[Paper Review] Mass-like invariants for asymptotically hyperbolic metrics

Julien Cortier, Mattias Dahl+1 more|arXiv (Cornell University)|Mar 25, 2016

Algebraic and Geometric Analysis28 references3 citations

TL;DR

This paper classifies mass-like invariants for asymptotically hyperbolic Riemannian metrics by identifying two families of invariants linked to finite-dimensional representations of the hyperbolic isometry group O↑(n,1). Using wave harmonic polynomials and polynomial solutions to the linearized Einstein equations in Minkowski space, it shows that the standard mass is just one example among these invariants, with a complete classification derived via representation theory and invariant quadratic forms on harmonic and Weyl tensor polynomials.

ABSTRACT

In this article, we classify the set of asymptotic mass-like invariants for asymptotically hyperbolic metrics. It turns out that the standard mass is just one example (but probably the most important one) among the two families of invariants we find. These invariants are attached to finite-dimensional representations of the group of isometries of hyperbolic space. They are then described in terms of wave harmonic polynomials and polynomial solutions to the linearized Einstein equations in Minkowski space.

Motivation & Objective

• To classify all asymptotic mass-like invariants for asymptotically hyperbolic metrics beyond the standard mass.
• To understand the role of the hyperbolic isometry group O↑(n,1) in generating these invariants through its representations.
• To establish a correspondence between mass invariants and polynomial solutions to the linearized Einstein equations in Minkowski space.
• To provide a geometric and algebraic characterization of these invariants using wave harmonic polynomials and invariant quadratic forms.
• To generalize the notion of mass in general relativity to asymptotically hyperbolic settings with improved invariance and representation-theoretic clarity.

Proposed method

• Classify invariants using the action of the Lorentz group O↑(n,1) on mass-aspect tensors via Lie algebra intertwining operators.
• Analyze the structure of jets of asymptotically hyperbolic metrics to extract asymptotic invariants at infinity.
• Use wave harmonic polynomials as a basis for constructing mass invariants, particularly in the context of finite-dimensional representations of O↑(n,1).
• Relate invariants to solutions of the linearized Einstein equations in Minkowski space, especially for the Ricci, Cotton-York, and Bach tensors.
• Apply invariant theory to harmonic polynomials and polynomial Weyl tensors, using invariant quadratic forms to distinguish positive and negative eigenspaces.
• Derive explicit formulas for the dimensions of positive and negative eigenspaces of the mass invariants via combinatorial expressions involving binomial coefficients.

Experimental results

Research questions

• RQ1 What are all possible asymptotic mass-like invariants for asymptotically hyperbolic metrics, beyond the standard mass?
• RQ2 How do the invariants transform under the action of the hyperbolic isometry group O↑(n,1)?
• RQ3 What is the relationship between mass invariants and polynomial solutions to the linearized Einstein equations in Minkowski space?
• RQ4 How do the invariants decompose into irreducible representations of O↑(n,1), and what are their dimensions?
• RQ5 Can the standard mass be understood as a special case within a broader family of invariants arising from geometric differential operators?

Key findings

• The standard mass is just one example among two distinct families of mass-like invariants, both arising from finite-dimensional representations of O↑(n,1).
• The invariants are fully classified using wave harmonic polynomials and polynomial solutions to the linearized Einstein equations in Minkowski space.
• For each degree $ p $, the number of positive and negative eigenvalues of the mass form is given by explicit formulas: $ n_+(p) = \frac{1}{2}(n^2 + (n+1)p + 3)\frac{(p+1)(p+n+2)}{(n-1)(p+n)}\binom{p+n}{p+3} $ and $ n_-(p) = \frac{1}{2}(np + 4n + p)\frac{(p+1)(p+n+2)}{(n-1)(p+n)}\binom{p+n}{p+3} $.
• The difference $ n_+(p) - n_-(p) = (-1)^p \frac{p+1}{2}(p+n+2)\binom{p+n-1}{p+3} $, which simplifies to $ \frac{1}{12}(n+2)(n-1)(n-2)(n-3) $ for $ p=0 $.
• The classification is achieved via invariant quadratic forms on the spaces of harmonic polynomials and polynomial Weyl tensors, revealing a deep link between geometry and representation theory.
• The invariants are invariant under asymptotic isometries, and their norm in the Minkowski metric is preserved, generalizing the positive mass theorem to this broader class.