Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: chiralSPO on 01/05/2018 17:50:19

Please note: This thread may sit on the edge of generally accepted science and speculation. I would like to keep it in this subforum (which is intended for discussions of generally accepted science), so let us agree to be clear about when we are speculating/hypothesizing. Thanks! ;D (also note: While I personally have a fairly advanced understanding of QM, I have only a modest understanding of general relativity, so please bear with me, and know that I am here to learn)
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Time is simultaneously (sorry) a simple concept and a difficult one to grasp. Clearly, our perception (as human beings) of time is not entirely quantitative. There are many different biological and psychological aspects of our perception that I do not want to delve in to. Rather, I would like to discuss purely physical and mathematical interpretations of time.
I have had many questions about time swirling in my head for a while now, and I think I can distill them into two primary questions:
(1) Is time the most fundamental "temporal" dimension, or is it a function of an unrecognized dimension?
This question comes about when trying to conceive of an absolute time scale. Every equation I have come across in my education in the physical sciences deals with dt or Δt, and so is really just dealing with relative time (t = 0 is arbitrary or internally defined, and any indefinite ∫f(t)dt will have that pesky "+ c" term).
Many people point to the Big Bang as the natural t = 0, but this raises at least as many problems as it addresses.
So is time a function of another more fundamental parameter t = G(α), and most of our equations are actually only valid for α, but are comfortably within experimental error of t?
For example let us imagine that α = t +ln(t)/t (unfortunately, there is no algebraic solution for the inverse, but we can solve it graphically, just by swapping the α and t axesit would be nice if someone can recommend a function that has similar properties but can be explicitly solved both ways):
Screen Shot 20180501 at 12.11.47 PM.png (36.26 kB . 570x563  viewed 1912 times)
(If we look at the blue curve pictured above, and consider the abscissa as the α axis and the ordinate is the t axis)
Clearly, consideration of t ≤ 0 gives only nonsensical results, but the function is welldefined for all values of α, and for large positive values of α (or t), α ≈ t. For example, t = 1000 returns α = 1000.006908 (an error of about 7 ppm), and for t = 10^{10} the error is only 2.3×10^{–19}, well beyond the temporal precision of any experiment I know of. And the convergence of dα and dt (or Δα and Δt) will come even faster.
This relation has the obvious shortcoming that it requires some explicit definition of the unit (you will get different answers using years, seconds, or any other arbitrary unit) ie we have traded the problem of an absolute zero for the problem of an absolute 1. Perhaps there may be a fundamental unit of time (and therefore a fundamental unit of space, based on the relationship defined by the speed of light)?
However, this may obviate some of the difficulties that come when considering inflation (the big bang type, and perhaps what we see now). How could we experimentally (or theoretically) test the validity of this type of model?
(2) Are there different "kinds" of time?
Just as there is inertial mass and gravitational mass, which are conceptually different, but appear to be the same, for reasons which (to the best of my knowledge) still elude us. Are there conceptually independent "kinds" of time?

Interesting thoughts.
Do we have the same problem with distance, there is no true origin. All measurements are Δx, Δy or Δz.
Similarly, units are arbitrary.

Interesting thoughts.
Do we have the same problem with distance, there is no true origin. All measurements are Δx, Δy or Δz.
Similarly, units are arbitrary.
Thanks Colin!
I guess I am trying to find a temporal dimension that is more like the spatial ones, where there is no (apparently) meaningful zero. Currently, by defining the Big Bang as t =0 we introduce problems associated with negative times and other nonsensical consequences. Perhaps instead, there is no true temporal origin, and the observations interpreted as the BB are just consequences of the divergence of t and α.

I am going to have to think a little about your questions. This is a doozy.

And following up from Colin's response:
I guess our current system would be similar to finding some point of reference (like a massive object), and defining our spatial coordinates as in spherical coordinates about it. In this case, time would be like r, for which there is nothing meaningful to be gaining by considering r ≤ 0. Spherical coordinate systems, of course, make the most sense when dealing with spaces that have some sort of spherical symmetry: like looking at orbits around a lonely planet, or orbitals about a lonely nucleusthis becomes increasingly cumbersome when considering multiple planets or atoms. (I know that internally vs externallydefined coordinate systems are a big deal when modeling molecules, and I presume that this is similar when looking at orbital systems containing multiple bodies)
Pure speculation: Drawing on the analogy to spherical coordinates, could there be additional dimensions akin to θ and φ? Are they what we think of as spatial dimensions, or are they distinct? Is this how trigonometric functions become involved in GR? And expanding the multibody problem of spherical coordinates: is there an external coordinate system that allows for multiple Big Bangs, each being a local origin?

First of all, is an indefinite integral unsuitable for functions of time? Surely only time intervals make sense so definite integrals are the way to go. I am also thinking about your function for α and t.

