⚛️ general relativity

Conformal geodesics and the hyperboloid at timelike infinity

This paper utilizes conformal geodesics to construct an extended hyperboloidal representation of timelike infinity in Minkowski spacetime, deriving field equations for massless scalar and spin-2 fields in this gauge to establish a geometric framework connecting late-time and null asymptotics for flat space holography.

Original authors: Achilleas V. Koukas, Mariem Magdy, Juan A. Valiente Kroon

Published 2026-10-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Achilleas V. Koukas, Mariem Magdy, Juan A. Valiente Kroon

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast, silent theater of the universe, gravity is the director, shaping the paths of everything from light beams to massive stars. For decades, physicists have relied on a specific way of mapping this stage, known as the Penrose diagram, to understand how the universe behaves at its very edges. In this map, the far future of any object moving slower than light—like a planet or a spaceship—is represented by a single, tiny point. This point, called future timelike infinity, is where all such journeys theoretically end. However, treating this destination as a mere dot creates a problem. It hides the details of how massive objects actually arrive, obscuring the relationship between their final state and the radiation that flies away at the speed of light. To understand how the universe scatters matter and energy, or to build a theory of quantum gravity that works in flat space, scientists need a map that shows the arrival of these massive objects in greater detail, not just as a point, but as a place with structure.

A team of researchers has now drawn a new map that replaces that single point with a rich, three-dimensional surface. By using a mathematical tool called conformal geodesics, they have shown that the end of time for massive particles is not a singular dot, but a hyperboloid—a shape resembling a cooling tower or a saddle that curves outward in all directions. This discovery, made by analyzing the geometry of empty space, provides a way to "blow up" the point of infinity into a full boundary. This new boundary meets the edge of the universe where light travels at a sharp corner, creating a complete picture of how massive matter and light interact at the very end of time.

The researchers began their work in the simplest possible setting: a universe with no matter or energy, known as Minkowski spacetime. In this empty void, they focused on the paths of objects moving at a constant speed. Traditionally, these paths are described by standard geometry. However, the team applied a different set of rules, known as conformal geometry, which allows the scale of space and time to change while preserving the angles between paths. They discovered that by carefully adjusting the speed at which they measured time along these paths, they could transform the standard straight lines of motion into a special family of curves called conformal geodesics. These curves have a unique property: they naturally carry a built-in clock that tells them exactly when to stop, effectively attaching a future endpoint to every single path.

When the team applied this method to a collection of paths starting from a single moment in time, something remarkable happened. Instead of all paths converging to a single point at the end of time, they spread out to fill a surface. This surface is a hyperboloid, a shape that exists in the mathematical description of the universe's edge. The researchers named this new boundary the hyperboloid at timelike infinity. It acts as a stage where the final state of every massive object can be recorded individually, rather than being crushed into a single, indistinguishable point. This construction also solved a long-standing difficulty in mapping the universe: it fixed the location of the edge where light travels. In their new coordinate system, the boundary for light is always found at a specific, unchanging place, making it much easier to study how light and matter interact as they reach the end of time.

To test if this new map was useful, the team examined how simple waves, representing massless particles like light or gravitational ripples, behaved on this new geometry. They traced the evolution of these waves as they traveled toward the new hyperboloid boundary and the corner where it meets the light-like edge. They found that the waves settle into a predictable pattern as they approach this corner. For the simplest waves, the value stabilizes to a constant number. For more complex waves, the value fades away to zero. This behavior suggests that the new geometric framework is stable and capable of describing how physical fields evolve all the way to the very end of time without breaking down. The researchers also showed that this method works not just for empty space, but for universes that are slightly disturbed, such as those containing small amounts of energy or gravitational waves. This means the new map is robust enough to handle the messy reality of a dynamic universe.

The implications of this work extend to the cutting edge of theoretical physics, particularly in the field of flat space holography. This field attempts to describe the physics of our three-dimensional universe using information stored on a two-dimensional boundary. A major hurdle in this approach has been the inability to properly describe the arrival of massive particles, which are essential for a complete theory of scattering. By replacing the point of infinity with a hyperboloid, the researchers have provided the missing geometric link. They have created a framework where the late-time behavior of massive objects can be studied alongside light, offering a clearer path toward understanding how the universe scatters matter and energy. While the full application to complex, real-world scenarios like black holes remains a task for the future, this work establishes a solid foundation. It proves that the end of time for massive objects is not a mystery hidden in a dot, but a structured landscape that can be explored, measured, and understood.

Technical Summary: Conformal Geodesics and the Hyperboloid at Timelike Infinity

Problem Statement
The paper addresses the geometric challenge of describing scattering in asymptotically flat spacetimes that includes both massless radiation and massive asymptotic states. A complete description requires a unified framework that controls the relationship between null infinity ( $\mathscr{I}^+$ ) and timelike infinity ( $i^+$ ). In the standard Penrose conformal compactification of Minkowski spacetime, $i^+$ is represented as a single point, which fails to resolve the dependence of massive-field asymptotic data on timelike directions. While previous works have resolved spatial infinity ( $i^0$ ) and proposed extended structures for $i^+$ (such as the line bundle $T_i$ over a hyperboloid), these constructions often lack a joint description that controls the regularity of fields where these asymptotic regions meet, particularly at the intersection of future null and timelike infinity.

