1 to frame 2 (orthogonal to the direction of v) is constant. No matter where a, b, c, and d are located on the x-axis of frame 1, when the photons are fired simulaneously towards the plane of frame 2, the photons will travel the same distance and at the same velocity c. If they leave frame 1 simultaneously, there is no way in a physical world where they can arrive at the plane of frame 2 other than simultaneously or the reflected photons arrive back in frame 1 other than simultaneously. The photons are traveling in the y direction in the framework of a Lorentz transform. There is no preference between them. There are no observers or relational photons involved in the x direction during the travel of the photons in this situation. There are just after the fact 'measurers' who measure the separation of e and f, g and h after those points have all been created.

I think the problem here is that you're unaware of one of the foundational concepts of the theory of relativity, the "relativity of simultaneity" (edit: never mind, I just realized the thread title refers to it so you're obviously aware of it, but maybe you don't understand the reasoning behind it). Relativity of simultaneity says that different reference frames have different definitions of what it means for two spatially separated events to happen simultaneously, so that if one frame says two such events happened at the same time, that guarantees that every other inertial reference frame will say the two events happened at different times. To understand this, you have to understand the physical procedure that each observer would use to synchronize two clocks at different locations which are at rest in his own frame. Einstein's suggestion was that each observer would synchronize his clocks using the assumption that light travels at the same speed in all directions in his own frame; so, if I set off a flash at the midpoint of two clocks, and each clock reads the same time at the moment the light from the flash reaches them, then I define them to be "synchronized" in my frame. But you can see that if different observers use this procedure, then clocks which are synchronized in one observer's frame will be out-of-sync in another's frame--if I see you flying by on a spaceship, and you set off a flash at the midpoint of the ship, then from my point of view the clock at the back of the ship is moving towards the point where the flash occurred while the clock at the front of the ship is moving away from that point, so if I assume the light travels at the same speed in both directions in my own frame, that means in my frame the light from the flash will reach the back clock before it reaches the front clock.

The importance of the relativity of simultaneity in SR can be seen from the fact that the first section of Einstein's 1905 to this topic of clock synchronization:

http://www.fourmilab.ch/etexts/einstein/specrel/www/

§ 1. Definition of Simultaneity

Let us take a system of co-ordinates in which the equations of Newtonian mechanics hold good.2 In order to render our presentation more precise and to distinguish this system of co-ordinates verbally from others which will be introduced hereafter, we call it the "stationary system.''

If a material point is at rest relatively to this system of co-ordinates, its position can be defined relatively thereto by the employment of rigid standards of measurement and the methods of Euclidean geometry, and can be expressed in Cartesian co-ordinates.

If we wish to describe the motion of a material point, we give the values of its co-ordinates as functions of the time. Now we must bear carefully in mind that a mathematical description of this kind has no physical meaning unless we are quite clear as to what we understand by "time.'' We have to take into account that all our judgments in which time plays a part are always judgments of simultaneous events. If, for instance, I say, "That train arrives here at 7 o'clock,'' I mean something like this: "The pointing of the small hand of my watch to 7 and the arrival of the train are simultaneous events.''3

It might appear possible to overcome all the difficulties attending the definition of "time'' by substituting "the position of the small hand of my watch'' for "time.'' And in fact such a definition is satisfactory when we are concerned with defining a time exclusively for the place where the watch is located; but it is no longer satisfactory when we have to connect in time series of events occurring at different places, or--what comes to the same thing--to evaluate the times of events occurring at places remote from the watch.

We might, of course, content ourselves with time values determined by an observer stationed together with the watch at the origin of the co-ordinates, and co-ordinating the corresponding positions of the hands with light signals, given out by every event to be timed, and reaching him through empty space. But this co-ordination has the disadvantage that it is not independent of the standpoint of the observer with the watch or clock, as we know from experience. We arrive at a much more practical determination along the following line of thought.

If at the point A of space there is a clock, an observer at A can determine the time values of events in the immediate proximity of A by finding the positions of the hands which are simultaneous with these events. If there is at the point B of space another clock in all respects resembling the one at A, it is possible for an observer at B to determine the time values of events in the immediate neighbourhood of B. But it is not possible without further assumption to compare, in respect of time, an event at A with an event at B. We have so far defined only an "A time'' and a "B time.'' We have not defined a common "time'' for A and B, for the latter cannot be defined at all unless we establish by definition that the "time'' required by light to travel from A to B equals the "time'' it requires to travel from B to A. Let a ray of light start at the "A time'' [tex]t_A[/tex] from A towards B, let it at the "B time'' [tex]t_B[/tex] be reflected at B in the direction of A, and arrive again at A at the "A time'' [tex]{t^'}_A[/tex].

In accordance with definition the two clocks synchronize if

[tex]t_B - t_A = {t^'}_A - t_B .[/tex]

We assume that this definition of synchronism is free from contradictions, and possible for any number of points; and that the following relations are universally valid:--

1. If the clock at B synchronizes with the clock at A, the clock at A synchronizes with the clock at B.
2. If the clock at A synchronizes with the clock at B and also with the clock at C, the clocks at B and C also synchronize with each other.

Thus with the help of certain imaginary physical experiments we have settled what is to be understood by synchronous stationary clocks located at different places, and have evidently obtained a definition of "simultaneous,'' or "synchronous,'' and of "time.'' The "time'' of an event is that which is given simultaneously with the event by a stationary clock located at the place of the event, this clock being synchronous, and indeed synchronous for all time determinations, with a specified stationary clock.

In agreement with experience we further assume the quantity

[tex]\frac{2AB}{t^{'}_A - t_A} = c[/tex]

to be a universal constant--the velocity of light in empty space.

It is essential to have time defined by means of stationary clocks in the stationary system, and the time now defined being appropriate to the stationary system we call it "the time of the stationary system.''

 

[arrell]

I'm trying to ascertain the answers posed, if possible, with clear constructs in the physical world without resorting to 'observers'. As I said, I would like to use 'measurers' rather than 'observers'. As you so aptly point out, 'measurers' are only meaningful if photon detection points are created simultaneously. (and, yes - I have been over and over the first couple sections of the 1905 paper)

As an aside, I like photons. The emission and detection of photons have to be among the 'pristine' events in the universe. And we are in a fascinating day and age where we can actually create the emission and detection of single photons.

If, in frame 2, the arrival of photons at g and h are not simultaneous, then the arrival at one point has to occur first and the arrival at the other point after that. If there is no intrinsic preferentiallity that can be ascribed to the photons emitted from c and d, what physical law(s) do we use to determine at which point g or h arrival/detection occurs first ??

As I say, I would really like to avoid observers for the moment and just try to deal first with this question of simulataneity from physical laws/properties.
--------------------------------

But, if you want to only consider answers based on observers, then the first thing I would want to consider would be the set of ALL observers in frame 2. I haven't spent time thinking about what all of those observers would 'observe', but if and only if we have to deal with observers, I would be wondering if there is a subset A of all observer(s) that would indeed see the arrival at g and h as simultaneous. And of the observers outside of subset A, would exactly half of them see the detection at g as occurring first and exactly the other half see the detection at h as occurring first ? If this were to be the case, then it would seem as though the only 'neutral' conclusion would be that the detections at g and h are intrinsically simultaneous in frame 2.

Of course, the whole theory in I. KINEMATICAL PART stands on the crux of the postulate:

§ 2. (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.