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| The arrow of time mystery |
Here, theory and experience part ways. Physicists and philosophers keep trying to explain—or to explain away—the arrow of time. Is it objective? Or merely subjective? And is its mystery somehow intertwined with that of consciousness?
Recently, I attended a seminar on the arrow of time, where one approach drew on the relational theory of Yu. S. Vladimirov. What I noticed was this: while the speaker aimed for an “objective explanation,” he could not avoid repeatedly saying “we.” I resisted the impulse to ask, “Who are these ‘we’ within the framework you propose?” I felt entitled to ask, having already confronted that very question myself—in a paper on the quantum theory of measurement.
So we have the arrow of time, we have irreversibility, and we have an unease about time loops—except, perhaps, in our rare dreams.
Here is the new version of my notes, with added Ch. 6.1 "Time-like streams", and 6.2. "Space-like streams," with new Figures 10,11,12.
In the next post we take a look at what kind of life can reside on the two infinity circles?
16-10-25 Laura posted on Substack
"The Quest for Truth: The Tangled Web of Epistemology"
08-11.25 Next Laura's post
Ark, I've also heard the talk about time's arrow that you mentioned in the post above. I've been familiar with this relational theory for a long time, and it seems weakly supported, but not without elegance, and in some ways reminds me of Julian Barbour's "The End of Time."
ReplyDeleteNow reading the new pages about streams, with pleasure, and trying to repeat the calculations.
The more I think about Flows or Streams, the more convinced I am that the eternal omnipresent motion implies transition across infinity. There, in the infinity, lies the "driving force" of any change.
Sorry, I have difficulties with normalization, formula (59).
ReplyDeleteWouldn't you please specify what is ξ′(s)1 and ξ′(s)2 there?
Added explicit formulas under (58). If you would like me to expand even further - just let me know.
DeleteThank you! It was not difficult to get the formulas for ξ′(s)_i,
Deleteand then I decided to check that the sum of their squares
(ξ′(s)_1)^2+(ξ′(s)_2)^2+(ξ′(s)_3)^2+(ξ′(s)_4)^2 is zero.
Now i see that it was not a good idea. Will try again using your prompts.
Not the sum, but
Delete(ξ′(s)_1)^2 + (ξ′(s)_2)^2 - (ξ′(s)_3)^2 - (ξ′(s)_4)^2 is zero.
You can use Reduce to check it, otherwise it is tedious. I used Mathematica for all tedious calculations.
Delete"Not the sum, but..."
DeleteYes, surely, the signature is (+1, +1, -1, -1), I am so inattentive.
Got formula (60) - exactly! One step to (61) and I'm done:)
The updated file has also 6.3 which I did not yet mentioned on the blog. I will deal with it in the next post.
Delete"We now take into account the formulas d cos(α(s))/dt = − sin(α(s))dα/ds,..." transition from (60) to (61):
DeleteDon't quite understand why these expressions for derivatives are needed. I reasoned like this:
The first pair of coordinates of X_t is
(-sin(a), cos(a)) multiplied by (cos(a)cos(b)),
the second pair is
(-sin(b), cos(b)) multiplied by (1 - sin(a)sin(b));
we can consider the stream components on the torus as velocities of motion along the small and large circles, and these components are (cos(a)cos(b)) and (1- sin(a)sin(b)), respectively.
Less rigorous, more intuitive.
Thanks. Added a line between (60) and (61) and fixed the line after. Should be more clear now.
DeleteNow it is perfectly clear, thank you!
DeleteThere is one remark concerning physical meaning of the shape of infinity:
"The blue and red lines constitute the “infinity” of M+p. Light rays cross the future infinity, continue through M−p, cross the past infinity, and return to the origin".
Such 'crossing infinity' can be performed by light rays only, but if we consider a massive object, what is its 'shape of infinity'?
It should be formed by the same blue and red lines, but the world line of a massive object never crosses these lines; instead, as the object approaches a point p (or -p), its velocity X_t approaches zero, which corresponds to time dilation, right?
Recently, I discovered the wonderful world of toric varieties for myself. Currently reading William Fulton's "Introduction to Toric Varieties." Ark, I suspect you're a leading expert in this field. Perhaps you'll dedicate a couple of posts to this topic if you deem it appropriate.
ReplyDelete