Does Inertial time dilation demonstrate that Time is not a dimension?

Question

If time is a dimension and 'now' simply an expression of your position with respect to that dimension, the progress of any object along that dimension should remain in step with all other objects. By this I mean that any object which moves in the time dimension by a specific amount will arrive at the same time as every other object which has moved by that specific amount. Two observers separated for any period of time and then brought together again would agree on the amount of time that had passed, regardless of any motions that either had taken in the interval, in order for them to have traveled the same distance along the time dimension.

The classic Twin Paradox however tells us that this is not the case. The twin that remains in the same inertial frame of reference during the interval will have experienced one duration, while the twin who traveled and returned through a variety of changes in inertial frames will have experienced a shorter duration. Both will agree upon being reunited that they are both present in the same 'now' but will disagree on the duration of their separation.

So if time dilation is real - which certainly appears to be the case - doesn't this require that we discard the notion of time as a dimension?

One objection I can see is that time itself is non-uniform, warped in much the same way that we believe space is, but given that objects which are spatially adjacent can experience different rates of time dilation I think that this cannot be the case. It seems that each object would have to exist in a separate temporal dimension distorted by time dilation, raising time to an unbounded number of loosely linked dimensions. No doubt friar William would object.

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Please note, I am not asking what time dilation is but rather what the existence of time dilation says about Time itself. Please don't mark this as a duplicate of this question since it (a) is about the nature of time dilation and (b) appears to conclude that time dilation is an observational phenomenon and the Twin Paradox does not actually result in different measured intervals:

So in in my frame the time interval measured on my clock while I move from $A$ to $C$ is $t$, but in your frame the time interval while I move from $A$ to $C$ is the distance $AD$ i.e. it is $\gamma t$. And since $\gamma t \gt t$ you see my time dilated in the same way as I see your time dilated. It’s just that we disagree about our start and end points.

I am not talking about the observational effects caused by the successively greater or smaller travel distances of photons between the observers, I'm specifically talking about the observed phenomenon that allows high speed muons to pass through the Earth's atmosphere in much greater numbers than should be possible considering their short halflife of 2.2µs and a travel time over 8µs. If time dilation were purely observational then those muons would not act as they are observed to do.

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I see that several people are still determined that this is a duplicate, but I honestly don't see the relationship, and the answers to that question certainly do not appear to answer this question. As such I strongly disagree with the duplicate question flag.

So, to expand on the question...

As I understand it there are three forms of time dilation that are often talked about:

1. Gravitational time dilation - caused by the direct warping of spacetime by mass.

2. Inertial time dilation - caused by or related to the inertia of the object being observed.

3. Observational time dilation - covers all forms of perceptual time dilation related to the Doppler effect on the mediators of observation, observation being defined as interaction with particles, waves, etc. carrying information about the observed object and being necessarily bound by the speed of light.

The third type is of no interest to me as it is an illusion created by the limited speed of information transfer between objects. Such anomalies of observation are interesting but ultimately only of use in explaining why time dilation doesn't actually occur in specific examples and so do not add anything of interest here.

I'll try to illustrate my meaning a little better...

Take two objects which are spatially adjacent in a region with minimal gravitational stress (not on a planet or in the vicinity of a significant external mass). Object A remains in its current reference frame with no acceleration applied for the duration of the experiment. Starting at T₀ object B is accelerated to an appreciable fraction of the speed of light, decelerated to relative rest then reverses course to return to its starting position adjacent to object A at T₁, with constant rates of acceleration throughout. At the moment that object B returns to its' starting position the duration measured by both objects is recorded and a comparison made.

Both objects start and finish in the same inertial frame, both are temporally existent within the same 'now' before the experiment begins and after the experiment completes. Object A will have experienced a longer duration than object B due to inertial time dilation experienced by object B, resulting in a larger duration measurement.

At T₀ and T₁ both objects will agree that they exist within the same small area of spacetime which we'll call 'now'. The time portion of the coordinate must be identical since they agree on the fact of their shared now, but one has experienced a much longer duration than the other between T₀ and T₁.

If time is a dimension then in order for the T₀ and T₁ now moments to match both objects would have to traverse the same distance along that dimension, i.e. they must have experienced the same duration.

Since this is clearly not the case, as object B has measured a shorter duration, does this not demonstrate that time is not an actual dimension?

Note that I am not contesting the utility of spacetime as a conceptual model that helps to understand what happens in this type of scenario, but the existence of a temporal dimension as an aspect of existent reality.

Answer 1

Spacetime is a 4d manifold. You have to label events to keep them different. If you called all you friends Bob, that would be confusing and problematic. If you had to assign the same spatial coordinates to all events even when they were far away. That would be equally problematic. And similarly you need time coordiabtes as well to distinguish events that are far away in time.

So you need to have coordinates to keep distinct far away things (in space or time) distinct and far away. But these are just labels and a sense of close. It's not a quantitative distance or duration.

For a quantitative theory you could choose the option that clocks and rulers have magic powers to just see these labels and tell you changes. But it's up to experiment to tell us whether this is the case.

And experiment tells us the oppsite. Experiment tells us that clocks tick differently depending on how the move. So they don't tell is the change in coordinates from event A to event B, they tell you something about a 4d path from A to B, the 4d path the clock actually took.

We can try to theorize that something is measured at each point along the path that adds up to what the clock reads along that path. Since the result is what is measured we could call that thing a metric. And we do.

So you decide there is a metric at every event and that clocks measure the metric along their 4d path. So now it comes down to making a theory about what the metric is at each point.

In SR there is one metric. And in GR there is a different metric. A warped or curved one.

So there is a four dimension, and it is time. But that's not what clocks measure. Clocks measure the metric along their path.