I don’t want to appear to be discouraging. We do know for a fact that the quantification of time uses some form of clock, and we know that the rate that a clock measures the passing of time is affected by the relative acceleration of that clock vs some presumed rest clock.
Unfortunately there is no absolute space in which to place our rest clock, and so there is no absolute measure of time. Therefore, time is relative to conditions of local energy density set up by differences in gravitational potential, relative motion, and/or applied forces between two clocks in relative motion. The conditions of each clock in relative motion are therefore different.
I don’t see how you can get to anything fundamental in regard to a time increment from those circumstances, so I personally settle for the concept that time simply passes, but the measure of the rate that time passes is relative to the differing conditions in which our clocks are located.
On the bright side, maybe there is some fundamental increment of energy based on some foundational minimum background energy level. If such an energy level could not be reduced below this as yet unknown natural limit, and if we could place our rest clock there, maybe all other frames could be compared to that clock.

I don’t want to appear to be discouraging. We do know for a fact that the quantification of time uses some form of clock, and we know that the rate that a clock measures the passing of time is affected by the relative acceleration of that clock vs some presumed rest clock.
Unfortunately there is no absolute space in which to place our rest clock, and so there is no absolute measure of time. Therefore, time is relative to conditions of local energy density set up by differences in gravitational potential, relative motion, and/or applied forces between two clocks in relative motion. The conditions of each clock in relative motion are therefore different.
I don’t see how you can get to anything fundamental in regard to a time increment from those circumstances, so I personally settle for the concept that time simply passes, but the measure of the rate that time passes is relative to the differing conditions in which our clocks are located.
On the bright side, maybe there is some fundamental increment of energy based on some foundational minimum background energy level. If such an energy level could not be reduced below this as yet unknown natural limit, and if we could place our rest clock there, maybe all other frames could be compared to that clock.
Thanks Bogie.
I think I may have caused some confusion using the terms "absolute" and "relative." I understand that different frames of reference will have different relative rates at which time passes, and that looking for any sort of Universal Clock is probably futile. My intended meaning of "absolute" and "relative" is more akin to Kelvin vs Celsius temperature scales, in which "absolute" scales are referenced to a meaningful zeropoint, whereas "relative" scales only concern themselves with the difference between two points. (My apologies, if you already understood my meaning, and the confusion is all on my endif so, please clarify).
Your points do still seem quite relevant to my question of whether there can be a meaningful unit of time (an absolute 1). I will have to think more about this aspect. Thank you.

Thanks Bogie.
I think I may have caused some confusion using the terms "absolute" and "relative." I understand that different frames of reference will have different relative rates at which time passes, and that looking for any sort of Universal Clock is probably futile. My intended meaning of "absolute" and "relative" is more akin to Kelvin vs Celsius temperature scales, in which "absolute" scales are referenced to a meaningful zeropoint, whereas "relative" scales only concern themselves with the difference between two points. (My apologies, if you already understood my meaning, and the confusion is all on my endif so, please clarify).
Your points do still seem quite relevant to my question of whether there can be a meaningful unit of time (an absolute 1). I will have to think more about this aspect. Thank you.
I didn't mean to imply you weren't entirely on top of the basics of measuring and comparing the rate that time passes in different frames. I see your point about absolute and relative scales now, and perhaps there is something to uncover along that avenue in regard to a fundament unit of time. Thanks for the thoughtful response.

I didn't mean to imply you weren't entirely on top of the basics of measuring and comparing the rate that time passes in different frames. I see your point about absolute and relative scales now, and perhaps there is something to uncover along that avenue in regard to a fundament unit of time. Thanks for the thoughtful response.
No offense taken :) Your response was entirely reasonable given my ambiguous word choice.

First of all, is an indefinite integral unsuitable for functions of time? Surely only time intervals make sense so definite integrals are the way to go.
Well, one can invoke indefinite integrals to make generic forms of useful functions. For instance, if we know that an object is accelerating at some function f(t), then we can describe the generic velocity as the indefinite integral of f(t)dt. However, after thinking about it more, I think that the example of the integral actually has little to do with my main point, and is largely tangential...
I am also thinking about your function for α and t.
Yes, please do. I would be very interested in your thoughts on this.

Here is an aside. Not directly related to your function but interesting none the less.
https://www.quora.com/WhatistheLaplacetransformoflogt
This also leads to the following.
http://mathworld.wolfram.com/EulerMascheroniConstant.html
Euler is likely the way to go in starting to analyse the equation.

We can arrive at imaginary time which is interesting since that is one way to arrive at negative energy.
http://m.wolframalpha.com/input/?i=integrate%5Bt%2B%28log+t%29%2Ft%5D

Let's break out the interesting part.
http://m.wolframalpha.com/input/?i=integrate%5B%28%28log+t%29%2Ft%29%5E1%5D
Now that is something you wouldn't want to explain to your granny.

