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[[General relativity]] does not offer a single definition of the term ''[[mass]]'', but offers several different definitions that are applicable under different circumstances. Under some circumstances, the mass of a system in general relativity may not even be defined. The subtlety of this definition stems from the fact that the energy and momentum in a gravitational field cannot be unambiguously localized.<ref name="mtw">{{cite book |last1=Misner |first1=Charles W. |last2=Thorne |first2=Kip S. |last3=Wheeler |first3=John A. |title=Gravitation |date=1973 |publisher=W. H. Freeman |location=New York |isbn=0-7167-0334-3}}</ref>{{rp|Ch. 20}} As such, rigorous definitions of ''mass'' in general relativity cannot be not local as they are in [[classical mechanics]] or [[special relativity]], but must make reference to the asymptotic nature of spacetime. A well-defined notion of mass exists for asymptotically flat spacetimes and for asymptotically [[anti-de Sitter space]]. However, these definitions must be used with care in other settings.

== Defining mass in general relativity: concepts and obstacles ==
In special relativity, the rest mass of a particle can be defined unambiguously in terms of its energy and momentum (see [[Mass in special relativity]]). Generalizing the notion of the energy and momentum to general relativity, however, is subtle. The main reason for this is that that gravitational field itself contributes to the energy and momentum. However, the "gravitational field energy" is not a part of the energy–momentum tensor; instead, what might be identified as the contribution of the gravitational field to a total energy is part of the Einstein tensor on the other side of Einstein's equation (and, as such, a consequence of these equations' non-linearity).  While  in certain situations it is possible to rewrite the equations so that part of the "gravitational energy" now stands alongside the other source terms in the form of the [[stress–energy–momentum pseudotensor]], this separation is not true for all observers, and there is no general definition for obtaining it.<ref>Cf. {{Harvnb|Misner|Thorne|Wheeler|1973|loc=§20.4}}</ref>

The total mass of a system{{snd}}a concept easily defined in classical mechanics{{snd}} may be defined for spacetimes which are [[asymptotically flat]] (roughly speaking, which represent some isolated gravitating system in otherwise empty and gravity-free infinite space) using the [[ADM formalism|ADM]] 3+1 split. As in the usual [[Hamiltonian mechanics|Hamiltonian formalism]], the time direction used in that split has an associated energy, which can be integrated up to yield a global quantity known as the [[ADM mass]] (or, equivalently, ADM energy).<ref>{{Harvnb|Arnowitt|Deser|Misner|1962}}.</ref> Alternatively, there is a possibility to define mass for a [[stationary spacetime|spacetime that is stationary]], in other words, one that has a time-like [[Killing vector field]] (which, as a generating field for time, is canonically conjugate to energy); the result is the so-called [[Komar mass]]<ref>Cf. {{Harvnb|Komar|1959}}</ref><ref name="Wald.11.2">For a pedagogical introduction, see {{Harvnb|Wald|1984|loc=sec. 11.2}}.</ref>  Although defined in a different way, it can be shown to be equivalent to the ADM mass for stationary spacetimes.<ref>This is shown in {{Harvnb|Ashtekar|Magnon-Ashtekar|1979}}.</ref> The Komar integral definition can also be generalized to non-stationary fields for which there is at least an asymptotic [[time translation symmetry]]; imposing a certain gauge condition, one can define the [[#ADM and Bondi masses in asymptotically flat space-times|Bondi energy]] at null infinity. In a way, the [[ADM energy]] measures all of the energy contained in spacetime, while the Bondi energy excludes those parts carried off by gravitational waves to infinity.<ref name="Wald.11.2"/> Effort has been expended on proving positivity theorems for the masses just defined, not least because positivity, or at least the existence of a lower limit, has a bearing on the more fundamental question of boundedness from below: if there were no lower limit to the energy, then no isolated system would be absolutely stable; there would always be the possibility of a decay to a state of even lower total energy. Several kinds of proofs that both the ADM mass and the Bondi mass are indeed positive exist; in particular, this means that [[Minkowski space]] (for which both are zero) is indeed stable.<ref>See the various references given on p. 295 of {{Harvnb|Wald|1984}}.</ref> While the focus here has been on energy, analogue definitions for global momentum exist; given a field of angular Killing vectors and following the Komar technique, one can also define global angular momentum.<ref>E.g. {{Harvnb|Townsend|1997|loc=ch. 5}}.</ref>

