Introduction

This work presents a conformal approach to representing infinite regions of spacetime, focusing on Minkowski and Schwarzschild spacetimes. By using conformal mapping, we aim to visualize and explore the entirety of these spacetime structures within finite boundaries while preserving their causal relationships. Such a visualization helps not only in understanding spacetime at infinity but also in studying trajectories, null geodesics, and event horizons in a more comprehensive manner. Our study contributes to the broader goal of constructing diagrams that allow us to represent infinities compactly, facilitating a better understanding of complex spacetime geometries.

Penrose Diagram (A Brief Introduction)

2.a Motivation Behind the Penrose Diagram

In the early 1960s, the Penrose diagram, originally introduced by Roger Penrose and republished in 2011 as "Conformal Treatment of Infinity," aimed to address a unique challenge: representing infinite spacetime on a finite plane. This approach was particularly relevant for analyzing Minkowski spacetime, where both time and spatial dimensions extend to infinity.

Consider a particle moving along an infinite trajectory from past to future and from negative to positive spatial infinity. Depicting this complete trajectory on a finite diagram without information loss is impossible using traditional representations. Thus, the problem arose of how to "compactify" infinity within a finite structure.

2.b The Challenge and Approach to Solution

Achieving a complete picture of infinity in finite form requires ensuring that no significant information is lost during compactification. The Penrose diagram addresses this by applying a conformal transformation, mapping a physical manifold, such as Minkowski spacetime, to an "unphysical" manifold. This transformation results in a finite boundary in the unphysical manifold corresponding to the infinite extent of the physical spacetime.

For Minkowski spacetime, the conformal factor, denoted Ω, approaches zero at the boundary of the unphysical manifold. By introducing null coordinates, we effectively achieve a representation where causal relationships remain intact, capturing the essential features of spacetime within finite bounds.

2.c Diagram Formulation

The transformation from null coordinates 𝑢 and 𝑣 to compactified coordinates achieves this compactification. Starting with the Minkowski metric:

\[ ds_0^2 = dt^2 - dr^2 -r^2(d\theta^2 + sin^2 \theta . d\phi^2) \] we perform a coordinate transformation to arrive at a metric suitable for compactification:

\[ ds^2 = dpdq - \frac{1}{4}sin^2(p-q)(d\theta^2 + sin^2 \theta . d\phi^2) \] where 𝑝 and 𝑞 are bounded and the conformal factor Ω becomes zero on the boundary. This approach gives a finite representation of infinity, facilitating a complete and accurate representation of null geodesics and causal structure.

Schwarzschild Spacetime

3.a Motivation for Studying Schwarzschild Spacetime

The Schwarzschild solution to Einstein's field equations is a key concept in understanding black hole physics, as it describes the spacetime surrounding a spherically symmetric mass. This spacetime geometry is asymptotically flat, resembling Minkowski space at infinity, and introduces new challenges near the event horizon, where conventional coordinates fail to represent the full structure without encountering singularities.

3.b Key Results and Challenges

For Schwarzschild spacetime, the line element is given by:

\[ ds^2 = \left( 1 - \frac{2GM}{r} \right)dt^2 - \left( 1 - \frac{2GM}{r} \right)^{-1} dr^2 - r^2.d\Omega^2 \] where 𝑟 = 2𝐺𝑀 denotes the event horizon, introducing a coordinate singularity. To resolve this, we perform a coordinate transformation, defining the tortoise coordinate 𝑟∗, which extends 𝑟 beyond the event horizon, followed by null coordinates 𝑢 and 𝑣. This transformation facilitates a more comprehensive analysis of both black holes (Region II) and white holes (Region III).

3.c Applying the Penrose Diagram to Schwarzschild Spacetime

To visualize Schwarzschild spacetime in a finite representation, we apply further transformations similar to those used in Minkowski spacetime, resulting in a finite, bounded diagram that accurately represents the entire Schwarzschild manifold. This conformal approach, introduced by Kruskal and Szekeres, maps the infinite 𝑡-𝑟 coordinates to finite 𝑇-𝑋 coordinates, avoiding the singularity at 𝑟=2𝐺𝑀 and preserving causal structures across regions.

Discussion

4.a General Applicability of Conformal Mapping

The conformal mapping technique used in this work is not universally applicable to all spacetime geometries but is effective for spacetimes where a differentiable boundary can be established. In Minkowski and Schwarzschild spacetimes, the technique preserves the causal structure while compactifying infinity within a finite boundary.

4.b Constraints and Limitations

In applying conformal transformations to general manifolds, it is essential that the manifold and metric tensor satisfy sufficient differentiability. Additionally, the conformal factor Ω must vanish at the boundary without introducing discontinuities. In cases where null geodesics originate and terminate at the boundary, this approach provides a coherent and finite representation of spacetime, allowing for a complete understanding of causal relationships and asymptotic behavior.

A Conformal Treatment of Minkowski and Schwarzschild Spacetime