Thanks Jeffrey! I had not come across the EulerMascheroni γ before. It will take some time to work through the first two links and digest the content, but looks well worth it!
As for the last two links you sent, it looks like the imaginary components only appear for negative values of t (and the solutions are purely real for values of t > 0). The functions also appear poorly behaved at 0 (jump discontinuity at t = 0 in the imaginary components, and not differentiable at t = 0 (or t = 1 for the second integral) in the real components).
My goal with the proposed transform from t to α was to find a dimension (α) in which the entire range of reals (–∞ to +∞) maps onto only the strictly positive values of t ( t > 0), thereby avoiding the difficulties associated with t ≤ 0.
Unfortunately, with the function I have chosen, there is no algebraic inverse (only numerical or graphical solutions), so determining indefinite integrals is a nonstarter (for definite integrals, numerical and graphical solutions are feasible).
Is there a function y = f(x) for which:
• limit of y as x → –∞ = 0
• limit of y as x → +∞ = x (I know this would be y = x = +∞, but I want x and y to nearly converge well before +∞)
• the function is continuous for all real values of x (–∞ to +∞) and y > 0
• an algebraic solution for f^{–1} can be found: x = f^{–1}(y)
Ideally this function would also be differentiable everywhere (probably would have to be to satisfy the above constraints, but if not, I will still consider solutions that are not differentiable everywhere, but still adhere to the above)

Well you don't ask for much. I'll think about it a while.

Thanks! :D

Pure speculation: Drawing on the analogy to spherical coordinates, could there be additional dimensions akin to θ and φ? Are they what we think of as spatial dimensions, or are they distinct?
Are you thinking that what we perceive as a single time dimension might be the resultant of other, hidden dimensions? Or are you thinking that those other dimentions are what we see as space?
Or something else? :)
Is this how trigonometric functions become involved in GR?
In relativity if you look at Minkowski ST diagrams the worldlines of uniformly accelerating particles are hyperbolic which links to GR.
The trig functions come about because the Lorentz transformations can be treated as hyperbolic rotations.

Quote from: chiralSPO on Yesterday at 18:45:43
Pure speculation: Drawing on the analogy to spherical coordinates, could there be additional dimensions akin to θ and φ? Are they what we think of as spatial dimensions, or are they distinct?
Are you thinking that what we perceive as a single time dimension might be the resultant of other, hidden dimensions? Or are you thinking that those other dimentions are what we see as space?
Or something else?
Yes... Maybe... I really don't know.
One could imagine a 4D spherical coordinate system in which there is one radial dimension (time), and 3 orthogonal angular dimensions [t, θ,φ,ψ]. It could be mapped onto the orthonormal basis set of [t, X, Y, Z], but it isn't immediately clear to me that it would be useful.
Quote from: chiralSPO on Yesterday at 18:45:43
Is this how trigonometric functions become involved in GR?
In relativity if you look at Minkowski ST diagrams the worldlines of uniformly accelerating particles are hyperbolic which links to GR.
The trig functions come about because the Lorentz transformations can be treated as hyperbolic rotations.
Yes, indeed. I am reading up on Minkowski spaces now (this was part of the inspiration for starting this thread). Obviously these are only defined for the observer, placing them at the origin (and only a good local approximation, once thinking about GR).
More pure speculation: If the other angles involved (even if it's just one) then this might (I haven't worked through the math yet) offer a way of performing Lorenzlike transformations for GR: if we treat time like the radius of the nsphere (from circle on up), and think of acceleration/gravitation (or other distortions) as a rotation that shrinks the projection of the radius (ie perceived time = t×cos(θ), where θ relates to the (gravitational) accelerationso in the absence of acceleration, perceived time = t, while for increasing θ the perception of time (elsewhere) contracts (efectively blue shifting the rest of the universe)
I don't know, I think this speculation might just be veering off topic and into New Theories territory....
I really like the idea of Minkowski space being centered at the observer (which it would have to be for relativity), with infinitely extending orthogonal [t, X, Y, Z] dimensions (and it appears to be experimentally valid). My main question is: none of the X, Y, or Z coordinates have any boundaries, so why should t? And therefore, is it possible that the way we have defined t is wrong, and actually we should be thinking in terms of [α, X, Y, Z], which eliminates all boundaries while still being essentially commensurate with what we observe.

In the Minkowski ST diagrams t doesn’t have a boundary, just the same as x.
I’ll have to think about the rest, I have enough problems imagining 4 orthoganal cartesian dimensions without imaging the same for spherical! 8)

We can arrive at imaginary time……
The “Ausies” got there ahead of us! Fred Alan Wolf says of dreamtime:
“Aboriginals believe in two forms of time. Two parallel streams of activity".
Interestingly, the aboriginal term “Alcheringa” is not best translated as “Dreamtime” (which was coined by research worker Frank Gillen). The term “The Dreaming” is preferred by the Indigenous Australian People as being a more accurate translation, and certainly involves two “dimensions” of time.
Forgive that unscientific aside.
Who was it who said: “The truth is that a scientist's understanding of a phenomenon is inversely proportional to the number of math equations he uses to describe it.”
I have a nasty feeling it might have been Louis Savain.
Interesting and valuable as the maths undoubtedly are, how well does this approach address the OP?
As far as we are aware, nothing can happen without time, in which it can happen. How fundamental is that?

This question comes about when trying to conceive of an absolute time scale.
Could it be that there is no “absolute time scale”?
We perceive our Universe as functioning in accordance with the “rules” of relativity, at least, on a scale that is meaningful in our everyday lives. Should we expect anything, other than “c”, to be “absolute”?

If we define a universe in which no forces exist then all frames are inertial. However, if all inertial frames are in relative motion with each other then any particles traveling along with the frame will eventually collide with another particle. Therefore, these collisions act like a force. We have time. If all frames are at rest in relation to one another the same particles still exist but don't move. Where has time gone?

