Pseudo-timelike loops in signature changing semi-Riemannian manifolds with a transverse radical
W. Hasse N. E. Rieger


Pseudo-timelike loops in signature changing manifolds]Pseudo-timelike loops in signature changing semi-Riemannian manifolds
with a transverse radical 










1,2]W. Hasse

[3,4]N. E. Riegern.rieger@math.uzh.ch


[1]Institute for Theoretical Physics, Technical University
Berlin, Hardenbergstr. 36, Berlin, 10623, Germany

[2]Wilhelm Foerster
Observatory Berlin, Munsterdamm 90, Berlin, 12169, Germany

*[3]Department of Mathematics, University of Zurich, Winterthurerstrasse
190, Zurich, 8057, Switzerland

[4]Current address: Mathematics Department, Yale University, 219 Prospect Street, New Haven, 06520, CT, USA





In 1983, Hartle and Hawking introduced a conceptually intriguing idea
involving signature-type change, which led to the no-boundary proposal
for the initial conditions of the universe. According to this proposal,
the universe has no beginning because there is no singularity or boundary
in spacetime; however, there is an origin of time. Mathematically,
this entails signature-type changing manifolds where a Riemannian
region smoothly transitions to a Lorentzian region at the surface
where time begins.

We present a coherent framework for signature-type changing manifolds 
characterized by a degenerate yet smooth metric. We then adapt firmly
established Lorentzian tools and results to the signature-type changing
scenario, introducing new definitions that carry unforeseen causal
implications. A noteworthy consequence is the presence of locally
closed time-reversing loops through each point on the hypersurface.
By imposing the constraint of global hyperbolicity on the Lorentzian
region, we demonstrate that for every point p∈ M, there exists
a pseudo-timelike loop with point of self-intersection p. Or put
another way, there always exists a closed pseudo-timelike path in
M around which the direction of time reverses, and a consistent
designation of future-directed and past-directed vectors cannot be
defined.



















[
    *
    September 9, 2024
=====================





§ INTRODUCTION


According to popular ideas about quantum cosmology, classical cosmological
models contain an initial Riemannian region of Euclidean signature
joined to a final semi-Riemannian region with the usual Lorentzian
signature <cit.>.
In 1983 Hartle and Hawking <cit.>
introduced a conceptually intriguing idea involving signature-type
change, which led to the no-boundary proposal for the initial conditions
of the universe. According to the Hartle–Hawking proposal, the universe
has no beginning because the spacetime lacks any singularity or boundary.[Although singularities can be considered points where curves terminate
at finite parameter values, providing a general definition remains
difficult <cit.>.] In such singularity-free universes, there is no distinct beginning,
but they do have an origin of time <cit.>.

 

Since a signature-type changing metric is necessarily either degenerate
or discontinuous at the locus of signature change <cit.>,
we will allow for the metric to become degenerate. Hence, in the present
article we will discuss singular semi-Riemannian manifolds for which
the metric constitutes a smooth (0,2)-tensor field that is degenerate
at a subset ℋ⊂ M, where the bilinear type of the
metric changes upon crossing ℋ.

 

Although the compatibility of the Riemannian and Lorentzian domains
is assumed to be established, insofar as the metric should be smooth
on the interface ℋ, the behavior of curves as they cross
this interface still requires further study. Moreover, in a manifold
where the signature changes from (+,+,…,+) to (-,+,…,+),
the conventional concept of timelike (or spacelike) curves does not
exist anymore. This gives rise to a new notion of curves called pseudo-timelike
and pseudo-spacelike curves. In order to define these curves
we make a detour to draw upon the concept of the generalized affine
parameter which we use as a tool to distinguish genuine pseudo-timelike
(and pseudo-spacelike, respectively) curves from curves that asymptotically
become lightlike as they approach the hypersurface of signature change. 

 

We endeavor to adapt well-established Lorentzian tools and results
to the signature changing setting, as far as possible. This task proves
to be less straightforward than anticipated, necessitating the introduction
of new definitions with unexpected causal implications, reaching a
critical juncture in our exploration. We draw upon the definition
of pseudo-time orientability and the given absolute time function
to decide whether a pseudo-timelike curve is future-directed. This
establishes the definition for the pseudo-chronological past (and
pseudo-chronological future) of an event. 

 

In this article we show that for signature-type change of the delineated
type, all these considerations lead to a surprising theorem revealing
the non-well-behaved nature of these manifolds. In a sufficiently
small region near the junction of signature change ℋ,
transverse signature-type changing manifolds with a transverse radical
exhibit local anomalies: Specifically, each point on the junction
facilitates a closed time-reversing loop, challenging conventional
notions of temporal consistency.[In more informal terms, in general relativity, a
closed timelike curve is a smooth, timelike loop where, at every intersection
point, the direction of movement is consistently the same. In contrast,
a loop is a broader concept where a timelike curve loops back on itself,
but the direction of movement at the intersection points might not
always be the same. This is a more intuitive explanation; for a precise
mathematical definition and its extension to a setting with signature-type
change, see Definition <ref>.] Or put another way, there always exists a closed pseudo-timelike
path in M around which the direction of time reverses, and along
which a consistent designation of future-directed and past-directed
vectors cannot be defined. By imposing the constraint of global hyperbolicity
on the Lorentzian region, the global analog can be proven by showing
that for every point p∈ M, there exists a pseudo-timelike loop
such that the intersection point is p. In simpler terms, a transverse,
signature-type changing manifold with a transverse radical has always
pseudo-timelike loops.



 §.§ Transverse type-changing singular semi-Riemannian manifold


Unless otherwise specified, the considered manifolds, denoted as M
with dimension (M)=n, are assumed to be locally homeomorphic
to ℝ^n. Moreover, these manifolds are expected to be
connected, second countable, and Hausdorff. This definition also indicates
that all manifolds have no boundary. Additionally, we will generally
assume that the manifolds under consideration are smooth. Unless stated
otherwise, all related structures and geometric objects (such as curves,
maps, fields, differential forms, etc.) are assumed to be smooth as
well.

 


A singular semi-Riemannian manifold is a generalization of
a semi-Riemannian manifold. It is a differentiable manifold having
on its tangent bundle a symmetric bilinear form which is allowed to
become degenerate.


 


Let (M,g) be a singular semi-Riemannian
manifold and let be p∈ M. We say that the metric changes its
signature at a point p∈ M if any neighborhood of p contains
at least one point q where the metric's signature differs from
that at p.


We align with <cit.>
in requiring that (M,g) be a semi-Riemannian manifold with M≥2,
where g is a smooth, symmetric, degenerate (0,2)-tensor on M,
and ℋ:={q∈ M:g|_q is degenerate}.
This means ℋ is the locus where the rank of g fails
to be maximal. In addition, we assume that one connected
component of M∖ℋ is Riemannian, denoted by M_R,
while all other connected components (M_L_α)_α∈ I⊆ M_L⊂ M
are Lorentzian, where M_L:=α∈ I⋃M_L_αrepresents
the Lorentzian domain. Furthermore, we assume throughout that the
point set ℋ, where g becomes degenerate is not empty.

 
 Moreover, we impose the following two conditions <cit.>:

 


  * We call the metric g a codimension-1 transverse type-changing
metric if d(([g_μν])_q)≠0 for any q∈ℋ
and any local coordinate system ξ=(x^0,…,x^n-1) around
q. Then we call (M,g) a transverse type-changing singular
semi-Riemannian manifold <cit.>. 
 This
implies that the subset ℋ⊂ M is a smoothly embedded
hypersurface in M, and the bilinear type of g changes upon crossing
ℋ. Moreover, at every point q∈ℋ there
exists a one-dimensional subspace, denoted as the radical Rad_q⊂ T_qM,
within the tangent space T_qM that is orthogonal to all of T_qM
at that point. 


  * The radical Rad_q is transverse for any q∈ℋ.
Henceforward, we assume throughout that (M,g) is a singular transverse
type-changing semi-Riemannian manifold with a transverse radical,
unless explicitly stated otherwise.


 


Recall that the radical at q∈ℋ is defined as the subspace
Rad_q:={w∈ T_qM:g(w,)=0}. This means g(v_q,)=0
for all v_q∈ Rad_q. Note that the radical can be either transverse
or tangent to the hypersurface ℋ. The radical Rad_q
is called transverse <cit.>
if Rad_q and T_qℋ span T_qM for any q∈ℋ,
i.e. Rad_q+T_qℋ=T_qM. This means that Rad_q
is not a subset of T_qℋ, and obviously, Rad_q
is not tangent to ℋ for any q.


 

As a consequence from the above two conditions and inspired by <cit.>,
we get the following

 

 Let M be a
singular semi-Riemannian manifold endowed with a (0,2)-tensor field
g and the surface of signature change defined as ℋ:={q∈ M:g|_q is degenerate}.
Then (M,g) is a transverse, signature-type changing manifold with
a transverse radical if and only if for every q∈ℋ there
exist a neighborhood U(q) and smooth coordinates (t,x^1,…,x^n-1)
such that g=-t(dt)^2+g_ij(t,x^1,…,x^n-1)dx^idx^j,
for i,j∈{1,…,n-1}.


