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An axiomatic foundation of
relativistic spacetime
Thomas Benda
National Yang Ming University
Taipei, Taiwan
Motive
Providing a precise foundation of an important physical theory,
the General Theory of Relativity.
Such a project may be (and has been) held to offer no benefit
since physics is fundamentally different from mathematics.
However, if we suppose that physics is about individuals, then
there is at least a description of the physical world in a formal
language. No superior theoretical reason is found not to
attempt to formalize physical theories. What is more, no reason
is found not to integrate physical and formal individuals.
Even so, formal theories miss out on something specifically
physical. Above critique has merit since it compels us to think
about what that something specifically physical is.
The theory ST
Our starting point is Zermelo’s set theory Z, which entails
(almost) all of mathematics. Z is modified to a set theory with
urelemente ZU. A conservative extension ST of ZU is made.
In ZU:
Additional defined predicate constant W, read “is a worldline”.
Weakened extensionality axiom, excluding worldlines.
Additional axiom: Vy Λx (xy  Wx)
In the language of ST:
Additional primitive ternary predicate constant ∤ .
x ∤z y is read as “y intersects x at z”.
Intended interpretation
The urelemente of ST, called “worldlines”, are spacetime
curves; going further: are free-falling particle paths in
spacetime.
Spacetime points are not primitive, but sets of worldlines.
We aim to show that, given the axioms of ST, spacetime
points form a manifold, in which a Lorentzian metric can be
defined.
Thus: The only primitive physical objects are worldlines. The
only primitive physical relation is intersection of a first
worldline with a second worldline at some real number.
The set-theoretical picture
impure sets
(phys.)
pure sets
(math.)
union
urelemente
(worldlines)

Proper axioms of ST
(P1)
Λvwx (v ∤x w  Wv  Ww  x  ℝ)
“w intersecting v at x implies that v and w are worldlines and x is
a real number.”
v
w
x
Proper axioms of ST
(P2)
Λvwx (v ∤x w  v  w)
“Intersection of worldlines is irreflexive.”
(P3)
Λvwxy (v ∤x w  Fin {z | v ∤z w  x  z  y})
“For any v, w, and any x, y, the number of times w has intersected
v between x and y is finite.”
v
w
Proper axioms of ST
(P4)
Λvwx (v ∤x w  V!y (Isnvw xy)
“For any worldlines v, w, and any real number x, if w intersects v
at x, then one and only one real number y exists, such that v and
w mutually intersect at x and y.”
with
Isnvw xy for v ∤x w  w ∤∤y v 
 Λu (u  v  u  w  (v ∤x u  w ∤y u))
v
w
u
“v and w mutually intersect at x and y.”
Proper axioms of ST
(P5)
Λvwx (Wv  Ww  v  w  x  ℝ  Vy (Leawv yx))
“For any two worldlines v, w and any real number x, there is a
real number y, such that a worldline exists that leads from y on
w to x on v.”
v
w
x
y
Proper axioms of ST
(P6)
Λvwx (Wv  Ww  v  w  (v ∤x w) 
 Vab (a  b  Λu (w ∤a u  w ∤b u  (v ∤x u))))
“For any two worldlines v, w and any real number x, with w not
intersecting v at x, two objects a, b exist, such that no worldline
intersecting w at a or b intersects v at x.”
v
w
b
no such worldline
x
a
no such worldline
Proper axioms of ST
(P7)
Λvwxy (Isnvwxy  Λab (a  x  y  b  Leavw ab))
“For any two worldlines v, w and all real numbers x, y, such that
v and w mutually intersect at x and y, and all real numbers a that
are smaller than x and all real numbers b that are greater than y, a
worldline exists which leads from a on v to b on w.”
v
w
Proper axioms of ST
(P8)
Λvwxy (Isnvwxy  Λb (y  b  Va (x  a  Leavw ab))
“For any two worldlines v, w and all real numbers x, y, such that
v and w mutually intersect at x and y, and all real numbers b that
are greater than y, there exist a real number a greater than x and a
worldline which leads from a on v to b on w.”
v
w
a
exist
Proper axioms of ST
(P9)
Λvwxy (Isnvwxy  Λa (a  x  Vb (b  y  Leavw ab))
“For any two worldlines v, w and all real numbers x, y, such that
v and w mutually intersect at x and y, and all real numbers a that
are smaller than x, there exist a real number b smaller than y and
a worldline which leads from a on v to b on w.”
v
w
b
exist
Proper axioms of ST
(P10) Λuvx (Wu  Wv  x  ℝ  (u ∤x v  u = v) 
 Vabz (a  x  b  v  z 
 Λydmnuab (y⊳  z  Vwz (y⊳∤y⊲w)) 
 Λw1w2y (w1  z  w2  z  w1 ∤y w2 
u
 <w1, y>  dmnuab) ))
b
densely
filled
a
“For all worldlines u, v and all real numbers x,
such that v intersects with u at x or is identical
with u, there is an a-b-diamond around u and a
set of worldlines z containing v, where for every
element of the a-b-diamond around u, its first
member, a worldline, is either contained in z or
is intersected by some element of z within the ab-diamond around u and no pair of worldlines in
z mutually intersect within the a-b-diamond
around u.”
