If not, can you please elaborate further what you meant to imply about time's reality/dimension etc in that comment about co-ordinates being "not real"?
Time is obvious -- but time is also personal. Equally personal is ones state of motion. So, I say that one may assert without contraction: time is a direction (alternately, a dimension) chosen by ones personal state of motion. This is the physical consequence of the generalized Galilean transform when K = c⁻².
In that not everyone's choice of time is parallel in any coordinate space, there is no reality to the concept of absolute time.
In that one may choose between alternate directions of time via changing ones state of motion, time is an inseparable part of space-time.
This is what Minkowski sought to convey to his audience in his 1908 address "Raum und Zeit".
https://en.wikisource.org/wiki/Space_and_Time_(1920_edition)
Because coordinates are not real -- they are man-made inventions to describe geometry in the language of number and algebra.
So, there is co-ordinate time and proper time.
Correct -- every inertial standard of rest implies a different way of assigning coordinate time to events. Every slower-than-light world line implies a way to assign a particular proper time to events along that worldline.
Thanks. We agree (though I suspect we come at that agreement from different perspectives/reasons).
That's the two things the generalized Galilean transform when K = c⁻² (i.e. the Lorentz transform) does: it relabels the coordinates of events and it changes the standard of rest.
But neither change affects space time or the physics that happen in space time -- thus neither coordinates nor choice of standard of rest are real things.
Since you say emphatically and unambiguously that co-ordinates are "not real", then by implication co-ordinate-time is also "not real"?
Correct. It's just the extent of a world line in a particular direction. Different choice of direction implies a different elapsed coordinate time, i.e. time dilation.
Again, thanks. Again, perhaps from difrerent directions.
Given two events along an inertial world line, one has a measure in any given coordinate system of a certain amount of elapsed coordinate time, $$\Delta t$$, and also a certain amount of change in spatial coordinates, $$\Delta \vec{x}$$. One is free to find any other inertial coordinate system where $$\Delta t' \geq \sqrt{(\Delta t)^2 - c^{-2} (\Delta \vec{x})^2}$$ and $$\left| \Delta \vec{x}' \right| \geq 0$$ such that $$c^{2} (\Delta t')^2 - (\Delta \vec{x}')^2 = c^{2} (\Delta t)^2 - (\Delta \vec{x})^2$$.
So that's quite a bit of freedom (but not total) one has to choose coordinate time for a given pair of events.
In the same way my analogy with coordinates for a sheet of paper, one has freedom to choose the x-axis to point in any direction.
Imagine a large, irregular sheet of paper. Alice and Bob each pick a different point as the origin of a 2-dimensional Euclidean coordinate system. Each coordinate system labels each point of the paper with two coordinates. A formula lets us convert Alice's coordinates (x,y) to Bob's coordinates (x',y').
$$\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} \frac{1}{\sqrt{1 + m^2}} & \frac{-m}{\sqrt{1 + m^2}} \\ \frac{m}{\sqrt{1 + m^2}} & \frac{1}{\sqrt{1 + m^2}} \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} \chi \\ \upsilon \end{pmatrix}$$
And coordinate differences transform like:
$$\begin{pmatrix} \Delta x' \\ \Delta y' \end{pmatrix} = \begin{pmatrix} \frac{1}{\sqrt{1 + m^2}} & \frac{-m}{\sqrt{1 + m^2}} \\ \frac{m}{\sqrt{1 + m^2}} & \frac{1}{\sqrt{1 + m^2}} \end{pmatrix} \begin{pmatrix} \Delta x \\ \Delta y \end{pmatrix}$$
or back the other way
$$\begin{pmatrix} \Delta x \\ \Delta y \end{pmatrix} = \begin{pmatrix} \frac{1}{\sqrt{1 + m^2}} & \frac{m}{\sqrt{1 + m^2}} \\ \frac{-m}{\sqrt{1 + m^2}} & \frac{1}{\sqrt{1 + m^2}} \end{pmatrix} \begin{pmatrix} \Delta x' \\ \Delta y' \end{pmatrix}$$
Naturally, if Alice measures out $$\begin{pmatrix} \Delta x \\ \Delta y \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$$, Bob measures this same line as $$\begin{pmatrix} \Delta x' \\ \Delta y' \end{pmatrix} = \begin{pmatrix} \frac{1}{\sqrt{1 + m^2}} \\ \frac{m}{\sqrt{1 + m^2}} \end{pmatrix}$$.
