[Paper Review] Gravitational energy-momentum in small regions according to Moller's tetrad expression

Lau Loi So, James M. Nester|ArXiv.org|Dec 11, 2006

Black Holes and Theoretical Physics4 references3 citations

TL;DR

This paper evaluates Møller's tetrad-based gravitational energy-momentum expression in small vacuum regions using Riemann normal coordinates and a corresponding orthonormal tetrad. It demonstrates that the expression yields a positive multiple of the Bel-Robinson tensor at second order, confirming its consistency with the strong physical requirement of positive gravitational energy in small regions.

ABSTRACT

Moller's tetrad gravitational energy-momentum "tensor" is evaluated for a small vacuum region using an orthonormal frame adapted to Riemann normal coordinates. We find that it does satisfy the highly desired property of being a positive multiple of the Bel-Robinson tensor.

Motivation & Objective

• To assess whether Møller’s tetrad-based gravitational energy-momentum expression satisfies the strong physical criterion of yielding a positive multiple of the Bel-Robinson tensor in small vacuum regions.
• To investigate the behavior of Møller’s expression in the small region limit, particularly its consistency with the equivalence principle and higher-order curvature behavior.
• To test whether Møller’s expression, which is frame-dependent, yields unambiguous results in the absence of a preferred frame by using Riemann normal coordinates and a natural orthonormal tetrad.
• To determine if Møller’s expression can pass the stringent test of reproducing the Bel-Robinson tensor in vacuum, a property not satisfied by classical pseudotensors.
• To establish the physical plausibility of Møller’s expression as a candidate for gravitational energy-momentum localization by verifying its positivity and correct curvature-order behavior.

Proposed method

• Adopting Møller’s tetrad expression for gravitational energy-momentum, derived from Einstein’s equations in differential form formalism.
• Using Riemann normal coordinates centered at a point in spacetime to expand the metric and connection coefficients to second order in spatial coordinates.
• Employing the associated orthonormal tetrad (normal tetrad) to express the coframe and connection one-forms in the local inertial frame.
• Expanding the total energy-momentum current 3-form ${\cal P}_\mu$ to second order in the radial coordinate $x^l$, isolating the gravitational contribution in vacuum.
• Applying curvature symmetry identities and the vacuum condition (Ricci tensor vanishing) to simplify the second-order terms.
• Integrating the resulting expression over a small spatial 3-sphere at $x^0 = 0$ using the identity $\int x^l x^m d^3x = \frac{1}{3}\delta^{lm} \int r^2 d^3x$, leading to a result proportional to the Bel-Robinson tensor.

Experimental results

Research questions

• RQ1 Does Møller’s tetrad energy-momentum expression yield a positive multiple of the Bel-Robinson tensor in the small vacuum region limit?
• RQ2 How does Møller’s expression behave in the small region limit, particularly in terms of its order of magnitude and dependence on curvature?
• RQ3 Can Møller’s expression satisfy the strong criterion of reproducing the Bel-Robinson tensor, a property that excludes most classical pseudotensors?
• RQ4 Is Møller’s expression unambiguous and physically meaningful in the small region limit despite its frame dependence?
• RQ5 Does the resulting energy-momentum value exhibit positivity, as required by physical consistency?

Key findings

• Møller’s tetrad energy-momentum expression produces a second-order contribution in small vacuum regions that is proportional to the Bel-Robinson tensor $B^{\nu}{}_{\mu lm}$.
• The total energy-momentum $P_\mu$ is found to be $P^\mu \simeq \frac{r^5}{240G}(E_{ab}E^{ab} + H_{ab}H^{ab}, -2\epsilon^{cab}E_{ad}H^{d}{}_{b})$, with $P^0 \geq |P^i| \geq 0$, confirming positivity.
• The result is $P^\mu \simeq \frac{1}{240G} B^{0}{}_{\mu 00} r^5$, explicitly showing proportionality to the Bel-Robinson tensor at leading order.
• The derivation confirms that Møller’s expression satisfies the highly desirable vacuum small region property, which is not satisfied by classical pseudotensors.
• The expression yields a positive energy density in the small region limit, consistent with the known positive energy proof for Møller’s formulation.
• The use of Riemann normal coordinates and the associated normal tetrad ensures a physically natural frame choice, making the result unambiguous in this context.