== Quasi-local quantities ==
The disadvantage of all the definitions mentioned so far is that they are defined only at (null or spatial) infinity; since the 1970s, physicists and mathematicians have worked on the more ambitious endeavor of defining suitable ''quasi-local'' quantities, such as the mass of an isolated system defined using only quantities defined within a finite region of space containing that system.  However, while there is a variety of proposed definitions such as the [[Hawking energy]], the [[Geroch energy]] or [[Roger Penrose|Penrose's]] quasi-local energy–momentum based on [[Twistor theory|twistor]] methods, the field is still in flux.  Eventually, the hope is to use a suitable defined quasi-local mass to give a more precise formulation of the [[hoop conjecture]], prove the so-called [[Penrose inequality]] for black holes (relating the black hole's mass to the horizon area) and find a quasi-local version of the laws of black hole mechanics.<ref>See the review article {{Harvnb|Szabados|2004}}.</ref>

== Types of mass in general relativity ==

=== Komar mass in stationary spacetimes ===
{{Main | Komar mass}}

A non-technical definition of a [[stationary spacetime]] is a spacetime where none of the metric coefficients <math>g_{\mu\nu}\,</math> are functions of time.  The [[Schwarzschild metric]] of a [[black hole]] and the [[Kerr metric]] of a [[rotating black hole]] are common examples of stationary spacetimes.

By definition, a stationary spacetime exhibits [[time translation symmetry]].  This is technically called a time-like [[Killing vector]].  Because the system has a time translation symmetry, Noether's theorem guarantees that it has a conserved energy.  Because a stationary system also has a well defined rest frame in which its momentum can be considered to be zero, defining the energy of the system also defines its mass.  In general relativity, this mass is called the Komar mass of the system.  Komar mass can only be defined for stationary systems.

Komar mass can also be defined by a flux integral.  This is similar to the way that [[Gauss's law]] defines the charge enclosed by a surface as the normal electric force multiplied by the area.  The flux integral used to define Komar mass is slightly different from that used to define the electric field, however{{snd}} the normal force is not the actual force, but the "force at infinity".  See the [[Komar mass|main article]] for more detail.

Of the two definitions, the description of Komar mass in terms of a time translation symmetry provides the deepest insight.

=== ADM and Bondi masses in asymptotically flat space-times ===
<!-- There is a link to this section from General relativity, section "Consequences of Einstein's theory", subsection "Global and quasi-local quantities".  Please do not change the title of this section or the link will be broken. -->

If a system containing gravitational sources is surrounded by an infinite vacuum region, the geometry of the space-time will tend to approach the flat [[Minkowski space|Minkowski geometry]] of special relativity at infinity. Such space-times are known as "asymptotically flat" space-times.

For systems in which space-time is [[asymptotically flat]], the [[ADM mass|ADM]] and Bondi energy, momentum, and mass can be defined.  In terms of Noether's theorem, the ADM energy, momentum, and mass are defined by the asymptotic symmetries at [[conformal infinity|spatial infinity]], and the Bondi energy, momentum, and mass are defined by the asymptotic symmetries at [[conformal infinity|null infinity]].  Note that mass is computed as the length of the energy–momentum [[four-vector]], which can be thought of as the energy and momentum of the system "at infinity".

The ADM energy is defined through the following flux integral at infinity.<ref name="mtw" /> If a spacetime is asymptotically flat this means that near "infinity" the metric tends to that of flat space. The asymptotic deviations of the metric away from flat space can be parametrized by 
:<math>g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}
</math>
where <math>\eta_{\mu \nu}</math> is the flat space metric.
The ADM energy is then given by an integral over a surface, <math>S</math> at infinity
:<math>
P^0 = {1 \over 16 \pi G} \int \left(\partial^k h_{j k} - \partial^j h_{k k} \right) d^2 S_j,
</math>
where <math> S_j </math> is the outward-pointing normal to <math>S</math>. The [[Einstein summation]] convention is assumed for repeated indices but the sum over k and j only runs over the spatial directions. The use of ordinary derivatives instead of covariant derivatives in the formula above is justified because of the assumption that the asymptotic geometry is flat.

