Mass in general relativity
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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 are applicable under different circumstances. Under some circumstances, the mass of a system in general relativity may not even be defined.
The reason for this subtlety is that the energy and momentum in the gravitational field cannot be unambiguously localized. (See Chapter 20 of [1].) So, rigorous definitions of the mass in general relativity are not local, as in classical mechanics or special relativity, but make reference to the asymptotic nature of the spacetime. A well defined notion of the 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 as described in the article on 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.[2]
How, then, does one define a concept as a system's total mass – which is easily defined in classical mechanics? As it turns out, at least for spacetimes which are
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
Types of mass in general relativity
Komar mass in stationary spacetimes
A non-technical definition of a stationary spacetime is a spacetime where none of the metric coefficients 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
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 – the normal force is not the actual force, but the "force at infinity". See the 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
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 geometry of special relativity at infinity. Such space-times are known as "asymptotically flat" space-times.
For systems in which space-time is
The ADM energy is defined through the following flux integral at infinity.[1] 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
where is the flat space metric. The ADM energy is then given by an integral over a surface, at infinity
where is the outward-pointing normal to . The
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 is expected to fall off as and the derivative with respect to r converts this into The area of the sphere at large radius also grows precisely as 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
where
Then the momentum is obtained by a flux integral in the asymptotically flat region
Note that the expression for 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:
When the components of the momentum vector of the system are zero, i.e. Pi = 0, the approximate mass of the system is just (E+Ebinding)/c2, Ebinding 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
The formulas for the total energy and momentum can be seen to arise naturally in this limit as follows.[1] In the linearized limit, the equations of general relativity can be written in the form
In this limit, the total energy-momentum of the system is simply given by integrating the stress-tensor on a spacelike slice.
But using the equations of motion, one can also write this as
where the sum over j runs only over the spatial directions and the second equality uses the fact that is anti-symmetric in and . Finally, one uses the
which coincides precisely with the formula for the total momentum given above.
History
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 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 principle. Noether's theorem associates conserved energies with time-translation symmetries. When the time-translation symmetry is a finite parameter 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
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 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
See also
- Mass in special relativity
- General relativity
- Conservation of energy
- Komar mass
- Hawking energy
- ADM mass
- Positive mass theorem
Notes
- ^ ISBN 0-7167-0334-3.
- ^ Cf. Misner, Thorne & Wheeler 1973, §20.4
- ^ Arnowitt, Deser & Misner 1962.
- ^ Cf. Komar 1959
- ^ a b For a pedagogical introduction, see Wald 1984, sec. 11.2.
- ^ This is shown in Ashtekar & Magnon-Ashtekar 1979.
- ^ See the various references given on p. 295 of Wald 1984.
- ^ E.g. Townsend 1997, ch. 5.
- ^ See the review article Szabados 2004.
References
- Ashtekar, Abhay; Magnon-Ashtekar, Anne (1979). "On conserved quantities in general relativity". Journal of Mathematical Physics. 20 (5). AIP Publishing: 793–800. ISSN 0022-2488.
- Komar, Arthur (1959). "Covariant Conservation Laws in General Relativity". Phys. Rev. 113 (3): 934–936. .
- Arnowitt, R.; Deser, S.; Misner, C. W. (1960-03-15). "Canonical Variables for General Relativity" (PDF). Physical Review. 117 (6). American Physical Society (APS): 1595–1602. S2CID 120715041.
- Arnowitt, Richard; Deser, Stanley; Misner, Charles W. (1962). "The dynamics of general relativity". In Witten, L. (ed.). Gravitation: An Introduction to Current Research. Wiley. pp. 227–265.
- Bondi, H.; Van Der Burg, M. G. J.; Metzner, A. W. K. (1962-08-21). "Gravitational waves in general relativity, VII. Waves from axi-symmetric isolated system". Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. 269 (1336). The Royal Society: 21–52. S2CID 120125096.
- Held (1980). General Relativity and Gravitation, One Hundred Years After the Birth of Einstein, Vol 2. Plenum Press. ISBN 978-0-306-40266-1.
- Szabados, László B. (2004). "Quasi-Local Energy-Momentum and Angular Momentum in GR". Living Rev. Relativ. 7 (1): 4. PMID 28179865.
- Townsend, P. K. (1997). "Black Holes (Lecture notes)". arXiv:gr-qc/9707012.
- ISBN 0-226-87033-2.
- Nakumura, Tadas K. (2005). "Covariant thermodynamics of an object with finite volume". Physics Letters A. 352 (3): 175–177. S2CID 14211373.
- Carlip, S. (1999). "Kinetic Energy and the Equivalence Principle". American Journal of Physics. 66 (5): 409–413. S2CID 119052544.
- "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
- Olson, D.W.; Guarino, R. C. (1985). "Measuring the active gravitational mass of a moving object". American Journal of Physics. 53 (7): 661. doi:10.1119/1.14280.