Methodology
The authors utilize the properties of conformal geodesics to construct an extended representation of timelike infinity in Minkowski spacetime. The core methodology relies on the following theoretical tools:

Conformal Geodesics in Einstein Spacetimes: The authors exploit the fact that in an Einstein spacetime (where the Ricci tensor is proportional to the metric), every non-null metric geodesic can be reparametrised as a conformal geodesic. This is achieved by introducing a covector field $\beta$ along the curve.
Conformal Gaussian Gauge Systems: By reparametrising a congruence of timelike geodesics starting from a Cauchy hypersurface (or a fiduciary hyperboloid), the authors construct a conformal Gaussian system. In this gauge, the associated conformal factor is known explicitly a priori, allowing for the precise location of the conformal boundary.
Hyperboloidal Initial Data: The construction is based on initial data prescribed on a hyperboloidal hypersurface (specifically, constant mean curvature hyperboloids). The authors analyze two specific spatial compactifications for these hyperboloids:
- The Poincaré ball model, where the hyperboloid is conformally mapped to the interior of a unit ball.
- Zenginoglu's gauge, an alternative spatial compactification frequently used in numerical relativity that avoids conformally rescaling the intrinsic metric to Euclidean space.
I-Fixing: The resulting gauges implement " $\mathscr{I}$ -fixing," a condition where the location of future null infinity ( $\mathscr{I}^+$ ) is fixed as the locus where a specific spatial coordinate vanishes.

Key Constructions and Results

Blow-up of $i^+$ : The reparametrisation of the timelike geodesics effectively "blows up" the point $i^+$ into a hyperboloid at timelike infinity ( $H^+$ ). This hyperboloid meets future null infinity ( $\mathscr{I}^+$ ) at a corner ( $C^+$ ). In the constructed conformal Gaussian coordinates, the metric is degenerate but smooth at $H^+$ .
Transport Equations and Asymptotics: The authors derive the evolution equations for massless fields in this new gauge:
- Scalar Field: They derive a transport equation for a massless scalar field along $\mathscr{I}^+$ . By decomposing the field into spherical harmonics, they analyze the asymptotic behavior of the modes near the corner $C^+$ . They find that for the monopole mode ( $\ell=0$ ), the field approaches a constant, while for higher modes ( $\ell \geq 1$ ), the solutions decay exponentially or oscillate with decaying amplitude as they approach $H^+$ .
- Linearised Gravity: The authors derive the explicit massless spin-2 equations (linearised gravity) in the Poincaré ball gauge. They observe that the equations exhibit mild singularities at $\mathscr{I}^+$ , suggesting that analytic treatment via $b$ -calculus tools may be required.
Projective Extensions and $T_i$ : The paper examines the relationship between their construction and projective extensions. They demonstrate that the standard boundary defining function used in their construction does not yield a regular projective extension for generic timelike geodesics. However, by introducing a new coordinate that blows up the corner $C^+$ , they relate their hyperboloid $H^+$ to the base of the extended Carrollian space $T_i$ described in previous literature (Ashtekar-Hansen, Friedrich). They show that $H^+$ is conformally related to the unit hyperboloid base of $T_i$ .
Nonlinear Stability: Leveraging Friedrich's semiglobal stability results for the Minkowski spacetime, the authors prove that their construction extends to nonlinear perturbations of Minkowski spacetime. They show that for spacetimes arising from hyperboloidal initial data, a conformal representation with $\mathscr{I}$ -fixing exists where timelike infinity is diffeomorphic to a unit solid ball in $\mathbb{R}^3$ .

Significance and Claims
The paper claims to provide a geometric framework for studying the connection between late-time (timelike) and null asymptotics. This framework is presented as a prerequisite for scattering constructions relevant to flat space holography, where understanding the boundary description of massive states is essential.

The authors emphasize that their work offers an alternative to the standard point-like representation of $i^+$ and the $T_i$ construction, demonstrating that there is no unique manner of obtaining a blow-up of timelike infinity. The construction is motivated by the need to disentangle structures associated with $i^+$ and to provide a setting where the regularity of fields at the intersection of null and timelike infinity can be investigated using the extended conformal Einstein field equations.

The paper positions itself as exploratory, providing the first step toward a broader analysis of timelike infinity within the extended conformal Einstein field equations. It explicitly states that a detailed analysis of the full transport equations, the treatment of massive fields, and applications to black hole spacetimes (Schwarzschild, Kerr) are beyond the scope of the current work and will be addressed in future publications.

More like this