Conical singularity in spacetimes with NUT is observer-dependent

This paper proposes an observer-dependent geometric definition of conical deficits in spacetimes with NUT parameters, demonstrating that the perceived conicity along the symmetry axis varies with the choice of timelike Killing vector and challenging the standard interpretation of conicity differences as indicators of acceleration-induced strings or rods.

The Gist

Imagine the universe as a giant, stretchy fabric. In the world of General Relativity, massive objects like stars and black holes warp this fabric, creating gravity. Usually, this fabric is smooth, but sometimes, it has "defects" or "kinks."

This paper is about two specific types of kinks in the fabric of spacetime: Conical Singularities (like a sharp point on a cone) and Torsion Singularities (like a screw or a twist).

Here is the simple breakdown of what the authors discovered, using some everyday analogies.

1. The Two Types of "Kinks"

The Cone (Conical Singularity):
Imagine taking a piece of paper, cutting out a slice of pizza, and taping the edges back together. You get a cone. The tip of the cone is a "conical singularity."

• What it means: If you walk around the tip, you don't make a full 360-degree circle; you make a smaller circle because a piece is missing.
• The old view: Physicists used to think this "missing piece" was a fixed, objective fact. It was like a permanent scar on the universe caused by a cosmic string (a super-thin, super-tight rope) pulling on space.

The Screw (Torsion Singularity / NUT):
Now, imagine taking that same piece of paper, but instead of just cutting it, you twist it before taping the edges. Or, imagine a spiral staircase that never ends. This is a "torsion singularity" (often called a Misner string in the Taub-NUT spacetime).

• What it means: As you get closer to the center axis, time itself gets "twisted." If you try to walk in a circle around the center, you don't just move sideways; you also move forward or backward in time.

2. The Big Discovery: "It Depends on Who You Are"

For a long time, scientists thought they could measure the "sharpness" of the cone (the conical deficit) just by looking at the math. They thought, "If there is a missing slice of pizza, everyone agrees it's missing."

The authors of this paper say: "Not so fast!"

They found that when you have a Torsion Singularity (the twist), the measurement of the cone's sharpness depends entirely on who is doing the measuring.

The "Moving Train" Analogy

Imagine you are standing on a platform watching a train go by.

• Observer A is standing still on the platform.
• Observer B is running alongside the train at the exact same speed.

If you ask them to measure the length of the train, they might get different answers depending on how they define "length" in a relativistic world.

In this paper, the "train" is the twisted spacetime.

• If you stand still (a specific type of observer), you might see a huge, sharp cone with a missing slice. You think, "Wow, there is a massive cosmic string pulling on this space!"
• But if you start moving (choosing a different "observer"), you might look at that same spot and say, "Actually, it's perfectly round. There is no missing slice at all."

The Shocking Result:
In spacetimes with a "NUT" parameter (the twist), there is always a special observer who sees zero conical deficit. To them, the axis is perfectly smooth. There is no "missing slice" of pizza.

3. Why This Matters

The "String" Problem:
Previously, physicists thought that if the top half of the axis had a different "missing slice" than the bottom half, it meant there was a physical force (like a string) pulling the black hole, causing it to accelerate. They used this difference to calculate how hard the black hole was being pulled.

The New Reality:
The authors show that because the measurement changes based on who is looking, you cannot use the "missing slice" to calculate physical forces anymore.

• If Observer A sees a difference between the top and bottom, Observer B might see them as identical.
• If the measurement changes based on the observer, it's not a fixed physical property of the universe; it's a property of the relationship between the observer and the space.

4. The "Canonical Observer" Dilemma

The paper asks: "Okay, so if everyone sees something different, is there one 'correct' observer we should use?"

They tried to find a "Goldilocks" observer—one that is naturally defined by the geometry itself.

• They tried picking the observer who is "stationary" far away.
• They tried picking the observer who is "orthogonal" to the axis.

The Conclusion: None of these choices worked perfectly. The "correct" observer changes depending on the specific type of black hole or spacetime you are looking at. There is no universal rulebook.