This question comes about when trying to conceive of an absolute time scale.
Could it be that there is no “absolute time scale”?
We perceive our Universe as functioning in accordance with the “rules” of relativity, at least, on a scale that is meaningful in our everyday lives. Should we expect anything, other than “c”, to be “absolute”?
Indeed. I suspect that there isn't an "absolute" time scale. I want to reiterate that my intended meaning of "absolute" and "relative" is with respect to whether the 0 is meaningful or not:
My intended meaning of "absolute" and "relative" is more akin to Kelvin vs Celsius temperature scales, in which "absolute" scales are referenced to a meaningful zeropoint, whereas "relative" scales only concern themselves with the difference between two points.
It would seem that when people invoke the Big Bang as "the beginning of time" that they are declaring some sort of absolute scale, in which everything that can be observed is t > 0, and that all t ≤ 0 is fanciful. My hope is to do away with this boundary condition by exploring the possibility of a temporal dimension that is not the same as what is currently defined as time, which has no bounds.

Hence the mapping. ;)

If we define a universe in which no forces exist then all frames are inertial. However, if all inertial frames are in relative motion with each other then any particles traveling along with the frame will eventually collide with another particle. Therefore, these collisions act like a force. We have time. If all frames are at rest in relation to one another the same particles still exist but don't move. Where has time gone?
Time has not necessarily gone anywhere, but without change (movement) how can we perceive it?
It might be true that there can be no change without time, but time without change? That's a cat of a different colour.

It would seem that when people invoke the Big Bang as "the beginning of time" that they are declaring some sort of absolute scale, in which everything that can be observed is t > 0, and that all t ≤ 0 is fanciful.
Agreed! If the BB was the beginning of time, how could it have happened?
“In the beginning there was nothing, which exploded.” (T. Pratchett). Of course, he would have the answer! :)
My hope is to do away with this boundary condition by exploring the possibility of a temporal dimension that is not the same as what is currently defined as time, which has no bounds.
If it has no bounds; wouldn’t that make it infinite?

If it has no bounds; wouldn’t that make it infinite?
Not necessarily... it just means there's no boundary. One could imagine a loop, or some other type of closed, but still unbounded time...
What I have been describing with my attempts at defining α does seem to be unbounded and open, which would indeed be infinite.

See attached article. I tried to post other articles but they're too big. This site is too stingy!!!

Not necessarily... it just means there's no boundary. One could imagine a loop, or some other type of closed, but still unbounded time...
If time were a loop, wouldn’t it repeat? If that were the case, it would be bounded by its own repetition. If anyone could experience the whole loop, he/she would be able to recognise the start/finish.
In principle, a featureless loop might be said to be boundless. No doubt someone can think of an example.
Good luck identifying infinite time.

Fascinating piece of "history", Pete. If it were not for the comment about dating the discussion some twenty years back, it would have been interesting trying to work out when it was written.
Shame about your other articles. Could you split them into postable size chunks?

Fascinating piece of "history", Pete. If it were not for the comment about dating the discussion some twenty years back, it would have been interesting trying to work out when it was written.
Shame about your other articles. Could you split them into postable size chunks?
Possible, yes.It'd be easier if I put them all on my website and post the URL to it.

Please see the flour articles listed below the title "Articles about time" at
http://www.newenglandphysics.org/other/other.htm

If I have this right we need two conditions for the function.
And
And the function should be continuous. Is this correct?

An interesting but not very pertinent function is f(t) = (t)^1.

An interesting but not very pertinent function is f(t) = (t)^1.
Is this not equivalent to f(x) = –x ? Perhaps there is a typo in your equation? f(x) = (–x)^{x} is definitely an odd one! (all x < 0 are well defined, but then only odd positive integers get real outputs)

If I have this right we need two conditions for the function.
And
And the function should be continuous. Is this correct?
Close! I would like:
And
And the function should be continuous, and f(α) > 0

Ok, got it. How soon do you want it?

An interesting but not very pertinent function is f(t) = (t)^1.
Is this not equivalent to f(x) = –x ? Perhaps there is a typo in your equation? f(x) = (–x)^{x} is definitely an odd one! (all x < 0 are well defined, but then only odd positive integers get real outputs)
Sorry that was just wrong. Should have been f(t) = (t)^{1}

I think Einstein pointed out that time is what prevents everything from happening at once.
So no problem with the Big Bang: it was the event when everything did indeed happen at once.

