[Paper Review] The group aspect in the physical interpretation of general relativity theory

S. Antoci, Dierck‐Ekkehard Liebscher|ArXiv.org|Oct 12, 2009

Relativity and Gravitational Theory6 references3 citations

TL;DR

This paper re-evaluates the physical significance of group theory in general relativity by arguing that the group of general coordinate transformations is physically irrelevant, as any theory can be made generally covariant. Instead, it identifies the Killing group—generated by infinitesimal isometries of the metric—as the physically meaningful group, whose structure determines the 'relativity postulate' of a spacetime. The key contribution is showing that intrinsic geometric singularities can arise from divergences in Killing vector curvature norms, even when Riemann invariants remain finite, as demonstrated in the Kerr-Newman solution.

ABSTRACT

When, at the end of the year 1915, both Einstein and Hilbert arrived at what were named the field equations of general relativity, both of them thought that their fundamental achievement entailed, inter alia, the realisation of a theory of gravitation whose underlying group was the group of general coordinate transformations. This group theoretical property was believed by Einstein to be a relevant one from a physical standpoint, because the general coordinates allowed to introduce reference frames not limited to the inertial reference frames that can be associated with the Minkowski coordinate systems, whose transformation group was perceived to be restricted to the Poincaré group. Two years later, however, Kretschmann published a paper in which the physical relevance of the group theoretical achievement in the general relativity of 1915 was denied. For Kretschmann, since any theory, whatever its physical content, can be rewritten in a generally covariant form, the group of general coordinate transformations is physically irrelevant. This is not the case, however, for the group of the infinitesimal motions that bring the metric field in itself, namely, for the Killing group. This group is physically characteristic of any given spacetime theory, since it accounts for the local invariance properties of the considered manifold, i.e., for its ``relativity postulate''. In the present chapter it is shown what are the consequences for the physical interpretation of some of these solutions whose relativity postulate is of intermediate content, when Kretschmann's standpoint is consistently adhered to.

Motivation & Objective

• To re-express the physical interpretation of general relativity by shifting focus from general coordinate invariance to the physically meaningful Killing group.
• To resolve the Kretschmann objection that general covariance is physically vacuous by identifying the Killing group as the true carrier of physical symmetry.
• To demonstrate that spacetime singularities can emerge from the behavior of Killing vector fields, even when standard curvature invariants remain finite.
• To show that solutions with nontrivial Killing groups—such as Kerr-Newman—exhibit intrinsic geometric features tied to the structure of their isometry group.

Proposed method

• Uses the mathematical tool of infinitesimal Killing vectors to characterize the isometry group of a spacetime manifold.
• Applies the Killing equation $\xi^{i;k} + \xi^{k;i} = 0$ to define symmetries of the metric tensor.
• Analyzes the squared norm of the first curvature of Killing congruences using the expression $\alpha^2 = -g_{ij}a^i a^j$, with $a^i = \frac{\xi^i}{N}_{;k}\frac{\xi^k}{N}$.
• Evaluates the behavior of $\alpha^2$ in the Kerr-Newman metric in Boyer-Lindquist coordinates, particularly near $r = r_0 = M + \sqrt{M^2 - J^2 - Q^2}$.
• Compares the physical content of the relativity postulate across spacetimes by analyzing the dimension and structure of their Killing groups.
• Uses invariant geometric quantities derived from Killing vectors to detect singularities independent of polynomial Riemann invariants.

Experimental results

Research questions

• RQ1 Why is the group of general coordinate transformations physically irrelevant despite its role in general covariance?
• RQ2 How does the Killing group serve as the true physical carrier of the relativity postulate in spacetime theories?
• RQ3 Can intrinsic geometric singularities arise from the behavior of Killing vector fields even when Riemann tensor invariants remain finite?
• RQ4 What is the physical significance of the divergence in the norm of the first curvature of Killing congruences in the Kerr-Newman spacetime?
• RQ5 How does the structure of the Killing group affect the physical interpretation of solutions to Einstein's field equations?

Key findings

• The group of general coordinate transformations is physically irrelevant because any theory can be written in generally covariant form, rendering it insufficient as a physical principle.
• The Killing group—the group of infinitesimal isometries of the metric—is the physically meaningful symmetry group, as it encodes the 'relativity postulate' of a spacetime.
• In the Kerr-Newman solution, the norm of the first curvature of certain Killing congruences diverges at $r = r_0 = M + \sqrt{M^2 - J^2 - Q^2}$, indicating an intrinsic geometric singularity.
• This singularity is invariant and intrinsic, as it arises from the behavior of Killing vector fields and is not reflected in polynomial invariants of the Riemann tensor.
• The existence of nontrivial Killing groups in solutions like Kerr-Newman leads to physical features—such as divergent accelerations of congruences—that are not captured by curvature scalars alone.
• Pasting together spacetimes with different Killing groups can produce divergent geometric quantities at boundaries, even if Riemann invariants remain finite, indicating a breakdown in smoothness independent of curvature singularities.