Some intuition for the formula above can be obtained as follows. Imagine that that we take the surface, S, to be a spherical surface so that the normal points radially outwards. At large distances from the source of the energy, r, the tensor <math>h_{i j}</math> is expected to fall off as <math>r^{-1}</math> and the derivative with respect to r converts this into <math>r^{-2}</math>. The area of the sphere at large radius also grows precisely as <math>r^2</math> and therefore one obtains a finite value for the energy.

It is also possible to obtain expressions for the momentum in asymptotically flat spacetime. To obtain such an expression one defines
:<math>
H^{\mu \alpha \nu \beta} = -\bar{h}^{\mu \nu} \eta^{\alpha \beta} - \eta^{\mu \nu} \bar{h}^{\alpha \beta} + \bar{h}^{\alpha \nu} \eta^{\mu \beta} + \bar{h}^{\mu \beta} \eta^{\alpha \nu}
</math>
where
:<math>
\bar{h}_{\mu \nu} = h_{\mu \nu} - {1 \over 2} \eta_{\mu \nu} h^{\alpha}_{\alpha}
</math>
Then the momentum is obtained by a flux integral in the asymptotically flat region
:<math>
P^{\mu} = {1 \over 16 \pi G}\int \partial_{\alpha} H^{\mu \alpha 0 j} d^2 S_j
</math>
Note that the expression for <math>P^0</math> obtained from the formula above coincides with the expression for the ADM energy given above as can easily be checked using the explicit expression for H.

=== The Newtonian limit for nearly flat space-times ===
In the Newtonian limit, for quasi-static systems in nearly flat space-times, one can approximate the total energy of the system by adding together the non-gravitational components of the energy of the system and then subtracting the ''Newtonian'' gravitational binding energy.

Translating the above statement into the language of general relativity, we say that a system in nearly flat space-time has a total non-gravitational energy E and momentum P given by:

:<math>E = \int_v T_{00} dV \qquad P^i = \int_V T_{0i} dV </math>

When the components of the momentum vector of the system are zero, i.e. P<sup>i</sup> = 0, the approximate mass of the system is just (E+E<sub>binding</sub>)/c<sup>2</sup>, E<sub>binding</sub> being a negative number representing the Newtonian gravitational self-binding energy.

Hence when one assumes that the system is quasi-static, one assumes that there is no significant energy present in the  form of "gravitational waves".  When one assumes that the system is in "nearly-flat" space-time, one assumes that  the metric coefficients are essentially [[Minkowskian]] within acceptable experimental error.

The formulas for the total energy and momentum can be seen to arise naturally in this limit as follows.<ref name="mtw" /> In the linearized limit, the equations of general relativity can be written in the form
:<math>
\partial_{\alpha} \partial_{\beta} H^{\mu \alpha \nu \beta} = 16 \pi G T^{\mu \nu}
</math>
In this limit, the total energy-momentum of the system is simply given by integrating the stress-tensor on a spacelike slice.
:<math>
P^{\mu} = \int T^{\mu 0} d^3 x
</math>
But using the equations of motion, one can also write this as
:<math>
P^{\mu} = {1 \over 16 \pi G} \int \partial_{\alpha} \partial_{\beta} H^{\mu \alpha 0 \beta} d^3 x = {1 \over 16 \pi G} \int \partial_{\alpha} \partial_j H^{\mu \alpha 0 j} d^3 x
</math>
where the sum over j runs only over the spatial directions and the second equality uses the fact that <math>H^{\mu \alpha \nu \beta}</math> is anti-symmetric in <math>\nu</math> and <math> \beta </math>.
Finally, one uses the [[Gauss law]] to convert the integral of a divergence over the spatial slice into an integral over a Gaussian sphere 
:<math>
{1 \over 16 \pi G} \int \partial_{\alpha} \partial_j H^{\mu \alpha 0 j} d^3 x = {1 \over 16 \pi G} \int \partial_{\alpha} H^{\mu \alpha 0 j} d^2 S_j
</math> 
which coincides precisely with the formula for the total momentum given above.

