Light shone in a train bouncing off mirrors

[Dale]

The physical solution to a physical problem cannot be frame-dependent.

Are you still talking about the fairness of the duel? How is "fairness" physical? That seems like a big stretch.

"Fairness" is even less physical than simultaneity, so what would be wrong with it being frame-variant as long as the duellers agree on the rules. In any case, if someone is so stupid as to duel are they going to be smart enough to understand the relativity of simultaneity anyway?

It might happen that, under normal, every-day circumstances, the proper times intervals are always identical, but, under extraordinary circumstances we are not accustomed to, we discover that the intervals are not identical.

In what specific extraordinary circumstances are you suggesting that the proper time could ever be frame varying?

 

[atyy]

It might happen that, under normal, every-day circumstances, the proper times intervals are always identical, but, under extraordinary circumstances we are not accustomed to, we discover that the intervals are not identical. Hence if we do not want to be blinded by prejudices, we must look at the physical causes. I can list one thousand examples of scientists who asked themselves these “why questions” and thus logically predicted that the experimental results would be different with better instruments or under different circumstances and such statements were later confirmed by better observations.

Yes, SR is an experimentally verified description of nature, and may be wrong at some level. However, we haven't reached that point yet, eg. Mattingly, http://relativity.livingreviews.org/Articles/lrr-2005-5/ .

There is also the caveat that maybe complex biological clcoks do not behave like ideal clocks. However, it is a reasonable assumption, since biological and atomic clocks are made of the same stuff, and it is the standard assumption in the twin paradox. Nonetheless, it is an assumption.

Therefore, I say: let us look at the causes. We have, on the one hand, the “warnings” (light signals) and, on the other hand, the “shots” (bullets). Tell me “something”, something at least about the physical nature of the interactions that cause the motion of the light signals and of the bullets. I do not need that we find here the ultimate explanation of those interactions. I just want to be told if, for you, the nature of the interaction is in both cases such that, for motion purposes, light signals and bullets behave identically. If the answer is yes, we can still discuss a little more why. If the answer is no, there may still be a (smalller) room for discussion.

Please look at the problem as follows: SR is telling me that, even if different observers observed the “warnings” to be behaving unequally, they also observed the “shots” to behave unequally, so that one thing compensates the other and in the end they all agree that the duellers disposed of the same number of “ticks” to do their “tricks”. I have no problem with that if the “warnings” and the “shots” are both made of the same thing (light). But when the “warnings” are light and the “shots” are bullets, if you wish to maintain the same solution, you must then hold that light and matter, at least in so far as motion pattern is concerned, are analogous or analogous under certain circumstances. It may be so. But please say it expressly.

But I don't believe that different observers observe relativistic invariants unequally. The proper time is a relativistic invariant. Yes, with respect relativistic invariants, light and matter behave analogously.

I know there is a different way of looking at it involving a frame-dependent concept like simultaneity, and having all sorts of frame-dependent quantities cancel out in different frames, but that is very difficult and always makes my head spin.

I am far from postulating that light moves through a medium (a hypothetical aether). Maybe it does not take the motion of the source for another reason, namely because it “self-accelerates” itself (the electric field creates a magnetic field, which in turn creates an electric field) and this self-acceleration follows a pattern that is independent of the motion of the emitting or reflecting matter. Both models may be analogous for practical purposes. But if you believe that any of them is applicable to the light “warnings”, then you have to explain if it is also applicable to the mechanical “shots”.

The reason is that all the known laws of physics have Lorentz invariance.

http://pdg.lbl.gov/2008/reviews/rpp2008-rev-qcd.pdf
http://pdg.lbl.gov/2008/reviews/rpp2008-rev-standard-model.pdf
http://arxiv.org/abs/gr-qc/9512024

It is true that we don't know how to calculate from the standard model many everyday phenomena. However, there are links. For example, from QCD we can get that quarks make up protons. From QED we know that protons and electrons bind into hydrogen atoms. From QED we can transition into relativistic QM from which we can transition to non-relativistic QM from which we can transition into condensed matter physics.

However, it is the Lorentz invariance, rather than any detailed reasoning that is important. For example, suppose I have spherically symmetric ball. If I turn it upside down, it will look the same. Do I need an explanation from the standard model about how the atoms in the ball interact with each other and with my non-spherically symmetric hands which inverted the ball, and with my non-spherically symmetric eyes and brain? Or can I just argue that the ball is spherically symmetric?