Summary in a Nutshell

1. Old Idea: A "twisted" spacetime (Taub-NUT) has a fixed, measurable defect (conical deficit) that tells us about physical forces.
2. New Idea: In these twisted spacetimes, the size of the defect changes depending on how fast or in what direction you are moving.
3. The Twist: There is always an observer who sees the defect as zero. To them, the universe is smooth.
4. The Takeaway: We can no longer treat these "defects" as simple physical rulers for measuring forces. We have to admit that in these strange, twisted universes, reality is relative to the observer.

The paper essentially tells us that in the presence of a "NUT" charge (a gravitational twist), the concept of a "conical deficit" is not an absolute fact of nature, but a perspective-dependent illusion.

Technical Summary

1. Problem Statement

The paper addresses a fundamental ambiguity in defining and measuring conical deficits (conicity) in stationary, axially symmetric spacetimes that contain torsion singularities (specifically the Misner string found in Taub–NUT spacetimes).

• Standard Context: In standard spacetimes (like the C-metric or cosmic strings), a conical singularity is defined by the limit of the ratio of the circumference of a small circle around the symmetry axis to $2\pi$ times its radius. This yields a constant "conicity" parameter $C$, where $C \neq 1$ indicates a deficit (tension) or excess (stress).
• The Issue: In spacetimes with a NUT parameter (torsion singularity), the orbits of the standard cyclic Killing vector near the axis do not shrink to zero length; instead, they remain finite due to the "twist" (time-shift) in the geometry. Consequently, the standard limit definition yields infinite conicity.
• The Gap: Previous attempts to generalize the definition involved choosing specific Killing vectors or integration segments, but these choices were often coordinate-dependent or lacked a rigorous, coordinate-independent geometric justification. The authors argue that the existing literature has overlooked the fact that the measured conicity in these spacetimes is inherently observer-dependent.

2. Methodology

The authors propose a new, rigorous geometric definition of conicity applicable to stationary, axially symmetric spacetimes with quasi-regular singularities (torsion singularities).

• Geometric Framework:
  • They consider a spacetime with a 2-dimensional Abelian group of symmetries generated by Killing vectors.
  • They distinguish between the cyclic Killing vector ($c$), which has closed orbits (periodicity $2\pi$), and the axial Killing vector ($a$), which vanishes at the axis (defining the radial distance).
  • In spacetimes with torsion, $c$ and $a$ are distinct. The cyclic vector $c$ is a linear combination of the axial vector $a$ and a stationary timelike Killing vector $t$.
• New Definition:
  • The conicity $C$ and time-shift $T$ are defined by decomposing the axial Killing vector $a$ in terms of the cyclic vector $c$ and an arbitrary stationary observer's timelike Killing vector $t$:
    $$a = \frac{1}{K}(c + Tt)$$
    Here, $C = |K|$ is the conicity, and $T$ is the time-shift.
  • Observer Dependence: Since the choice of the stationary observer $t$ is arbitrary, $C$ and $T$ become functions of $t$. The authors demonstrate that changing the observer (a different linear combination of $t$ and $c$) transforms the measured conicity and time-shift.
• Generalized Limit:
  • They extend the standard limit definition ($L/2\pi\rho$) by replacing the circular orbit of $c$ with a segment of the orbit of $a$.
  • To define the endpoints of this segment, they introduce an observer $t$. The segment is bounded by two intersections with the observer's worldlines (constant spatial position). This "spiral-segment" integration recovers the geometric definition $C = |K|$.
• Boundary Construction: They utilize the concept of a generalized boundary (specifically the $g$-boundary) to rigorously define the "axis" as a set of equivalence classes of incomplete geodesics, ensuring the definition holds even where the manifold is not extendable.