Ok, got it. How soon do you want it?
No real rushbut it is driving me nuts. :)

Here is something that I haven't attempted to graph yet but may be interesting. It is α = t^{2} + log^{2}(t)t^{2}

This is now interesting. Looks like a gravity well.
http://www.wolframalpha.com/input/?i=t%5E2%2Blog(t%5E2)t%5E2%3D0

The non homogeneous solution.
http://www.wolframalpha.com/input/?i=t%5E2%2Blog(t%5E2)t%5E2%3Du

.....I have discussed this elsewhere (https://www.thenakedscientists.com/forum/index.php?topic=73163.0)
I've had another look at this thread. Now I recall why Alan's quote was the only one I remembered. :)
I have to ask one question, though. If you replaced the word "time" with "change" in all the appropriate places; how much difference would it make to the arcane maths?
(mod edit: you forgot to close the quote, so it was hard to know where the question started. I think I fixed it correctly)

Change and time are not equivalent. Time is a dimension, and change refers to differences within one dimension.
This is most well defined within calculus (sometimes called the math of change), where the symbol ∂ (sometimes d) is used to denote microscopic change, often relating the change of one thing with respect to the change of another.
In this way, we can define instantaneous (microscopic) change of position along a spatial dimension (x) with respect to instantaneous (microscopic) change in time (t) as ∂x/∂t which is equivalent to the velocity in the x direction.
Although time is a very common dimension (variable) to compare changes against, it is not the only one. For instance we can look at the change in altitude with respect to change in forward distance traveled ∂y/∂dx (slope) or the change in cost of making chocolate with respect to the change in the cost of labor ∂$_{chocolate}/∂$_{labor} etc. etc.
What I am trying to learn in this thread is whether our concept of velocity (or any other rate of change with respect to apparent time) is best described as a ∂x/∂dt, or if there is a more fundamental variable (I believe I called it α earlier in the thread) such that ∂t/∂α is welldefined, and ∂x/∂α holds up better to the boundary condition that the big bang appears to impose on ∂t.

(mod edit: you forgot to close the quote, so it was hard to know where the question started. I think I fixed it correctly)
Spoton, thanks.

Thanks Chiral. I knew I was in the wrong thread. :)
I'm fine with your explanation, but I guess we all have a "mumpsimus", or two, lurking somewhere. One of mine is the idea that time is just a measure of rate of change, like inches miles etc are measures of distance.

I had thought about this a lot last year. I came to the conclusion that a continuous function would be impossible because it would always end up with a zero denominator at some point. I may have another look now that I have not been thinking about it for a while.

I'm fine with your explanation, but I guess we all have a "mumpsimus", or two, lurking somewhere. One of mine is the idea that time is just a measure of rate of change, like inches miles etc are measures of distance.
My lack of background maths and physics means that, sometimes, even my “eureka” moments are ephemeral, and this may be an example, but thinking about my post (#47) I spotted an “unconformity”.
I’m comparing time with inches and miles. The real comparison should be between time and length (distance) on the one hand, and (e.g.) seconds and millimetres, on the other. Thus, time by association, can be considered as a dimension.

If we take f(α) to be 1/α then at α = ∞ we have our zero value. You stated that a value of 1 for time was a consideration. At α = 1 we have our value of 1. All negative fractions from α = 1 and approaching zero give an infinite positive sequence that can map to time. Infinite time then terminates at this α = 0 boundary. Crossing into positive α then gives us negative time which descends back to zero. You still have a discontinuity.

I’m comparing time with inches and miles. The real comparison should be between time and length (distance) on the one hand, and (e.g.) seconds and millimetres, on the other. Thus, time by association, can be considered as a dimension.
This is how I understand it to be.

There is a singularity separating positive time from negative time. One time 'direction' will never be able to detect the other.

If we take f(α) to be 1/α then at α = ∞ we have our zero value. You stated that a value of 1 for time was a consideration. At α = 1 we have our value of 1. All negative fractions from α = 1 and approaching zero give an infinite positive sequence that can map to time. Infinite time then terminates at this α = 0 boundary. Crossing into positive α then gives us negative time which descends back to zero. You still have a discontinuity.
Indeed... t = 1/α will not fit the necessary criteria.
Is there a function t = f(α) for which:
• t is strictly positive for all values of α (t > 0)
• limit of t as α → –∞ = 0
• limit of t as α → +∞ = α (I know this would be t = α = +∞, but I want α and t to nearly converge well before +∞)
• the function is continuous for all real values of α (–∞ to +∞)
• an algebraic solution for f^{–1} can be found: α = f^{–1}(t)
Ideally this function would also be differentiable everywhere (probably would have to be to satisfy the above constraints, but if not, I will still consider solutions that are not differentiable everywhere, but still adhere to the above)
I think that t = ln(1+e^{α}) works!
(I found it by realizing that the derivative of the desired curve followed a sigmoid curve, going monotonically from zero to 1, and that therefore the integral of a sigmoid would have the desired properties.)
t = ln(1+e^{α})
• is continuous and differentiable and strictly positive across all reals
• limit of t as α → –∞ = ln(1) = 0
• limit of t as α → +∞ = ln(e^{α}) = α
• has the inverse function α = ln(e^{t} – 1)
Thank you to Bill S for reviving this thread, and to jeffreyH for prompting me to revisit the math!

This is how I understand it to be.
Eureka!!! Mumpsimus not completely exorcised, but still a lot of thinking to do. Thanks for the patience.
There is a singularity separating positive time from negative time. One time 'direction' will never be able to detect the other.
That’s one of the things that will need some thought!