== History ==
{{uncited section|date=April 2023}}
In 1918, [[David Hilbert]] wrote about the difficulty in assigning an energy to a "field" and "the failure of the energy theorem" in a correspondence with [[Felix Klein|Klein]].  In this letter, Hilbert conjectured that this failure is a characteristic feature of the general theory, and that instead of "proper energy theorems" one had 'improper energy theorems'.

This conjecture was soon proved to be correct by one of Hilbert's close associates, [[Emmy Noether]].  [[Noether's theorem]] applies to any system which can be described by an [[Action (physics)|action principle]]. Noether's theorem associates conserved energies with time-translation symmetries.   When the time-translation symmetry is a finite parameter [[lie group|continuous group]], such as the [[Poincaré group]], Noether's theorem defines a scalar conserved energy for the system in question.  However, when the symmetry is an infinite parameter continuous group, the existence of a conserved energy is not guaranteed.  In a similar manner, Noether's theorem associates conserved momenta with space-translations, when the [[symmetry group]] of the translations is finite-dimensional.  Because General Relativity is a [[diffeomorphism]] invariant theory, it has an infinite continuous group of symmetries rather than a finite-parameter group of symmetries, and hence has the wrong group structure to guarantee a conserved energy.  Noether's theorem has been influential in inspiring and unifying various ideas of mass, system energy, and system momentum in General Relativity.

As an example of the application of Noether's theorem is the example of stationary space-times and their associated Komar mass.(Komar 1959).  While general space-times lack a finite-parameter time-translation symmetry, stationary space-times have such a symmetry, known as a [[Killing vector]].  Noether's theorem proves that such stationary space-times must have an associated conserved energy.  This conserved energy defines a conserved mass, the Komar mass.

ADM mass was introduced (Arnowitt et al., 1960) from an initial-value formulation of general relativity.  It was later reformulated in terms of the group of asymptotic symmetries at spatial infinity, the SPI group,  by various authors. (Held, 1980).  This reformulation did much to clarify the theory, including explaining why ADM momentum and ADM energy transforms as a 4-vector (Held, 1980).  Note that the SPI group is actually infinite-dimensional.  The existence of conserved quantities is because the SPI group of "super-translations" has a preferred 4-parameter subgroup of "pure" translations, which, by Noether's theorem, generates a conserved 4-parameter energy–momentum.  The norm of this 4-parameter energy–momentum is the ADM mass.

The Bondi mass was introduced (Bondi, 1962) in a paper that studied the loss of mass of physical systems via gravitational radiation.  The Bondi mass is also associated with a group of asymptotic symmetries, the [[Bondi–Metzner–Sachs group|BMS group]] at null infinity.  Like the SPI group at spatial infinity, the BMS group at null infinity is infinite-dimensional, and it also has a preferred 4-parameter subgroup of "pure" translations.

Another approach to the problem of energy in General Relativity is the use of [[pseudotensor]]s such as the [[Landau–Lifshitz pseudotensor]].(Landau and Lifshitz, 1962). Pseudotensors are not gauge invariant{{snd}} because of this, they only give consistent gauge-independent answers for the total energy when additional constraints (such as asymptotic flatness) are met.  The gauge dependence of pseudotensors also prevents any gauge-independent definition of the local energy density, as every different gauge choice results in a different local energy density.

== See also ==
* [[Mass in special relativity]]
* [[General relativity]]
* [[Conservation of energy]]
* [[Komar mass]]
* [[Hawking energy]]
* [[ADM mass]]
* [[Positive mass theorem]]