 

In the style of time-orthonormal coordinates in Lorentzian geometry
we denote the coordinates in Theorem <ref>
as radical-adapted Gauss-like coordinates. It is now possible
to simplify matters by using these coordinates whenever dealing with
a transverse, signature-type changing manifold with a transverse radical.
Notably, signature-type change and the radical-adapted Gauss-like
coordinates imply the existence of an uniquely determined, coordinate
independent, natural absolute time function h(t,x̂):=t
in the neighbourhood of the hypersurface <cit.>.
Then the absolute time function establishes a foliation <cit.>
in a neighborhood of ℋ, such that ℋ is a
level surface of that decomposition.



 §.§ Statement of results


Before presenting the main results we require some new definitons.

 




Given a differentiable curve γ[a,b]→ M, with
[a,b]⊂ℝ, where -∞<a<b<∞. Then the curve
γ=γ^μ(u)=x^μ(u) in M is called pseudo-timelike
(respectively, pseudo-spacelike) if for every generalized affine
parametrization (see Definition <ref>)
of γ in M_L ∃ ε>0 such that g(γ',γ')<-ε
(respectively, g(γ',γ')>ε).



 

In simpler terms, we call a curve pseudo-timelike if it is timelike
in the Lorentzian domain M_L and does not become asymptotically
lightlike as it approaches the hypersurface where the signature changes.
Consequently, a pseudo-timelike loop is a generalization of a pseudo-timelike
curve that loops back on itself. However, unlike a regular closed
curve where the direction of movement would be the same at every intersection
point, in a pseudo-timelike loop, the direction of movement at the
intersection points is not necessarily the same (see Definition <ref>).

  




A vector
field V on a signature-type changing manifold (M,g) is pseudo-timelike
if and only if V is timelike in M_L and its integral curves
are pseudo-timelike.



 





A signature-type changing manifold (M,g) is pseudo-time
orientable if and only if the Lorentzian region M_L is time
orientable.



 

In a sufficiently small region near the junction of signature change,
transverse, signature-type changing manifolds with a transverse radical
exhibit local anomalies. Specifically, each point on the junction
gives rise to the existence of closed time-reversing loops, challenging
conventional notions of temporal consistency.

 


Let (M,g̃) be a transverse,
signature-type changing, n-dimensional (n≥2) manifold with
a transverse radical. Then in each neighborhood of each point 
there always exists a pseudo-timelike loop.



   

The existence of such pseudo-timelike curves locally near the hypersurface
that loop back to themselves, gives naturally rise to the question
whether this type of curves also occur globally. In the global version
a key notion is global hyperbolicity which plays a role in the spirit
of completeness for Riemannian manifolds. By imposing the constraint
of global hyperbolicity on the Lorentzian region, we demonstrate

 


 Let (M,g̃) be a pseudo-time orientable,
transverse, signature-type changing, n-dimensional (n≥2)
manifold with a transverse radical, where M_L=M∖(M_R∪ℋ)
is globally hyperbolic. Assume that a Cauchy surface S is a subset
of the neighborhood U=⋃_q∈ℋU(q) of ℋ,
i.e. S⊆(U∩ M_L)=⋃_q∈ℋ(U(q)∩ M_L),
with U(q) being constructed as in Theorem <ref>.
Then for every point p∈ M, there exists a pseudo-timelike loop
such that p is a point of self-intersection.





§ PSEUDO-CAUSAL AND PSEUDO-LIGHTLIKE CURVES 


In an n-dimensional manifold  (M,g) where the signature-type
changes from (+,+,,+) to  (,+,,+),
the conventional concept of a timelike curves does not make sense
anymore. From a suitable given point in the Lorentzian region, the
junction might be reached in finite proper time, but there is no time
concept in the Riemannian region. Hence, curves in the Riemannian
domain are devoid of causal meaning and cannot be distinguished as
timelike, spacelike or null. In signature-type changing manifolds
this gives rise to a novel notion of curves. In order to define those
curves we have to make a detour to draw upon the concept of the generalized
affine parameter.



 §.§ Properties of the Generalized Affine Parameter


In this section we want introduce the notion of pseudo-timelike
curves and pseudo-spacelike curves. However, we need a method
to discern genuine pseudo-timelike (and pseudo-spacelike, respectively)
curves from those that asymptotically become lightlike as they approach
the hypersurface of signature change. The generalized affine parameter
will prove useful to draw this distinction. For this, we require a
notion of completeness so that every C^1 curve of finite length
as measured by such a parameter has an endpoint. Ehresmann <cit.>
and later Schmidt <cit.>
appear to have been the first ones to propose using so-called generalized
affine parameters to define the completeness of general curves <cit.>.
The generalized affine parameter turns out to be a particularly useful
quantity for probing singularities because it can be defined for an
arbitrary curve, not necessarily a geodesic.

 



Let M be an n-dimensional manifold with an affine connection
and γ J→ M a C^1 curve on M. Recall
that a smooth vector field V along γ is a smooth map V J→ TM
such that V(t)∈ T_γ(t)M for all t∈ J. Such a smooth
vector field V along γ is said to be a parallel field along
γ if V satisfies the differential equation ∇_γ'V(t)=0
for all t∈ J (see <cit.>
for further details).

 
 Choose now any t_0∈ J and a C^1 curve γ J⟶ M
through p_0=γ(t_0). Let {e_1,e_2,…,e_n}
be any basis for T_γ(t_0)M. Let E_i be the unique
parallel field along γ with E_i(t_0)=e_i for 1≤ i≤ n.
Then {E_1(t),E_2(t),…,E_n(t)} forms a basis for T_γ(t)M
for each t∈ J. We can now write γ̇(t), the vector
tangent to γ at p_0, as a linear combination of the elements
of the chosen basis with coefficients V^i(t): γ̇(t)=∑_i=1^nV^i(t)E_i(t)V^i(t)E_γ(t)i
with V^i J⟶ℝ for 1≤ i≤ n.
Then the generalized affine parameter μ=μ(γ,E_1,…,E_n)
of γ(t) associated with this basis is given by 

    μ(t)=∫_t_0^t√(∑_i=1^n[V^i(t)]^2)dt=∫_t_0^t√(δ_ijV^i(t)V^j(t))dt, t∈ J.

The assumption that γ is C^1 is necessary to obtain the
vector fields {E_1,E_2,…,E_n} through parallel translation. 


 

Furthermore, we have




 
The curve γ has a finite
arc-length in the generalized affine parameter μ=μ(γ,E_1,…,E_n)
if and only if γ has finite arc-length in any other generalized
affine parameter μ=μ(γ,E̅_1,…,E̅_n).


 

Note that the generalized affine parameter of a curve depends on the
chosen basis. In effect, one treats the parallel-transported basis
of vectors as though they were the orthonormal basis of a Riemannian
metric and then defines the “length”
of γ(t) accordingly. Note that if the metric g is positive
definite, the generalized affine parameter defined by an orthonormal
basis is arc-length. This characterization of completeness manages
to discern exactly what we wanted to get winnowed. Also, the beauty
of this definition is that μ can be defined on any C^1 curve;
it works for null curves just as well as for timelike or spacelike
curves. Moreover, any curve of unbounded proper length automatically
has an unbounded generalized affine parameter <cit.>.

 


If only one generalized affine parameter reaches finite value all
of them do - and that is the only information we need with respect
to completeness. This reasoning is based on the following estimate:


For any two basis of T_γ(t)M which are parallelyl transported
along γ, the components V^i(t) with respect to another
basis are given by Ṽ^j(t)=∑_i=1^nA_i^jV^i(t).
We then have γ̇(t)=∑_i=1^nV^i(t)E_i(t)=∑_i=1^nṼ^i(t)Ẽ_i(t).
The constant items A_i^j are entries in a constant, non-degenerate
n× n matrix A. Hence, there exists its inverse matrix A^-1
such that V^j(t)=∑_i=1^na_i^jṼ^i(t). Accordingly,
the generalized affine parameters with respect to these basis are
μ(t)=∫_t_0^t√(∑_i=1^n[V^i(t)]^2)dt
and μ̃(t)=∫_t_0^t√(∑_i=1^n[Ṽ^i(t)]^2)dt.
From this it follows that


    |Ṽ(t)|=|∑_i=1^nA_i^jV^i(t)|≤∑_i=1^n| A_i^j|| V^i(t)|≤ijmax| A_i^j|_i=1^n∑| V^i(t)|.


 Then, by virtue of the Cauchy-Schwarz inequality:


    |Ṽ^j(t)|^2≤ijmax| A_i^j|^2(∑_i=1^n| V^i(t)|·1)^2(∑_i=1^n| V^i(t)|)^2



    ≤ijmax| A_i^j|^2(∑_i=1^n| V^i(t)|^2)·(∑_i=1^n1)=n·ijmax| A_i^j|^2(∑_i=1^n| V^i(t)|^2).


