null-strut-grid △▽△▽△▽△▽△▽△▽△▽△▽△▽△▽△▽△▽△▽△▽△▽△▽△▽△▽△▽△▽△▽△▽△▽△▽ "All our assertions of spatio-temporal relations are reducible to individual observers' judgements of having collected several perceptions either together, or separately; i.e. determinations of space-time coincidence"
The drawing is of an (arbitrarily extended) "octet truss" 3D space frame ( open access taken from https://www.mdpi.com/2075-4701/12/3/410 ); see also tetrahedral-octahedral honeycomb.
ReplyDeleteInterpreted as illustration of a (3+1) dimensional null-strut-grid:
- each individual vertex depicts the individual time-like world line of a suitably selected "material point" (in the sense of [Einstein-1916]; [Minkowski-1908] ...); a.k.a. a "constituent" (of the null-strut-grid being shown), and
- each edge segment depicts continuous signal (front) exchange between relevant pairs of those material points; implying the causal structure of a "photon-2-surface", see [gr-qc/0306042];
or, being only a limited segement, rather: a "photon ribbon";
and thereby also a certain set of "null-struts" (in the sense of [Kheyfets-1988]).
In the corresponding (arbitrarily extended) null-strut-grid, for each constituent these null-struts are required (and selected such that) ping-coincidence relations are found and satisfied
- wrt. the 12 nearest neighbors,
- wrt. the 6 next-to-nearest neighbors,
- wrt. the 12 next-to-next-to-nearest neighbors, etc.
Each constituent is thereby also identified as "the middle between" (in the sense of [Comstock-1910] and of [Einstein-1917]) six different (and even disjoint) pairs of its nearest neighbors (where the two constituents making up any such pair are in turn next-to-next-to-nearest neighbors of each other).
off-topic: draft for https://scirate.com/arxiv/2605.21660 1/n
ReplyDeleteDavid Garfinkle promises to clear up misconceptions about special relativity --
and he succeeds at least to the extent to allow being very specific about how he fails (almost expectedly) in section VI.
First a general remark: Garfinkle employs a generous share of what in certain circumstances are disparagingly called "weasel words", especially concerning topics of section VI (incl. their introduction in prior sections) -- viz. “you think [this and that], but I think [such and such]”; "that each observer thinks that the other’s [...] "; "seems to contradict the hard won notion of the previous section that [...]".
Any presentation might be more palatable to retired engineers, and arguably even closer to Einstein's own broader pedagogical intentions, by ever more establishing and concentrating on **what is** and what is consequently unanimously agreed by everybody involved (as "being so, and not otherwise"); i.e. in the applicable terminology: invariants.
Consequently, the length of the ruler discussed in section V is unambiguously \(L_0\), the length of the thing itself, i.e. the distance of its actual material ends from each other (provided they are both members of the same inertial frame;
where the worldline of one of them is labelled \(O^{\prime}\) in Fig. 5);
while the material points whose distance from each other has the value \(L = \frac{L_0}{\gamma}\) are members of a completely different inertial frame
(where the worldline of one of them is labelled \(O\) in Fig. 5);
and the actual ends of the ruler itself are ("only") projected on those other points.
Correspondingly, the duration of the lifetime of a (free) muon **is** about \(2.2~{\rm \mu s}\)
If the event which involved the production of a specific muon ("in the upper ahmosphere"), as well as the event with the decay ("the death") of this muon are both projected onto another time-like, straight world-line, e.g. on the world-line of a constituent of "the laboratory" (who didn't itself take part in both these aforementioned events -- therefore the need for projecting at least one, or both of these events onto the worldline of this laboratory constituent) then the arc-length \(\gamma \tau\) of the accordingly delimited world-line segment is systematically longer than the duration \(\tau\) of the lifetime of the muon itself.
For due distinction and correct attribution, the former longer duration can be called the "occupancy duration of the laboraory, by the muon", for instance.