Does SR postulate that the mechanical shots follow a Galilean pattern of motion when they are at rest in a frame and, as their speed increases, their motion pattern progressively conforms to light’s motion pattern?

It is clear that in vacuum light is faster than anything else, while in media it may not be. But the question is: As the light warning is slowed down by the medium, does it mean that it progressively adapts its motion pattern to conform to the mechanical bullet’s motion pattern?

I don't understand what "their motion pattern progressively conforms to light’s motion pattern" means. The shots always obey Lorentzian relativity even at low speeds. Light in vacuum, and light in media and bullets always obey Lorentzian relativity.

 

[Saw]

Are you still talking about the fairness of the duel? How is "fairness" physical? That seems like a big stretch.

"Fairness" is even less physical than simultaneity, so what would be wrong with it being frame-variant as long as the duellers agree on the rules. In any case, if someone is so stupid as to duel are they going to be smart enough to understand the relativity of simultaneity anyway?
In what specific extraordinary circumstances are you suggesting that the proper time could ever be frame varying?

Hello DaleSpam. I had missed your posts. We were not used to so authoritative visits here, in this chaotic thread, since long ago. Maybe I should write a recapitulation. I'll try in the next post.

 

[altonhare]

Full information about what? About every physical aspect of the problem? Certainly specifying the velocities of two objects doesn't specify everything physical about them (it doesn't specify their positions for example). And what's more, when you say both must agree about which velocity is "greater", you only seem to be talking about one aspect of the velocity and ignoring other aspects like the angles of the two velocity vectors.

It's not even clear what you mean by "greater" when talking about two velocity vectors--you obviously aren't just talking about the norm of each vector since that would just be the speed which is always positive, but before you argued that an object with negative velocity on the x-axis had a "smaller velocity" than one at rest. If we have only a single spatial dimension, then all velocity vectors are parallel to each other so we can just talk about positive or negative velocities on this axis, but what if we have 2 or three spatial dimensions and non-parallel velocities? You can take the component of each velocity vector which lies parallel to a particular axis and then the components will each be either positive or negative, but in this case, which has the greater velocity depends on what axis you want to use.


For example, suppose we have an x-y-z coordinate grid, and we break down two velocity vectors into their x and y and z components. For example, object A has Vx = 5 meters/second and Vy = -3 meters/second and Vz = 0 meters/second, while object B has Vx = 4 meters/second and Vy = 12 meters/second and Vz = 0 meters/second. Obviously object B has a greater speed in this frame, but which has a "greater velocity"? The x-component of A's velocity is greater than B's, but the y-component of B's velocity is greater than A's. So do you claim there is some absolute truth about whether A or B has a "greater velocity" here, where "greater velocity" does not just mean "greater speed" (i.e. greater norm of the velocity vector)? If so, what is it? Please give me a specific answer to this question about whether A or B has a greater velocity here.

By "full information" I mean the observers incorporate all their data into their qualitative conclusions. If their qualitative conclusions conflict the only reason is that they neglected to measure a critical attribute.

About the velocity question. We must be careful about what we mean when we invoke "direction". Observers in rotated coordinate frames concluded that the object had this X extent in a direction parallel to a line through another object while Dx away from that object and Y extent in a direction parallel to a line through another object while Dy away from that object. The two other objects were necessary to define the observer's coordinate system and the distances were critical components. Different rotated observers had data containing apples and oranges because their extents were all reported along with Dx's and Dy's, which were not the same.

The same occurs when an observer uses a rotated coordinate system to measure velocity. The velocity is measured in a direction defined by some reference object. An observer that "turns around" will be talking about velocity in the direction of x2 while the original observer will be talking about velocity in the direction of x1. They are, again, comparing apples and oranges. This is not to say, of course, that they cannot combine their quantitative data and resolve the issue, it just means they cannot draw logically comparable qualitative conclusions without quantitatively accounting for "reference frame". In this case they are not really qualitatively comparing so much as normalizing their data to have the same units.

In the train/embankment scenario we are not talking about rotated coordinate systems. The two observer's qualitative conclusions are directly comparable, they should both be apples. Both observers should conclude that the first apple is redder than the second apple. If they were using "rotated" coordinate systems then they would be asking each other if O1's apple is redder than O2's orange (not quite a perfect analogy here but I use it nonethless to avoid being overly pedantic).