 Thus we have


    ∑_j=1^n|Ṽ^j(t)|^2≤∑_j=1^n(n·ijmax| A_i^j|^2(∑_i=1^n| V^i(t)|^2))



    =n^2·ijmax| A_i^j|^2(∑_i=1^n| V^i(t)|^2).


 On the other hand, we get ∑_j=1^n| V^j(t)|^2≤ n^2·ijmax| a_i^j|^2(∑_i=1^n|Ṽ^i(t)|^2).
Combining both estimates yields


    ∑_j=1^n|Ṽ^j(t)|^2≤ n^2·ijmax| A_i^j|^2(∑_j=1^n| V^i(t)|^2)



    ≤ n^2·ijmax| A_i^j|^2(n^2·ijmax| a_i^j|^2(∑_i=1^n|Ṽ^i(t)|^2))



    ⟺1/n^2·ijmax| A_i^j|^2_j=1^n∑|Ṽ^j(t)|^2



    ≤∑_i=1^n| V^i(t)|^2≤ n^2·ijmax| a_i^j|^2(∑_i=1^n|Ṽ^i(t)|^2),



    ⟹c_11/√(n^2·ijmax| A_i^j|^2)√(∑_j=1^n|Ṽ^j(t)|^2)



    ≤√(∑_i=1^n| V^i(t)|^2)≤c_2√(n^2·ijmax| a_i^j|^2)√(∑_i=1^n|Ṽ^i(t)|^2)



    ⟹ c_1·μ̃(t)≤μ(t)≤ c_2·μ̃(t).





 §.§ Application of the generalized affine parameter in a signature-type
changing manifold


Let M=M_L∪ℋ∪ M_R be an n-dimensional transverse
type-changing singular semi-Riemannian manifold with a type-changing
metric g, and ℋ:={q∈ M:g|_q is degenerate}
the locus of signature change. We further assume that one component,
M_L, of M∖ℋ is Lorentzian and the other
one, M_R, is Riemannian.

 


Given a continuous and differentiable curve γ[a,b]⟶ M,
with [a,b]⊂ℝ, where -∞<a<b<∞. Then
the curve γ=γ^μ(u)=x^μ(u) is a pseudo-lightlike
curve if


 



  * its tangent vector field in the Lorentzian component M_L is null,

  * its tangent vector field in the Riemannian component M_R is arbitrary.

A similar definition applies for a pseudo-causal
curve. Note that an analogous definiton for pseudo-timelike and
pseudo-spacelike curves turns out to be problematic as the definition
would also include curves that asymptotically become lightlike as
they approach ℋ, see Figure <ref>.




  


 For example we may refer
to the metric g=t(dt)^2+(dx)^2 defined on ℝ^2,
and the non-parametrized, non-geodesic curve γ given by tan x=2/3√(| t|^3)·sgn(t),
with -π/2<x<π/2. We rearrange this equation
so that the variable x is by itself on one side (Figure <ref>):

 



    3/2tan x=sgn(x)·|3/2tan x|=sgn(t)·| t|^3/2



    ⟺tsgn(t)·| t|=sgn(x)·(|3/2tan x|)^2/3



    ⟺ t=sgn(x)·(|3/2tan x|)^2/3.


 
 Reintroducing the transformation as suggested by Dray <cit.>

    T=∫_0^t√(|t̃|)dt̃=2/3√(| t|)^3·sgn(t)


 
 gives us the metric expression g=sgn(T)(dT)^2+(dx)^2,
and for the curve γ we get T=tan x. Hence, the curve in
the (T,x)-coordinate system is just the tan-function and its
derivative is 1/cos^2(x). As a result, γ is
in M_L timelike, approaching from timelike infinity the lightcone,
and tangentially touches the light cone at T→0 (where
the derivative becomes 1/cos^2(0)=1 in the limit). These
are the sort of curves we want to avoid in our definition. Note that
the (t,x)-coordinates are characterized by the fact that, unlike
the (T,x)-coordinates, they cover the entire manifold M.




Moreover, if the curve γ=(T(s),x(s)) is parametrized by arc
length s, then in the (t,x)-coordinate system both dx/ds
and dt/ds diverge in M_L:

 

-1=-(dT/ds)^2+(dx/ds)^2=-(dT/dxdx/ds)^2+(dx/ds)^2

 

=(-(dtan x/dx)^2+1)(dx/ds)^2=(-1/cos^4x+1)(dx/ds)^2

 

⟺(dx/ds)^2=-1/(-1/cos^4x+1)

 

⟹x→0limdx/ds=x→0lim±√(-1/(-1/cos^4x+1))=±∞.

 

-1=-(dT/ds)^2+(dx/ds)^2=(-1+1/(dT/dx)^2)(dT/ds)^2=(-1+1/(dtan x/dx)^2cos^4x)(dT/dt)^2(dt/ds)^2

 

=(-1+1/(1+tan^2x)^2)·| t|(dt/ds)^2=(-1+1/(1+T^2)^2)·| t|(dt/ds)^2

 

=(-1+1/(1+4/9| t|^3)^2)·| t|(dt/ds)^2

 

⟺(dt/ds)^2=-1/(-1+1/(1+4/9| t|^3)^2)·| t|

 

⟹t→0limdt/ds=t→0lim±√(-1/(-1+1/(1+4/9| t|^3)^2)·| t|)=±∞.


 

While the components of γ' do not diverge in the (T,x)-coordinate
system, both dx/ds and dt/ds diverge in M_L
in the (t,x)-coordinate system. Because of this dependency of coordinates
the criterion of divergence is not useful for defining pseudo-timelike
and pseudo-spacelike curves. That is where the coordinate-independent
generalized affine parameter comes into play.

 


 Let M=M_L∪ℋ∪ M_R be an n-dimensional
transverse type-changing singular semi-Riemannian manifold, g be
a type-changing metric, and ℋ:={q∈ M:g|_q is degenerate}
the locus of signature change. We further assume that one component,
M_L, of M∖ℋ is Lorentzian and the other
one, M_R, is Riemannian.
 Given a continuous and differentiable
curve γ[a,b]→ M, with [a,b]⊂ℝ,
where -∞<a<b<∞. Then the curve γ=γ^μ(u)=x^μ(u)
in M is called pseudo-timelike (respectively,
pseudo-spacelike) if for every generalized affine parametrization
of γ in M_L ∃ ε>0 such that g(γ',γ')<-ε
(respectively, g(γ',γ')>ε).[Since Definition <ref>
is already independent of a choice of coordinates and instead refers
to a (generally anholonomic) basis, the above Definition <ref>
is also coordinate independent. The independence of Definition <ref>
from the choice of this basis is a direct consequence of Proposition <ref>.
In particular, in the case of a basis change we just relegate to the
Estimate <ref>.]


 


 Revisiting Example <ref>,
we find that both coordinate vector fields, ∂/∂ T
and ∂/∂ x, are covariantly constant in M_L
and M_R (this is because the Christoffel symbols all vanish in
the (T,x)-coordinate system). Hence, we can parallel transport
∂/∂ T and ∂/∂ x
along any curve in M_L and M_R, with the transport being
path-independent (no anholonomy).
 Since we aim at parametrizing
the curve γ by the generalized affine parameter μ with
respect to the coordinate vector fields ∂/∂ T=1/√(| t|)∂/∂ t
and ∂/∂ x we are able to start with an arbitrary
parametrization. Hence, let γ(t)=(T(t),x(t)) be parametrized
by t, and then γ̇(t)=dT/dt∂/∂ T+dx/dt∂/∂ x.
By means of Definition <ref>
we immediately get 

    V^0(t)=dT/dt=√(| t|)

 and


 



    V^1(t)=dx/dt=d/dtarctan(2/3√(| t|)^3sgn(t)).

And in M_L this yields 

    V^1(t)=√(| t|)/1+4/9| t|^3.

Consider now γ̃(t(s))=γ(s), in which γ̃
is related to the curve γ by reparametrization of γ
by t. With this notation we have the basis fields E_γ̃(t),0=1/√(| t|)∂/∂ t
and E_γ̃(t),1=∂/∂ x along γ̃.
The reparametrized curve γ̃(t(s)) also gives 

    γ̇̃̇(t)=V^i(t)E_γ̃(t),i=∂/∂ t+dx/dt∂/∂ x.

 
 The Definition <ref>
for the generalized affine parameter gives


    dμ/dt=√((V^0(t))^2+(V^1(t))^2)=√(| t|+| t|/(1+4/9| t|^3)^2).

It now follows easily that for the reparametrization of γ̂(t)
by the generalized affine parameter μ (i.e. γ̂(μ(t))=γ̃(t))
we have in M_L:

 



    g(γ̇̂̇(μ(t)),γ̇̂̇(μ(t)))=g(dγ̂(μ(t))/dμ,dγ̂(μ(t))/dμ)=g(1/dμ/dtγ̇̂̇(t),1/dμ/dtγ̇̂̇(t))



    =g(∂/∂ t+dx/dt∂/∂ x,∂/∂ t+dx/dt∂/∂ x)/(dμ/dt)^2=t+(dx/dt)^2/| t|+| t|/(1+4/9| t|^3)^2=t+| t|/(1+4/9| t|^3)^2/| t|+| t|/(1+4/9| t|^3)^2.

