An Axiomatic Road to General Relativity€¦ · IntroSpecRelAccRelGenRel An Axiomatic Road to...
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Intro SpecRel AccRel GenRel
An Axiomatic Road to General Relativity
Gergely Székely
MTA Alfréd Rényi Institute of Mathematics 1
2014.02.21. MTA Wigner Research Centre for Physics
1 joint work with H.Andréka, J.X.Madarász, I.Németi
Gergely Székely
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Intro SpecRel AccRel GenRel
General goal:
�Investigate/understand the logical structure of relativity theories.�
In more detail:
Explore the tacit assumptions and make them explicit.
Axiomatize relativity theories (in the sense of math. logic).
Derive the predictions from a few natural basic assumptions.
Analyze the relations between assumptions and consequences.
Gergely Székely
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Intro SpecRel AccRel GenRel
Terminology
Axioms: Starting/basic assumptions.(Things that we don't prove from other assumptions.)They are NOT �nal/basic truths.
Theory: A list of axioms.
Model: A (mathematical) structure, from which we candecide whether it satis�es the axioms or not.
Model of the axioms: A model satisfying the axioms.
Gergely Székely
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Intro SpecRel AccRel GenRel
Axiomatization in general:
Ax.1.
Ax.2.
Ax.3.
Axioms: Theorems:
Thm.1.
Thm.2.
Thm.3.
Etc. Etc.
Streamlined
Economical
Transparent
RichComplex
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Intro SpecRel AccRel GenRel
S.R.
G.R.
SpecRel
AccRel
GenRel
Gergely Székely
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Intro SpecRel AccRel GenRel
Relativity theory is axiomatic (in its spirit) since its birth.
Two informal postulates of Einstein (1905):
Principle of relativity: �The laws of nature are the same forevery inertial observer.�
Light postulate: �Any ray of light moves in the 'stationary'system of co-ordinates with the determined velocity c , whetherthe ray be emitted by a stationary or by a moving body.�
Corollary: �Any ray of light moves in all the inertial systems ofco-ordinates with the same velocity.�
Gergely Székely
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Intro SpecRel AccRel GenRel Language Axioms Theorems Meta-theorems
SpecRel
Gergely Székely
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Intro SpecRel AccRel GenRel Language Axioms Theorems Meta-theorems
Logic Language: {B, IOb,Ph,Q,+, ·,≤,W }
B 〈Q,+, ·,≤〉
PhIOb
W
B! Bodies (things that move)IOb! Inertial Observers Ph! Photons (light signals)
Q! Quantities +, · and ≤! �eld operations and ordering
W! Worldview (a 6-ary relation of type BBQQQQ)
Gergely Székely
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Intro SpecRel AccRel GenRel Language Axioms Theorems Meta-theorems
W(m, b, x , y , z , t)! �observer m coordinatizes body b atspacetime location 〈x , y , z , t〉.�
m t
x
y
b
〈x , y , z , t〉
Worldline of body b according to observer m
wlinem(b) = {〈x , y , z , t〉 ∈ Q4 : W(m, b, x , y , z , t)}
Gergely Székely
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Intro SpecRel AccRel GenRel Language Axioms Theorems Meta-theorems
AxPh :
For any inertial observer, the speed of light is the same in every
direction everywhere, and it is �nite. Furthermore, it is possible to
send out a light signal in any direction.
m t
x
y
p
x
y
space2(x , y)
time(x , y)2
∀m(IOb(m)→ ∃c
[c > 0 ∧ ∀x y
(∃p[Ph(p) ∧W(m, p, x)
∧W(m, p, y)]↔ space2(x , y) = c2 · time(x , y)2
)])Gergely Székely
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Intro SpecRel AccRel GenRel Language Axioms Theorems Meta-theorems
AxOField :
The structure of quantities 〈Q,+, ·,≤〉 is an ordered �eld,
Rational numbers: Q,
Q(√2), Q(
√3), Q(π), . . .
Computable numbers,
Constructable numbers,
Real algebraic numbers: Q ∩ R,Real numbers: R,Hyperrational numbers: Q∗,Hyperreal numbers: R∗,Etc.
Gergely Székely
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Intro SpecRel AccRel GenRel Language Axioms Theorems Meta-theorems
AxEv :
Inertial observers coordinatize the same events (meetings of
bodies).
m t
x
y
b1b2 m′ t
x
y
b1 b2b2
∀mm′x IOb(m)∧IOb(m′)→[∃x ′ ∀b W(m, b, x)↔W(m′, b, x ′)
].
Gergely Székely
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Intro SpecRel AccRel GenRel Language Axioms Theorems Meta-theorems
AxSelf :
Every Inertial observer is stationary according to himself.
t
x
y
wlinem(m)
∀mxyzt(IOb(m)→
[W(m,m, x , y , z , t)↔ x = y = z = 0
]).