3. Key Contributions

1. Observer-Dependent Conicity: The primary contribution is the proof that in spacetimes with a non-vanishing NUT parameter (torsion singularity), the conical deficit is not an intrinsic, observer-independent property of the spacetime geometry. It depends entirely on the choice of the timelike Killing vector (the observer).
2. Existence of "Regular" Observers: The authors prove that for any spacetime with a torsion singularity, there exist exactly two specific observers (defined by specific linear combinations of the cyclic and stationary Killing vectors) who measure zero conical deficit ($C=1$) along the axis.
3. Vanishing Conicity Difference: In spacetimes with NUT parameters, the axis is typically split into two semi-axes (top and bottom). The difference in conicity between these two semi-axes ($\Delta C$), often interpreted as the cause of black hole acceleration, is also observer-dependent. The authors show there always exist observers who measure no difference in conicity between the two semi-axes ($\Delta C = 0$).
4. Unified Formalism: They provide a unified geometric definition that applies to the entire Plebański–Demiański class of solutions, including the recently discovered accelerated Taub–NUT solution.

4. Results and Applications

The authors applied their definition to several specific spacetimes:

• Spinning Cosmic String:
  • They derived explicit formulas for conicity $C(t)$ and time-shift $T(t)$.
  • Confirmed that for a spinning string ($s \neq 0$), $C$ depends on the observer. There are two observers who see no deficit.
• Plebański–Demiański Metric:
  • Analyzed the general case with parameters for mass, charge, rotation, NUT charge, acceleration, and cosmological constant.
  • Showed that the parameters $s$ (Manko–Ruiz parameter) and $b$ (conicity scaling) govern the distribution of the torsion singularity.
  • Demonstrated that for non-zero NUT charge ($l \neq 0$), the time-shifts on the top ($+$) and bottom ($-$) semi-axes are generally different ($T_+ \neq T_-$), making the singularity ineliminable globally, though movable between axes.
• C-Metric (Acceleration without NUT):
  • As a consistency check, they showed that when the NUT parameter is zero ($l=0$), the time-shift vanishes ($T=0$).
  • In this case, the conicity becomes observer-independent, recovering the standard interpretation where the conicity difference drives the black hole acceleration.
• Taub–NUT Spacetime:
  • Explicitly calculated that the standard observer ($t = \partial_t$) measures no conical deficit, but other observers do.
  • Identified specific observers who measure equal conicity on both semi-axes, challenging the interpretation of conicity differences as the sole driver of dynamics.
• Accelerated Taub–NUT:
  • Applied the formalism to the Petrov Type D accelerated Taub–NUT solution.
  • Found that switching on the NUT parameter in the C-metric changes the conicity from a fixed physical quantity to an observer-dependent one that can be measured as zero.

5. Significance and Implications

• Reinterpretation of Acceleration: The standard interpretation of the C-metric posits that the difference in conicity between the two semi-axes represents the tension of cosmic strings/struts causing the black hole to accelerate. The authors' results challenge this for spacetimes with NUT parameters. If the conicity difference is observer-dependent, it cannot be uniquely identified as the physical force causing acceleration without specifying a "canonical" observer.
• Black Hole Thermodynamics: The first law of black hole thermodynamics for accelerating black holes often includes terms related to conical deficits and acceleration. The observer-dependence of these quantities suggests that thermodynamic quantities in NUT spacetimes may need to be re-evaluated relative to specific observers (e.g., those measuring no deficit).
• Lack of Canonical Observer: The authors attempted to find a "canonical" observer (e.g., one that remains timelike at infinity or satisfies specific orthogonality conditions near the axis) but concluded that no such natural geometric condition exists that yields a unique, physically meaningful conicity for all cases.
• Conclusion: The paper concludes that in spacetimes with torsion singularities, conicity is a kinematical property dependent on the observer's state of motion, not an intrinsic geometric invariant. Therefore, attributing physical forces (like string tension) directly to conicity differences in NUT spacetimes is problematic without a rigorous definition of the observer.

In summary, the paper fundamentally shifts the understanding of conical singularities in NUT spacetimes from a static geometric defect to a dynamic, observer-relative phenomenon, necessitating a re-evaluation of how acceleration and thermodynamics are modeled in these complex gravitational fields.