In cases of length, colocal "simultaneity", and velocity observers comparing apples reach the same qualitative conclusions. In the case of noncolocal "simultaneity" they do not. It is my argument that apples are always comparable and, when we find that observers are coming to different qualitative conclusions when comparing apples, we need to reassess. It appears that noncolocal simultaneity is simply meaningless because it leads to logical contradiction. Local simultaneity does not need the "simultaneity" at all, we can simply say that A and B are local i.e. occupy the same location (or came in contact).

If you claim there is some objective truth about which object has the greater velocity along the x-axis, then once again it seems you must believe in some sort of ghostly "true" x-axis.

Indeed I do not believe this. In the most essential fundamental sense observers talk about motion of an object in a direction toward another object. They may pick 2 or 3 objects which they determine to be along perpendicular lines of site for convenience. The objective truth is that object X is moving faster than Y toward one (or more) reference object(s). If two observers are using different reference object(s) then they are not comparing apples.

Your comment about "greater velocity" seems unclear as I discussed above. And in SR different coordinate systems do disagree about which of two objects has a greater extent in a specific direction, because of length contraction. Even if you think there is an absolute truth about which frame's judgment is "really" correct, do you disagree that according to the standard way of defining SR coordinate systems, disagreements between coordinate systems about which of two objects has a greater "length" are quite possible?

Only if the observers are superficial and care not to actually think about what they mean by "length". They are referring to extent in a specific direction. I went over this a few times. In rotated coordinate systems observers are not comparing apples. Additionally they will know they are not comparing apples and will not be led astray to coming to contradictory conclusions.

But you don't believe that certain quantities, such as speed, represent "objective realities", and thus you don't believe there need to be objective truths about which object has the greater speed--is that right? If so, why can't you accept the possibility that quantities like velocity or length may also fail to represent "objective realities"?

It's not that speed isn't an "objective reality", it's simply not the whole story. Two observers would be foolish to come to qualitative conclusions based on excising certain data. Or you could argue that they are trying their best to incorporate everything relevant. If they are but they still arrive at a qualitative contradiction, the only answer is that they did not include everything relevant. Just as with the metal block and the brick. Two observers thought that, as long as their rulers were identical, they would come to the same conclusions about whether the brick or the metal was bigger. When they came to qualitatively contradictory conclusions did they throw up their hands and say,"That's just how it is sometimes!" or did they look for a reason?

In the brick/metal example the two observers left out temperature. In the "speed" example they leave out direction.

The "procedure I discussed above" was just the standard one for constructing inertial coordinate systems in SR, and the result is that the coordinates assigned to the same event by different observers are related by the Lorentz transformation. Do you not understand that under the Lorentz transformation, even if two coordinate systems have their spatial axes oriented the same way, if the two coordinate systems are in motion relative to one another they can disagree about which of two objects has a greater length? If so I can give you a numerical example, if that's what you're asking for. But if you're asking me to justify something else, please be specific about what it is.

So there are no rotations of coordinate axes. O1 and O2 are just watching A and B fly away (B moving faster relative to O1 and O2 at the outset and they are identical in size). Or O2 is standing on B. There is no way that, without rotating their axes, they will come to contradictory conclusions about which one is bigger. O2 can stand on B and turn around but must now use negative coordinates. i.e. O1 sees A and B flying away from him/her while O2 sees B approaching. O2 will not conclude on a length contraction of A relative to B but rather a length expansion:

Va1 = xa1*i + ya1*j
Vb1 = xb1*i + yb1*j

with (xa1²+ya1²)^1/2 < (xb1²+yb1²)^1/2

Va2 = (xb1-xa1)*i+(yb1-ya1)*j
Vb2 =0

In the length contraction formula the normed velocity of a will be less than the normed velocity of b for both observers, leading to the conclusion that B is shorter than A for both observers. When O2 turns around on B s/he is staring at "coordinate system" full of negative numbers. A will appear to be moving away from B, Va<0, the conclusion is that B is moving faster than A and that A is longer than B.

But by "visualizable" do you mean actually being able to form a visual image of the shape in your mind's eye, or do you just mean that the shape can be defined using the language of mathematics?

