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RELATIVISTIC COMPUTING
by Péter Németi (joint work with Hajnal Andréka, István Németi and Gergely Székely)
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THE IDEA
Church Thesis was formulated in Newtonian worldview
Turing Machine concept incorporates “ABSOLUTE TIME”
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G. R. you can MANIPULATE TIME (just like space)
after Black Hole Physics breaking the Turing Barrier becomes conceivable
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PLAN OF TALK
Black hole computing
Wormhole computing (astrophysicist Igor Novikov)
Comparison, realisticity issues
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GRAVITY CAUSES SLOW TIME
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G. R. Gravity causes slow time
New physics brings in new horizons, new possibilities and breaks old barriers.
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GRAVITATIONAL TIME DILATION IN EVERYDAY LIFE
+45 μs/day because of gravity-7 μs/day because of the speed of sat.
approx. 4 km/s (14400 km/h)
Relativistic Correction: 38 μs/day approx. 10 km/day
GPSGlobal Positioning System
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COMPUTING BEYOND THE TURING BARRIER
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We want to use the GR effect that gravity causes slow time to decide an arbitrary recursively enumerable set.
We will show the idea in two levels of abstraction.1. In Cartoon-like pictures2. The same idea in spacetime diagrams
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SPEEDING UP TIME USING GRAVITY
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SPEEDING UP TIME USING GRAVITY
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SPEEDING UP TIME USING GRAVITY
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SECOND LEVEL: SPACETIME DIAGRAMS
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Let us see in more mathematical detailhow this fairy tale can be implementedin terms of GR spacetime diagrams.
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ORDINARY BLACK HOLE – COMPUTER’S VIEW
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ORDINARY BLACK HOLE – PROGRAMMER’S VIEW
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ROTATING BLACK HOLE
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Malament-Hogarth event!
Etesi-Nemeti paper: computationsInfinite space inside
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WORRIES – ONE BY ONE
1. Blue shift (communication)2. Evaporation of BH3. Strong Cosmic Censor Hypothesis4. Instability of Cauchy Horizon5. Blue shift (cosmic, distant galaxies)6. Unlimited tape7. Predictability, Repeatability, Radiation from
singularity8. Decay of protons9. Heat Death
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Cauchy horizon
outer event horizon
worldline of programmer
worldline of computer
photon signal communication messenger communication
already solved in [ND], Fig.5, sec.5.3.2, p.133
WORRIES: 1. COMMUNICATION BLUE SHIFT
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WORRIES: 2. EVAPORATION OF BH Shielding (only gradually) CMB Apostolos’ book, p.146 Evaporation is only hypothetical Penrose 2004, p.848:
“… BH evaporation is an entirely theoretical … might be that Nature has other ideas for future of BH’s.”
Papers: Nikolic, H., Black holes radiate but do not evaporate.
arXiv:hep-th/0402145v3, Aug 2005. Helfer,A. D., Do black holes radiate? arXiv:gr-qc/0304042v1, Apr 2003.
Therefore since we are using a huge BH for hypercomputing, Hawking radiation should cause no problem.
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WORRIES: 3. – 4. – 5.
3. Strong Cosmic Censor 4. Instability of Cauchy Horizon 5. Blue shift (cosmic, distant galaxies)
- Expansion of Universe forever
- Asymptotically de Sitter background implies stability
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FOREVER EXPANDING UNIVERSE
worldlines of galaxies
photon geodesicsphoton geodesics
worldlines of galaxies
t
cosmological event horizon
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KERR-NEWMANN WORMHOLE IN FOREVER EXPANDING UNIVERSE
r = 0 singularityr = 0
I+I+ BH interior distant galaxies
(Olbers paradox)
worldline of computer
1
23
1: Inner event horizon(Cauchy horizon)
2: Outer event horizon
3: Cosmological eventhorizon
Kerr-Newmann-deSitter space-time
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WORRIES: 3. – 4. – 5.
Papers for “fall of censor”: Joshi, P.S., Do naked singularities break the
rules of physics? Scientific American, January 2009.
Chambers, C. M., The Cauchy horizon in black hole-de Sitter spacetimes. arXiv:gr-qc/9709025v1, Sep 1997.
German,W.S., Moss, I.A., Cauchy horizon stability and cosmic censorship. arXiv:gr-qc/0103080v1, Mar 2001.
Lobo, F.X. N., Exotic solutions in general relativity: traversable wormholes and “warp drive” spacetimes. arXiv:0710.4474v1 Oct 2007.
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WORRIES: 3. – 4. – 5.
Weak Energy Condition is not LAW: Barcelo, C., Visser, M., Twilight for the
energy conditions?, Int. J. Mod. Phys. C 11 (2002), 1553. arXiv:gr-qc/0205066.
Cosmology accelerated expansion, dark energy
Wormholes’ boom implies new kinds of MH-regions
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WORRIES: 6. – 7. – 8.