Gergely Székely
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Intro SpecRel AccRel GenRel Language Axioms Theorems Meta-theorems
AxSym :
Inertial observers agree as to the spatial distance between two
events if these two events are simultaneous for both of them.
Furthermore, the speed of light is 1.
p
e1e2
m
e1
e2
k
Gergely Székely
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Intro SpecRel AccRel GenRel Language Axioms Theorems Meta-theorems
What follows from SpecRel?
AxPh
AxEv
AxOField
AxSelfAxSym
SpecRel: Theorems:
?
??
???
Etc.
Gergely Székely
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Intro SpecRel AccRel GenRel Language Axioms Theorems Meta-theorems
Theorems of SpecRel
SpecRel = AxOField + AxPh + AxEv + AxSelf + AxSym
Theorem:
SpecRel⇒ �Worldlines of inertial observers are straight lines.�
Theorem:
SpecRel− AxSym⇒ �No inertial observer can move FTL.�
Theorem:
SpecRel⇒
�Relatively moving clocks slow down.�,
�Relatively moving spaceships shrink.�
etc.
Gergely Székely
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Intro SpecRel AccRel GenRel Language Axioms Theorems Meta-theorems
Theorems of SpecRel
Theorem:
SpecRel⇒ �The worldview transformations between inertial
observers are Poincaré transformations.�
o
wordview of o ′
o
wordview of o
Gergely Székely
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Intro SpecRel AccRel GenRel Language Axioms Theorems Meta-theorems
Theorems about SpecRel
Theorem: (Consistency)
SpecRel is consistent.
Theorem: (Independence)
No axiom of SpecRel is provable from the rest.
Theorem: (Completeness)
SpecRel is complete with respect to the �standard model of SR�,
i.e., the Minkowski spacetimes over ordered �elds.
Gergely Székely
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Intro SpecRel AccRel GenRel Language Axioms Theorems Meta-theorems
AxPh
AxEv
AxOField.
AxSelfAxSym
SpecRel: Theorems:
Slowing down of clocks, etc.
@ FTL observers
Poincaré transformationsetc.
Gergely Székely
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Intro SpecRel AccRel GenRel Language Axioms Theorems
AccRel
Gergely Székely
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Intro SpecRel AccRel GenRel Language Axioms Theorems
GenRelSpecRel AccRel
S.R. G.R.
Gergely Székely
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Intro SpecRel AccRel GenRel Language Axioms Theorems
The language is the same.
B 〈Q,+, ·,≤〉
Ob
PhIOb
W
B! Bodies (things that move)IOb! Inertial Observers Ph! Photons (light signals)
Q! Quantities+, · and ≤! �eld operations and ordering
W! Worldview (a 6-ary relation of type BBQQQQ)
Observers: Ob(k)def⇐⇒ ∃xyzt b W(k , b, x , y , z , t)
Gergely Székely
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Intro SpecRel AccRel GenRel Language Axioms Theorems
AxCmv :
At each moment of its life, every observer coordinatizes the nearby
world for a short while in the same way as an inertial observer does.
∀x
∃m ∈ IOb
∀k ∈ Ob
∀k ∈ Ob ∀x ∈ wlinek(k) ∃m ∈ IOb dxwmk = Id , where
dxwmk = Ldef⇐⇒ ∀ε > 0 ∃δ > 0 ∀y |y − x | ≤ δ
→ |wmk(y)−L(y)| ≤ ε|y − x |.
Gergely Székely
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Intro SpecRel AccRel GenRel Language Axioms Theorems
AxEv− :
Any observer encounters the events in which he was observed.
AxSelf− :
The worldline of an observer is an open interval of the time-axis, in
his own worldview.
AxDi� :
The worldview transformations have linear approximations at each
point of their domain (i.e., they are di�erentiable).
Gergely Székely
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Intro SpecRel AccRel GenRel Language Axioms Theorems
CONT :
Every de�nable, bounded and nonempty subset of Q has a
supremum.
Rational numbers: Q,
Q(√2), Q(
√3), Q(π), . . .
Computable numbers,
Constructable numbers,
Real algebraic numbers: Q ∩ R,Real numbers: R,Hyperrational numbers: Q∗,Hyperreal numbers: R∗,etc.
Gergely Székely
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Intro SpecRel AccRel GenRel Language Axioms Theorems
SpecRel
AxCmv
AxEv−
AxSelf−
AxDi�
CONT
AccRel: Theorems:
?
??
???etc.
