The point being made isn't simply: since Albert's reference frame, which labels him as "at rest" is consistent with the metaphysical statement "Albert is at absolute rest", it must mean Albert's reference frame is making a metaphysical statement about absolute rest.
The point being made is that, given the fact that a photon will inherit the horizontal velocity component of a clock that is in motion, then only a position of absolute rest satisfies the conditions for Einsteinian relativity.
The issue can be distilled down to one foundational question (and possibly a follow up question):
Is a system which is not at absolute rest, necessarily in motion?
If not, why not?
You are confusing Einsteinian relativity with Galilean relativity. Einsteinian relativity says all coordinate systems measure the speed of light to be c, the laws of physics are the same under all coordinate systems, and all coordinate systems are related by lorentz transformations. Inheriting horizontal velocity is not a problem.
Consider the three relevant events:
A photon leaves the emitter.
A photon bounces off a mirror.
A photon hits the detector.
Now consider two coordinate systems S and S'. The apparatus is stationary in S, and is moving at a speed of 0.5 c in S'. The distance between the detector and the mirror in S is 1 lightsecond. Now let us look at the positions of the three events (and hence the speed of the photon) in each case. First, we consider the Galilean transformation.
The time (T) dimension is on the Y axis, and position (X) is on the X axis (both in natural units). The red line is the photon path between the three events in S. The green line is the photon path in S'. Note that, in diagrams like these, velocity is represented by the inverse of slope of the line (speed/time). You can see that the slope is different for S and S'. This is because of the inherited velocity of the photon. It is travelling faster in S' than it is in S. But now consider lorentz transformations.
Note that the slopes are the same in each case. This is because, even though the photon inherits the velocity of the apparatus, lorentz transformations tell us S and S' will not label events with the same time stamps (unlike galilean transformations).
Hence, the issue of absolute rest doesn't even come up. It is entirely superflous.
Oh yes, I should mention that the diagrams shown are for an apparatus oriented along the direction of motion. I can do the same for apparatus oriented perpendicular to the direction of motion, though the diagrams will require an extra dimension.
I may have it wrong; I may not be explaining it correctly; conditioning in Einsteinian relativity might be making it difficult to see; or it could be a combination of the latter two; but hopefully I can state this lucidly enough to try and negate the latter two, or highlight where my misunderstanding lies, and we can hopefully resolve it.
It might initially seem like I am mistaking Einsteinian relativity of Galilean, but I'll try to illucidate why this isn't the case; this will, largely, be done by considering only the Einsteinian thought experiment, and highlighting where the relevant Galilean, Einsteinian, or Lorentzian conclusions would apply.
"not at absolute rest"
Firstly, we need to establish whether a reference frame that is not at "absolute rest" is, by necessity, "in motion"; I would like to say that I could take the fact that you didn't answer the question, directly, as an acceptance of that point, but I don't think I can; however, I think it's a matter of definition, so I think it is a fair point; unless you reason otherwise.
If the above point is accepted, and if "at rest" doesn't imply "at absolute rest", then we can reason that Albert's labeling system (or reference frame), which labels him as "at rest" is actually in motion. This means that the photon in Albert's clock will be imparted with a horizontal velocity component.
Now, under Galilean relativity, this would indeed lead to the vertical velocity component being less than c, such that Albert would measure the speed of light to be lower than c.
However, if this isn't the case, and Albert measures the speed of light to be c, as Einsteinian relativity states, then Albert's measuring instruments must have contracted by an amount unknown to himself, due to the motion of his "not at absolute rest" reference frame. This, however, is a Lorentzian interpretation.
relative to what?
The question is, relative to what is Alberts' "not at absolute rest" reference frame in motion? Presumably it must be an undetectable reference frame, because Albert's, arbitrary, co-ordinate labeling system labels him as at rest relative to all other relatively moving, detectable reference frames (itself somewhat contradictory, but that isn't necessary here).
This too would be a Lorentzian interpretation.
