[chiralSPO (OP)]

Please note: This thread may sit on the edge of generally accepted science and speculation. I would like to keep it in this sub-forum (which is intended for discussions of generally accepted science), so let us agree to be clear about when we are speculating/hypothesizing.  Thanks! (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 axes--it would be nice if someone can recommend a function that has similar properties but can be explicitly solved both ways):

Screen Shot 2018-05-01 at 12.11.47 PM.png (36.26 kB . 570x563 - viewed 3730 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 well-defined 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 = 10¹⁰ 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?

[Colin2B]

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.

[chiralSPO (OP)]

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.

[jeffreyH]

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

[chiralSPO (OP)]

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 nucleus--this becomes increasingly cumbersome when considering multiple planets or atoms. (I know that internally- vs externally-defined 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 multi-body problem of spherical coordinates: is there an external coordinate system that allows for multiple Big Bangs, each being a local origin?

[jeffreyH]

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.

[Bogie_smiles]

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.

[chiralSPO (OP)]

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 zero-point, 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 end--if 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.

[Bogie_smiles]

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 zero-point, 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 end--if 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.

[chiralSPO (OP)]

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.

[chiralSPO (OP)]

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.

[jeffreyH]

Here is an aside. Not directly related to your function but interesting none the less.
https://www.quora.com/What-is-the-Laplace-transform-of-log-t
This also leads to the following.
http://mathworld.wolfram.com/Euler-MascheroniConstant.html
Euler is likely the way to go in starting to analyse the equation.

[jeffreyH]

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

[jeffreyH]

Let's break out the interesting part.
http://m.wolframalpha.com/input/?i=integrate%5B%28%28log+t%29%2Ft%29%5E-1%5D
Now that is something you wouldn't want to explain to your granny.

[chiralSPO (OP)]

Thanks Jeffrey! I had not come across the Euler-Mascheroni 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)

[jeffreyH]

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

[chiralSPO (OP)]

Thanks! 

[Colin2B]

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.

[chiralSPO (OP)]

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 4-D 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 Lorenz-like transformations for GR: if we treat time like the radius of the n-sphere (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) acceleration--so 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.

[Colin2B]

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!