[Paper Review] The limiting behavior of the Liu-Yau quasi-local energy

Pin Yu|ArXiv.org|Jun 7, 2007

Black Holes and Theoretical Physics4 references14 citations

TL;DR

This paper analyzes the limiting behavior of Liu-Yau's quasi-local energy in general relativity, showing it correctly recovers matter energy density in the small-sphere limit and Bondi mass/energy loss at null infinity. In vacuum, however, an unexpected extra term appears in the small-sphere limit, suggesting sensitivity to the reference embedding, though the large-sphere limit remains physically consistent with known results.

ABSTRACT

The small- and large-sphere limits of the quasi-local energy recently proposed by Liu and Yau are carefully examined. It is shown that in the small-sphere limit, the non-vacuum limit of the Liu-Yau quasi-local energy approaches the expected value \frac{4π}{3} r^3 T(e_0, e_0)$. Here, T is the energy-stress tensor of matter, e_0 \in T_p M is unit time-like and future-directed at the point p located at the center of the small sphere of radius $r$ in the limit r o 0. In vacuum, however, the limiting value of the Liu-Yau quasi-local energy contains the desired limit \frac{r^5}{90} B(e_0, e_0, e_0, e_0), where B is the Bel-Robinson tensor, as well as an extra term. In the large-sphere limit at null infinity, for isolated gravitational sources, the Liu-Yau quasi-local energy is shown to recover the Bondi mass and Bondi news flux, in space-times that are asymptotically empty and flat at null infinity. The physical validity of the Liu-Yau model in view of these results is discussed.

Motivation & Objective

• To assess the physical validity of Liu-Yau's quasi-local energy by examining its limiting behaviors.
• To verify whether the quasi-local energy correctly reduces to known physical quantities in extreme limits such as small spheres and large spheres at null infinity.
• To investigate the role of the reference embedding in the Liu-Yau construction, particularly in vacuum small-sphere limits.
• To explore the feasibility of generalizing the quasi-local energy to non-isolated gravitational sources, using the C-metric as a test case.

Proposed method

• Analytical computation of the Liu-Yau quasi-local energy in the small-sphere limit using normal coordinates and Taylor expansion around a point p.
• Use of the null-cone reference space-time to define the embedding of the 2-surface S into a reference Minkowski spacetime.
• Application of the blow-up technique to handle singular behavior in the small-sphere limit, particularly in vacuum.
• Evaluation of the large-sphere limit at null infinity in asymptotically flat spacetimes, comparing with Bondi mass and news flux.
• Analysis of the C-metric as a non-isolated source model to test the applicability of the quasi-local energy in non-vacuum, non-asymptotically-flat scenarios.
• Use of the Bel-Robinson tensor and energy-stress tensor to compare the leading-order terms in the small-sphere expansion.

Experimental results

Research questions

• RQ1 Does the Liu-Yau quasi-local energy recover the expected matter energy density in the small-sphere limit in non-vacuum spacetimes?
• RQ2 Does the Liu-Yau quasi-local energy correctly reproduce the Bondi mass and energy loss at null infinity in radiating, asymptotically flat spacetimes?
• RQ3 Why does the small-sphere limit in vacuum spacetimes contain an extra term beyond the known Bel-Robinson contribution?
• RQ4 How does the choice of reference embedding affect the small-sphere limit of the Liu-Yau quasi-local energy?
• RQ5 Can the Liu-Yau quasi-local energy be meaningfully generalized to non-isolated gravitational sources such as the C-metric?

Key findings

• In the non-vacuum small-sphere limit, the Liu-Yau quasi-local energy approaches the expected value $\frac{4\pi}{3}r^{3}{\mathfrak{T}}(e_{0},e_{0})$, where ${\mathfrak{T}}$ is the energy-stress tensor and $e_0$ is a unit future-directed timelike vector at the center.
• In the vacuum small-sphere limit, the leading term is $\frac{r^{5}}{90}{\mathfrak{B}}(e_{0},e_{0},e_{0},e_{0})$, where ${\mathfrak{B}}$ is the Bel-Robinson tensor, but an additional term appears that is not accounted for in the standard expectation.
• The extra term in the vacuum small-sphere limit arises due to the dependence of the Liu-Yau construction on the choice of reference embedding, indicating sensitivity to the embedding scheme.
• In the large-sphere limit at null infinity in asymptotically flat spacetimes, the Liu-Yau quasi-local energy correctly recovers the Bondi mass and the news flux, confirming consistency with standard gravitational energy loss.
• The analysis of the C-metric shows that non-isolated sources pose technical challenges for large-sphere limits due to the need for multiple coordinate charts, making practical evaluation of Bondi quantities difficult.
• Despite the extra term in vacuum, the overall consistency with known physical limits suggests the Liu-Yau model remains physically viable, though the embedding dependence warrants further scrutiny.