6. Unlimited tape information ≠ energy
7. Predictability Repeatability Radiation from singularity
New orbit for programmer 8. Decay of protons
Small enough particles do not decay (electron, positron, neutrino)
9. Heat Death
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PART II: WORMHOLE COMPUTING
We reviewed BH computing, now comes WH computing.
This is another kind of thought-experiment, based on a different GR space-time
Both have advantages and disadvantages. WH computing: can compute more complex problems, is smaller, no need for Noahs Ark
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AB
WORMHOLE (AS SPACE-STRUCTURE)
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A
BLong path (outside)
Short path (inside)unfold
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AB
WORMHOLE IN SPACETIME DIAGRAM
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Short path (inside)
AB
t
Long path (outside)
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ORDINARY BLACK HOLE – PROGRAMMER’S VIEW
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MANIPULATING TIME...
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t
1
1
2
3
4
5
6
2
3
B Ar
r = 0
Signal outside WHFrom A to B
Signal inside WHFrom B to A
Event horizon of BH
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MANIPULATING TIME...
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t
1
1
2
3
4
5
6
2
3
B Ar
r = 0
Signal outside WHFrom A to B
Signal inside WHFrom B to A
Event horizon of BH
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INDUCING A TIME SHIFT
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t
2
4
6
8
10
12
B Ar
Signal outside WHFrom A to B
Signal inside WHFrom B to A
1
3
5
7
9
11
2
4
6
8
1
3
5
7
9
BlackHole disappears
BlackHole appears
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Designing the wormhole computer
Task is: decide for any recursive set S whether S is finite or infinite.
S is specified by fixing a number theoretic 0 -formula Φ(x)
S x : Φ(x)
We will check whether there is a greatest x with Φ(x).
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ENTITIES OF THE WH COMPUTER Ca – Turing machine (TM) at mouth A (see its
program later) Ma – Mirror at mouth A (acts either as a mirror or a
prism) Mirror: reflects signals coming from the wormhole towards
Mb outside the wormhole
Prism: filters red light, reflects green towards Mb
Cb – TM at mouth B (see its program later) Mb – Mirror at mouth B
Reflects light signals coming from Cb or Ma into the
wormhole towards Ma
Pr – Programmer (checks incoming signals on Ma till TMH) Red and Green light means there are finitely many x such
that Φ(x) Only Red light means there are infinitely many x such that
Φ(x) No signals means there is no x such that Φ(x)
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Σ2 THOUGHT EXPERIMENT
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Wait TMH time
Φ(x) & Flash
Infinite many x Φ(x)
Finite many x
Φ(x)
Pr Ca Ma Mb Cb
Wormhole time gap
(Malament-Hogard event)
Inside WHOutside WH
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PROGRAM OF CB
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X := 0
Φ(x)?Message to Mb
(red light)Y
Flash to Ca
X := X+1
N
Wait TMH time Pr Ca Ma Mb Cb
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PROGRAM OF CA
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F := 0
Ma := Prism
Red light till TMH? F := 1Y
Wait till TMH
Ma := Mirror
Wait for Flash from Cb
Ma := Mirror N
F = 1
Message to Ma
(green light)
True
HALT
False
Pr Ca Ma Mb Cb
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WORMHOLES CAN COMPUTE MORE
How much more? Beyond-Turing complexity issues: Jiri
Wiedermann, Philip Welch, Marek Suchenek, Mark Hogarth
BH decidable problems are all
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WORMHOLES CAN COMPUTE MORE
Theorem (P. Welch): BH can compute all problems if and only if one can create a computer which able to receive an arbitrary large (but finite) amount of information.
WH can decide all problems.
Question: can WH decide all n problems by using perhaps n WH-s?
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COMPARISON BETWEEN BH AND WH COMPUTING
BHs observedBlue shift problem:
BHs are focusing lenses
No time travel
We need big BHs
Programmer travels
BH can compute
less
WHs not yet
observedWHs are defocusing
lenses (no bs roblem)
Time travel
WHs may be small
No need for Noahs
Ark
WH can compute
more
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THE WAY TO GET THROUGH THE TURING BARRIER
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THE END
OR THE FUTURE ? …
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1.ST QUESTION
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Physics General Relativity
Cosmic censor
conjectureExotic Matter
Closed time-like curves
Malament-Hogart Spacetimes
Cosmology
SingularitiesRELATIVISTICCOMPUTING
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QUESTIONS…You can read more on Istvan Nemeti’s homepage:
http://www.renyi.hu/~nemeti/
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PEOPLE, HISTORY
History of relativistic computing
1987 - start:Andréka & Németi Univ. Ames USA Lecture Notes
1992 - Independent start:Hogarth (Cambridge)
Other authors:Pitowsky (Israel), Shagrir (Israel), Earman (Pittsburgh), Norton
(Pittsburgh), Malament (USA), Etesi (Hungary, Dept. Phys.), Dávid (Hungary, Dept. Phys.), Tipler, Barrow, Jiří Wiedermann, Philip Welch, Marek Suchenek, Chris Wüthrich, B. Gyenis (Pittsburgh)
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