This brings me back to "P does not imply Q" does not mean "P implies notQ". That we cannot say a reference frame implies absolute rest, does not mean we can say a reference frame implies absolute motion. You can metaphysically suppose absolute rest, or you can metaphysically reject absolute rest. It will not have any effect on the coordinate descriptions of Einstein's relativity.
Under Galilean relativity, the vertical velocity component would be unchanged, but one frame describes the photon as also having a non-zero horizontal velocity, and hence a speed greater than c. This does not happen under lorentz transformations because, under lorentz transformations, time is transformed as well as space.
This relates back to the above: "We do not say Albert is at absolute rest" is not the same as "We say Albert is not at absolute rest." You can suppose absolute rest, or you can suppose no absolute rest. The physics won't change.
That depends on what P and Q are though; when dealing with the law of the excluded middle, as I believe we are, not P does imply Q, just as not Q does imply P.
As far as I can see it is a matter of definition; if something is "not at absolute rest" it must, by necessity, be "in motion".
OK, I presume that this is taking the assumption that Albert will measure the speed of light to be c, relative to the carriage. In the case above you mentioned that the vertical velocity component would be less than c, this would be the case without the assumption I take it.
The implications of the co-ordinate labeling system are a little more explicit, however, because they make positive statements about Albert, as opposed to negative statements i.e. they say what Albert is, not what he isn't.
The co-ordinate labeling system labels Albert with a zero velocity and labels him as "at rest", despite the fact that there is motion relative to another reference frame. The question is, does this "at rest" mean at absolute rest, or not? Not expressly stating which it is, is fine; but we are free to deduce what it must mean i.e. what the tacit assumption (or consequence) must be.
If Albert's reference frame, which is labelled as "at rest" is not at absolute rest, then as above, according to the law of the excluded middle, it must mean that Albert's reference frame is necessarily in motion. Which begs the question, relative to what is it in motion? It must be an undetectable reference frame because his co-ordinate labeling system labels all relatively moving, detectable reference frames as "in motion"; it ascribes 100% of the velocity to them, and labels Albert as "at rest" with a zero velocity.
If his reference frame is "in motion", then, if he measures the speed of light to be c, his instruments must be contracted by an amount unknown to himself, due to this motion relative to the undetectable reference frame.
As mentioned, this is the conclusion of Lorentzian relativity.
I think you may be over complicating the notion of absolute rest roosh. Relativity says there is no absolute rest in the sense that there exists no experiment which all observers can agree on which would assign a velocity of v = 0 to some object. This is unlike acceleration, where everybody can agree if an object has a=0, since it will have no force exerted on it.
It is not at absolute rest. Relativity has no absolute rest. Rather one says Albert is inertial. That is, since he has "a = 0", something all observers agree on, there exists a frame with respect to which he is at rest.
This is not valid reasoning. There is no absolute rest in relativity and neither is there absolute motion. All one says is that Albert is at rest in some frames and in motion in others. You are missing the point that if you can not absolutely say Albert is at rest, you cannot absolutely say he is in motion.
To echo what Enkidu has said, you are making a logic error.
Analogously, you are trying to infer statements about the matrix from Newton's laws.
You are not invoking the law of the excluded middle correclty, for two reasons. Firstly, in our case
P: There exists a coordinate system where Albert is "at rest".
Q: Albert is at absolute rest.
What I am saying is
P → (Q ∨ ¬Q)
Hence, no implicit assumption is made about Q.
Secondly, the law of the excluded middle is
(Q ∨ ¬Q)
Either Albert is at absolute rest (Q), or Albert is not at absolute rest (¬Q). If we introduce another proposition M "Albert is in absolute motion", then we have the relations
Q → ¬M
M → ¬Q
¬Q → (M ∨ ¬M)
¬M → (Q ∨ ¬Q)
You are introducing a false dichotomy (Q ∨ M), and trying to justify it with the law of the excluded middle. Your dichotomy assumes absolute space exists, when the very message of relativity is that it is physically meaningless.
The point being made isn't so much that there is, or must be, an absolute rest frame, it is that the co-ordinate labeling system used in Einsteinian relativity treats specific reference frames, or observers, as being at absolute rest. For example, Albert's co-ordinate labeling system labels him as having a zero velocity, just as Henry's labels Henry as having a zero velocity.