Think of decoherence Chiral. Then treat time as if it would be a equivalence, scaling up Quantum objects to a macroscopic plane gaining that 'local clock'. In that case a Big Bang, no matter where and how it comes to be, indeed is t=0.
But it also tells you that t must be 0 at 'all times'. Making it a choice of from where you measure. You can't measure it at a same scale (QM) though so you become restricted by your (macroscopic) wrist watch, which never stops ticking.
And I think you are correct in splitting it into two concepts, that's what I do too. Time should be a 'property' of SpaceTime, with the 'local clock' being a expression of our macroscopic reality. In relativity it indeed is a 'dimension', that you need for a SpaceTime.
As for " Perhaps there may be a fundamental unit of time (and therefore a fundamental unit of space, based on the relationship defined by the speed of light)? "
Yes, your local clock is a equivalence to 'c', but the question of it thereby also being a 'fundamental unit' seems to me to belong to whether SpaceTime has a 'granularity'.
Is 'c' a granularity?
I like the idea of 'c' being a 'constant' much better, and defining ones local clock as a equivalence then makes it into a constant too, aka 'Proper Time'.
It is of a outmost importance to understand how we define a repeatable experiment' and so 'c'. It's local definitions agreeing with each other that gives us the base from where we find this 'global' definition of a universe. When turned around into ' time dilation's ' and ' length contractions ' by relativity you will need to take a new look at that base. Failing to do so makes you unprepared.
=
Thinking of it this way a 'Big Bang' is the beginning of a cone widening out into 'a local clock'. And presuming that a Big Bang has no defined localization. or turned around, started 'everywhere' as can be proven by using a 'isotropic and homogeneous universe'. Then the 'cone', aka 'Big Bang', isn't something you just can 'time reverse' from the macroscopic to the microscopic finding a 'origin', neither is the 'arrow of time' what you can use to back track it. It's a 'co existence'.
It makes the quantum regime into something entirely different from the way we define a SpaceTime. We use four dimensions for it, length, width, height and 'time'. Exchange 'time' for 'local clocks' macroscopically and make our wristwatches what give us a direction. The 'one way arrow'. That way your clock is no 'illusion' and 'proper time' is the base of our universe. Time becomes very real treating it like this, without it nothing exist.
Another way to look at those 'cones' would then be to use each ones 'history' to give us a 'time scale' of when a 'Big Bang' started. Then as we compare my local 'cone' to yours, and find us in agreement, we come to a same conclusion as we do finding a 'repeatable experiment' to work. It holds at all times and wherever you go to test it.
The 'cone' I'm using is to its nature two folded, it has no arrow in one sense as it is about QM and our macroscopic reality co existing. But we can't get away from our local macroscopic clock so in that sense it is possible to define histories to each 'cone'. And it makes 'scaling' what you use to go from one concept to the other.
It's not that difficult a concept, and it fits. But it makes the 'universe' a even stranger place than what we thought before.
==
PS: I suspect one might be able to do the same for the rest of our 'dimensions'. Leaving them to be a result of, what I like to call, 'scale dependencies'. It makes 'scales' into something different than what we usually define them from, as we do it from presuming 'dimensions' to be their origin. And that one is quite worth thinking of.
It's also a primary reason for trying to be clear from where you define something, locally or 'globally'. People tend to miss doing so, but it's just as important as telling what 'frame of reference' one define it from.

thanks yor_on, there is a lot to digest there, and I want to think more before I respond to your post.
As far as the function I posted earlier, it appears to be one of a class of functions that will all meet the criteria I had posited earlier.
t = log_{β}(1 + β^{α}) meet all of the criteria, for any β > 1
As β increases, α and t become more similar to each other (converging at lower values of α (or t))
For instance:
If we evaluate with the natural base, e, then α and t converge to error < 1ppm by α = 12
If we evaluate in base 10, then α and t converge to < 1ppm by α = 5
While if we evaluate in base 2, then α and t converge to < 1ppm by α = 17
Again, one must be aware that we don't know what the units of α and t are... is this 17 seconds, or 17 quintillion years?
Speculation: Is there a way to fit to z values of Hubble shift to see if instead of expansion, it's actually this deviation? This might help us nail down what β is and what the actual fundamental units of time are?...

Bill: the jargon is the distinction between quantities and units. "Length" is a quantity, "meters" is a unit.
So no problem with time: it's a quantity. Just as length separates points, time separates events, and "seconds" are a unit.
As all the units are defined in terms of the number of cycles of an atomic clock, we have a universal scale for each  now including mass following its SI redefinition.
The most fundamental quantities that are unrelated to each other are also called dimensions, usually signified by square brackets [M], [L] and [T] being the most often encountered, but you can usefully add charge [Q] to produce a fairly complete armoury for engineering analysis.
So back to the question: how fundamental is time? As I see it, it is as fundamental as the other dimensions, none of which requires an absolute frame or origin. If we consider mass to be "that which is subject to gravitational attraction" then zero mass is no more conceptually difficult than zero length or zero time  no need for an origin or any embodiment of negative mass.
Or am I being too naive to see the problem?

time separates events
Are there any scenarios where such separations do not exist? Even one example?
If not can we say that the separation is measured (not caused which you are probably not saying) by time ?
Ad do some of us need to keep reminding ourselves that time is not a "thing" ,more a description?