Further, Garfinkle's specific phrase "that each observer thinks that the other’s clock is running slow" (which surely appears in previous writing, too) may not even grammatical to begin with;
an arguably more careful wording is
»that each observer thinks that the other’s clock has been running **slower than** [the own clock]«.
(And never mind the repeated spelling mishap on p. 18)
off-topic: draft for https://scirate.com/arxiv/2605.21660 2/2
ReplyDeleteTaking then a closer look at section VI, Garfinkle describes his variant of the well-familiar "twin experiment" (cmp. "FIG. 7. Twins take different paths through spacetime [...]), where (for continuation of the preparatory section V, but also for distinction) we shall stick to calling \(O\) the twin who had a straight worldline from participating in event \(A\) until participating in event \(B\), but call the other twin \(P\), whose worldline took a detour via event \(B\) (rather than attempting even more acrobatics with primes and/or apostrophes).
The relevant **readings** assigned to the (indications of the) twins at the three relevant events \(A\), \(B\), and \(C\) shall accordingly be denoted as
- the two applicable readings \(t_A^O\) and \(t_C^O\) of twin \(O\)
(instead of writing \(t_A\) and \(t_C\), in the exact notation of section V), and
- the three applicable readings \(t_A^P\), \(t_B^P\) and \(t_C^P\) of twin \(P\)
(instead of writing primes, trying to match the exact notation of section V).
By Garfinkle's detailed prescription, the values of these readings can be taken (unambiguously enough) as the numbers of candles on their respective birthday cakes at those respective events:
\(t_A^O = 25 = t_A^P\),
\(t_B^P = 33\),
\(t_C^O = 45\),
\(t_C^P = 41\).
Now, helpfully, Garfinkle does imply values of invariants which characterize the relevant worldline sections, namely the respective **durations**, a.k.a. arc-lengths of timelike worldline segments
(here in notation being a synthesis of p. 8, p. 21, and the above naming of the two twins):
\(\Delta \tau^O_{AC} = 20 \text{ years}\),
\(\Delta \tau^P_{AB} = 8 \text{ years}\),
\(\Delta \tau^P_{BC} = 8 \text{ years}\).
(For his actual explicit prescription, along with Fig. 7, Garfinkle has used, or implied, apparently referring to both twins likewise:
»units where time is measured in years and distance in light years«.)
Consequently also \(\Delta \tau^P_{AC} = 16 \text{ years}\), as the duration of twin \(P\)'s life path from having attended event \(A\) until having attended event \(C\) (a.k.a. the arc-lenght of \(P\)'s entire relevant worldline from event \(A\) to event \(C\)).
But (distressing to me, never mind retired engineers) Garfinkle goes on to conclude that thereby:
> »we have done a direct comparison between the two clocks«
... explicitly: "birthday-cake-candle clocks" ...
> »and one of them is definitely slower.«
??? No!
The comparison shows instead that twin \(P\)'s birthday-cake-candle clock **had run shorter** than twin \(O\)'s birthday-cake-candle clock, on their respective separate worldlines, getting from the jointly attended event \(A\) to the jointly attended event \(C\)).
But the comparison thereby also shows that both these clocks **had run equally fast**, i.e. with equal (average) clock rates:
\(\nu^O := \frac{(t_C^O - t_A^O)}{\Delta \tau^O_{AC}} = \frac{20}{20 \text{ years}} \),
\(\nu^P := \frac{(t_C^P - t_A^P)}{\Delta \tau^P_{AC}} = \frac{16}{16 \text{ years}} \),
thus \( \nu^O = \nu^P \),
where notably the corresponding inverses of these equal (average) clock rates are in turn equal to the »unit« which Garfinkle had prescribed (likewise for both twins):
\(\frac{1}{\nu^O} = \frac{1}{\nu^P} = 1 \text{ year}}. \)
Hopefully, the specific described misconceptions about special relativity which had still afflicted David Garfinkle article is thus resolved; perhaps even to the satisfaction of retired engineers.