== Notes ==
{{reflist|2}}

== References ==
*{{cite journal | last1=Ashtekar | first1=Abhay | last2=Magnon-Ashtekar | first2=Anne | title=On conserved quantities in general relativity | journal=Journal of Mathematical Physics | publisher=AIP Publishing | volume=20 | issue=5 | year=1979 | issn=0022-2488 | doi=10.1063/1.524151 | pages=793–800| bibcode=1979JMP....20..793A }}
*{{Cite journal
 | last=Komar
 | first=Arthur
 | title=Covariant Conservation Laws in General Relativity
 | doi=10.1103/PhysRev.113.934
 | journal=Phys. Rev.
 | volume=113
 | issue=3
 | pages=934–936
 | date=1959
 |bibcode = 1959PhRv..113..934K }}
*{{cite journal | last1=Arnowitt | first1=R. | last2=Deser | first2=S. | last3=Misner | first3=C. W. | title=Canonical Variables for General Relativity | journal=Physical Review | publisher=American Physical Society (APS) | volume=117 | issue=6 | date=1960-03-15 | issn=0031-899X | doi=10.1103/physrev.117.1595 | pages=1595–1602| bibcode=1960PhRv..117.1595A | s2cid=120715041 | url=https://authors.library.caltech.edu/88200/1/PhysRev.117.1595.pdf }}
*{{Cite book | first1=Richard | last1=Arnowitt | first2=Stanley | last2=Deser |first3=Charles W. |last3=Misner | year=1962 | chapter=The dynamics of general relativity | editor-last=Witten | editor-first=L. | title=Gravitation: An Introduction to Current Research | publisher=Wiley | pages=227–265}}
*{{cite journal | title=Gravitational waves in general relativity, VII. Waves from axi-symmetric isolated system | journal=Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences | publisher=The Royal Society | volume=269 | issue=1336 | date=1962-08-21 | issn=2053-9169 | doi=10.1098/rspa.1962.0161 | pages=21–52| bibcode=1962RSPSA.269...21B | last1=Bondi | first1=H. | last2=Van Der Burg | first2=M. G. J. | last3=Metzner | first3=A. W. K. | s2cid=120125096 }}
*{{cite book | title = General Relativity and Gravitation, One Hundred Years After the Birth of Einstein, Vol 2 | last = Held | date = 1980 | publisher = Plenum Press | isbn=978-0-306-40266-1}}
*{{Cite journal
  | last=Szabados
  | first=László B.
  | title=Quasi-Local Energy-Momentum and Angular Momentum in GR
  | journal=Living Rev. Relativ.
  | volume=7
  | issue=1
  | pages=4
  | date=2004
  | doi=10.12942/lrr-2004-4
  | doi-access=free
 | pmid=28179865
  | pmc=5255888
  | bibcode=2004LRR.....7....4S
  }}
*{{Cite arXiv|last=Townsend|first=P. K.|year=1997|title=Black Holes (Lecture notes)|eprint=gr-qc/9707012}}
*{{Cite book|last=Wald|first=Robert M.|author-link=Robert M. Wald|year=1984|title=[[General Relativity (book)|General Relativity]]|location=Chicago|publisher=University of Chicago Press|isbn=0-226-87033-2}}
*{{cite journal | title= Covariant thermodynamics of an object with finite volume | last=Nakumura |first=Tadas K.| date=2005 | arxiv = physics/0505004 | doi=10.1016/j.physleta.2005.11.070 | volume=352 | issue=3 | journal=Physics Letters A | pages=175–177| bibcode=2006PhLA..352..175N | s2cid=14211373 }}
*{{cite journal | title = Kinetic Energy and the Equivalence Principle | last=Carlip|first=S. | date=1999 | arxiv = gr-qc/9909014 | doi=10.1119/1.18885 | volume=66 | issue=5| pages=409–413| journal=American Journal of Physics | bibcode=1998AmJPh..66..409C | citeseerx=10.1.1.340.3195| s2cid=119052544}}
*[http://math.ucr.edu/home/baez/physics/Relativity/BlackHoles/black_fast.html "If you go too fast, do you become a black hole?"] Updated by Don Koks 2008. Original by Philip Gibbs 1996. ''The Original Usenet Physics FAQ''
*{{cite journal | title = Measuring the active gravitational mass of a moving object | journal = American Journal of Physics | doi=10.1119/1.14280 | last = Olson | first = D.W. | author2 = Guarino, R. C.| date = 1985 | volume = 53 | issue = 7 | page = 661 |bibcode = 1985AmJPh..53..661O }}

== External links ==
* [http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html "Is energy conserved in General Relativity?]

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