While Einsteinian relativity might not expressly say anything about absolute rest, or might even claim that there is no such thing, we can examine it critically and see that it is a tacit assumption, or consequence, of the co-ordinate labeling system.
Again, we can critically examine the consequences of different scenarios and deduce that it is a tacit assumption, or consequence, of the co-ordinate labeling system. That it isn't expressly stated in the formalism of Einsteinian relativity is immaterial.
Albert's co-ordinate labeling system labels him and "his" train (or platform) as having a zero velocity; that is, his reference frame is ascribed a zero velocity, despite there being motion relative another reference frame. 100% of the relative velocity is ascribed to the relatively moving reference frame à la an absolute rest frame.
We need to move beyond the fact that Einsteinian relativity doesn't expressly include an absolute reference frame, or even says that such doesn't exist; we need to apply reason to the possible scenarios and see that it can be deduced that it is a tacit assumption, or consequence, of the co-ordinate labeling system.
Taking the simplistic example, Albert's co-ordinate labeling system will always label him as having a zero velocity, regardless of the motion relative to another reference frame. If a co-ordinate labeling system labels Albert as having a zero velocity, then that co-ordinate labeling system treats him as though he were at absolute rest.
If Albert's "at rest" label doesn't mean "at absolute rest" then it must mean he is in motion relative to an undetectable reference frame, because his co-ordinate labeling system applies 100% of the relative velocity to all relatively moving, detectable reference frames.
Apologies, my formulation may have been a bit sloppy. Firstly though, the concept of "absolute rest" is indirectly referenced through the Galilean Principle of Invariance (PoI) and the special Principle of Relativity (PoR), in the stated consequence that relatively moving observers cannot determine, by experiment, which one is moving*.
The stated consequence of the Galilean PoI, is that an inertial observer cannot determine if they are "in motion" or "at rest"; in the Galilean case it is indisputable that the concept of "absolute rest" is what is being referenced.
The special Principle of Relativity (PoR), together with the equivalence principle, is just an extension of this principle to include accelerating reference frames, such that two relatively moving observers cannot determine which one is moving. Again, this implies that one of the observers might not be moving.
The question is, how can an observer not be moving, when there is relative velocity between them and another observer? The answer is: only if they are at "absolute rest" - a very old, and pretty well understood, concept in scientific philosophy.
Put another way
If we take the example of two lone observers, in the universe, at rest relative to each other; there is no experiment which they can conduct to determine if they are moving; this leaves two options: either they are moving, or they aren't.
If they are moving, and it isn't relative to each other, then it must be relative to an undetectable reference frame.
Clearly they can conduct numerous experiments to determine if the are moving relative to each other, so the only alternative is that they are at "absolute rest" - as per the well understood concept.
Assumption of absolute space
Secondly, we don't need to assume that absolute space exists; the concept of absolute rest is a fairly well understood concept which has existed in scientific philosophy for hundreds of years, even if it has been rejected. All we need to do is to compare the Einsteinian thought experiment to this already existing concept and see what comparisons can be drawn. "Absolute rest" can remain a purely abstract, mathematical construct for the purpose of examination.
What was intended with the point of the excluded middle is that something is either "at absolute rest" or it is "in motion"; if it isn't one, then, by definition, it is the other.
You are saying that Albert's "at rest" doesn't imply he is at absolute rest, and you're right, it doesn't imply either absolute rest or absolute motion, but, logically, it must be one or the other.; it is either "at absolute rest" or it is "in motion". If it is "in motion" then it must be relative to an undetectable reference frame, and Albert's instruments must be contracted by an amount unknown to himself due to this motion - which would be the contention of Lorentzian relativity, not Einsteinian.
*Note: this is not supposed to be an expression, or interpretation, of either principle, it is the often stated consequence of the same.
It is referenced to highlight that relativity says nothing about it. The same way the laws of physics say nothing about metaphysical claims about the matrix.