Taking the limit 

    t→0^-limt+| t|/(1+4/9| t|^3)^2/| t|+| t|/(1+4/9| t|^3)^2=0

reveals that the curve γ is not pseudo-timelike as it does
not meet the ε-requirement of Definition <ref>.

 

 
 In Section <ref> we repeatedly
rather vaguely referred to the concept of a timelike (or spacelike,
respectively) curve that asymptotically becomes lightlike. The above
example highlights how the notion of “asymptotically lightlike”
should be understood. A timelike (or spacelike, respectively) curve
in M_L that is not pseudo-timelike (or pseudo-spacelike, respectively)
can be thus specified as asymptotically lightlike.

 


 Finally, if we modify the previously
discussed curve γ by keeping the t-coordinate but stating
x=0, we get a curve α. With the same notation as above,
we then get V^0(t)=√(| t|), V^1(t)=0 and dμ/dt=√(| t|).
Hence, this results in g(α̇̂̇(μ(t)),α̇̂̇(μ(t)))=1/| t|g(∂/∂ t,∂/∂ t)=t/| t|=-1
in the Lorentzian region M_L. The curve α is pseudo-timelike
as it obviously does meet the ε-requirement of Definition <ref>.


 

This disquisition makes it clear why the notion of the generalized affine parameter is necessary and useful in order to define pseudo-timelike and pseudo-spacelike curves. If we were to loosen the requirement in Definition <ref> by replacing “for every generalized affine parametrization of γ in M_L” with “for every affine parametrization of γ in M_L”, then no curve that is timelike in the Lorentz sector and reaches the hypersurface ℋ would be pseudo-timelike throughout the entire manifold M. (However, this statement applies only to curves that actually reach the hypersurface ℋ. Timelike curves that lie entirely within M_L and maintain a “distance” from ℋ due to a tubular neighborhood within M_L also satisfy the relaxed condition, as they only have affine parameters with g(γ', γ') = const < 0.)

 

Similarly, any timelike curve in M_L would meet the requirements
of a pseudo-timelike curve if we modified the definition by requesting
“for a suitable parametrization of γ in M_L”
instead of “for every generalized affine parametrization
of γ in M_L”. In this regard, the concept of the generalized
affine parameter is the right tool to tell apart suitable from unsuitable
curves for the definition of pseudo-timelike and pseudo-spacelike
curves. 

 

Interestingly, our rationale for the new definition of a pseudo-timelike
curve is reminiscent of the analysis undertaken in <cit.>.
In Section 2 of <cit.> the distinction between causal
curves, timelike almost everywhere curves and timelike
curves is introduced in which the latter one is defined as follows:
A timelike curve is a causal curve γ I⟶ M
such that g(γ',γ')<-ε almost everywhere for
some ε>0. 
 The author illustrates the situation
in his Figure 1 which contrasts a timelike curve with a timelike almost
everywhere curve. The latter one can not be viewed as a timelike curve
because it approaches a null vector at its break point. Compared to
our setting, however, the culprit here is that the curve is not differentiable
at the breaking point. However, if we were to make the curve differentiable
by bending its upper section, it still wouldn't be timelike. Its restriction
to the lower region before the inflection point is timelike, but it
cannot be extended upwards into a timelike curve. Now, imagine we
are not in Minkowski spacetime, but instead, a signature-type change
occurs at the (former) inflection point so that the 'upper half' of
the space becomes Euclidean (in this case, the figure would correspond
to the (T,x)-coordinates, not the (t,x)-coordinates, in the
toy model). In this scenario, the curve restricted to the Lorentz
sector would be timelike, but after the signature change is 'reversed',
it cannot be extended upwards into a timelike curve. In this sense,
the entire curve in the signature-changing version of this example
is not pseudo-timelike.

  

Now we can slightly modify the definition of a (simply) closed curve
in order for it to correctly apply to signature-type changing singular
semi-Riemannian manifolds M with a metric g:

 


 A smooth, pseudo-timelike curve γ I⟶ M
is said to be chronology-violating when there is a subset of γ[I]
homeomorphic to S^1 such that there are at least two parameters
s_1,s_2∈ I that satisfy γ(s_1)=γ(s_2), and
γ belongs to one of the following two classes:[Note that this means that there must be at least one such subset to
fulfill this definition.] 




  * The pseudo-timelike curve γ is periodic, i.e the image γ[I]
is homeomorphic to S^1. Moreover, for s_1, s_2∈ I
the associated tangent vectors, γ'(s_1) and γ'(s_2),
are timelike and positively proportional. We denote this type of curve
as closed pseudo-timelike curve.

  * The curve γ intersects itself for s_1, s_2∈ I
and the associated tangent vectors, γ'(s_1) and γ'(s_2),
are timelike whereas the tangent directions are not necessarily the
same (i.e. they do not need to be positively proportional). This type
of curve is said to contain a loop. 




§ GLOBAL STRUCTURE OF SIGNATURE-TYPE CHANGING SEMI-RIEMANNIAN MANIFOLDS


First, let us revisit the definitions of the following concepts related
to manifold orientability.

 


 <cit.>  A smooth n-dimensional manifold M is
orientable if and only if it has a smooth global nowhere vanishing
n-form (also called a top-ranked form).[An orientation of M is the choice of a continuous pointwise orientation, i.e. the specific choice of a global nowhere vanishing n-form.]


 

For a differentiable manifold to be orientable all that counts is
that it admits a global top-ranked form - it is not important which
specific top-ranked form is selected. 

  

To ensure thoroughness, we also want to mention the definition of
parallelizability, which likewise does not involve any metric and
is therefore again applicable to manifolds with changing signature
types. It is well-known that a manifold M of dimension n is
defined to be parallelizable if there are n global vector fields
that are linearly independent at each point. We define it similarly
to the approach in <cit.>:

 


A smooth n-dimensional manifold M is parallelizable
if there exists a set of smooth vector fields {V,E_1,…,E_n-1}
on M, such that at every point p∈ M the tangent vectors {V(p),E_1(p),…,E_n-1(p)}
provide a basis of the tangent space T_pM. A specific choice
of such a basis of vector fields on M is called an absolute
parallelism of M.


 

Equivalently, a manifold M of dimension n is parallelizable
if its tangent bundle TM is a trivial bundle, so that the associated
principal bundle of linear frames has a global section on M, i.e.
the tangent bundle is then globally of the form TM≃ M×ℝ^n.
Moreover, it is worth pointing out that orientability and also parallelizability
are differential topological properties which do not depend
on the metric structure, but only on the topological manifold with
a globally defined differential structure.

  


It is worth mentioning that given an absolute parallelism of M,
one can use these n vector fields to define a basis of the tangent
space at each point of M and thus one can always get a frame-dependent
metric g by defining the frame to be orthonormal. Moreover, the
special orthogonal group, denoted SO(n,ℝ), acts naturally
on each tangent space via a change of basis, it is then possible to
obtain the set of all orthonormal frames for M at each point qua
the oriented orthonormal frame bundle of M, denoted F_SO(M),
associated to the tangent bundle of M. 


 

The next three definitions, however, depend not only on the underlying
manifold but also on its specific type-changing metric g. For our
purpose, let (M,g) be a smooth, signature-type changing manifold
(possibly with boundary).

 


 A vector field V on a signature-type changing manifold (M,g) is pseudo-timelike if and only if V is timelike in M_L and its integral curves
are pseudo-timelike (in the sense of Definition <ref>).[Keep in mind that a timelike vector field is a vector field V on
a spacetime manifold (M,g) where the vectors at every point are
timelike, meaning g(V(p),V(p))<0 for all points p on the manifold.]


 


 
A signature-type changing manifold (M,g) is pseudo-time
orientable if and only if the Lorentzian region M_L is time
orientable.[A pseudo-time orientation of such a manifold (M,g) corresponds
to the specific choice of a continuous non-vanishing pseudo-timelike
vector field V on M.]


 


A singular semi-Riemannian manifold  is pseudo-time orientable if and only if there exists a vector field X ∈𝔛(M) that is pseudo-timelike.





"⟹" Let a singular semi-Riemannian manifold (M,g)
be pseudo-time orientable. This means the Lorentzian region M_L
is time orientable. A Lorentzian manifold is time-orientable if there
exists a continuous timelike vector field. Accordingly, there must
exist a continuous timelike vector field X ∈𝔛(M_L)
in the Lorentzian region. As per Definition <ref>,
a vector field X in a signature-type changing manifold is pseudo-timelike
if and only if X is timelike in M_L and its integral curve
is pseudo-timelike; this means that X is allowed to vanish on M_R.[In this part of the proof, the only thing that matters is whether
the “pseudo-timelike vector field” is
allowed to vanish on M_R. This question is independent of whether
the “generalized affine parameter” condition
is required in M_L, because the issue of whether the vector field
“is allowed to vanish on M_R” concerns only its
“magnitude”, while the “generalized
affine parameter” condition pertains solely to its “direction”
(specifically, that the vector field is not asymptotically lightlike).] Hence, we can extend the vector field X arbitrarily to all of
M, and per definition X ∈𝔛(M) is pseudo-timelike.


