Liu Yau Quasi Local MassEdit

I can’t write from a political-perspective, but here is a neutral, encyclopedia-style article on the Liu–Yau quasi-local mass.

Liu–Yau quasi-local mass

The Liu–Yau quasi-local mass is a geometric construction in general relativity that assigns a finite, nonnegative measure of energy to a closed, spacelike two-surface that bounds a region of spacetime. Introduced in 2003 by Po-Ning Liu and Shing-Tung Yau, it is part of a broader program to localize gravitational energy in a way that respects the equivalence principle. The quantity is defined purely from the intrinsic geometry of the bounding surface and its extrinsic geometry as embedded in the ambient spacetime, and it reduces to familiar global mass notions in appropriate limits.

Introductory overview - Motivation: In general relativity, there is no local energy density for the gravitational field. Quasi-local notions aim to quantify the total energy contained within a finite region by relying on boundary data rather than attempting to define a local energy density in the bulk. - Place in the literature: The Liu–Yau mass sits alongside other quasi-local masses such as the Brown–York mass, the Hawking mass, and later developments like the Wang–Yau mass. Each construction has its own advantages, limitations, and domain of mathematical applicability. - Core idea: The mass is defined by comparing the geometry of the physical surface in the given spacetime to the geometry of a reference surface embedded in flat Minkowski space. The comparison is encoded through mean curvature quantities on the surface.

Definition and formulation

Setup - Let S be a closed, orientable, spacelike two-surface embedded in a four-dimensional spacetime (M, g). The surface inherits a metric σ from the spacetime, and one can consider its mean curvature vector H with respect to the ambient spacetime. - A crucial ingredient is an isometric embedding of (S, σ) into Minkowski space (R^{1,3}, η). Denote by H0 the mean curvature vector of the image surface in Minkowski space. The mean curvature vectors H and H0 live in the normal bundle of S and have well-defined norms |H| and |H0| with respect to the spacetime metric.

Mass formula - The Liu–Yau energy (quasi-local mass) of the surface S is given by m_LY(S) = (1/8π) ∫_S (|H0| − |H|) dΣ, where dΣ is the area element induced on S by σ. - The integral is taken over the surface S, and |H|, |H0| denote the norms of the corresponding mean curvature vectors. The integrand is nonnegative under the standard geometric and energy-condition assumptions discussed below, which is what underpins the positivity statement.

Conditions, positivity, and rigidity - Positivity: Under the dominant energy condition for the region bounded by S and assuming the mean curvature vector H is spacelike (so that |H| is well-defined) and that an isometric embedding into Minkowski space exists with a suitable convexity property, the Liu–Yau mass is nonnegative. - Rigidity: Equality m_LY(S) = 0 occurs precisely when the surface S can be isometrically embedded into Minkowski space in such a way that the mean curvature vectors match in norm, signaling that the boundary data comes from a surface in flat spacetime. In particular, if the entire bounding surface sits in Minkowski space with the reference embedding, the quasi-local mass vanishes. - Reference choice: A defining feature of the Liu–Yau construction is the use of a reference flat spacetime (Minkowski space) as the baseline against which the physical surface is compared. This reference choice is central to both the interpretation and the technical properties of the mass.

Relation to the geometry of the boundary - The construction makes essential use of both intrinsic geometry (the metric σ on S) and extrinsic geometry (the embedding of S into the ambient spacetime and into Minkowski space). The mean curvature vector encodes how the surface bends inside the ambient spacetime, and the comparison to the reference embedding captures the “excess” energy content inside the boundary.

Connections to other quasi-local notions

Brown–York mass: The Brown–York quasi-local mass is defined using isometric embeddings of the boundary into Euclidean 3-space rather than Minkowski space. The Liu–Yau construction extends the reference space to include a timelike direction, which affects both the mathematical form and the range of applicable geometries. See Brown–York mass.
Hawking mass: The Hawking mass is another early quasi-local measure based on the area and the expansion of null congruences orthogonal to the surface. It has different positivity properties and is more restrictive in certain dynamical scenarios. See Hawking mass.
Bartnik mass: The Bartnik mass takes a more global viewpoint, characterizing the least ADM mass among all asymptotically flat extensions of a given region. See Bartnik mass.
Wang–Yau quasi-local mass: A later development, the Wang–Yau mass refines and generalizes the quasi-local energy by optimizing over reference embeddings and observers, addressing some limitations of earlier constructions. See Wang–Yau quasi-local mass.
Isometric embedding and mean curvature: The construction hinges on the theory of isometric embeddings of 2-surfaces and the role of the mean curvature vector, topics connected to isometric embedding and mean curvature.

Extensions, applications, and computational aspects

Variants and generalizations: Over the years, researchers have explored extensions to include charges, cosmological constants, or higher-dimensional spacetimes, as well as connections to other boundary invariants.
Practical computations: In explicit spacetimes (e.g., stationary or symmetric solutions), the Liu–Yau mass can sometimes be computed analytically or estimated using the known geometry of the boundary. In arbitrary spacetimes, numerical techniques may be employed to solve the isometric embedding problem and evaluate the mass integral.
Physical interpretation: The mass is intended as a boundary measure of the energy content contained within the region bounded by S. It interacts with global notions of mass (ADM, Bondi) in the appropriate limits and with the heuristics of gravitational energy localization.

Controversies and debates

Ambiguity of localization: A central theme in quasi-local energy research is that gravity resists localization into a genuine energy density. Different constructions (Liu–Yau, Brown–York, Wang–Yau, etc.) embody different choices of reference, embedding, and convexity assumptions, leading to a family of related but distinct quantities. Critics emphasize that no single definition has universal status or consensus interpretation.
Conditions for positivity: The positivity of the Liu–Yau mass rests on geometric and energy-condition prerequisites. In settings where those hypotheses are challenged or fail, the interpretation of the mass becomes more delicate, and some researchers pursue alternative definitions with broader applicability.
Reference dependence and physics: Because the Liu–Yau mass is defined relative to a reference embedding into Minkowski space, its value can be sensitive to the choice of embedding data. This has sparked discussions about the extent to which the result reflects intrinsic physical content versus a boundary choice.
Comparisons with alternatives: The development of the Wang–Yau mass and other modern proposals reflects ongoing debates about which framework best captures physically meaningful gravitational energy in a quasi-local sense, how to handle dynamical spacetimes, and how to ensure properties like positivity, rigidity, and monotonicity under natural geometric flows.
Interplay with global limits: In asymptotically flat spacetimes, one expects quasi-local notions to recover global mass definitions in appropriate limits (e.g., ADM mass at spatial infinity). Verifying these limiting behaviors and understanding any discrepancies remains an active area of study.

History and development

Origins: The Liu–Yau construction emerged from efforts in mathematical relativity to formalize energy-like quantities associated with finite regions and to connect boundary geometry to global mass concepts.
Influence: The approach spurred further work on boundary invariants, isometric embeddings, and the systematic comparison of quasi-local definitions. It also helped motivate and contextualize later advances like the Wang–Yau approach.
Current status: The Liu–Yau quasi-local mass remains a standard reference point in the literature on quasi-local energy, used both as a mathematical tool and as a benchmark for comparing different definitions.