Proper axioms of ST
(P11) Λvw (Wv  Ww  v  w  Λx (souwvx  ) 
 Vf (f: ℝ→ℝ  Smm f  Λxf◄ (f (x) = inf inawvx))
“For any two worldlines v, w, the infimum of the inaccessible
section on w of any real number x on v, if the source on w of x on
v is nonempty, varies monotonously and smoothly with x.”
v
x
w
inaccessible
from x on v
Proper axioms of ST
(P12) Λvwuxz (z    z = inawv x 
 Vuyab (Λcd (a  c  a 
 (d  1)2  (b  1)2  lincd z = inawu y)))
“For any two worldlines v, w and all z, x, where z is the
inaccessible section on w of x on v, any linear shift of z by a
limited offset and a stretching factor having a limited deviation
from 1 results in an inaccessible sections on w of some real
number on some worldline.”
v
w
x
inaccessible
from x on v
Proper axioms of ST
(P13) Λuxvw (Cwlvwux 
 Vab (Λyzdmnuab (Isn y⊳z⊳y⊲z⊲ 
 inavy⊳y⊲ = inavz⊳z⊲  inawy⊳y⊲ = inawz⊳z⊲ 
 y⊳ = z⊳  y⊲ = z⊲)))
“For any real number x on a worldline u and any two coordinate
worldlines v, w thereof, there is an a-b-diamond around u, which
contains x on u, such that the inaccessible sections on v and w of
any two elements of the a-b-diamond around u differ.”
v
u
x
w
uniquely determine
x on u
Some simple consequences
x
Inaccessible
from x on v
v
v
w
y2
x2
y1
x1
w
x1 < x2  y1 < y2
Spacetime points and topology
Spacetime points are defined as sets of worldlines.
u
v
p2
w
p
p
p1
Alexandroff topology
The spacetime manifold
From the axioms of ST, we obtain:
For each spacetime point p, a four-dimensional spacetime
coordinate system around p exists.
x2
A standard spacetime atlas is definable,
x4
which is an oriented atlas of smoothly
overlapping four-dimensional
x3
p
x1
coordinate systems of the
spacetime Alexandroff-topology.
The spacetime Alexandroff-topology is a four-dimensional
smooth oriented Hausdorff manifold.
(Th. Benda, Journal of Philosophical Logic 2008)
Tangent vectors
Now the following are defined:
The set fst of smooth spacetime functions from the set of
spacetime points to ℝ4.
The set vst of tangent vector fields: mappings from fst to fst
which are linear and have the Leibniz property.
vst is a module over the ring fst. In each coordinate system c, the
derivations c|0, c|1, c|2, c|3 form a basis of vst.
Curves in spacetime, particularly, curves along worldlines,
which we call “worldline curves”. Velocities of worldline
curves are tangent vector fields which are restricted to the
worldline curves.
Again: worldline curves are sets of spacetime points. Spacetime
points are sets of worldlines (our primitives).
Connections
We consider a composition of tangent vector fields:
t: vst  vst→vst with t (t1, t2) = t1∘t2. It is fst-linear in the first
argument, but only ℝ-linear in the second argument .
Connections are now defined as generalizations of such
compositions, that is, mappings vst  vst→vst with the same
linearity properties.
No metric has been defined yet.
A unique connection
With above definitions, the following is obtained:
Let t be a tangent vector field with only positive coordinates
(in coordinate systems in the standard spacetime atlas).
Then there is a unique connection d, such that
d (t, t) = 0
for all such t.
In particular and roughly speaking,
d-shifting the velocity vector t of any
worldline curve h along h
keeps t unchanged, that is,
d makes all worldline
curves geodesics. We call d
“geodesic connection”.
Defining a metric
We define 0-n tensor derivations as ℝ-linear mappings on the
set of 0-n tensors, where 0-n tensors are fst-linear mappings
from vstn to fst.
With that, each tangent vector field and each connection (with
their first argument being merely a parameter) is a 0-0 and 01 tensor derivation, respectively. 0-2 tensor derivations are
obtained by a product rule (in obvious notation)
d00(g(t1, t2)) = (d02g)(t1, t2) + g(d01t1, t2) + g(t1, d01t2) .
Given the set of tangent vector fields and the geodesic
connection, four 0-2 tensor derivations d1, d2, d3, d4 are
uniquely defined.
We now define a metric g as a 0-2 tensor with Lorentz
signature, such that d1g, d2g, d3g, d4g all vanish. g is unique
up to a constant g0.
Towards the Einstein equation
The Einstein tensor r – ½ s g is obtained from the metric g by
a formal construction (Lovelock 1971; Navarro 2010). In the
present approach, it is determined up to a constant g0 times
the scalar curvature s. This indeterminacy differs from the
cosmological constant by the factor s. That may provide a
testable condition.
The present geometric approach suggests to view the righthand term in the Einstein equation
r–½sg=e
as being defined.
e is the energy-momentum tensor, supposedly an empirically
accessible magnitude.
A unified mathematical-physical ontology
The theory ST is motivated by the conviction that physics, as
far as it speaks about individuals, can be formalized.
Assuming--somewhat generously--that ST covers all of
physics, a unified mathematical-physical ontology arises
naturally.
What distinguishes physical from mathematical objects is
their being impure rather than pure sets.
This picture does not cover experience, the indispensable
ingredient of any physical, as opposed to mathematical
theory. But then again, arguably, no known axiomatization of
(parts of) physics does.