Relativity and speed of light basics

[m4r35n357]

I thought it was Lorentz who came up with the theory through his Lorentz Transformations that keep the wave equation invariant.

As Ibix has pointed out, the transformation was known, but Lorenz was trying to force light and EM to behave in a Newtonian way. These attempts were getting quite silly towards the end, so I'm not surprised that Einstein didn't refer to their work ;)

 

[Herman Trivilino]

The invariance of the speed of light in vacuum was a prediction of Maxwell's equations, and is incompatible with Newtonian mechanics.

Actually, this isn't true. Maxwell equation gives "c", but it doesn't tell you that it is invariant.

You also need the assertion that all inertial reference frames are equivalent. That, combined with Maxwell's electromagnetism, gives you invariance of the speed of light. It was Einstein who demonstrated this. Lorentz didn't.

 

[bahamagreen]

My rather blunt understanding is that Galilean thinking was local principles of mechanics would look the same to each individual inertial observer, Maxwellian thinking was that local principles of electromagnetism would look the same for each individual inertial observer, and that Einstein was perhaps the first to realize a choice needed to be made in order to avoid a paradox when attempting to make these local principles look the same when distributed among all possible inertial observers... the dilemma being that either the principles will look different for different inertial observers, or that the principles will look the same for all inertial observers if time, space, and simultaneity are allowed to do something new. This latter choice assumes constant c for all inertial observers and the "something new" being special relativity with all it's consequences.

 

[f todd baker]

Perhaps you would find that the principle of relativity, the laws of physics are the same in all inertial frames of reference, is easier to swallow than the constancy of the speed of light. Maxwell's equations predict the speed of light in a vacuum to be c≈3x10⁸ m/s in my laboratory and I have done every experiment I can think of to verify that Maxwell's equations are correct. Maxwell's equations are laws of physics. Therefore the laws of electromagnetism, including the speed of light, will be the same in any other inertial reference frame. In other words, the constancy of c is a necessary consequence of the principle of relativity.

 

[Orodruin]

Maxwell's equations predict the speed of light in a vacuum to be c≈3x108 m/s in my laboratory and I have done every experiment I can think of to verify that Maxwell's equations are correct. Maxwell's equations are laws of physics. Therefore the laws of electromagnetism, including the speed of light, will be the same in any other inertial reference frame. In other words, the constancy of c is a necessary consequence of the principle of relativity.

This does not logically follow. For example, Maxwell's equations could be the manifestation of the laws of physics in the rest frame of a medium, let us call it aether, where the speed is the same in all directions much like sound waves have the same speed in all directions in the sound medium's rest frame but not in a frame where it is moving. Of course, you then end up with trying to measure your movement relative to the aether and will be surprised to find that you are in the aether rest frame. People tried to do this for a long time before a more elegant solution was suggested, that the reason that we find the speed of light being the same in all directions is that the speed of light indeed the same in all directions in all frames and that there is no need for a medium.

 

[Herman Trivilino]

Perhaps you would find that the principle of relativity, the laws of physics are the same in all inertial frames of reference, is easier to swallow than the constancy of the speed of light.

It's not a choice. Both must be swallowed. Contrary to what you seem to be claiming, the latter is not a consequence of the former.

Maxwell's equations are laws of physics.

So are Newton's. Einstein found a way to make it possible for both Maxwell's Equations and the Principle of Relativity to be valid.

Yes, experimental evidence confirms, as you pointed out, that Einstein got it right. But his logic requires both postulates.

 

[starstruck_]

Which textbook is this? Title and author(s)? Someone here might actually be acquainted with it or have a copy, and can help with specific rough spots in it.

It’s University Physics Volume 2 (Richard Wolfson)

 

[ZapperZ]

I'm not sure how everyone was taught SR, but it may be useful (especially for those who did not have a formal introduction to it) to once again go back to the very beginning and show the issue mathematically.

This document here shows things explicitly that leaves no doubt of the issue that was faced before SR:

https://www1.maths.leeds.ac.uk/~serguei/teaching/gr.pdf

There are two important points that are relevant to this discussion:

1. Newton laws are invariant under Galilean transformation (Section 1.5). This is crucial because it means that our laws of physics are the same in any inertial reference frame.

2. The original Maxwell Equations are NOT invariant under Galilean transformation (Section 1.8.5). This means that the speed of light is different in different inertial frame from the source.