Hence, relativity says nothing about the claim in blue.
It might be relevant (not sure yet) to point out that relativity does permit a form of absolute motion in the context of spacetime, if motion is defined as a non-geodesic spacetime path. But such a definition of motion and rest is only possible in the context of spacetime, and very different to the concept you are advocating. Hence Minkowski's famous quote:
"Henceforth, space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality."
The above includes the tacit assumption that absolute space is meaningful. You are tacitly assuming absolute space to argue that relativity assumes absolute space. The only thing the law of the excluded middle allows you to say is "They are at absolute rest or they are not". I does not allow you to say "If they are not at absolute rest they must be in absolute motion".
Comparisons can indeed be drawn. And such comparisons show that relativity is independent of such a concept. You can suppose it, or you can reject it.
What you are tendering as the law of the excluded middle is
¬Q → M
which is not the law of the excluded middle. If something is not at absolute rest, it means either A) Something is in absolute motion, or B) Absolute space does not exist.
The concept of absolute rest is referenced in Galilean principle of invariance, not to demonstrate that Einsteinian relativity says nothing about it - for obvious historical reasons - but because the concept of absolute rest was a central concept in Galilean relativity.
The special principle of relativity is just an extension of the Galilean principle - I'm not sure you could argue that it is referenced to demonstrate that Einsteinian relativity says nothing about it. Also, I'm not familiar with any principle of relativity that indirectly references "the matrix" for the purpose of highlighting that nothing is said about it.
I don't think re-ordering the paragraphs, below, should affect the point, but let me know if it does.
Just to address the point of the tacit assumption first.
If we start with the stated consequences first, and then consider a number of applicable scenarios, we can see what can be deduced, as opposed to assumed:
Consequence: Inertial observers cannot determine, by experiment, if they are "in motion" or "at rest" (or if they are moving or not); this applies equally to Galilean relativity as it does Einsteinian, if I'm not mistaken.
Scenario: Two non-accelerating observers at rest relative to each other .
Conclusions: The observers can determine, by experiment, if they are at rest relative to each other - so this cannot be what is meant by the stated consequence above.
If they cannot determine if they are "in motion" or at "rest" it means that they are either "in motion" or "at rest"; if they are "in motion" but not relative to each other then they must be in motion relative to an undetectable reference frame.
If they are "at rest" then, relative to what, are they "at rest"? It can't mean relative to each other, for the reason stated above.
If they are "in motion", relative to what are they "in motion"; obviously it isn't relative to each other?
The point being made is that it references the well-known concept of "absolute rest"; not least because it appears to resemble it in every way.
Consequence: Two inertial observers, moving relative to each other, cannot determine, by experiment, which one is moving.
Scenario: As above.
Conclusions: The consequence here can be fleshed out to include the one above; each observer cannot determine if they are "in motion" or "at rest", but because there is relative motion between them, at least one of them has to be "in motion", but they cannot determine which one it is.
This allows for one of the observers being "at rest", while one is "in motion"; again, the question is, relative to what, is the observer "at rest"? This is where the claim in blue comes in, because the only other alternative is that both are "in motion" and neither is "at rest" i.e. one of the possibilities is that one of the observers is not moving. That Einsteinian relativity doesn't expressly state anything about the claim doesn't prevent deductions being made.
Again, the point being made is that here "absolute rest" is tacitly referenced.
I was expecting this point to be made more forcefully; tbh, I'm not sure of the relevance either. I would say, though, that as long as the thought experiment referenced here is representative of Einsteinian relativity, the points should hold. I think if we treat reference frames as being at absolute rest, according to their own co-ordinate labeling system, then the conclusion we would draw would be similar, if not identical to, minkowski spacetime.
I think the comparisons show that the Einsteinian co-ordinate labeling systems resemble it in almost every way.
Apologies, my use of terminology is not precise at the best of time, but I think you know the point that is being made.
I'm not sure that the concept of absolute space is required for the concept of absolute rest; it may indeed have arisen in that context, but I'm not sure it is required, in the Galilean or Newtonian sense. I think the fact that we cannot determine if we are "at rest" or "in motion" demonstrates that, because it is as true under Minkowskian spacetime as it is under absolute space and time.
