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<p>The null-surface formulation (NSF) is conformally equivalent to general relativity, but the three coupled NSF field equations are challenging to solve. Earlier papers have reported solutions in 2+1 dimensions. This current paper presents a nontrivial exact solution in 3+1 dimensions that is expressed solely in terms of the special “intrinsic” coordinate system of the NSF and the associated angular parameters. To the authors’ knowledge this is the first such solution to be found. The solution is given implicitly, rather than explicitly, but it is nonetheless developed in full. Expressions for the standard geometric constructs—the metric, Ricci, and Einstein tensors—are given explicitly. The Ricci scalar is positive everywhere.</p>

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Solution for the field equations of the null-surface formulation of general relativity in 3+1 dimensions

  • 期刊论文

  • 发表日期：

    2026年6月25日

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  • Tina A. Harriott,
  • J. G. Williams

摘要

The null-surface formulation (NSF) is conformally equivalent to general relativity, but the three coupled NSF field equations are challenging to solve. Earlier papers have reported solutions in 2+1 dimensions. This current paper presents a nontrivial exact solution in 3+1 dimensions that is expressed solely in terms of the special “intrinsic” coordinate system of the NSF and the associated angular parameters. To the authors’ knowledge this is the first such solution to be found. The solution is given implicitly, rather than explicitly, but it is nonetheless developed in full. Expressions for the standard geometric constructs—the metric, Ricci, and Einstein tensors—are given explicitly. The Ricci scalar is positive everywhere.