[Paper Review] Spacetime Exterior to a Star: Against Asymptotic Flatness

Mark D. Roberts|ArXiv.org|Nov 28, 1998

Cosmology and Gravitation Theories12 references3 citations

TL;DR

This paper challenges the assumption of asymptotic flatness in the spacetime exterior to a star by deriving a geometric-thermodynamic relation, N = 1/h, from fluid conservation laws under spherical symmetry and the shift-free ADM formalism. It shows that the requirement of asymptotic flatness—N → 1 and h → 0 simultaneously—is incompatible with this relation, implying that asymptotic flatness cannot hold in realistic stellar models.

ABSTRACT

In many circumstances the perfect fluid conservation equations can be directly integrated to give a Geometric-Thermodynamic equation: typically that the lapse $N$ is the reciprocal of the enthalphy $h$, ($ N=1/h$). This result is aesthetically appealing as it depends only on the fluid conservation equations and does not depend on specific field equations such as Einstein's. Here the form of the Geometric-Thermodynamic equation is derived subject to spherical symmetry and also for the shift-free ADM formalism. There at least three applications of the Geometric-Thermodynamic equation, the most important being to the notion of asympotic flatness and hence to spacetime exterior to a star. For asymptotic flatness one wants $h o 0$ and $N o 1$ simultaneously, but this is incompatible with the Geometric-Thermodynamic equation. Observational data and asymptotic flatness are discussed. It is argued that a version of Mach's principle does not allow asymptotic flatness.

Motivation & Objective

• To re-express the fluid conservation equations in a geometric-thermodynamic form independent of specific field equations.
• To investigate the implications of the derived relation N = 1/h for the spacetime geometry exterior to a star.
• To challenge the physical validity of asymptotic flatness in stellar exterior solutions.
• To assess whether observational data and Machian principles support asymptotic flatness.
• To explore the consistency of the geometric-thermodynamic equation with general relativity and astrophysical constraints.

Proposed method

• Derives the geometric-thermodynamic equation N = 1/h from the perfect fluid conservation equations under spherical symmetry.
• Applies the shift-free ADM formalism to analyze the spacetime geometry and lapse function N.
• Uses the relation N = 1/h to examine the behavior of N and h at spatial infinity.
• Analyzes the asymptotic limit where N → 1 and h → 0, showing incompatibility with the derived equation.
• Evaluates observational and conceptual constraints, including Mach's principle, to assess the plausibility of asymptotic flatness.
• Compares the derived relation with standard solutions like the Schwarzschild metric to highlight inconsistencies.

Experimental results

Research questions

• RQ1 Can the fluid conservation equations alone yield a geometric-thermodynamic relation independent of Einstein's equations?
• RQ2 Is the assumption of asymptotic flatness compatible with the derived relation N = 1/h in spherically symmetric spacetimes?
• RQ3 What are the implications of the N = 1/h relation for the exterior geometry of a star?
• RQ4 Does observational data or Machian principles support the validity of asymptotic flatness?
• RQ5 Can a spacetime exterior to a star be both physically realistic and asymptotically flat under the derived constraints?

Key findings

• The geometric-thermodynamic relation N = 1/h is derived directly from fluid conservation laws under spherical symmetry and the shift-free ADM formalism.
• The relation N = 1/h implies that N → 1 and h → 0 cannot occur simultaneously, contradicting the requirement for asymptotic flatness.
• Asymptotic flatness is therefore incompatible with the derived geometric-thermodynamic equation in the context of a perfect fluid star.
• The paper argues that observational data and Machian principles do not support the assumption of asymptotic flatness.
• The result suggests that standard exterior solutions like the Schwarzschild metric may not be physically consistent with the fluid dynamics of a realistic star.
• The analysis implies that the spacetime exterior to a star cannot be both asymptotically flat and governed by the derived N = 1/h relation.