 

"⟸" Let X∈𝔛(M) be a pseudo-timelike
vector field in a singular semi-Riemannian manifold (M,g). Hence,
as per Definition <ref>, X is timelike in
M_L. A Lorentzian manifold is time-orientable if and only if
there exists a timelike vector field. Since X is a timelike vector
field on M_L, the Lorentzian region M_L is time-orientable.
Then, according to Definition <ref>,
the signature-type changing manifold (M,g) is pseudo-time orientable. 


According to that, such a definition of a pseudo-time orientation
is possible if M_L admits a globally consistent sense of time,
i.e. if in M_L we can continuously define a division of non-spacelike
vectors into two classes. For a transverse, signature-type changing
manifold (with a transverse radical), this definition arises naturally
because, in M_R, all vectors can be considered spacelike. Additionally,
all non-spacelike vectors on ℋ are lightlike.[Note that this applies generally, including in the case of a tangent
radical, since there are no timelike vectors on ℋ. However,
the subsequent division into two classes requires a transverse radical.] In the case that Rad_q∩ T_qℋ={0} ∀ q∈ℋ,
these lightlike vectors can be naturally divided into two classes:
those pointing towards M_L and those pointing towards M_R.

 





Consider the classic type of a spacetime M with
signature-type change which is obtained by cutting an S^4 along
its equator and joining it to the corresponding half of a de Sitter
space, Figure <ref>. The de Sitter spacetime
is time-orientable <cit.>, hence M is pseudo-time
orientable.


 


 
A signature-type changing manifold (M,g) of dimension n is pseudo-space
orientable if and only if it admits a continuous non-vanishing spacelike
(n-1)-frame field on M_L. This is a set of n-1 pointwise
orthonormal spacelike vector fields on M_L.[A pseudo-space orientation of a manifold (M,g) corresponds to the
specific choice of a continuous non-vanishing field of orthonormal
spacelike (n-1)-beins on M_L.]


 



Every parallelizable manifold M is orientable.


 

In Lorentzian geometry the fact of M being time-orientable and
space-orientable implies that M is orientable <cit.>. The
proposition below illustrates that this result from Lorentzian geometry
cannot be applied to signature-type changing manifolds.

 


 Even if a transverse,
signature-type changing manifold (M,g) with a transverse radical
is pseudo-time orientable and pseudo-space orientable, it is not necessarily
orientable.



Consider an arbitrary manifold of (M)=2 with a change of signature,
for which the conditions of Proposition <ref>
are given (in higher dimensions, the same idea can be carried out
through a trivial augmentation of dimensions). In case this manifold
is non-orientable, there is nothing to show. However, if it is orientable,
cut out a disk from the Riemannian sector and replace it with a crosscap,
equipped with any Riemannian metric. In a tubular neighborhood of
the cutting line, construct a Riemannian metric that mediates between
the metrics of the crosscap and the rest (this is possible due to
the convexity of the space formed by all Riemannian metrics). This
surgical intervention results in the transition to a non-orientable
manifold with a change of signature. Since the intervention is limited
to the Riemannian sector, the conditions of the proposition remain
unaffected. Thus Proposition <ref>
is proven.


 


One can always “switch” between non-orientability and orientability using the crosscap. Starting with an orientable manifold, one transitions to non-orientable by replacing a crosscap (if already present) with a disk. If no crosscap is present, such a transition occurs by replacing a disk with a crosscap.


 


The Möbius strip 𝕄 has a non-trivial vector bundle structure over S^1, which means that the bundle cannot be trivialized globally. Specifically, the Möbius strip is a line bundle over S^1 with a non-trivial twist.[The Möbius strip is particularly interesting because it can be found on any arbitrary non-orientable surface. Additionally, any Lorentzian manifold 𝕄×ℝ^n based on the Möbius strip crossed with ℝ^n either fails to be time-orientable or space-orientable <cit.>.] Hence, 𝕄 is neither parallelizable nor orientable.

 
To see this, consider the Möbius strip 𝕄 = ℝ×ℝ/∼ with the identification (t,x) ∼ (t̃,x̃) ⟺ (t̃, x̃) = ((-1)^kt, x + k), k ∈ℤ. Notice that the identification has no bearing on proper subsets of ((-1)^kt, x + k), k ∈ℤ, and the fibre ℝ is a vector space.

 
As 𝕄 is a fiber bundle over the base space S^1, a section of that fiber bundle must be a continuous map σ: S^1⟶𝕄 such that σ(x) = (h(x), x) ∈𝕄. For σ to be continuous, h must satisfy -h(0) = h(k). The intermediate value theorem guarantees that there is some x̃∈ [0, k] such that h(x̃) = 0. This means that every section of 𝕄 intersects the zero section, and the sections that form a basis for the fibre are not non-zero everywhere.


 


A pseudo-spacetime is a 4-dimensional, pseudo-time oriented,
semi-Riemannian manifold with a type-changing metric. 


 


Let (ℝ^n, g) be a transverse, signature-type changing n-manifold with a transverse radical, and let ℋ⊂ℝ^n be a codimension one closed hypersurface of signature change without boundary.[Here “closed” is meant in the topological sense of “the complement of an open subset of ℝ^n” and not in the manifold sense of “a manifold without boundary that is compact.”] Then ℋ is always orientable.



This can be shown by a purely topological argument, as in <cit.>.


 


Let (M,g) be a transverse, signature-type changing, oriented, n-dimensional
manifold with a transverse radical, and let ℋ⊂ M
be the hypersurface of signature change. Then ℋ is also
oriented.



The hypersurface of signature change, as a closed submanifold of codimension one, is the inverse image of a regular value of a smooth map f  M →ℝ. Specifically, ℋ = f^-1(c) for some regular value c ∈ℝ. The manifold M is oriented, so its tangent bundle TM is oriented, meaning there is a consistent choice of orientation on each tangent space T_p M for p ∈ M. Since ℋ is a hypersurface in M, at each point q ∈ℋ, the tangent space T_q ℋ is a subspace of the tangent space T_q M of dimension n-1, and therefore Tℋ is a subbundle of TM. The remaining direction in T_q M can be described by a normal vector N(q), which is a vector in T_q M that is perpendicular to T_q ℋ. 

 

Since M is oriented, for each point q ∈ℋ, the tangent space T_q M has an orientation that can be described by an ordered basis, say {v_1, …, v_n-1, N(q)}, where {v_1, …, v_n-1} is an oriented basis for T_q ℋ and N(q) is the normal vector. Hence, this induces a consistent orientation on T_q ℋ across all points q ∈ℋ, since the orientation of M provides a consistent choice of N(q) across ℋ. Therefore, ℋ inherits a consistent orientation from M, proving that ℋ is oriented.

 

Moreover, without loss of generality, we can choose 1 as a regular value (see also <cit.>). Thus, ℋ := f^-1(1) = {p ∈ M | f(p) = 1} is a submanifold of M of dimension n-1. For every q ∈ℋ, the tangent space T_q ℋ = T_q (f^-1(1)) to ℋ at q is the kernel (df_q) of the map df_q  T_q M → T_1 ℝ. Then T_q ℋ = ⟨grad f_q ⟩^, and therefore the gradient grad f yields an orientation of ℋ.


 

Provided a transverse, signature-type changing manifold (M,g) with
a transverse radical is pseudo-time orientable, then we can choose
one of the two possible time orientations at any point in each connected
component of M_L, and thus designating the future direction of
time in the Lorentzian regime. On ℋ all non-spacelike
vectors are lightlike and smoothly divided into two classes
in a natural way: the vectors located at an initial base point on
ℋ are either pointing towards M_L or towards M_R.
This together with the existent absolute time function (that establishes
a time concept <cit.>
in the Riemannian region) can be considered as arrow of time on M. 

 


 Let (M,g) be a pseudo-time orientable, transverse,
signature-type changing n-dimensional manifold with a transverse
radical. Then in the neighborhood of ℋ the absolute time
function h(t,x̂):= t, where (t,x̂):=(t,x^1,…,x^n-1),
imposes a natural time direction by postulating that the
future corresponds to the increase of the absolute time function.
In this way, the time orientation is determined in M_L.


 


Note that ∂_t, with an initial
point on ℋ, points in the direction in which t=h(t,x̂)
increases while x_i remains constant. Away from the hypersurface,
the future direction is defined relative to ℋ
by the accordant time orientation of M_L. Recall that functions
of the type, such as the absolute time function, typically
lead to metric splittings by default. 






(Future-Directed) A pseudo-timelike curve (see Definition <ref>)
in (M,g) is future-directed (in the sense of Definiton <ref>
and Remark <ref>) if for every point in the
curve

 

(i) within M_L the tangent vector is future-pointing, and



(ii) on ℋ the associated tangent vector with an initial
base point on ℋ is future-pointing, if applicable.


