[chiralSPO (OP)]

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

No real rush--but it is driving me nuts.

[jeffreyH]

Here is something that I haven't attempted to graph yet but may be interesting. It is α = t² + log²(t)t^-2

[jeffreyH]

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

[jeffreyH]

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

[Bill S]

.....-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)

[chiralSPO (OP)]

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 well-defined, and ∂x/∂α holds up better to the boundary condition that the big bang appears to impose on ∂t.

[Bill S]

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

Spot-on, thanks.

[Bill S]

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.
 

[jeffreyH]

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.

[Bill S]

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.

[jeffreyH]

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.

[chiralSPO (OP)]

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.

[jeffreyH]

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

[chiralSPO (OP)]

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!

[Bill S]

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!

[yor_on]

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.

[chiralSPO (OP)]

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?...

[alancalverd]

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?

[geordief]

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?

[yor_on]

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