The notion of absolute space comes, I think, from the notion that absolute rest and velocity could, theoretically be measured; however, that would be a contradiction in terms, as measurement is, by it's very nature, relative. Absolute motion is a simple 'yes' or 'no' answer to the question, is there motion? Or, with regard to specific objects or observers, is the object/observer in motion, "yes" or "no"?
If we consider the ancient belief that the earth was the centre of the universe and that everything was in motion around it, the idea of absolute space isn't really necessary. Every observable entity in the universe can be "in motion" while the earth doesn't move; of course, there is relative motion between the earth and everything else, but the earth is ascribed a velocity of zero and the relative velocity is ascribed to everything else - this is partly what makes geocentrism "technically plausible" even today.
But, all that being said, we are still left with the question of what Albert's "at rest" means. If we consider just Albert and Henry alone in the universe, Albert's co-ordinate labeling system will still label him as "at rest" despite the relative motion between him and Henry. The question is, relative to what is Albert "at rest"?
Is this "at rest" different to the "at rest" that Albert cannot determine through experiment?
Einstein's relativity is an extension of Galileo's relativity insofar as Galilean relativity was incompatible with electromagnetism (The laws of electromagnetism are not Galilean invariant). Einstein's principle of general covariance explicitly states that coordinate systems are constructs, and not fundamental properties. I.e. Physics is independent of any supposition regarding absolute rest.
The bit in blue is where the problem is. Absolute rest does indeed require an absolute space to define itself. So it is not a case of "at absolute rest" or "in absolute motion". Taking your second scenario as an example, you can say there is no coordinate label which says both observers are at rest. This does not mean, however, that one of the reference frames has to be incorrect, or "less representative of reality". What physically matters is that both observers' frames are inertial, and the fact that no coordinate system has both as "at rest" does not mean we can infer that at least one has to be in motion with respect to some absolute space. Hence, absolute space is not assumed.
And it also resembles a spacetime structure where no such absolute exists.
It was more the relativity principles of each that was meant.
It is probably worth noting that in our deductive process we are speaking solely about relative motion, so the asserted problem with the bit in blue doesn't apply.
A stated consequence of the PoR is that inertial observers cannot determine, by experiment, if they are "in motion" or "at rest".
This leaves us with the possibilities that they are "in motion" relative to something or they are "at rest" relative to something. The question we are asking is, what is that something?
Given that they can determine, by experiment, that they are at rest relative to each other, what is the something, relative to which they cannot determine their motion, or lack thereof?
Similarly, inertial observers moving relative to each other cannot determine if they are moving or if their counterpart is moving; given that they can determine, by experiment, that they are moving relative to each other, relative to what can they not determine their motion?
Absolute motion is a simple "yes" or "no" answer to the question, is there motion? There is no need for absolute space to define it; equally, there is no need for absolute space to define absolute rest, it is, similarly, a "yes" or "no" answer to the question, is something moving. That we cannot determine if one thing or another is moving, does not mean we cannot determine that there is absolute motion; relative motion is proof of absolute motion.
For two relatively moving observers is there motion? Yes, of course there is, so there must be absolute motion; we may not be able to determine which one is absolutely moving, but we can reason that one must absolutely be moving. If neither observer was in motion, then there would be no relative motion, they would be at rest relative to each other.
If we take the example of two observers at rest relative to each other; then relative motion occurs. In order for this relative motion to occur, at least one of the observers has to move; if both observers remained at rest then there would be no relative motion, in anyone's reference frame. So, if both observers say that they themselves didn't move, that it was their counterpart that moved, then one of them must be wrong; that we cannot determine which one is right and which one is wrong is immaterial, we can reason that one of them must, absolutely, be wrong.
With two inertial observers at rest relative to each other, the principle of relativity says they cannot determine
Indeed, which is indistinguishable from what we would conclude if all observers were treated as being at absolute rest by their co-ordinate labeling systems.