Time is the most 'real' thing you will meet in your life. It gives you a existence and it ends with your death. Doesn't matter if you're a mathematician. It's here, and can't be transformed locally.
=
Gravity can, not your local clock

@chiralSPO I am impressed! What exactly are your thoughts on a connection with the Hubble shift? Are you assuming a connection to the convergence of α and t? Also, what started you down this path?

Bill: the jargon is the distinction between quantities and units. "Length" is a quantity, "meters" is a unit.
So no problem with time: it's a quantity. Just as length separates points, time separates events, and "seconds" are a unit.
As all the units are defined in terms of the number of cycles of an atomic clock, we have a universal scale for each  now including mass following its SI redefinition.
So far, so good. Add "jargon" to my list of "deficiencies". :)
The most fundamental quantities that are unrelated to each other are also called dimensions,
I’ve been looking for a good definition of dimensions. Thanks.
usually signified by square brackets [M], [L] and [T] being the most often encountered, but you can usefully add charge [Q] to produce a fairly complete armoury for engineering analysis.
Next step, time permitting: Get to grips with dimensional analysis.
So back to the question: how fundamental is time? As I see it, it is as fundamental as the other dimensions, none of which requires an absolute frame or origin. If we consider mass to be "that which is subject to gravitational attraction" then zero mass is no more conceptually difficult than zero length or zero time  no need for an origin or any embodiment of negative mass.
Or am I being too naive to see the problem?
Alan, I suspect you are not the one who is being naïve. It’s much more likely that my questions are naïve, and that you, in common with many other experts, sometimes don’t see the extent of that naivety.
“Conceptually”, I have little difficulty with zero mass, zero length or zero time. I then need to try to relate these concepts to the physical world. This leads to questions like: If something has no mass, or no length, can it be said to exist, other than as a mental concept? Similarly, if something exists for zero time, is it ever “there”?
Then we come back to:
A point on a line has no length. A point on a timeline has no duration. What's the problem?
I have no problem with either of those, in principle. However, in practice, can you show me a point that has no length, but is still there?
And that tends to be as far as it goes.

So back to the question: how fundamental is time? As I see it, it is as fundamental as the other dimensions, none of which requires an absolute frame or origin. If we consider mass to be "that which is subject to gravitational attraction" then zero mass is no more conceptually difficult than zero length or zero time  no need for an origin or any embodiment of negative mass.
Or am I being too naive to see the problem?
I would certainly not say you are being naive.
I liked the analogy to mass at first glance, but after some thought, I am not sure it is a good one. In my mind (happy to be shown wrong), an object with zero mass would be more akin to an object that existed for a duration of 0 time, or that had a radius of 0 length. However, unlike mass, both space and time appear to be descriptors of location within spacetime (I am open to discussion of massspaces, but that may require another thread...)
Our own perception of time is quite unusual, and certainly prone to mistakes. That we have found ways of making devices to measure it certainly has helped a lot, but I still do wonder if we are correctly interpreting what they are measuring. I will use another analogy that involves mass: we perceive weight directly, and it was quite an advancement for society, when it was realized that weight is not a fundamental quantity, and instead is a function of mass and our gravitational environment. My question regarding the fundamentality of time is along the lines of asking whether time is more masslike (fundamental) or weightlike (can be distilled further).

@chiralSPO I am impressed! What exactly are your thoughts on a connection with the Hubble shift? Are you assuming a connection to the convergence of α and t? Also, what started you down this path?
I am still trying to see if it fits (or show that it cannot be, which is equally useful, but a little less interesting...)
I started down this path because I wondered if the apparent expansion (current and ancient) and big bang both could be explained in a way different from the currently accepted model. Questions like "what happened before the big bang?" can be answered more easily as "nothing" because the big bang would be an infinitely long period (in α time), which is identified only by how different the α and t values are. The concept of "rapid inflation" immediately after the big bang could potentially be reinterpreted as a period of time for which α and t were very different.
If "time" had been going more slowly before, then light emitted longer ago could appear to have lower frequency, even if there isn't any expansion (ongoing or ancient). This model could account for red shifts: ∂t/∂α < 1 for all α (and all t), and ∂t/∂α increases monotonically, so as α (or t) increases, so does ∂t/∂α. However, this model does not account for accelerating expansion (indeed, it was designed to converge such that α = t for sufficiently large values, and assuming that we are in that sufficiently advanced region). There are ways to tweak this model, but of course, once we start adding parameters in willy nilly, we can make a model that will reproduce anything...
I am not optimistic that this model will turn cosmology upside down, but I am interested to see how far this line of reasoning can go before it fails theoretical or experimental tests.

Go for it!

Consider a photon to be an object with zero mass. Some of those reaching us now, were launched umpteen billion years ago, so they don't exist for zero time!

Consider a photon to be an object with zero mass. Some of those reaching us now, were launched umpteen billion years ago, so they don't exist for zero time!
I don't think you understood what I meant.
If I understood correctly, you had said something along the lines of, 'why is zero time any harder to understand than zero mass?' To which I responded, 'that analogy is not quite right: An object existing for a length of zero time is not that same as an object existing at t = 0"
Do you see why I differentiate between the usage of time as it pertains to the duration of an event or existence, and the usage of time as it pertains to a location of an event on a timeline?
Something that has the property of zero mass, is analogous to something that has the property of zero volume (not analogous to being at the origin of a spatial coordinate system), and is analogous to something that has the property of zero duration (not being at the origin of a timeline).