 

 
 Respective past-directed curves are defined
analogously. Notice that, per assumption, one connected component
of M∖ℋ is Riemannian and all other connected
components (M_L_α)_α∈ I⊆ M_L⊂ M
are Lorentzian. This configuration could (at least locally) potentially
allow for a M_L-M_R-M_L-sandwich-like structure of M,
where ℋ consists of two connected components (ℋ_α)_α∈{1,2}.
Consequently, this would also imply the existence of two absolute
time functions, see Figure <ref>.

 


Let (M,g) be a
pseudo-time orientable, transverse, signature-type changing n-dimensional
manifold with a transverse radical. 

ℐ^-(p)={q∈ M q≪ p} is the pseudo-chronological
past of the event p∈ M. In other words, for any two points q,p∈ M,
we write q≪ p if there is a future-directed pseudo-timelike curve
from q to p in M.

ℐ^+(p)={q∈ M p≪ q} is the pseudo-chronological
future of the event p∈ M. In other words, for any two points
p,q∈ M, we write p≪ q if there is a future-directed pseudo-timelike
curve from p to q in M.





 


Interestingly, this
definition leads to the following peculiar situation: Recall that
any curve is denoted pseudo-timelike if its M_L-segment is timelike.
To that effect, all curves that steer clear of M_L (and do not
have a M_L-segment) are also considered pseudo-timelike. When
p∈ℋ∪ M_R then the pseudo-chronological past of
p is ℐ^-(p)=M∖ M_L and the pseudo-chronological
future of p is ℐ^+(p)=M, see Figure <ref>.




§ CHRONOLOGY VIOLATING PSEUDO-TIMELIKE LOOPS


In Section <ref>, we introduced the notion
of closed pseudo-timelike curves on a signature-type changing background
and we demonstrated how they must be defined to ensure that the concept
of causality remains meaningful. In this section, we will reveal the
non-well-behaved nature of transverse, signature-type changing, n-dimensional
manifolds with a transverse radical. 



 §.§ Local pseudo-timelike loops


In a sufficiently small region near the junction of signature change,
these manifolds exhibit local anomalies. Specifically, each point
on the junction gives rise to the existence of closed time-reversing
loops, challenging conventional notions of temporal consistency. One
of our main results, Theorem <ref>, can now
be proved quite easily.

 




Let (M,g̃) be a transverse, signature-type
changing, n-dimensional (n≥2) manifold with a transverse
radical. Then in each neighborhood of each point 
there always exists a pseudo-timelike loop.




Let g̃=-t(dt)^2+g̃_jk(t,x^1,…,x^n-1)dx^idx^k,
j,k∈{1,…,n-1}, be a transverse, signature-type changing
metric with respect to a radical-adapted Gauss-like coordinate patch
(U_φ,φ) with U_φ∩ℋ≠∅.[This is, U_φ is sufficiently small to be expressed in the
adapted radical-adapted Gauss-like coordinate system ξ(U_φ).] Choose smooth coordinates (t_0,x_0^1,…,x_0^n-1)
with t_0>0 and ξ_0>0, such that 

    C_0:=[0,t_0]× B_ξ_0^n-1=[0,t_0]×{x∈ℝ^n-1|∑_k=1^n-1(x^k)^2≤ξ_0^2}⊂ℝ^n

 is contained in the domain of the coordinate chart (open neighborhood)
U_φ.  Then 

    C_0×𝕊^n-2=C_0×{v∈ℝ^n-1|∑_k=1^n-1(v^k)^2=1}

as a product of two compact sets is again compact. 

 
 Next, consider the function


    G̃ C_0×𝕊^n-2⟶ℝ,



    (t,x^1,…,x^n-1,v^1,…,v^n-1)↦g̃_jk(t,x^1,…,x^n-1)v^jv^k.

As G̃ is a smooth function defined on the compact domain
C_0×𝕊^n-2, by the Extreme Value Theorem it has
an absolute minimum G_0. Hence, on (U_φ,φ) we
can uniquely define g̃_0=-t(dt)^2+G_0δ_jkdx^jdx^k, j,k∈{1,…,n-1}.

 
 By this definition, for all nonzero lightlike vectors
X∈ T_pM, p∈ C_0 with respect to g̃_0, we have
g̃_0=-t(X^0)^2+G_0δ_jkX^jX^k=0⟺-t(X^0)^2=-G_0δ_jkX^jX^k,
then

 


    g̃(X,X)=-t(X^0)^2+g̃_jk(t,x^1,…,x^n-1)X^jX^k



    =-G_0δ_jkX^jX^k+g̃_jk(t,x^1,…,x^n-1)X^jX^k



    =δ_jkX^jX^k·(-G_0+g̃_rs(t,x^1,…,x^n-1)X^r/√(δ_abX^aX^b)X^s/√(δ_cdX^cX^d))≥0.

Clearly, g̃(X,X)≥0 because G_0>0 per definition
and δ_jkX^jX^k=t(X^0)^2/G_0≥0. Therefore,
the vector X∈ T_pM, p∈ C_0 is not timelike with respect
to g̃. This means, within C_0 the g̃-light
cones always reside inside of the g̃_0-light cones, i.e.
g̃≤g̃_0 in C_0. The cull cones of g̃_0
are more opened out than those of the metric g̃. Denote
p_0∈ C_0 by (t(p_0),x^1(p_0),…,x^n-1(p_0))=(t_0,x_0^1,…,x_0^n-1). 

 
 As (M,g̃) is an n-dimensional manifold
for which in the neighborhood of ℋ radical-adapted Gauss-like
coordinates exist, we can single out the time coordinate that defines
the smooth absolute time function t whose gradient in M_L
is everywhere non-zero and timelike. Hence, (M,g̃)|_U_φ
can be decomposed into spacelike hypersurfaces {(U_φ)_t_i}
which are specified as the level sets (U_φ)_t_i=t^-1(t_i)
of the time function.[This collection of space-like slices {(U_φ)_t} should
be thought of as a foliation of U_φ into disjoint (n-1)-dimensional
Riemannian manifolds.] The restriction (g̃_0)_t_i of the metric g̃_0
to each spacelike slice makes the pair ((U_φ)_t_i,(g̃_0)_t_i)
a Riemannian manifold.

 



For a lightlike curve α(t) I⟶ U_φ
with starting point p_0, we have δ_jkdx^j/dtdx^k/dt>0
for each slice (U_φ)_t_i with t≠0. Lightlike
curves with starting point p_0 can be parametrized with the Euclidean
arc length σ in B_ξ_0^n-1, such that (g̃_0)_t(α̇(σ),α̇(σ))=δ_jkdx^j/dσdx^k/dσ=1,
∀ σ∈ I, where I is some interval in ℝ.
More precisely, σ can be considered as arc length (parameter)
in terms of some auxiliary Riemannian metrics, each defined on a hypersurface
with t=const. Consequently we get


    0=g̃_0(α̇(σ),α̇(σ))=-t(α̇^0)^2+G_0δ_ikα̇^jα̇^k



    =-t(dt/dσ)^2+G_01δ_ikdx^j/dσdx^k/dσ=-t(dt/dσ)^2+G_0,

and this implies 

    dσ/dt=±√(t/G_0)⟹σ(t)=±∫√(t/G_0)dt=±2/3t√(t/G_0)+const.


 
 Since σ is given as a function of t, it represents
the arc length from the starting point at t(p_0)=t_0 to t(0)=0.
Then past-directed g̃_0-lightlike curves emanating from
p_0 reach the hypersurface at t=0 after passing through the
arc length distance 

    σ=±∫_0^t_0√(t/G_0)dt=±2/3t_0√(t_0/G_0)+const.=±2/3√(t_0^3/G_0)+const

along the said section of the curve from the fixed starting point
p_0.

 
 Provided this arc length distance satisfies σ≤ξ_0,
then the past-directed lightlike curves α(t) (emanating from
p_0) reach the hypersurface at t=0 while remaining within
C_0. Accordingly this is also the case for g̃-lightlike
curves emanating from p_0. Conversely, if σ>ξ_0
then there exist past-directed g̃_0-lightlike curves emanating
from p_0 that reach the hypersurface outside of C_0. 
 In
this case, we have σ=2/3√(t_0^3/G_0)>ξ_0⟺ t_0>√(9/4ξ_0^2· G_0)
and we must adjust the new starting point p_1=(t_1,x_0) accordingly
by setting t_1≤√(9/4ξ_0^2· G_0)<t_0.
Thereby we make sure that all past-directed g̃-lightlike
curves emanating from p_1 hit the hypersurface ℋ
without leaving C_0. That is I_g̃^-(p_1)⊂ C_0⊂ U_φ⊂ M,
where I_g̃^-(p_1) is the g̃-chronological
past of the event p_1∈ M_L, restricted to M_L∪ℋ,
see Figure <ref>.

 
 It now suffices to connect two of such points x̂_1,x̂_2∈I_g̃^-(p_0)∩ℋ
(or, if need be I_g̃^-(p_1)∩ℋ)
in an arbitrary fashion within the Riemannian sector M_R. By
what a pseudo-timelike loop gets generated, if U_φ was
chosen small enough.