This could start venturing far off topic, but I mean the former. In this case it's an object, but it doesn't exist because it lacks location.

We can't picture such 4D objects visually because our brains have become adapted to find 3D space intuitive, but I imagine if you could somehow take the brain of a newborn and feed it sensory inputs from a simulated 4-dimensional body in virtual reality, as it grew up it would be able to visualize 4-dimensional shapes.

I don't buy any of this. If you're going to believe such things you may as well also believe in Santa Claus, the Flying Spaghetti Monster, and the Tooth Fairy.

Visualization isn't a very rigorous criterion in any case because it depends on the contingent details of our biology and history, whereas mathematics allows us to define the notion of "shape" in a completely rigorous way that doesn't depend on what we can visualize (and arguably a blind person can't 'visualize' any shapes at all, although I suppose they could imagine what it would feel like to run their hands over it).

Mathematics was created by humans and definitely depends on our biology/history directing our cognition.

Shape is a static concept. It is the primary, most essential quality of an object. I am not talking about the quantitative description of it. I'm talking about the primary quality a thing has independent of any other things. Other attributes we assign like color, age, smooth, rough, etc. are relational, they depend upon a comparison to another object. Shape is primitive. An object doesn't have shape by comparison with objects that lack shape because there are no objects that lack shape. An object has shape because if it didn't it would be nothing, i.e. it would not be an object. An object has shape even if it is the only object in the universe.

Before one can use mathematics to quantitatively describe/characterize an object, one must point to it, or at least a model of it. This is the only test. Without this crucial component the equations may refer to an object, but maybe not. I don't have a problem with equations by themselves, I have a problem with the proposing of physical interpretations/explanations which are literally unimaginable. These are the same kinds of interpretations and explanations provided by traditional religion. In fact, traditional religion (older times) at least proposed the anthropomorphic God, and other entities with shape, although the acts these hypothetical entities performed were no less than supernatural. Today religion has forsaken these "God objects" for "God the concept" i.e. "he" is everywhere and nowhere, etc. The point is, we are asked to accept it on faith rather than on the ability to visualize the real thing ourselves, which we are disbarred from. I don't care if you predict the weather perfectly for the next week or year, and have equations that show it quantitatively, if you tell me it was an unimaginable mechanism behind it you don't understand any better than I do. You've just done the requisite research to produce a good correlative model. I have no reason to believe humans are somehow precluded from understanding something about the universe, we have the capacity to understand anything.

If you are the observer that sees the ball going 15 m/s down and see your counter part going 25 m/s down, wouldn't you agree that your counterpart would measure the ball going at 10 m/s up relative to him? Don't you agree that in this case there is no quantitative disagreement once the observers clearly define the context of their measurements to one another.

There's never quantitative disagreement once observers apply the correct equations to the situation so that they are using common units (identical ticks of a clock and identical meter-sticks). This has nothing to do with what I'm talking about, which are qualitative conclusions. In fact, I took the effort before to distinguish between quantitative and qualitative.

Another example: Imagine looking at a 2 dimensional square in three space. There are two observers, one(1) looks on perpendicular to the plane the square is in. The other observer(2) is looking at an angle so he sees an odd looking rectangle. (2) would necessarily compute a smaller area than (1) since his view is screwed up. Say observer (2) gets .5 m² and (1) gets 1 m². You may think that (1) has the correct area, but notice. If (1) did some math to determine how his answer would change if he were looking from (2)'s position, don't you agree that he would get .5 m² there by agreeing that (2) measured the correct value based on his view? So the clear contridiction in their quantitative measurements is reconciled once you transform properly to between the two frames in this case. Agree so far?

Again, of course observers will quantitatively agree once they apply the correct equations. This has never been in doubt.

Hello all.

With regards to simultaneity. While browsing past threads I came accros Simultaneity which has the secondmost number of replies (280) and the thirdmosty munber of views (16,900). In #18 Dalespam says:-

--- The point is that simultaneity is an artificial construct arising from the definition of a coordinate system, not something objectively real in its own right. Fundamentally it appears that the universe doesn't care about simultaneity, only about causality. Two simultaneous events cannot be causally connected, so what does it matter if one happened before the other? On the other hand, a cause should always come before an effect, and this is exactly what we see in relativity. A cause will preceed the effect in all reference frames, and for the rest it doesn't really matter. -----

This is quite good, and is in the spirit of what I've been saying in the sense that the universe only cares about causality, i.e. did A hit B or not, not about simultaneity.
This (Dale's) is a very good way of putting it.

