The Hawking energy or Hawking mass is one of the possible definitions of

mass in general relativity The concept of mass in general relativity (GR) is more subtle to define than the concept of mass in special relativity. In fact, general relativity does not offer a single definition of the term mass, but offers several different definitions that ...

. It is a measure of the bending of ingoing and outgoing rays of

light Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 tera ...

that are

orthogonal In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...

to a 2-

sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

surrounding the region of space whose mass is to be defined.

Definition

Let

(\mathcal^3, g_)

be a 3-dimensional sub-manifold of a relativistic spacetime, and let

\Sigma \subset \mathcal^3

be a closed 2-surface. Then the Hawking mass

m_H(\Sigma)

\Sigma

is defined to be :

m_H(\Sigma) := \sqrt\left( 1 - \frac\int_\Sigma H^2 da \right),

where

H

is the

mean curvature In mathematics, the mean curvature H of a surface S is an ''extrinsic'' measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space. The ...

\Sigma

Properties

In the

Schwarzschild metric In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assumpti ...

, the Hawking mass of any sphere

S_r

about the central mass is equal to the value

m

of the central mass. A result of Geroch implies that Hawking mass satisfies an important monotonicity condition. Namely, if

\mathcal^3

has nonnegative scalar curvature, then the Hawking mass of

\Sigma

is non-decreasing as the surface

\Sigma

flows outward at a speed equal to the inverse of the mean curvature. In particular, if

\Sigma_t

is a family of connected surfaces evolving according to :

\frac = \frac\nu(x),

where

H

is the mean curvature of

\Sigma_t

and

\nu

is the unit vector opposite of the mean curvature direction, then :

\fracm_H(\Sigma_t) \geq 0.

Said otherwise, Hawking mass is increasing for the

inverse mean curvature flow In the mathematical fields of differential geometry and geometric analysis, inverse mean curvature flow (IMCF) is a geometric flow of submanifolds of a Riemannian or pseudo-Riemannian manifold. It has been used to prove a certain case of the R ...

. Hawking mass is not necessarily positive. However, it is asymptotic to the ADM or the Bondi mass, depending on whether the surface is asymptotic to spatial infinity or null infinity.Section 2 of

Definition

Properties

See also

References

Further reading