Gergely Székely
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Intro SpecRel AccRel GenRel Language Axioms Theorems
Twin paradox TwP
Theorem:
AccRel− AxDi� ⇒ TwP
Th(R) + AccRel− CONT 6⇒ TwP
m
k k
Gergely Székely
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Intro SpecRel AccRel GenRel Language Axioms Theorems
SpecRel
AxCmv
AxEv−
AxSelf−
AxDi�
CONT
AccRel: Theorems:
Twin paradox
etc.
Gergely Székely
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Intro SpecRel AccRel GenRel Language Axioms Theorems
GenRel
Gergely Székely
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Intro SpecRel AccRel GenRel Language Axioms Theorems
The language is the same.
B 〈Q,+, ·,≤〉
Ob
PhIOb
W
B! Bodies (things that move)IOb! Inertial Observers Ph! Photons (light signals)
Q! Quantities+, · and ≤! �eld operations and ordering
W! Worldview (a 6-ary relation of type BBQQQQ)
Observers: Ob(k)def⇐⇒ ∃xyzt b W(k , b, x , y , z , t)
Gergely Székely
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Intro SpecRel AccRel GenRel Language Axioms Theorems
�Let all observers be equal at the level of axioms.� (Einstein)
AxPh
AxEv
AxSelf
AxSym
AxPh−
AxEv−
AxSelf−
AxSym−
AxCmv
AxDi�
For example: AxPh,AxCmv⇒ AxPh−.
Gergely Székely
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Intro SpecRel AccRel GenRel Language Axioms Theorems
AxPh− :
The instantaneous velocity of light signals is 1 in the moment when
they are sent out according to the observer who sent them out, and
any observer can send out a light signal in any direction with this
instantaneous velocity.
m
p
x
Gergely Székely
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Intro SpecRel AccRel GenRel Language Axioms Theorems
AxSym− :
Any two observers meeting see each others' clocks behaving in the
same way at the event of meeting.
m
k
k
m
wmk
wkm
Gergely Székely
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Intro SpecRel AccRel GenRel Language Axioms Theorems
AxPh−
AxEv−
AxOField
AxSelf−
AxSym−
AxDi�
CONT
GenRel: Theorems:
?
??
???
etc.
Gergely Székely
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Intro SpecRel AccRel GenRel Language Axioms Theorems
Theorem:
SpecRel =| AccRel |= GenRel
GenRelSpecRel
AccRel
Gergely Székely
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Intro SpecRel AccRel GenRel Language Axioms Theorems
Theorem:
GenRel⇒ ∀m, k ∈ Ob ∀x ∈ wlinem(k) ∩ wlinem(m)→ � wmk is
di�erentiable at x and dxwmk is a Lorentz transformation.�
Theorem: (Completeness)
GenRel is complete with respect to the �standard models of GR�,
i.e., Lorentzian manifolds over real closed �elds.
M
Qd
Qd
Qd
ψi ψk
ψj
wik
wij wjk
Gergely Székely
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Intro SpecRel AccRel GenRel Language Axioms Theorems
Def. (Geodesic):
The worldline of an observer is called timelike geodesic if it �locally
maximizes measured time.�
k ∀h
∀x
∀δ > 0
COMPR :
For any parametrically de�nable timelike curve in any observers
worldview, there is another observer whose worldline is the range of
this curve.
GenRel+ = GenRel + COMPR
Gergely Székely
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Intro SpecRel AccRel GenRel Language Axioms Theorems
In GenRel+ the notion of geodesics coincides with its standardnotion. Via geodesics, we can de�ne the other notions of generalrelativity, such as Riemann curvature tensor.
Einstein's �eld equations: Rij −1
2Rgij = Tij .
De�nition or axiom? No real di�erence.
Gergely Székely
![Page 39: An Axiomatic Road to General Relativity€¦ · IntroSpecRelAccRelGenRel An Axiomatic Road to General Relativity Gergely Székely MTA Alfréd Rényi Institute of Mathematics 1 2014.02.21.](https://reader033.fdocuments.net/reader033/viewer/2022050203/5f5699a84a5d1373d04f8b63/html5/thumbnails/39.jpg)
Intro SpecRel AccRel GenRel Language Axioms Theorems
AxPh−
AxEv−
AxOField
AxSelf−
AxSym−
AxDi�−
CONT
COMPR
GenRel+: Theorems:
Loc. Lorenz transf.
Completeness
Geodetics
etc.
Gergely Székely
![Page 40: An Axiomatic Road to General Relativity€¦ · IntroSpecRelAccRelGenRel An Axiomatic Road to General Relativity Gergely Székely MTA Alfréd Rényi Institute of Mathematics 1 2014.02.21.](https://reader033.fdocuments.net/reader033/viewer/2022050203/5f5699a84a5d1373d04f8b63/html5/thumbnails/40.jpg)
Intro SpecRel AccRel GenRel Language Axioms Theorems
Thank you for your attention!
Background materials:www.renyi.hu/~turms
Gergely Székely