In such scenarios as those under discussion, simultaneity seems to matter in the sense that we have introduced a human element of fairness/right and wrong.

Deciding if two spatially separated events are simultaneous is merely a case of applying the agreed definition. A problem which arises in some of these proposed puzzles is that "making two events happen simultaneously" can only be done by making them both causally connected to a single event, such as the throwing of a switch, to set the chain of events in motion. The decision on the simultaneity of two spatially saparated events is an artificial construct and "constucting" the simultaneity of events is engineered. You cannot engineer a situation where things happen without having control of them, and this implies a causal connection. So I think it may be fair to say that unless the events happen to be simultaneous by chance, you cannot "construct" the simultaneity of two events unless the simultaneity is engineered from a single event/cause, or by some other mechanism constucted by a conscious entity. The last proviso is added as a get out clause should my belief that simultaneity cannot be manufactured in other ways than from a single event causally connected to both the events that are required to be simultaneous is wrong.

Note that in Eistein's train and embankment thought experiment the lightning strikes just happen to be simultaneous, they are not "consructed" to be so.

Matheinste.[/QUOTE]

I don't think you need the last proviso. Simultaneity is about the spatial locality of objects and nothing else. An event is simultaneous by definition and, indeed, the term "simultaneous" is generally superfluous except in human endeavors involving duels and trials and such. Nature doesn't seem to care about what we think of as "temporal simultaneity" but rather only about spatial locality and causality.

 

[JesseM]

By "full information" I mean the observers incorporate all their data into their qualitative conclusions. If their qualitative conclusions conflict the only reason is that they neglected to measure a critical attribute.

About the velocity question. We must be careful about what we mean when we invoke "direction". Observers in rotated coordinate frames concluded that the object had this X extent in a direction parallel to a line through another object while Dx away from that object and Y extent in a direction parallel to a line through another object while Dy away from that object. The two other objects were necessary to define the observer's coordinate system and the distances were critical components. Different rotated observers had data containing apples and oranges because their extents were all reported along with Dx's and Dy's, which were not the same.

The same occurs when an observer uses a rotated coordinate system to measure velocity. The velocity is measured in a direction defined by some reference object. An observer that "turns around" will be talking about velocity in the direction of x2 while the original observer will be talking about velocity in the direction of x1. They are, again, comparing apples and oranges. This is not to say, of course, that they cannot combine their quantitative data and resolve the issue, it just means they cannot draw logically comparable qualitative conclusions without quantitatively accounting for "reference frame". In this case they are not really qualitatively comparing so much as normalizing their data to have the same units.

In the train/embankment scenario we are not talking about rotated coordinate systems. The two observer's qualitative conclusions are directly comparable, they should both be apples. Both observers should conclude that the first apple is redder than the second apple. If they were using "rotated" coordinate systems then they would be asking each other if O1's apple is redder than O2's orange (not quite a perfect analogy here but I use it nonethless to avoid being overly pedantic).

In cases of length, colocal "simultaneity", and velocity observers comparing apples reach the same qualitative conclusions. In the case of noncolocal "simultaneity" they do not. It is my argument that apples are always comparable and, when we find that observers are coming to different qualitative conclusions when comparing apples, we need to reassess. It appears that noncolocal simultaneity is simply meaningless because it leads to logical contradiction. Local simultaneity does not need the "simultaneity" at all, we can simply say that A and B are local i.e. occupy the same location (or came in contact).
Indeed I do not believe this. In the most essential fundamental sense observers talk about motion of an object in a direction toward another object. They may pick 2 or 3 objects which they determine to be along perpendicular lines of site for convenience. The objective truth is that object X is moving faster than Y toward one (or more) reference object(s). If two observers are using different reference object(s) then they are not comparing apples.