And if Bob measures out $$\begin{pmatrix} \Delta x' \\ \Delta y' \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$$, Alice measures this same line as $$\begin{pmatrix} \Delta x \\ \Delta y \end{pmatrix} = \begin{pmatrix} \frac{1}{\sqrt{1 + m^2}} \\ \frac{-m}{\sqrt{1 + m^2}} \end{pmatrix}$$.
This does not mean that Alice and Bob both have a x-dilation of $$\frac{1}{\sqrt{1 + m^2}}$$ with respect to each other -- that is just a small part of the whole story. It is a piece so small as to be a gross distortion if substituted for a complete understanding. The whole story includes
- $$\sqrt{(\Delta x)^2 + (\Delta y)^2} = \sqrt{(\Delta x')^2 + (\Delta y')^2}$$ for all straight lines,
- that in terms of this invariant the straight line between two points has the lowest total of any path connecting the points ,
- the relation is a rotation
- what is rotated can be rotated back, and
- this is a relative rotation not connected with any absolute coordinate system, but only the choices made by Alice and Bob.
$$\begin{pmatrix} \frac{1}{\sqrt{1 + m^2}} & \frac{-m}{\sqrt{1 + m^2}} \\ \frac{m}{\sqrt{1 + m^2}} & \frac{1}{\sqrt{1 + m^2}} \end{pmatrix} = \begin{pmatrix} \cos \; \tan^{\tiny -1} m & - \sin \; \tan^{\tiny -1} m \\ \sin \; \tan^{\tiny -1} m & \cos \; \tan^{\tiny -1} m \end{pmatrix}$$
$$ \begin{pmatrix} \frac{1}{\sqrt{1 + m^2}} & \frac{m}{\sqrt{1 + m^2}} \\ \frac{-m}{\sqrt{1 + m^2}} & \frac{1}{\sqrt{1 + m^2}} \end{pmatrix} \begin{pmatrix} \frac{1}{\sqrt{1 + m^2}} & \frac{-m}{\sqrt{1 + m^2}} \\ \frac{m}{\sqrt{1 + m^2}} & \frac{1}{\sqrt{1 + m^2}} \end{pmatrix} = \begin{pmatrix} \frac{1 + (m)(m)}{1 + m^2} & \frac{m -m}{1 + m^2} \\ \frac{m - m}{1 + m^2} & \frac{(-m)(-m) + 1}{1 + m^2} \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} $$
The whole story of [special] relativity includes
- $$\sqrt{(c \Delta t)^2 - (\Delta x)^2 - (\Delta y)^2 - (\Delta z)^2} = \sqrt{(c \Delta t')^2 - (\Delta x')^2- (\Delta y')^2- (\Delta z')^2}$$ for all inertial paths,
- that in terms of this invariant the inertial path between two events has the highest total of any path connecting the events ,
- the relation is a hyperbolic rotation
- what is hyperbolic rotated can be hyperbolic rotated back, and
- this is a relative hyperbolic rotation not connected with any absolute coordinate system, but only the choices made by Alice and Bob.