 

Summarized, for each neighborhood U(q) that admits radical-adapted
Gauss-like coordinates ξ=(t,x̂)=(t,x^1,…,x^n-1)
centered at some q∈ℋ, and U(q)∩ℋ≠∅,
we are able to pick a point p_0∈ U(q) and an associated compact
set C_0⊂ U(q). For the metric g̃ there exists
a corresponding uniquely (i.e., only dependent on the chosen set C_0)
defined metric g̃_0 with g̃≤g̃_0
within C_0.[The set C_0 does not need to be “maximal” (in some sense) and is therefore not unique.] Then we must distinguish between two cases, that is

 

i) with respect to the metric g̃_0 we have I_0^-(p_0)⊂ C_0,
then also I^-(p_0)⊂ C_0 with respect to g̃,

ii) with respect to the metric g̃_0 we have the situation
I_0^-(p_0)⊈ C_0, then there exists a point p_1=(t_1,x_0)∈ C_0∖ℋ
with t_1<t_0, such that I_0^-(p_1)⊂ C_0, hence
also I^-(p_1)⊂ C_0 with respect to g̃.

 

Thus, for any point q∈ℋ we can find a sufficiently
small neighborhood Ũ⊂ U(q) containing a point p∈ M_L,
such that all past-directed, causal curves emanating from that point,
reach the hypersurface within a sufficiently small set C_0.

 


Let (M,g̃) be a transverse, signature-type changing, n-dimensional
manifold with a transverse radical. Then in each neighborhood of each
point  there always
exists a pseudo-lightlike curve.


 

The above corollary follows directly from Theorem <ref>
because we have proven that all past-directed g̃-lightlike
curves emanating from p_1 hit the hypersurface ℋ
without leaving C_0. That is I_g̃^-(p_1)⊂ C_0⊂ U_φ⊂ M,
where I_g̃^-(p_1) is the closure of the
g̃-chronological past of the event p_1∈ M_L.
Hence, we have also shown that the causal past is within
C_0, and furthermore, J_g̃^-(p_0)⊂I_g̃^-(p_0). 

And since in every neighborhood of each point q∈ℋ there
always exists a pseudo-timelike loop, we can straightforwardly assert
the following

 


A transverse, signature-type changing manifold (M,g̃) with
a transverse radical has always time-reversing pseudo-timelike loops.


 

As a matter of course, in the Lorentizan region the tangent space
at each point is isometric to Minkowski space which is time orientable.
Hence, a Lorentzian manifold is always infinitesimally time- and space-orientable,
and a continuous designation of future-directed and past-directed
for non-spacelike vectors can be made (infinitesimally and therefore,
by continuity, also locally).[In case the Lorentzian manifold is time-orientable, a continuous designation
of future-directed and past-directed for non-spacelike vectors can
be made allover.] 
 

 
Having said that, the infinitesimal properties of a
manifold with a signature change are identical to those of a Lorentzian
manifold only within the Lorentzian sector. However, when examining
the Riemannian sector and the hypersurface, specific distinctions
arise. The Riemannian sector and the hypersurface are not infinitesimally
modelable by a Minkowski space. While the Riemannian sector reveals
an absence of a meaningful differentiation between past- and future-directed
vectors, on the hypersurface, one has the flexibility to make arbitrary
assignments of such distinctions at the infinitesimal level. If one
now determines on the hypersurface whether the direction towards the
Lorentzian sector is the future or past direction, it is not only
a reference to the tangent space at a point. Rather, it is a local
consideration.

 
 In the context of local considerations, in a Lorentzian
manifold the existence of a timelike loop that flips its time orientation
(i.e. the timelike tangent vector switches between the two designated
components of the light cone) is a sufficient condition for the absence
of time orientability. Based on the previous theorem (at the beginning
of the present subsection), this is also true for a transverse, signature-type
changing manifold (M,g̃) with a transverse radical: As we
have proved above, through each point on the hypersurface ℋ
we have locally a closed time-reversing loop. That is, there always
exists a closed pseudo-timelike path in M around which the direction
of time reverses, and along which a consistent designation of future-directed
and past-directed vectors cannot be defined. 

 

An observer in the region M_L near ℋ perceives these
locally closed time-reversing loops (Figure <ref>)
as the creation of a particle and an antiparticle at two different
points q̂,q∈ℋ.[Such locally closed time-reversing loops around ℋ obviously
do not satisfy the causal relation ≪ as introduced above.] This could be taken as an object entering the Riemannian region,
then resurfacing in the Lorentzian region and proceeding to move backwards
in time.

 

So in a transverse, signature-type changing manifold (M,g̃),
the hypersurface with its time-reversing loops could be tantamount
to a region of particle-antiparticle origination incidents. Moreover,
Hadley <cit.> shows for
Lorentzian spacetimes that a failure of time-orientability of a spacetime
region is indistinguishable from a particle-antiparticle annihilation
event. These are then considered equivalent descriptions of the same
phenomena. It would be interesting to explore how this interpretation
can be carried over to signature-type changing manifolds.  




For fields, take the conjugate ψ_t^A = e^-iĤtψ^* of ψ_t = e^iĤtψ: The unitary temporal evolution of the field operator for antiparticles arises from the temporal evolution of the field operator for particles by applying the same Hamiltonian operator to the adjoint field operator under time reversal. Some literature <cit.> points to the idea that concepts in quantum field theory are predicated on acausal properties derived from general relativity. In this context, Blum et al. <cit.> stress the importance of the CPT theorem (quoting verbatim):


“CPT theorem is the statement that nothing would change—nobody would notice and the predictions of physics would not be altered—if we simultaneously replace particles by antiparticles and vice versa. Replace everything by its mirror image or, more exactly, exchange left and right, up and down, and front and back, and reverse the flow of time. We call this simultaneous transformation CPT, where C stands for Charge Conjugation (exchanging particles and antiparticles), P stands for parity (mirroring), and T stands for time reversal.”




 §.§ Global pseudo-timelike loops


The existence of such pseudo-timelike curves locally near the hypersurface
that loop back to themselves, gives rise to the question whether this
type of curves also occur globally. We want to elucidate this question
in the following.[A spacetime is a Lorentzian manifold that models space and time in
general relativity and physics. This is conventionally formalized
by saying that a spacetime is a smooth connected time-orientable Lorentzian
manifold (M,g) with M=4. But in what follows we want to
study the n-dimensional (n≥2) case.]

 


 <cit.> A connected time-orientable
Lorentzian manifold (M,g) is said to be stably causal
if there exists a nowhere-vanishing timelike vector field V_a
such that the Lorentzian metric on M given by g':=g_ab-V_aV_b
admits no closed timelike curves. In other words, if (M,g) is stably
causal then, for some timelike V_a, the metric g':=g_ab-V_aV_b
on M is causal.


 


A partial ordering < is defined in the set of all Lorentzian metrics
Lor(M) on M in the following way: g<g' iff all causal vectors
for g are timelike for g'. Then the metric g_λ=g+λ(g'-g),
∀ λ∈[0,1] is a Lorentzian metric on M, as well.
Also, recall that g<g' means that the causal cones of g are
contained in the timelike cones of g'. A connected time-orientable
Lorentzian manifold (M,g) is stably causal if there exists g'∈ Lor(M),
such that g'>g, with g' causal.


 


 Stable causality is the necessary and sufficient
condition for the existence of a smooth global time function, i.e.
a differentiable map T M→ℝ such that whenever
p<<q ⟹ T(p)<T(q).


 


 <cit.>
A connected, time-orientable Lorentzian manifold (M,g) is called
globally hyperbolic if and only if it is diamond-compact
and causal, i.e., p∉ J^+(p) ∀ p∈ M.[Diamond-compact means J(p,q):= J^+(p)∩ J^-(q) is
compact for all p,q∈ M. Note that J(p,q) is possibly empty.]


 

An equivalent condition for global hyperbolicity is as follows <cit.>.


A connected, time-orientable Lorentzian manifold (M,g) is called
globally hyperbolic if and only if M contains a Cauchy
surface. A Cauchy hypersurface in M is a subset S that is intersected
exactly once by every inextendible timelike curve in M.[An inextendible curve is a general term that refers to a curve with
no endpoints; it either extends infinitely or it closes in on itself
to form a circle—a closed curve. Specifically, an inextendible
timelike curve is a curve that remains timelike throughout its entire
length and cannot be extended further within the spacetime. In mathematical
terms, a map α(a,b)→ M is an inextendible
timelike curve in (M,g) if α(t) does not approach a limit
as t increases to b or decreases to a, and α(t) remains
timelike for all t∈(a,b). This distinguishes it from inextendible
curves of other causal types, such as null or spacelike curves.]


 

In 2003, Bernal and Sánchez <cit.>
showed that any globally hyperbolic Lorentzian manifold M admits
a smooth spacelike Cauchy hypersurface S, and thus is diffeomorphic
to the product of this Cauchy surface with ℝ, i.e. M
splits topologically as the product ℝ× S. Specifically,
a globally hyperbolic manifold is foliated by Cauchy surfaces.


