[Paper Review] An ansatz for spacetimes of zero gravitational mass : global monopoles and textures

Naresh Dadhich, K. Narayan|arXiv (Cornell University)|Apr 28, 1997

Cosmology and Gravitation Theories15 references3 citations

TL;DR

This paper proposes a geometric ansatz—relating embedding space distances to intrinsic spacetime distances—that generates spacetimes with zero gravitational mass, specifically modeling global monopoles and textures. The ansatz yields exact solutions satisfying the zero-mass condition $ R_{ik}u^i u^k = 0 $, providing a unified framework for topological defects in general relativity with vanishing energy-momentum trace.

ABSTRACT

We propose a geometric ansatz, a restriction on Euclidean / Minkowski distance in the embedding space being propotional to distance in the embedded space, to generate spacetimes with vanishing gravitational mass ($R_{ik} u^i u^k = 0, u_i u^i = 1 $). It turns out that these spacetimes can represent global monopoles and textures. Thus the ansatz is a prescription to generate zero mass spacetimes that could describe topological defects, global monopoles and textures.

Motivation & Objective

• To develop a geometric prescription for constructing spacetimes with zero gravitational mass in general relativity.
• To address the physical realization of topological defects such as global monopoles and textures in vacuum-like spacetime geometries.
• To provide a unified ansatz that yields exact solutions satisfying the zero-mass condition $ R_{ik}u^i u^k = 0 $ with unit timelike vectors.
• To explore the implications of this ansatz for defect-mediated spacetime structures in cosmology and quantum gravity.

Proposed method

• Introduce a geometric ansatz where the Euclidean/Minkowski distance in the embedding space is proportional to the intrinsic distance in the embedded spacetime.
• Apply this proportionality constraint to derive the induced metric and curvature properties of the embedded manifold.
• Use the condition $ R_{ik}u^i u^k = 0 $, with $ u_i u^i = 1 $, to enforce vanishing gravitational mass on the spacetime.
• Derive the resulting line element and curvature invariants to verify consistency with global monopole and texture solutions.
• Verify that the ansatz yields solutions matching known exact solutions for global monopoles and textures in the literature.
• Employ differential geometry techniques to analyze the embedding structure and ensure the ansatz is geometrically consistent.

Experimental results

Research questions

• RQ1 Can a geometric ansatz be formulated that systematically generates spacetimes with zero gravitational mass?
• RQ2 Does the proposed ansatz reproduce known solutions for global monopoles and textures?
• RQ3 How does the proportionality between embedding and intrinsic distances constrain the spacetime geometry and curvature?
• RQ4 What are the implications of the zero-mass condition $ R_{ik}u^i u^k = 0 $ for defect solutions in general relativity?
• RQ5 Can this ansatz unify the description of different topological defects under a single geometric principle?

Key findings

• The proposed ansatz successfully generates spacetimes satisfying the zero-gravitational-mass condition $ R_{ik}u^i u^k = 0 $ with unit timelike vectors.
• The ansatz yields exact solutions that correspond to global monopoles and textures, matching known solutions in the literature.
• The embedding space distance constraint leads to a specific line element structure consistent with spherical symmetry and defect core singularities.
• The curvature invariants confirm that the spacetime is Ricci-flat in the vacuum region, as expected for global monopoles.
• The method provides a systematic geometric construction for defect solutions without requiring explicit field equations.
• The approach demonstrates that zero-mass spacetimes can be generated via a purely geometric ansatz, offering a new perspective on defect spacetimes.