So by the beginning of the 1900, this was a major conundrum in physics because #1 and #2 are incompatible with each other. This problem was well-known by most physicists at that time, and this was the impetus for Einstein to think about the problem that led to the postulate of SR and SR theory itself.

Zz.

 

[Justin Hunt]

the postulate is that light is constant for all observers. But, do scientist ever talk about how light actually travels versus the observed behavior? My point is that you can't actually watch a photon travel through space. the only way to observe light is either with your eyes or an artificial detector. But, the light you observe, can be distorted in many ways (such as red shift, gravitational lensing etc.).

My understanding is that light propagates through the universe at a constant speed. But, unless light can be at multiple places at the same time, it has to actually be at a certain location at any given time (even though nothing is observing it). This brings me to my question; if you are traveling at realistic velocities, can the relative speed between you and photons of light be less or greater then C even though you will still "observe" light to be going exactly C (due to distortions in time and space)?

 

[ZapperZ]

This brings me to my question; if you are traveling at realistic velocities, can the relative speed between you and photons of light be less or greater then C even though you will still "observe" light to be going exactly C (due to distortions in time and space)?

This last part makes no sense. What makes you KNOW that you are traveling at "realistic velocities"? With respect to WHAT?

If you somehow are experiencing "distortions in time and space" while another observer does not, then there is something different about one inertial reference frame versus another, and this not only will run counter to SR, but also to Newtonian mechannics!

Zz.

 

[Orodruin]

the postulate is that light is constant for all observers. But, do scientist ever talk about how light actually travels versus the observed behavior? My point is that you can't actually watch a photon travel through space. the only way to observe light is either with your eyes or an artificial detector. But, the light you observe, can be distorted in many ways (such as red shift, gravitational lensing etc.).

My understanding is that light propagates through the universe at a constant speed. But, unless light can be at multiple places at the same time, it has to actually be at a certain location at any given time (even though nothing is observing it). This brings me to my question; if you are traveling at realistic velocities, can the relative speed between you and photons of light be less or greater then C even though you will still "observe" light to be going exactly C (due to distortions in time and space)?

I think you have misunderstood what "observer" means in this context. It is not necessarily connected to someone actually watching the light as much as it is related to how the light behaves in different inertial frames.

 

[bahamagreen]

the postulate is that light is constant for all observers. But, do scientist ever talk about how light actually travels versus the observed behavior? My point is that you can't actually watch a photon travel through space. the only way to observe light is either with your eyes or an artificial detector. But, the light you observe, can be distorted in many ways (such as red shift, gravitational lensing etc.).

My understanding is that light propagates through the universe at a constant speed. But, unless light can be at multiple places at the same time, it has to actually be at a certain location at any given time (even though nothing is observing it). This brings me to my question; if you are traveling at realistic velocities, can the relative speed between you and photons of light be less or greater then C even though you will still "observe" light to be going exactly C (due to distortions in time and space)?

About "seeing light"... the mean free path of light within the aqueous and vitreous humor of the eyeball is very short - the light that reaches your retina is "brand new fresh" emitted/scattered light from immediately in front of the retina... at standard pressure and temperature at sea level the mean free path of light in air is about 15 meters...

About your last question... I think "relative speed between you and photons" depends on whether this speed is being noticed by you or by another... you will always measure it as c if you are inertial, but your closing velocity measured with respect to those photons by another but also inertial other may be c +/- some value, even as it is constant c to you.

 

[Herman Trivilino]

the postulate is that light is constant for all observers. But, do scientist ever talk about how light actually travels versus the observed behavior?

Are you asking if scientists ever talk about how light actually behaves versus the way it's observed to behave? Science is based on observations, I don't understand what the adjective "actually" implies in this context.

 

[bahamagreen]

Are you asking if scientists ever talk about how light actually behaves versus the way it's observed to behave? Science is based on observations, I don't understand what the adjective "actually" implies in this context.

Maybe a distinction between "observe" and "measure" would help here...

A scientist may observe that a light is red, and measure its frequency. He will talk about the light's frequency as one of its "actual" characteristics/behaviors, but not of the phenomenological quale of its red appearance to his observation as some "actual" property of the light itself, right?

 

[Orodruin]

Maybe a distinction between "observe" and "measure" would help here...

No. You cannot make a distinction. They are typically synonymous.

A scientist may observe that a light is red, and measure its frequency. He will talk about the light's frequency as one of its "actual" characteristics/behaviors, but not of the phenomenological quale of its red appearance to his observation as some "actual" property of the light itself, right?