It seems you are changing your claim somewhat then. You are no longer claiming there is any objective truth about which of two objects has a "greater velocity", but just that there is an objective truth about which of two objects has a greater velocity towards some third "reference object". But if so, how are you defining "velocity towards the reference object"? Is it just the rate at which the distance between one object and the reference object is growing smaller, so it would only be negative if the object was moving away from the reference object? If so, consider the following example. Suppose in frame #1, object C is the "reference object" which is at rest in this frame, object A is approaching it at 0.6c in the +x direction, and object B is approaching at 0.8c in the -x direction. In this frame B would have a larger velocity towards the reference object according to the definition above. But now transform into frame #2 which has its x-axis oriented the same way but is moving at 0.6c relative to frame #1, in the same direction as object A. In frame #2 A is at rest while C is moving at 0.6c in the -x direction, and using the relativistic velocity addition formula, we find that in frame #2 object B has velocity (0.8c + 0.6c)/(1 + 0.8*0.6) = 0.946c in the -x direction. So in this frame, the distance between A and C is shrinking at a rate of 0.6c, while the distance between B and C is shrinking at a rate of 0.946c - 0.6c = 0.346c, meaning in this frame it is A that has a larger velocity towards the reference object using the definition above.

It's not that speed isn't an "objective reality", it's simply not the whole story. Two observers would be foolish to come to qualitative conclusions based on excising certain data. Or you could argue that they are trying their best to incorporate everything relevant. If they are but they still arrive at a qualitative contradiction, the only answer is that they did not include everything relevant.

And why can't I say that simultaneity is not the whole story either, and therefore it would be foolish to say that logic forces us to conclude that there must be a single truth about whether two events at different locations are simultaneous or not without more specification of the context (like what inertial frame we are using)?

The "procedure I discussed above" was just the standard one for constructing inertial coordinate systems in SR, and the result is that the coordinates assigned to the same event by different observers are related by the Lorentz transformation. Do you not understand that under the Lorentz transformation, even if two coordinate systems have their spatial axes oriented the same way, if the two coordinate systems are in motion relative to one another they can disagree about which of two objects has a greater length? If so I can give you a numerical example, if that's what you're asking for. But if you're asking me to justify something else, please be specific about what it is.

So there are no rotations of coordinate axes. O1 and O2 are just watching A and B fly away (B moving faster relative to O1 and O2 at the outset and they are identical in size). Or O2 is standing on B. There is no way that, without rotating their axes, they will come to contradictory conclusions about which one is bigger.

There is if O1 and O2 use coordinate systems that are related by the Lorentz transformation. If O1 assigns an event some coordinates x,y,z,t, then the Lorentz transformation tells us that O2 should assign it these x',y',z',t' coordinates:

x' = gamma*(x - vt)
y' = y
z' = z
t' = gamma*(t - vx/c^2)
where gamma = $$1/\sqrt{1 - v^2/c^2}$$

Here we are assuming that the x',y',z' axes of O2's coordianate system are oriented parallel to the x,y,z axes of O1's coordinate system (no spatial rotation), that the origin of O2's system is moving at velocity v in the +x direction of O1 (which means the origin of O1's system is moving at velocity v in the -x' direction of O2), and that the origins of the two coordinate systems coincide at t=0 in O1's system and t'=0 in O2's system. Let's also assume for the sake of argument that v=0.6c in this example, which means gamma = 1/0.8 = 1.25.

In this case, consider two rods which are oriented along the x and x' axes of the two coordinate systems, with rod A being at rest in O1's frame and rod B being at rest in O2's frame. Suppose that in O1's frame, rod A is 10 light-seconds long while rod B is only 8 light-seconds long, and that the left end of both rods lies at the origin at t=0 in this frame. If we label rod A's left end as "AL" and the right end as "AR", then AL's x-coordinate as a function of time in this frame is x(t) = 0, and AR's x-coordinate as a function of time is x(t) = 10 (both are constants since A is at rest in this frame). Meanwhile, if we use "BL" and "BR" to label the left and right ends of rod B, then in O1's frame BL's x-coordinate as a function of time is x(t) = t*0.6 and BR's x-coordinate as a function of time is x(t) = t*0.6 + 8. Agreed so far?

But now suppose we want to know x'(t') for each of these four rod endpoints in the O2 frame. In this case I'd say that AL's function is x'(t') = t'*(-0.6) and AR's function is x'(t') = t'*(-0.6) + 8, meaning that at any given t' coordinate the distance between the two ends of A is 8 light-seconds. I'd also say that BL's function is x'(t') = 0 and BR's function is x'(t') = 10, meaning that the distance between the two ends of B is 10 light-seconds. You can check yourself that these functions are correct according to the Lorentz transform. For example, pick any event on the worldline of AR whose coordinates satisfy x(t) = 10 in the O1 frame, and then find the corresponding coordinates in the O2 frame, you'll find that the coordinates in the O2 frame do always satisfy x'(t') = t'*(-0.6) + 8. For example, try x=10 and t=20; in this case, the Lorentz transform gives:

x' = 1.25 * (10 - 0.6*20) = 1.25*(-2) = -2.5
t' = 1.25 * (20 - 0.6*10) = 17.5

And t'*(-0.6) + 8 = 17.5*(-0.6) + 8 = -10.5 + 8 = -2.5, so it does work. More generally, if you pick x=10 and t=T, where T can be absolutely any number, you get:

x' = 1.25 * (10 - 0.6*T) = 12.5 - 0.75T, which gives T = 16.666... - 1.333...*x'
t' = 1.25 * (T - 0.6*10) = 1.25*T - 7.5, which gives T = 6 + 0.8t'

Combining the two gives -1.333...*x' = 0.8t' - 10.666..., dividing both sides by -1.333... gives x' = -0.6*t' + 8.

You can check that the other functions for position as a function of time are equivalent too. So, this shows that the two frames disagree on which has a greater length even though their spatial axes are all parallel and pointing in the same directions.

Of course, context is important here too--the coordinates each observer assigns to things are based on their own system of rulers and clocks, and each observer says the other observer's rulers are shrunk relative to their own. So in a way you could say that comparing the claims of the two observers about which rod is longer is another "apples and oranges" comparison. But this is exactly the point, there is no objective truth about which rod is longer in any absolute sense, only different contextual truths that are defined relative to a particular coordinate system.

We can't picture such 4D objects visually because our brains have become adapted to find 3D space intuitive, but I imagine if you could somehow take the brain of a newborn and feed it sensory inputs from a simulated 4-dimensional body in virtual reality, as it grew up it would be able to visualize 4-dimensional shapes.

I don't buy any of this. If you're going to believe such things you may as well also believe in Santa Claus, the Flying Spaghetti Monster, and the Tooth Fairy.

You don't believe that a newborn brain hooked up to different types of sensory input might adapt to be able to make sense of it? For example, if you hooked it up to artificial eyes which could detect a broader spectrum of electromagnetic frequencies, you don't think it would experience more colors than we see? If you hooked it up to something like the echolocation system of a bat or dolphin, you don't think it would come to experience this just as intuitively as we experience vision? The brain is quite adaptable, after all. And if you accept any of these possibilities, I don't see what's so hard to accept about the idea of adapting to navigate 4-dimensional geometry, which is just as consistent mathematically as 3-dimensional geometry. Indeed, we can create A-life creatures in virtual worlds with neural networks that adapt to the task of controlling evolved virtual bodies within simulated 2D or 3D worlds (see this video for example), hopefully you'd agree that we could do something similar with evolved virtual creatures in a 4D simulation; of course such simple simulated neural networks probably have little in the way of inner experience, but unless you are some kind of dualist or vitalist who thinks that no possible AI could have true consciousness, there shouldn't be any fundamental reason why it would be harder to create an intelligent AI "native" to a simulated 4D world as opposed to a simulated 3D world.

Mathematics was created by humans and definitely depends on our biology/history directing our cognition.

I disagree that mathematics depends on our biology (although what areas of mathematics we find interesting may depend on biology)--do you imagine that intelligent creatures with a different biology might disagree that if you take one discrete object and add it to a collection of two other discrete objects, the result will be 3 discrete objects? ('discrete' here assumes no splitting or merging)

Shape is a static concept. It is the primary, most essential quality of an object.

Why do you believe that? We can have objects with simulated shapes in virtual environments, but clearly in this case the shape is just a secondary result of the web of causes and effects going on inside the computer as it performs its calculations (since after all we can run the same simulation on two different computers with different spatial relationships between the computing elements, but the pattern of cause and effect as both run the same program will be the same); why couldn't it be true in the real world too that what's fundamental is something more like a collection of "events" linked by a particular pattern of cause and effect? This would be closer to the relational view of space and time postulated by the philosopher (and co-inventor of calculus) Leibniz.