An axiomatic foundation of relativistic spacetime

Thomas BendaNational 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 (xy  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

urelemente(worldlines)

pure sets (math.)

impure sets (phys.)

pow

er s

et

union

pow

er s

et

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.”

wv

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 and w mutually intersect at x and y.” u

vw

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 exists

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

xb

a

no such worldline

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

exists

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.”

vw

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.”

b

v

w

exist

Proper axioms of ST

(P10) Λuvx (Wu  Wv  x  ℝ  (u ∤x v  u = v)   Vabz (a  x  b  v  z 

 Λydmnuab (y⊳  z  Vwz (y⊳∤y⊲w))  Λw1w2y (w1  z  w2  z  w1 ∤y w2 

 <w1, y>  dmnuab) ))

“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 a-b-diamond around u and no pair of worldlines in

densely z mutually intersect within the a-b-diamond

filled around u.”

ub

a

Proper axioms of ST

(P11) Λvw (Wv  Ww  v  w  Λx (souwvx  )   Vf (f: ℝ→ℝ  Smm f  Λxf◄ (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.”

wv

x inaccessiblefrom 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.” wv

xinaccessiblefrom x on v

Proper axioms of ST

(P13) Λuxvw (Cwlvwux 

 Vab (Λyzdmnuab (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.”

uniquely determine

x on u

wu

x

v

Some simple consequences

x1 < x2 y1 < y2

v w

x Inaccessiblefrom x on v

v w

x1

x2

y1

y2

Spacetime points and topology

Spacetime points are defined as sets of worldlines.

Alexandroff topology

p

u v w p2

p1

p

The spacetime manifold

From the axioms of ST, we obtain:

For each spacetime point p, a four-dimensional spacetime coordinate system around p exists.

A standard spacetime atlas is definable,which is an oriented atlas of smoothly overlapping four-dimensional 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)

x1

x2

x3

x4

p

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 0-1 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 right-hand term in the Einstein equation

r – ½ s g = 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.