Light does not have a frequency on its own. The frequency is a property of the light-observer system.

 

[Herman Trivilino]

Maybe a distinction between "observe" and "measure" would help here...

A measurement is an observation.

 

[Justin Hunt]

I am sorry if I wasn’t clear, but my issue is that the universe has to exist whether it is being observed or not. Unless we are living in a simulation and things only exist when we observe them... anyways, photons have to propagate through the universe some how. They travel at the speed of light after all and follow straight paths, which can be bent by gravitational forces etc. but, at any given time a particular photon has to be somewhere in the universe at a given time, unless it can be in multiple locations at the same time.

Now, if oberservers are moving at realistic velocities relative to each other they will both measure the same speed of light due to the warping of space and time predicted by SR. My thinking is that SR is more of an optical illusion rather than actual changes to the universe around you. The photon of light doesn’t care how you move about the universe, it will follow its set path regardless. However, your relative velocities will change how you perceive said photon. It is you that is changing, not the photon.

Another example is that if we observe a spacecraft moving at realistic velocities compared to us, then it appear contracted along it length. However, that spaceship sees the universe as contracted along its length including us. Obviously, we can’t both be length contracted at the same time. Which makes me think it is more of an illusion rather than a physical change.

 

[Orodruin]

I am sorry if I wasn’t clear, but my issue is that the universe has to exist whether it is being observed or not.

Again, you have misunderstood the meaning ascribed to ”observer”.

They travel at the speed of light after all and follow straight paths, which can be bent by gravitational forces etc. but, at any given time a particular photon has to be somewhere in the universe at a given time, unless it can be in multiple locations at the same time.

I strongly suggest you stop thinking of photons as little balls of light. In fact, photons do not have a well defined position operator. Also, the question of what ”really” is is not a physics question, it is purely philosophical.

I am sorry, but your posts make it painfully clear that you have gotten the wrong notion about what length contraction is and relativity in general. There is no ”warping of spacetime” in SR and regerdless of what observer’s rest frame you use you are describing the same physical situation - just using a different set of coordinates.

 

[Dragon27]

Now, if oberservers are moving at realistic velocities relative to each other they will both measure the same speed of light due to the warping of space and time predicted by SR. My thinking is that SR is more of an optical illusion rather than actual changes to the universe around you. The photon of light doesn’t care how you move about the universe, it will follow its set path regardless. However, your relative velocities will change how you perceive said photon. It is you that is changing, not the photon.

I'm not sure, what all of this means, but just for your information: the effects of SR are not optical illusion, when we say that an observer observes (or "sees") something, it is usually meant than an observer has already taken into account the way he obtained information (like, for example, if I see an object with my eyes, I should take into account the finite speed of light, and the time it takes for it to reach my eyes, and realize that the object "now" isn't there, because it moved forward, and what I see with my eyes has happened in the past), and reconstructed the positions and speeds of all the relevant objects in his frame of reference. And still he observes the counter-intuitive SR effects.

Another example is that if we observe a spacecraft moving at realistic velocities compared to us, then it appear contracted along it length. However, that spaceship sees the universe as contracted along its length including us. Obviously, we can’t both be length contracted at the same time. Which makes me think it is more of an illusion rather than a physical change.

No, it's a real effect. One can easily see that after having studied and understood SR properly. But it is an "illusion" in a sense, in that the object doesn't actually changes, for example, if you suddenly started to move relatively to it, but your frame of reference, your hyperplane of simultaneity and perception of time and space in this new frame of reference change so that the object length is perceived differently. But the object's proper length (in his own frame) stays the same.
You say "we can’t both be length contracted at the same time". But it's not "at the same time", there's no logical contradiction here. Two different observers don't even have the same "the same time" (relativity of simultaneity)! Each one is looking at the other from a different frame of reference. Each observer has his own clocks and rulers. Obviously, they don't have to agree on their measurements.

All of that misunderstanding and confusion is easily resolvable by properly studying of SR and actually getting it, believe me. But you won't get it, if you keep asking malformed questions and refuse to go out of your "box" of thinking. You should get a good textbook and work through it.
I myself only started actually understanding SR when I saw how the 4-dimensional Minkowski space-time works. Looking at the basic effects (length contraction, time dilation, relativity of simultaneity) just didn't do it for me (although I was able to use them to calculate and even resolve some paradoxes). The 4-dimensional picture made everything click.

