Mass in general relativity

The concept of mass in general relativity (GR) is more complex than the concept of mass in special relativity. In fact, general relativity does not offer a single definition for the term mass, but offers several different definitions which are applicable under different circumstances. Under some circumstances, the mass of a system in general relativity may not even be defined.

Review of mass in special relativity

Main article: Mass in special relativity

In special relativity, the invariant mass (hereafter simply "mass") of an isolated system, can be defined in terms of the energy and momentum of the system by the relativistic energy-momentum equation. $m={\frac {\sqrt {E^{2}-\left(pc\right)^{2}}}{c^{2}}}$
Where:

$E$ is the total energy of the system

$p$ is the total momentum of the system

$c$ is the speed of light

Concisely, the mass of a system in special relativity is the norm of its energy-momentum four vector.

Obstacles in applying definitions from special relativity

In generalizing this definition to general relativity, one faces two main problems. The first problem is that it is not clear how to find the total energy and momentum of a system. In flat space-time, one can simply integrate, adding together the energy-momentum four-vectors of the components of the system to find the total energy-momentum four-vector of the entire system.

Unfortunately, this simple procedure does not directly generalize to general relativity, as the four-vectors themselves exist in different tangent spaces, and cannot be covariantly added together. The second problem that one faces in defining mass in general relativity is that in order to maintain energy as a conserved quantity, one must somehow account for the energy in the "gravitational field".

Unfortunately, energy conservation in general relativity turns out to be much less straightforward than it is in other theories of physics. In other classical theories, such as Newtonian gravity, electromagnetism, and hydrodynamics, it is possible to assign a definite value of energy density to fields. For instance, the energy density of an electric field E can be considered to be 1/2 ε₀ E².

This is not the case in general relativity. It turns out to be impossible in general to assign a definite location to "gravitational energy". (Misner et al, 1973 chapter 20 section 4).

The modern approach to the problem of energy conservation in general relativity is to avoid the concept of a "gravitational field" altogether, and to explain the conservation of energy as a consequence of time translation symmetry. