I don’t think you need the concept of time dimension here nor the two dimensional car example here. The equation is simply saying “as I see your motion through space increase I see your motion through time decrease”. Decreasing motion through time is precisely the concept of time dilation.

I think this approach is a good one. The reason is, I believe, not so much that a spacetime diagrams as such is confusing – I don’t think it is. What is difficult and non-intuitive to understand (in my view) is why you would want/need to move from an orthogonal to a non-orthogonal coordinate system and most of us are only used to moving from one orthogonal coordinate system to another (rotated) coordinate system. Doing the Lorentz transformation theory first provides the understanding why the transformation is from an orthogonal to a non-orthogonal coordinate system. Understanding the consequences of moving from an orthogonal coordinate system to a non-orthogonal coordinate system is what is key here.

It is not space that is physically contracting. The Lorentz contraction formula really defines the expansion of the metric used to measure space (distance). A metre is defined as the physical distance light travels per second. If I am on the train the seconds on the platform are longer (because the platform is moving with respect to me) so the metre used by someone on the platform is longer and so they will measure the distance to be shorter (that is fewer metres) than what I would measure being on the train.

Up to now we have been working exclusively with uniform motion. All the theory has been based on this. I don’t see how we know that any of the results we have obtained can be applied to the non-uniform motion of the rotating spokes.

From George’s perspective Gracie seconds are longer so Gracies meters are longer and if Gracie’s meters are longer fewer meters will fit in a particular physical distance (in this case the train) so Gracie will measure the train to be shorter. Again the metric is changing (how much physical distance is actually in a metre) due to the definition a meter: light covers a constant amount of distance in a given time. If the time unit changes (in this case a second becomes longer) light will cover more physical distance in that time unit and so the metre becomes longer. With a longer metre you will measure less metres for any given physical distance. The physical length hasn’t change just the metric for measuring the length has.

Consider this thought experiment. A rocket is launched. Attached the rocket is a the end of a tape measure. The tape measure roll remain on Earth. On the rocket is another tape measure roll which is attached to the Earth. As the rocket travels into space both tape measures unroll. At any point in time the same physical amount of tape is unrolled: both tapes are stretched taut and are adjacent to each other so it is clear that the same physical amount of tape has been unrolled on each tape measure so the physical distance between Earth and the rocket must be the same whether you are on the rocket or on Earth. So how do you account for Lorentz contraction? The answer is that as the tape measure unrolls on the rocket, observers on the rocket will see more metres (as measured on the tape) go by per second on the tape measure because their second has become longer. From their viewpoint the metres on the tape measure measure a distance that is too short.

Yes, as long as you know the equation of motion of the baseball.

“Speed is distance divided by duration which is space divided by time”. So if we learn that the speed of light is constant we also learn that time is not constant so that means that space must in some way compensate for the non-constant aspects of time in order that the ratio stays the same allowing the speed of light stays unchanged. So, in order to ensure, that the speed of light space must adjust itself in tandem with time so that the ratio for light stays fixed. If we consider that time is not constant therefor space must change too in relation to motion so that the ratio of space over time is such that the speed of light remains unchanged.”

In my view, this is NOT what is going on. Space is NOT “adjusting itself”. Distances are NOT shortening, objects do NOT get shorter. Yet we DO measure shorter distances and we DO measure objects to be shorter. What is changing is not the physical distance but the metric we are using to measure that distance. When our time “slows down” what this means is that the amount of time between two clock ticks increases and as light has a constant speed in space so too does the distance covered between two ticks. Since a meter is defined as the distance light travels in 1/299 792 458 of a second then if that second is longer clearly the meter will end up being longer as light will travel further in the additional time between the two clock ticks. If meters become longer then less meters will fit into any given physical distance and so the measurement of the distance (using meters) will end up giving a smaller value for the measured distance. This is no different to the case of changing from a millimetre to a centimetre metric or from grams to kilograms. In both cases we will measure different values for the same distance or mass.

In short, it is not space that is adjusting itself but the metric is (by definition) adjusting itself because it is based on a time unit that is variable. As time slows down the meter by its very definition becomes longer. Again, it is simply the metric that is changing, nothing is happening to physical space. Because the metric changes we measure different values for the same physical entities.

This interpretation immediately resolves a number of difficulties that would need to be explained if true distance shortening (Lorentz contraction) took place. If it really is physical distance that is shortened then one would need to account for apparent size. Apparent size is a property of distance: when you get closer to an object that object will appear larger. If Lorentz contraction really shortens the physical distance then as Lorentz contraction takes place we would get physically closer to the object. By travelling ever closer to the speed of light we could get arbitrarily close. We should then see the object at the distance dictated by the Lorentz contraction. If Lorentz contraction really equates to shorter distances then theoretically, we could even observe a star as being a mile away or even closer (with all its detail!) that is a billion years distant in the rest frame. Something that is very difficult to accept.

There are also other properties affected by (true) distance. Radiation intensity and gravitational pull to name two. If we are truly at just 1 mile distance from a star if we accelerate to the appropriate velocity then if we believe in true space Lorentz contraction we will experience the star just as if it were 1 mile away in a non-accelerated situation. We would be both fried and experience an extremely strongly gravitational pull.