$$\begin{pmatrix} \frac{1}{\sqrt{1 - \beta^2}} & \frac{\beta}{\sqrt{1 - \beta^2}} \\ \frac{\beta}{\sqrt{1 - \beta^2}} & \frac{1}{\sqrt{1 - \beta^2}} \end{pmatrix} = \begin{pmatrix} \cosh \; \tanh^{\tiny -1} \beta & \sinh \; \tanh^{\tiny -1} \beta \\ \sinh \; \tanh^{\tiny -1} \beta & \cosh \; \tanh^{\tiny -1} \beta \end{pmatrix}$$
Which is an essential part to get if you want to differentiate between the structure of coordinate system and the structure of a geometry. When physics uses mathematical models, it is saying that the structure of the mathematical model precisely models the observable structure of some aspect of reality. It is not a statement of mechanical underpinnings, but to the the extent that there may be underpinnings they replicate the behavior of the mathematical model. Euclidean geometry is a good model for the surface of paper provided we don't look close enough to see the texture of the paper or the thickness of the dots we put on it. Hyperbolic geometry, so experiments that favor K = c⁻² over K = 0 tell us, is a good model for space-time.
While proper time is a direct derivation of the proper physical processes which are actually occurring in the proper frame,
No that's a proper clock -- a physical way to measure proper time. Actual proper time does not depend on the existence of those physical processes, but you are almost on the right track.
the co-ordinate time is purely part of an abstract non-physical analytical/observational frame construct that is "not real"?
All coordinate systems are "not real" in this sense. They are man-made systems to label points in space-time so one may use the language of algebra to conduct geometry.
Not at all. The 'proper time' is directly related to/derived from the proper processes however they may vary under varying conditions. The proper processes/cycles are of the proper energy-space features in a proper dynamics of their own parameters/contents/flows. At no time is the proper process a 'time' process' unless we introduce that 'derived' parameter FROM those observed processes. The 'clock processes' of processing matter-energy does exist; the 'time' derived from he analysis of the dynamics/processes is just that: derived, not fundamental/proper to the processes irrespective of the rate at which the environment/frame velocities etc dictate they proceed at compared to processes in other enviromnents.
The 'dilation' of time is a further comparison-derived value of a comparison-derived differential between the two processes under study/comparison.
So unless you admit an ABSOLUTE energy-space THIRD FACTOR present in all processes whatever their frame/rate/location etc, then the processes are the determinants of what 'times' and 'dilations' that are 'derived' from observations/comparisons of those processes.
You have asserted many different things, but have not modelled them so have no way to communicate their physicality.
First of all you need to define what you mean by a proper frame and explain how it can have any physical processes, and which of those physical processes are "proper" and then explain how to derive what you call proper time. Next you must fully document how and what conditions vary the "proper" processes and which processes are cycles. Then let's hear you separate energy-space features and dynamics into those that are proper and improper. Having defined these terms and I suspect others, you need to demonstrate that they are physically reasonable in that they at least correspond in detail and precision to the physics your listeners know.
Then having demonstrates that your world-view is physically motivates, you may build a logical argument that some sort of "absolute energy-space" is needed.
But you haven't argued for any of this, you have only asserted statements about relations between undefined entities and asserted that I have some sort of admission due you. This is
argumentum ex culo, or bafflegab to be polite.
Is that what you meant to imply? Ie, that the only 'real' (proper) time
I never said there was
one proper time. However, given a particular slower-than-light world line, that line allows us to speak of proper time along its extension. If you have two slower than light world-lines meeting in two places then that allows the possibility that the proper time between the two intersections is dependent on which path is used -- this is realized to the maximal extent when only one of the two world-lines is
inertial as in the common expression of the twin so-called paradox.
is a 'dependent dimension' (not something fundamental in its own right) derived from proper frame processes;
You keep on saying processes. Coordinate frames (being imaginary) don't have any processes.
while the non-real (co-ordinate/mathematical) time is an 'abstract dimension' purely introduced as a convenient fiction in the use of non-real co-ordinate frame SR/GR perspectives/interpretations of the actual physical proper frame processes?
Minus the bafflegab, that's largely the point I have been making. You seem to say it less clearly the more you write.
Less clearly because the existing 'explanation' is being scrutinized under a new perspective which is not quite in synch with the view you assume is the right view. So naturally any novel perspective will seem bafflegab if the current orthodoxy hasn't the lexicon/concepts to convey this new perspective in the way you would prefer.