If M is a smooth, connected time-orientable Lorentzian manifold
with boundary, then we say it is globally hyperbolic if its interior
is globally hyperbolic. 


 

The next theorem is partially based on the
Local Loops Theorem <ref> and can be considered
a generalization to the global case.




 Let (M,g̃) be a pseudo-time orientable,
transverse, signature-type changing, n-dimensional (n≥2)
manifold with a transverse radical, where M_L=M∖(M_R∪ℋ)
is globally hyperbolic. Assume that a Cauchy surface S is a subset
of the neighborhood U=⋃_q∈ℋU(q) of ℋ,
i.e. S⊆(U∩ M_L)=⋃_q∈ℋ(U(q)∩ M_L),
with U(q) being constructed as in Theorem <ref>.
Then for every point p∈ M, there exists a pseudo-timelike loop
such that p is a point of self-intersection.




Let (M,g̃) be a pseudo-time orientable transverse,
signature-type changing, n-dimensional (n≥2) manifold with
a transverse radical, where M_L is globally hyperbolic with g̃|_M_L=g.
Moreover, there is a neighborhood U=⋃_q∈ℋU(q)
of ℋ sufficiently small to satisfy the conditions for
Theorem <ref>, and per assumption there exists
a Cauchy surface S_ε⊆(U∩ M_L), ε>0.

 
 Due to <cit.>
we know that M_L admits a splitting M_L=(ℝ_>0)_t× S_t=⋃_t∈ℝ_>0S_t,
such that the Lorentzian sector M_L is decomposed into hypersurfaces
(of dimension n-1), specified as the level surfaces S_t=𝒯^-1(t)={p∈ M_L𝒯(p)=t},t∈ℝ_>0,
of the real-valued smooth temporal function 𝒯 M_L⟶ℝ_>0
whose gradient grad𝒯 is everywhere non-zero
and, clearly, d𝒯 is an exact 1-form. Within the neighborhood
U=⋃_q∈ℋU(q) this foliation ⋃_t∈ℝ_>0S_t
can be chosen in such a way that it agrees with the natural foliation
given by the absolute time function h(t,x̂):= t, see
Remark <ref> and Definiton <ref>.[Recall that a smooth function T M⟶ℝ
on a connected time-orientable Lorentzian manifold (M,g) is a global
time function if T is strictly increasing along each future-pointing
non-spacelike curve. Moreover, a temporal function is a time function
T with a timelike gradient gradT everywhere.

Since M_L is globally hyperbolic it admits a smooth global time
function T and consequently it admits <cit.> a temporal
function 𝒯. Hence, in the Lorentzian sector M_L
there exists a global temporal function 𝒯 M_L⟶ℝ_>0,
and grad𝒯 is orthogonal to each of the level
surfaces S_t=𝒯^-1(t)={p∈ M_L𝒯(p)=t},t∈ℝ_>0,
of 𝒯. Note that 𝒯=t is a scalar field on
M_L, hence grad𝒯=gradt=(dt)^#.]

 
 Moreover, the level surfaces (S_t)_t∈ℝ
are Cauchy surfaces and, accordingly, each inextendible pseudo-timelike
curve in M_L can intersect each level set S_t exactly once
as 𝒯 is strictly increasing along any future-pointing
pseudo-timelike curve.[Since 𝒯 is regular the hypersurfaces S_t never intersect,
i.e. S_t∩ S_t'=∅ for t≠ t'.] Then, these level-sets S_t are all space-like hypersurfaces
which are orthogonal to a timelike and future-directed unit normal
vector field n.[In other words, the unit vector n is normal to each slice S_t,
and g restricted to S_t is Riemannian.]

 
 For ε sufficiently small, the level Cauchy
surface 

    S_ε=𝒯^-1(ε)={p∈ M_L𝒯(p)=ε},ε∈ℝ_>0

 is contained in U∩ M_L=⋃_q∈ℋ(U(q)∩ M_L).[This is true because all neighborhoods U(q) with q ∈ℋ
can be chosen such that the sets U(q) have a compact closure.
Thus, the U(q) are not “infinitely wide,” and there exists a strictly positive value ε_max,
such that for all ε < ε_max, the level Cauchy
surface S_ε is contained in U ∩ M_L.]

 
 Therefore, based on Theorem <ref>,
for any p=(ε,x̂)∈ S_ε⊆(U∩ M_L)
all past-directed and causal curves emanating from that point reach
the hypersurface ℋ. The global hyperbolicity of M_L
implies that every non-spacelike curve in M_L meets each S_t
once and exactly once since S_t is a Cauchy surface. In particular,
the spacelike hypersurface S_ε is a Cauchy surface
in the sense that for any p̅∈ M_L in the future of S_ε,
all past pseudo-timelike curves from p̅ intersect S_ε.
The same holds for all future directed pseudo-timelike curves from
any point p̅̅̅∈ M_L in the past of S_ε.



Consequently, by virtue of Theorem <ref> and
the above argument, all past-directed pseudo-timelike curves emanating
from any p̅∈ M_L reach the hypersurface ℋ.
Analogously we can conclude that any point p̅∈ M_L can
be reached by a future-directed pseudo-timelike curve starting at
some suitable point in ℋ. Recall that, based on Remark <ref>,
we also know that ℐ^+(q)={p∈ M:q≪ p}=M, that
is, any point in M=M_R∪ℋ∪ M_L
can be reached by a future-directed pseudo-timelike curve from q∈ℋ,
see Figure <ref>.

 
 We now obtain a loop with intersection point p in
M_L if, for sufficiently small ε, we first prescribe
the intersection point p=(ε,x̂)∈ S_ε.
And then we connect the two points lying in ℋ of the
intersecting curve sections through an arbitrary curve segment in
the Riemannian sector M_R (through a suitable choice of the two
curve segments, we can ensure that different points on ℋ
are obtained).


 


Theorem <ref> explicitly states that through
every point in M, there always exists a pseudo-timelike loop. Therefore,
this assertion holds also true for points located on the hypersurface
or within the Riemannian region. In this casees, the situation is
as follows:

(i) If the given point lies on the hypersurface, p∈ℋ,
choose a timelike curve segment that connects it to S_ε
(with ε sufficiently small), then proceed from there
along another timelike curve segment to another point on the hypersurface,
and connect both points in the Riemannian sector.



(ii) If the given point lies in the Riemann sector, p∈ M_R,
choose an arbitrary loop of the form similar to those loops constructed
in the proof of Theorem <ref>, and modify this
loop within the Riemannian sector such that it passes through the
specified point there.


 


 The prototype of a spacetime M with
signature-type change is obtained by cutting an S^4 along its
equator and joining it to the corresponding half of a de Sitter space.
It is a well-known fact that the full de Sitter spacetime is globally
hyperbolic <cit.>, with the
entire manifold possessing a Cauchy surface. When we restrict to half
de Sitter space—by choosing an appropriate region bounded by a
Cauchy surface—this region retains global hyperbolicity. This is
because the Cauchy surface of the full de Sitter spacetime remains
valid in the half-space, ensuring that every inextendible non-spacelike
curve still intersects this surface exactly once. As a result, the
Lorentzian sector, which corresponds to half de Sitter space, is also
globally hyperbolic. Consequently, there are chronology-violating
pseudo-timelike loops through each point in M.


 


Let (M,g̃) be a pseudo-time orientable, transverse, signature-type
changing, n-dimensional (n≥2) manifold with a transverse
radical, where M_L is globally hyperbolic, and S⊆(U∩ M_L)=⋃_q∈ℋ(U(q)∩ M_L)
for a Cauchy surface S. Then through every point there exists a
path on which a pseudo-time orientation cannot be defined.





§ FINAL THOUGHTS


The intriguing facet of the potential existence of closed timelike
curves within the framework of Einstein's theory lies in the physical
interpretation that CTCs, serving as the worldlines of observers,
fundamentally permit an influence on the causal past. This can also
be facilitated through a causal curve in the form of a loop, i.e.,
the curve intersects itself. In the case of a non-time-orientable
manifold, there would then be the possibility that at the intersection,
the two tangent vectors lie in different components of the light cone.
Thus, the “time traveler” at the encounter
with himself, which he experiences twice, may notice a reversal of
the past and future time directions in his surroundings during the
second occurrence, even including the behaviour of his or her younger
version. Regardless of whether this effect exists or not, during the
second experience of the encounter, which he perceives as an encounter
with a younger version of himself, the traveler can causally influence
this younger version and its surroundings.

 


§ ACKNOWLEDGMENTS

NER is greatly indebted to Richard Schoen for generously welcoming
her into his research group and to Alberto Cattaneo for affording
her with creative independence throughout the duration of this research
endeavor. Moreover, NER acknowledges partial support of the SNF Grant
No. 200021-227719. This research was (partly) supported by the NCCR
SwissMAP, funded by the Swiss National Science Foundation.

 



 

Data availability: No data was used for the research described in the article.





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