Chiral, you write : " If "time" had been going more slowly before, then light emitted longer ago could appear to have lower frequency " But I'm not sure relative what? light?
Then light need one invariant 'clock rate', or 'wavelength/Frequency' that then gets manipulated by this otherwise 'universal time change' you're wondering about, wouldn't it?
=
Or you're thinking of early light then being misinterpreted as it reach us in 'fast time'?
I'm not sure, but it's a interesting idea.
=
Let me translate it to this. Isn't that the same as to suggest that 'c' is a variable of sorts, although at all times keeping its 'proportion' relative all frames of reference, meaning that 'c' is 'c', both then and now. Seems a hard thing to test.

Chiral, you write : " If "time" had been going more slowly before, then light emitted longer ago could appear to have lower frequency " But I'm not sure relative what? light?
Then light need one invariant 'clock rate', or 'wavelength/Frequency' that then gets manipulated by this otherwise 'universal time change' you're wondering about, wouldn't it?
=
Or you're thinking of early light then being misinterpreted as it reach us in 'fast time'?
I'm not sure, but it's a interesting idea.
=
Let me translate it to this. Isn't that the same as to suggest that 'c' is a variable of sorts, although at all times keeping its 'proportion' relative all frames of reference, meaning that 'c' is 'c', both then and now. Seems a hard thing to test.
The second one (misinterpretation by us now in "fast time") is how I am trying to think about it.
I am definitely struggling with what it would mean for time to change. Would the frequency of the light shift as well, keeping up with time, and ultimately giving no difference? How can I reconcile this theory with the photon's point of view, in which its journey is instantaneous?
I think I am out of my league!

I think you have a point actually. The first thing you need to prove univocally would be that we actually have a locally slow versus fast times depending on mass and speeds. But that's me :)
After that you will need to prove that it is a 'field' with excitations. Think of it undulating and count in 'time' as part of that field. It would actually be easier if you stayed away from the idea of locally fast and slow time and instead referred to it the way I do, as 'proper time' being special.
You don't want 'propagation' for it Chiral, what you want is something expressing itself as a propagation, proportional to the 'time rate' you set. That takes care of 'speeds'. It's a field you're after in where those excitations observer dependently express themselves as waves or 'photons', but always proportionally to the 'time rate' you set.
Then let it undulate.
Now you just need to prove that this will produce your red shift at this 'faster' proper time/'undulation'.
Maybe :)
A hard one to prove I think, but very nice all the same.
In my terms 'slow' and 'fast' time is when considering it from a 'global perspective'. If you stick with the local definition then 'proper time' is what you use for proving it. Because you're discussing a red shift, right? Not a 'universal globally explainable' TOE. Those two are like mirrors of each other, with the local interpretation being the simplest. Actually logic tells me that if you get it right locally there will be a way to translate it 'globally' too, but that one will be a lot more complicated.
Think of using a local interpretation as doing away with observer dependencies. There are no observer dependencies if you just stay local ignoring global interpretations. The rest becomes a question of making it fit.
At a quantum regime it becomes trickier when considering super positions. Then again, there are no subtleties to it if you just accept that outcomes too then must be a function of time.
As for you worrying about " How can I reconcile this theory with the photon's point of view, in which its journey is instantaneous? " You don't have a frame of reference for a photon, as you well know Chiral :) It has a 'origin/recoil' and a 'absorption'. Any proof to otherwise will be a indirect proof, colored by expectations. It's 'c' and as long as the 'proportions' relative all other constants etc are correct it shouldn't be noticeable as differing with time. Unless we now presume the cosmological red shift being a 'proof' of your ideas.

Chiral, you write : " If "time" had been going more slowly before, then light emitted longer ago could appear to have lower frequency " But I'm not sure relative what? light?
Then light need one invariant 'clock rate', or 'wavelength/Frequency' that then gets manipulated by this otherwise 'universal time change' you're wondering about, wouldn't it?
=
Or you're thinking of early light then being misinterpreted as it reach us in 'fast time'?
I'm not sure, but it's a interesting idea.
=
Let me translate it to this. Isn't that the same as to suggest that 'c' is a variable of sorts, although at all times keeping its 'proportion' relative all frames of reference, meaning that 'c' is 'c', both then and now. Seems a hard thing to test.
The second one (misinterpretation by us now in "fast time") is how I am trying to think about it.
I am definitely struggling with what it would mean for time to change. Would the frequency of the light shift as well, keeping up with time, and ultimately giving no difference? How can I reconcile this theory with the photon's point of view, in which its journey is instantaneous?
I think I am out of my league!
The deeper the source of light is in a gravity well the stronger the effect of time dilation. We have no problem with that idea of slower time. The wavelength is affected on its way out into 'faster' time. It is the same for your idea I would imagine. If you think you are out of your depth move to shallower waters.