New lexicon, undefined and unpublished in the scientific literature, with no demonstrated value is bafflegab.
Anyway, we already agree that co-ordinate frames are not real, so I haven't changed that view; but it is you that seem to imply that just because co-ordinate frames are not real then processes are not real either.
Processes of unreal things are unreal. You asserted above that "proper frames" have "proper processes" but that doesn't define either frame or process or proper. Bafflegab.
Perhaps you should separate the two. Processes are fundamental and real irrespective of the views from non-real co-ordinates or otherwise. The frame does not change the fact that processes per se exist.
Neither can the frame (in standard physics a synonym for a choice of inertial coordinate system) render processes "proper" or "improper" with any real meaning. It's your bafflegab, it's your responsibility. But if you drop the "proper" there are a large number of processes, electromagnetic, weak, and strong and gravitational, which can with high precision track (i.e. act as clocks) the elapsed proper time of any object for which they are co-moving. And as elapsed coordinate time equals elapsed coordinate time for an inertial object which is not moving in that coordinate system, these physical clocks can be used as arguments that an inertial coordinate system can give precise times to events even in vacuum where none of these clocks actually exists. That clocks of such wildly different physical laws respect the same sense of time indicates that time is somehow more fundamental than each of them. That the clocks would report the time if they existed indicates that space-time has a mathematical structure (a symmetry) which is not tied to the existence of clocks.
This is why I think geometry is the best way to address this structure of space-time. If the symmetry is ever found to be flawed, then it might be high time to dig deeper, but until you find a flaw in a bearing it is a perfect sphere and until you find a flaw in Lorentz symmetry, the geometry of space-time is perfectly Lorentzian.
It is the interpretations from SR only based on non-real frames/co-ordinates; the processes are not subject to such interpretations. They exist and have a rate of their own depending on the energy-space environmental factors and flows in a dynamics of their own, irrespective of whatever frames we use. That is the point. Time is derived from these fundamentals. Time 'dilation' is derived by comparisons between processes in same conditions and between frames under man-made non-real co-ordinate constructs.
Incorrect. And with "energy-space environmental factors" you fall back into bafflegab thicker than quantum woo.
Also, so far, you have ignored that some coordinate systems are better than others. Inertial Cartesian coordinates are particularly beautiful to work with in the gravityless SR because Newton's law of inertia becomes the law of straight lines.
All co-ordinates are equal in the sense they are "not real" in the sense you admit. As such, they are not the arbiters of anything except what our theory interprets given that non-reality analysis.
Everything else flows from that:
- There is proper process; from which we derive proper time.
- There is proper process 'acceleration/decleration' factor from which we may further derive an 'imaginary' time 'dilation' factor for convenience in analysis/interpretations in non-real SR 'geometric/mathematical' construct.
That's all that can be taken from what you and I agree on and from what you maintain as SR non-real co-ordinates construct/perspective.
Empty assertions and non sequiturs.
In GR, you are free to use any smooth coordinates you want to so long as they map the necessary region of the space-time manifold to $$\mathbb{R}^4$$ in a one-to-one manner. But even here there are special coordinates that cause geodesics through a point in the space-time manifold to be mapped to straight-ish lines through the origin in $$\mathbb{R}^4$$. (A consequence of all manifolds being locally flat.)
But since co-ordinates are not real as you and I agreed, then all that is a limited non-real perspective regardless of extension of geometry from euclidian to Einsteinian etc. The fact remains that non-real persectives may be useful, but they 'explain' nothing regarding the PROPER and FUNDAMENTAL aspects which are yet to be treated in a less non-real way by future extension/improvement/completion of current 'non-reality' based theory.
Really, thanks for your reply, rpenner. It was helpful and constructive to objective scientific discourse of pioneering revisions of what are the uses and limitations of non-real geometric perspectives of the reality. Cheers.
Coordinates aren't geometric perspectives -- they are algebraic perspectives until you interpret them geometrically, as in algebraic geometry.