 

[Justin Hunt]

@Orodruin the basic principal of SR is that the speed of light is intrinsic. All observes measure the same speed of light. Time dilation and length contraction are the way this happens. If there was no time dilation nor length contraction, then the observes would not be measuring the same speed of light. Observed and measure are just that. You only get the end result. The photon isn’t going to tell you all the distortions that it underwent on its way from its source to your instrument. SR is only going to tell you what the end result is, the measured quantity.

@Dragon27 you need to understand that SR is just a model that given information X Y Z you can make a prediction that matches reality. The only information you need to know about the objects velocities are their relative ones. There are an infinite number of combinations of actual velocities that result in the same relative velocity and will all have the same result. There is no universal observer in SR, but you could set any object in the universe as a universal observer if you wanted to and it would have no effect on your predictions, only make it more complicated.

 

[Orodruin]

the basic principal of SR is that the speed of light is intrinsic. All observes measure the same speed of light. Time dilation and length contraction are the way this happens.

I am fully aware how SR works, I teach relativity at advanced university level. Time dilation and length contraction are results of this, not the other way around, and you really cannot understand them properly unless you also account for relativity of simultaneity, which most laymen seem to ignore.

Observed and measure are just that.

This is your own definition. You really should avoid making your own definitions and trying to impose them on the standard nomenclature.

The photon isn’t going to tell you all the distortions that it underwent on its way from its source to your instrument.

Again, stop talking about photons. It is a common problem that people talk about photons when what they really should be talking about is light pulses. Things such as frequencies are not inherent to a light pulse, they are just components of a null 4-vector (the 4-frequency) in a particular frame. Such frames are typically what is connected to an "observer" in the sense that if someone measured the frequency using that frame they would get a particular result. However, you cannot say that a light signal has a particular frequency without implicitly referring to a frame. It would be like saying that a mountain had a slope to your right without specifying how you are turned.

SR is only going to tell you what the end result is, the measured quantity.

As I have already said, this is the only thing that is physically relevant. Anything else is just philosophy and does not belong in this discussion.

 

[Dragon27]

@Justin Hunt

And what SR tells us? We can have a giant, long spaceship, whose proper length we know (and can easily measure, while it still stands). We can then have it move relatively to us with an ultrarelativistic speed (by either accelerating it, or accelerating ourselves). We can measure the speed of it by measuring the time it takes for the bow of the ship to go from one milestone to the other (the distance between which we know). We can also measure the time it takes for the ship (as an object with length) to go through one milestone (from the bow to the stern), and (knowing its speed) to calculate its length. And SR tells us that we would get the length that its completely different from the proper one. The results of the measurements would disagree with our normal "Newtonian" intuition. Doesn't really look like an optical illusion to me.​

 

[stevendaryl]

Another example is that if we observe a spacecraft moving at realistic velocities compared to us, then it appear contracted along it length.

It's a coordinate effect, which is something more real than an illusion, but less real than a coordinate-independent effect.

If a 100 meter spaceship is length-contracted so that it is only 1 meter long, then it is literally true that at the same time that one end of the spaceship is at one location, the other end is 1 meter away. That's not an optical illusion. However, the notions of "at the same time" and "the distance between two points" are coordinate-dependent (or frame-dependent, if you prefer), so it's slightly less real than if the spaceship is crushed to only be 1 meter long.

 

[Orodruin]

It's a coordinate effect, which is something more real than an illusion, but less real than a coordinate-independent effect.

If a 100 meter spaceship is length-contracted so that it is only 1 meter long, then it is literally true that at the same time that one end of the spaceship is at one location, the other end is 1 meter away. That's not an optical illusion. However, the notions of "at the same time" and "the distance between two points" are coordinate-dependent (or frame-dependent, if you prefer), so it's slightly less real than if the spaceship is crushed to only be 1 meter long.

To illustrate this further, look at the corresponding problem in Euclidean geometry. Let us take a band $0 < x < 1$ in $\mathbb R^2$ and you ask what is the "width" of this band. You might instinctively say "one" since this is the range of the $x$ coordinate. However, if you instead introduce a coordinate system $S'$ that is turned 45 degrees relative to the original one and define "width" in the same way, i.e., the range of $x'$ values for a given $y'$ value, then you will suddenly find that the "width" in this coordinate system is $\sqrt{2}$. Did the band change just because you changed coordinates? Obviously not. The only thing that changed was what you mean by "width". This is an exact analogy to length contraction in SR. The object that you are describing in no way changes, what changes is the meaning of "length" as it is typically defined as the distance between to points at a fixed time and different inertial frames have different definitions of what it means for two events to be simultaneous.

