[Paper Review] Time machine as four-dimensional wormhole

Alexandr K. Guts|arXiv (Cornell University)|Dec 25, 1996

Black Holes and Theoretical Physics6 references3 citations

TL;DR

This paper proposes a theoretical model of a time machine based on a four-dimensional wormhole formed by embedding a spacetime leaf in a five-dimensional Lorentzian manifold. If the foliation has a non-zero Godbillon-Vey class, resilient leaves allow closed timelike curves via a wormhole connecting distant past events, enabling time travel with large energy and electric charge requirements as per conformal Kaluza-Klein theory.

ABSTRACT

The following mechanism of action of Time machine is considered. Let space-time $$ be a leaf of a foliation F of codimension 1 in 5-dimensional Lorentz manifold $$. If the Godbillon-Vey class $GV(F) eq 0$ then the foliation F has resilient leaves. Let $V^4$ be a resilient leaf. Hence there exists an arbitrarily small neighborhood $U_a \subset V^5$ of the event $a \in V^4$ such that $U_a \cap V^4$ consists of at least two connected components $U_a^1$ and $U_a^2$. Remove the four-dimensional balls $B_a\subset U_a^1, B_b\subset U_a^2$, where an event $b\in U_a^2$, and join the boundaries of formed two holes by means of 4-dimensional cylinder. As result we have a four-dimensional wormhole C, which is a Time machine if b belongs to the past of event a. The past of a is lying arbitrarily nearly. The distant Past is more accessible than the near Past. It seems that real global space-time V^4 is a resilient one, i.e. is a resilient leaf of some foliation F. It follows from the conformal Kaluza-Klein theory that the movement to the Past through four-dimensional wormhole C along geodesic with respect to metric G_{AB} requires for time machine of large energy and electric charge.

Motivation & Objective

• To explore the geometric and topological conditions under which time machines can emerge from higher-dimensional spacetime structures.
• To investigate whether the global structure of spacetime could naturally support closed timelike curves via foliation resilience.
• To model a time machine as a four-dimensional wormhole formed by joining disconnected components of a resilient spacetime leaf in a 5D manifold.
• To examine the energy and charge requirements for traversing such a wormhole, based on conformal Kaluza-Klein theory.
• To establish a framework linking topological invariants like the Godbillon-Vey class to the existence of time travel mechanisms.

Proposed method

• Embed a four-dimensional spacetime $<V^4, g_{ik}>$ as a leaf in a five-dimensional Lorentzian manifold $<V^5, G_{AB}>$ with codimension-1 foliation.
• Utilize the Godbillon-Vey class $GV(F)$ to determine if the foliation has resilient leaves, implying non-trivial topology.
• Identify an event $a \in V^4$ with a neighborhood $U_a \subset V^5$ such that $U_a \cap V^4$ splits into at least two connected components $U_a^1$ and $U_a^2$, one containing event $b$ in the past of $a$.
• Remove four-dimensional balls $B_a \subset U_a^1$ and $B_b \subset U_a^2$, and join their boundaries via a 4-dimensional cylindrical manifold to form a wormhole $C$.
• Ensure that the wormhole $C$ allows closed timelike curves when $b$ lies in the past of $a$, enabling time travel to the distant past.
• Apply conformal Kaluza-Klein theory to estimate the energy and electric charge required for traversing the wormhole along geodesics in the 5D metric $G_{AB}$.

Experimental results

Research questions

• RQ1 Under what topological conditions in a 5D spacetime can a 4D spacetime leaf support closed timelike curves?
• RQ2 How does the non-vanishing Godbillon-Vey class of a foliation lead to the existence of resilient leaves that enable time travel?
• RQ3 Can a four-dimensional wormhole be constructed from disconnected components of a spacetime leaf in a higher-dimensional manifold?
• RQ4 What are the physical constraints—specifically energy and electric charge—required for traversing such a wormhole according to conformal Kaluza-Klein theory?
• RQ5 Is it possible for real global spacetime to be a resilient leaf of a foliation, thereby naturally allowing time machine behavior?

Key findings

• A non-zero Godbillon-Vey class $GV(F) \neq 0$ implies the existence of resilient leaves in the foliation, enabling topological structures that support time travel.
• The construction of a four-dimensional wormhole via joining boundaries of removed 4D balls in disconnected components of a resilient leaf creates a closed timelike curve when the destination event $b$ lies in the past of $a$.
• The distant past becomes more accessible than the near past due to the topological structure of the resilient leaf and the wormhole's geometry.
• Traversing the wormhole along a geodesic in the 5D metric $G_{AB}$ requires large energy and electric charge, as predicted by conformal Kaluza-Klein theory.
• The model suggests that real spacetime $V^4$ may inherently be a resilient leaf of a foliation, implying that time machines could be a natural consequence of spacetime topology.
• The mechanism provides a geometric and topological foundation for time machines without requiring exotic matter, relying instead on global differential topology and higher-dimensional embedding.