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I actually asked the following question at MathSE but didn't receive any response. My question is really about why the definition (2) below can be derived from the definition (1). Specifically, I don't know what the phrase convergence of the outer/inner null normal means. What on earth is this all about? How do I get to know that? Seriously, I don't see how a purely geometrical definition like (2) can be linked to the relativistic one, (1).

An answer to Heuristics for the Hawking mass appears to address my confusion, but the author used the phrase divergences along the two null directions, which doesn't quite get me further information. Can someone please help me? Thank you.

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Recently, I came across a strange definition of Hawking quasi-local mass, which states that given a surface $S$ in the spacetime, the Hawking mass of $S$ is defined as $$m(S)=\sqrt{\frac{\mathrm{Area}(S)}{16\pi}}\left(1-\frac{1}{4\pi}\int_S\rho\mu\right),\tag{1}$$ where $\rho$ is the convergence of the outer null normal and $-\mu$ is the convergence of the inner null normal. The definition is included in Some remarks on the quasi-local mass by Christodoulou and Yau. I call it strange because I used to learn about Hawking mass by defining $$m(S)=\sqrt{\frac{\mathrm{Area}(S)}{16\pi}}\left(1-\frac{1}{16\pi}\int_S H^2\right)\tag{2}$$ with $H$ as the mean curvature of $S$. This latter definition requires only knowledge of Riemannian geometry, and I'm thinking, is there possibly a way to show that it is in fact equivalent to the former definition, the more relativistic one?

I have to admit that in the first definition, there are many technical terms with which I have little acquaintance, so I grabbed a book titled Semi-Riemannian geometry with applications to relativity and authored by Barrett O'Neill. Does this book help to bridge the gap between these two definitions? On its 56th page, I'm told that a tangent vector $v$ is null if and only if $\langle v,v\rangle=0$ and $v\neq 0$. If this is what the null means, then what does it mean as to the convergence of the outer/inner null normal? Is there something that can be said to converge in the sense of (mathematical) limits?

Can anyone help me out? Thank you.

Edit. I hope it adds more information. According to Wikipedia, a null hypersurface is a hypersurface on which the normal vector takes on a null value. But I'm not sure if the surface $S$ in question is a null hypersurface because Christodoulou and Yau did not explicitly define $S$ in their remarks.

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  • $\begingroup$ Can you give the reference to eq. 1? $\endgroup$
    – VaibhavK
    Commented Dec 7, 2023 at 4:38
  • $\begingroup$ @VaibhavK Christodoulou, D., Yau, S.T.: Some remarks on quasi-local mass. In: Contemporary Mathematics 71, Mathematics and General Relativity, J. Isenberg, ed. Providence RI: Amer. Math. Soc., 1988, pp. 9–14 $\endgroup$
    – Boar
    Commented Dec 7, 2023 at 7:01
  • $\begingroup$ @VaibhavK Here's another reference (see its 57th page): Szabados, L.B. Quasi-Local Energy-Momentum and Angular Momentum in General Relativity. Living Rev. Relativ. 12, 4 (2009). doi.org/10.12942/lrr-2009-4 $\endgroup$
    – Boar
    Commented Dec 7, 2023 at 7:04
  • $\begingroup$ Is a part of your question that you don't know what the convergence of null normals is? The surface for which you are defining Hawking mass is always a spacelike codimension-2 surface. I think the null aspects pop in when you want to, say, compute the Bondi energy at null infinity, in which case you take the limits of the Hawking mass along a null foliation. I am confused by what you mean by ```... the more relativistic one" -- aren't both the same geometric quantities? The calculation of the mean curvature $H$ is via the computations of the trace of the congruence expansions. $\endgroup$
    – VaibhavK
    Commented Dec 8, 2023 at 7:03
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    $\begingroup$ I am adding an answer to point this out more explicitly. $\endgroup$
    – VaibhavK
    Commented Dec 8, 2023 at 7:03

2 Answers 2

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Specifically, I don't know what the phrase convergence of the outer/inner null normal means. What on earth is this all about? How do I get to know that? Seriously, I don't see how a purely geometrical definition like (2) can be linked to the relativistic one, (1).

This has to do with the null expansions. We will employ the general orthogonal basis $\chi $ and $\bar{\chi }$ and their corresponding traces for the ingoing and outgoing expansions. In this way, one would have a pair of surface orthogonal congruences for some codimension-2 spacelike surface $\sigma $. The terms $\rho $ and $\mu $ essentially correspond to $\text{tr } \chi $ and $\text{tr }\bar{\chi }$ respectively.

The definition is included in Some remarks on the quasi-local mass by Christodoulou and Yau. I call it strange because I used to learn about Hawking mass by defining... with $H$ as the mean curvature of $S$. This latter definition requires only knowledge of Riemannian geometry ...

Well, I fail to see how you compute mean curvature without understanding the role of the null expansions; both are essentially the same things in a different approach. In general, one would mention the expansions in order to make sense of the embedding of the surface in a larger family of surfaces or a foliation. For instance, the limit of Hawking mass of a marginally trapped surface drawn to say $\mathcal{I}^{+}$ in Schwarzschild geometry along a future null foliation gives the Bondi energy $E_{B}^{+}$. On the other hand, the limit of Hawking mass of a marginally trapped surface drawn to $i^{0}$ along a spacelike foliation gives the ADM mass. Of course, Hawking mass in general requires asymptotically round foliations to work, but you get the picture. What you are mentioning essentially relates to the mass aspect function associated to a foliation. Both are computing the same thing in two different settings, one where you require the mean curvature, which is the more generally used one, against the other, which explicitly requires you to compute the expansions for a larger reason of doing things on a foliation.

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According to my not so good friend Bard, your question regarding the equivalence of different definitions for the Hawking quasi-local mass highlights an important and subtle topic in general relativity. It's understandable that the presence of unfamiliar terms like "convergence of the outer/inner null normal" can be confusing.

Here's a breakdown to help you understand the connection between the two definitions:

Definition 1: Relativistic Definition (Area and Mean Curvature)

$m(S) = √(Area(S) / 16π) (1 - 1 / 16π ∫_S H^2)$

This definition relies on the concept of mean curvature (H) of the surface S, which is a purely geometric quantity. It is calculated using the intrinsic geometry of the surface and doesn't require the full machinery of general relativity.

Definition 2: Geometric Definition (Convergence of Null Normals)

$m(S) = √(Area(S) / 16π) (1 - 1 / 4π ∫_S (ρ - μ))$

This definition introduces the concept of "convergence of the outer/inner null normal". These terms are related to the extrinsic geometry of the surface, specifically how it curves within the larger spacetime.

Key Concepts:

Null vector: A vector in spacetime whose squared norm is zero (⟨v, v⟩ = 0). Null hypersurface: A surface in spacetime where all normal vectors are null. Outer/inner null normal: For a timelike surface S, there exist two null vectors normal to S pointing outwards and inwards, respectively. These are called the outer and inner null normals. Convergence: This refers to the behavior of the null normals as they approach the boundary of the surface S. In this context, it specifically means how the null vectors "diverge" or "focus" as they approach the boundary. Connecting the Definitions:

The two definitions are not simply different ways to calculate the same thing. They represent different approaches to defining the Hawking mass. Definition 1 focuses on the intrinsic geometry of the surface, while definition 2 uses the extrinsic geometry and relates the mass to the behavior of null normals.

The equivalence of these definitions can be proven using a combination of tools from differential geometry and general relativity. This typically involves analyzing the null geodesics associated with the surface S and their relationship to the null normals.

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