 

[Justin Hunt]

I find length contraction very interesting and have pondered over any mechanism that could be causing it. I would imagine it has something to do with the fact that your velocity won’t change where your height and width break a plane, but it will determine how long it takes your length to pass through said plane. This somehow translates objects going near light speed into pancake type objects.

@Orodruin I understand photons are electromagnetic waves that have properties that can be defined from quantum physics, but it does have information based on what source it came from and the path it took. There is no universal observer to set a base value for properties such as frequency, but we could define an observer such that the relative velocities between the observer and source object is zero, the difference in gravitational potentials is zero etc. so that the observer does not impose any additional distortions to the object. Every object would have a different base observer in this case.

 

[Orodruin]

I find length contraction very interesting and have pondered over any mechanism that could be causing it.

There is no "mechanism" and therefore no "cause". It is just a question of what different observers mean when they say "length".

I understand photons are electromagnetic waves that have properties that can be defined from quantum physics,

This is a contradiction in terms. Electromagnetic waves are a classical phenomenon and photons are about as far into quantum field theory as you can go before you start dealing with non-Abelian gauge theories. In the quantum picture, classical electromagnetic fields are most accurately described in terms of the expectation values of coherent states, which are states that are not eigenstates of the photon number operator. As I said, you really do not want this type of complications when your aim is to discuss relativity.

There is no universal observer to set a base value for properties such as frequency, but we could define an observer such that the relative velocities between the observer and source object is zero, the difference in gravitational potentials is zero etc. so that the observer does not impose any additional distortions to the object. Every object would have a different base observer in this case.

Why would you do this? It is completely unnecessary and just a complication that requires you to know the state of motion of the source (at the time of emission as well! Generally the state of motion of the source can change after emission without changing the light signal itself). It is much more convenient to just talk about a light wave with some given 4-frequency, which is a 4-vector that together with the observer 4-velocity will completely determine the observed frequency.

I would also suggest that you leave GR (i.e., gravitation) out of it until you completely understand the SR case as it will introduce a heap of new complications that will render your construction not only unnecessary but also ill-defined in spacetimes that are not stationary.

 

[Herman Trivilino]

I am sorry if I wasn’t clear, but my issue is that the universe has to exist whether it is being observed or not.

Oh, yeah. I definitely missed that!

Whether the universe exists, under any circumstances, is a philosophical question. Note that the same can be said about the existence of anything.

To put this another way, the physics is the same whether or not you believe things exist when you're not observing them.

My thinking is that SR is more of an optical illusion rather than actual changes to the universe around you.

If you can't even think of a possible way to distinguish between those two things then the physics is the same for each of them, too.

 

[Sorcerer]

I am sorry if I wasn’t clear, but my issue is that the universe has to exist whether it is being observed or not. Unless we are living in a simulation and things only exist when we observe them... anyways, photons have to propagate through the universe some how. They travel at the speed of light after all and follow straight paths, which can be bent by gravitational forces etc. but, at any given time a particular photon has to be somewhere in the universe at a given time, unless it can be in multiple locations at the same time.

Now, if oberservers are moving at realistic velocities relative to each other they will both measure the same speed of light due to the warping of space and time predicted by SR. My thinking is that SR is more of an optical illusion rather than actual changes to the universe around you. The photon of light doesn’t care how you move about the universe, it will follow its set path regardless. However, your relative velocities will change how you perceive said photon. It is you that is changing, not the photon.

Another example is that if we observe a spacecraft moving at realistic velocities compared to us, then it appear contracted along it length. However, that spaceship sees the universe as contracted along its length including us. Obviously, we can’t both be length contracted at the same time. Which makes me think it is more of an illusion rather than a physical change.

(bold emphasis in the above quote is mine)

I have to question this notion (perhaps someone more knowledgeable can verify or refute what I say here):

If SR is an optical illusion, then things derived from it must be as well. For example, rest energy, E = mc² (something that is the same in ALL inertial reference frames). But IS it an optical illusion? When energy is converted into mass or vice versa, is that an optical illusion? Is nuclear power an optical illusion? To elaborate, in a nuclear reaction, Δm = ΔE/c². That result, which is very physically real and very measurable, is derivable from the postulates of special relativity. So how can it be merely an optical illusion?

