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MatPlus.Net Forum General Abusing Special Relativity for fun and (problem) profit

### Abusing Special Relativity for fun and (problem) profit

You all will know my SCHWALBE article on chess and physics,
which draw a few analogies between fundamental notions
of physics and chess. Here is some more stuff. If a creative
way to look at moves gives you new ideas, I already succeeded.

Definitions.

Let the "future light cone" (FLC) of a position A be all positions
that can be (help)played from A.
Let the "past light cone" (PLC) of a position A be all positions
that can be (help)played into A.
(Obviously, if A is in the FLC of B, then B is in the PLC of A.)
If A is in the FLC of B and B in the FLC of A, call their relation
"spacelike". Otherwise "timelike": going from (say) A to B is irreversible
(B in FLC of A, but not A in FLC of B), and position A must come
before B in a game.

Let a "big bang" (BB) be a position A that provably occurs for the
first time in a game. Obvious reasons for a BB are pawn moves,
captures and castling.
Let a "big crunch" (BC) be a position A that provably occurs for the
last time in a game. Obvious reasons for a BC are mate, stalemate, and
the automatic draw rules (50, 3fold, dead).
Evidently, BB and BC can only come together - either a position A can be
repeated or not. (The 50 move rule doesn't change that, and neither does
castling/ep because "position" includes castling/ep rights!)

Let a "white hole" (WH) be a position A that has a finite PLC.
In established terms we call this a retroillegal position
(in orthodox chess, it happens that the beginning position is no BB).
(Danger: WH is sharper than retroillegal. A position A is already
retroillegal if the starting position is not in the PLC of A.
Challenge: A WH with as many moves until you're stuck as possible.
Let a "black hole" (BH) be a position A that has a finite FLC.
In established terms we call this a "problem without words".
(I.e. we need no stipulation. BTW, does one know a published PWW where
the outcome is variable? Here is an ad hoc example...
(= 4+3 )

Challenge: More than 2 outcomes.)

We now can ask things like "Can a move be timelike from A to B without
the trivial solution of B being a BB/BC?" (Which is just the
"Irreversible Chess" question - see e.g. Noam Elkies on StackOverflow
or Nikola Predrag here - rephrased understandably :-) Record as now:
(= 2+4 )

The only reversible moves are repeating, so if the last move was
say Qe2-f3+, it was irreversible.

Can FLC(A)=BLC(A)? (Answer no, since you can always uncapture into the past.
You need a condition like "Ohneschlag".)
It's possible that FLC is a subset of BLC, just take two kings.

Feel free to combine the above notions for interesting retro challenges
for 2020 :-)

Hauke

All positions are WH, because there are a finite number of positions, so a PLC is always finite. Or am I misinterpreting the definitions?

(3) Posted by Jakob Leck [Wednesday, Jan 1, 2020 22:15]; edited by Jakob Leck [20-01-01]

Joost's remark was also my first thought after reading the definition of WH.

But the whole analogy seems a bit dodgy to me. For example, future and past are, in our everyday understanding of those terms and in special relativity as well, disjoint, whereas your FLCs and PLCs have non-empty intersections in most positions. This leads to rather strange notions of time-like or space-like "relations" (defining distances or even geodesics seems impossible here). And the fact that you are writing about irreversible help-play to me suggests an analogy to thermodynamics rather than special relativity.

@Jakob: Indeed, in chess we can have closed timelike curves.
It's no bug, it's a feature :-)
@Joost: Yes, with an automatic 50 move rule every position
is a black hole (and a white hole too).

But let us rather concentrate on...
(= 4+7 )

1.Q~ s=
1.Qxg2 s#
1.Qxf2 Rxf2 w#
1.Qxf2 Pxf2/Bxf2 and 1.Qxg3 Rxg3 w=
(and even 1...exf3 2.exf3 s= if BTM; if you want length/variants
instead, make Pe3 black, with 1.Qxf2 exf2 2.e3 s=)

All three results still possible :-)

Hauke

But, Hauke, if we have closed time-like curves in chess then the analogy is clearly faulty: In special relativity no closed time-like (differentiable!) curves are possible because the space-time manifold is simply connected.

In (special or general) relativity you start with a semi-Riemannian manifold of dimension four with a metric of index one which you use to define space-like and time-like directions in tangent space. Only then can you start using terms like future and past when you have a globally defined time-like vector field which you can use to globally define a future time direction.
Your approach to the chess analogy sort of starts skipping the first steps and you use the natural idea of making moves or taking back moves to introduce future and past. After having obtained future and past light cones you then go back to the first step in relativity to define space-like and time-like separation.
But the fact that a move, played regularly or taken back, takes you to future or past suggests that all curves in chess would be time-like, wouldn't it?

Maybe part of the problem is our understanding of positions as the analogy to events. Is a position
1. a diagram,
2. a diagram and information on who is to move,
3. a diagram, info on who is to move, and a move counter?

Option 3 would be the best choice to keep checking the 50 move rule.

Btw. in physics, watching a reversible process backwards you would not be able to tell that you are seeing it backwards. In chess, interestingly, there are situations where you could: Sg1-f3 from the initial game array cannot be a move taken back due to a parity argument and, while it is a reversible move, you would see a time direction.