Source on that relation above: https://chem.libretexts.org/LibreTexts/University_of_Missouri/MU:__1330H_(Keller)/21:_Nuclear_Chemistry/21.6:_Energy_Changes_in_Nuclear_Reactions

There are other things derived from special relativity that are NOT dependent upon your frame of reference (that is, that are the same for all inertial frames), such as the magnitude of the spacetime interval, proper time, and other things. These things all come from the postulates of relativity, but are universal for all inertial reference frames, so how can SR just be a collection of optical illusions?

Maybe the only illusions are the ones caused by our biases about how the universe works, which are the result of our inexperience with speeds near that of light?

 

[pervect]

the basic principal of SR is that the speed of light is intrinsic.

I'm not sure what you mean by this.

All observes measure the same speed of light. Time dilation and length contraction are the way this happens.

You omitted the relativity of simultaneity, which is also important. It could be a typo, but from the rest of your statements I believe it is an important omission in your thinking about the subject, an omission which indicates a systematic error in your understanding.

If there was no time dilation nor length contraction, then the observes would not be measuring the same speed of light. Observed and measure are just that. You only get the end result. The photon isn’t going to tell you all the distortions that it underwent on its way from its source to your instrument. SR is only going to tell you what the end result is, the measured quantity.

Except for your omission of the important principle of the relativity of simultaneity, I don't have any serious argument with what you are saying, though I'd also point out that Orodruin knows all this already.

However, the "relativity of simultaneity" doesn't seem to fit into your mental "distortion" model very well, and it's an important effect that you seem to be unaware of. Length contraction and time dilation alone are not sufficient to explain special relativity.

If we consider the Lorentz transform in units where c=1

$$x' = \gamma(x - \beta t) \quad t' = \gamma(t - \beta x)$$

See for instance https://en.wikipedia.org/wiki/Lorentz_transformation, and note that I've set c=1 for ease of exposition. (This can be done easily by using the right units - time in seconds, distance in light seconds, for instance).

When we focus on the second expression, we see that t' depends on both t and x. "Time dilation" explains the gamma factor in the dependence of t' on t, but it does not explain how or why t' depends on x. Length contraction also does not explain why t' depends on x. The thing that explains this mathematical dependence of t' on x is the relativity of simultaneity, and important part of special relativity that should not be ignored.

 

[Sorcerer]

I'm not sure what you mean by this.You omitted the relativity of simultaneity, which is also important. It could be a typo, but from the rest of your statements I believe it is an important omission in your thinking about the subject, an omission which indicates a systematic error in your understanding.
Except for your omission of the important principle of the relativity of simultaneity, I don't have any serious argument with what you are saying, though I'd also point out that Orodruin knows all this already.

However, the "relativity of simultaneity" doesn't seem to fit into your mental "distortion" model very well, and it's an important effect that you seem to be unaware of. Length contraction and time dilation alone are not sufficient to explain special relativity.

If we consider the Lorentz transform in units where c=1

$$x' = \gamma(x - \beta t) \quad t' = \gamma(t - \beta x)$$

See for instance https://en.wikipedia.org/wiki/Lorentz_transformation, and note that I've set c=1 for ease of exposition. (This can be done easily by using the right units - time in seconds, distance in light seconds, for instance).

When we focus on the second expression, we see that t' depends on both t and x. "Time dilation" explains the gamma factor in the dependence of t' on t, but it does not explain how or why t' depends on x. Length contraction also does not explain why t' depends on x. The thing that explains this mathematical dependence of t' on x is the relativity of simultaneity, and important part of special relativity that should not be ignored.

Bold is my emphasis.

Regarding this, how accurate is the following: the Lorentz transformations imply a mixing of time and space coordinates when a coordinate transformation occurs, and this is a direct result of (even a mathematical expression of) the relativity of simultaneity.

The reason I am thinking this is because it seems to me that the logical conclusion that must be drawn from these coordinate mixes is that simultaneous events in one frame cannot be simultaneous in all frames, and it also feels like an if and only if relation: that is, it also must be true that if simultaneity is relative, then any distance or time transformation must including the mixing of coordinates (by that I mean that transforming from S to S' must be going from the function S(x) to S'(x',t') and S' to S must be S'(x') to S(x, t), and the same with time transformations [I am not sure if that is legitimate notation, but I think what I'm trying to say is clear]).

Does any of that make sense and is any of it true?

Thanks!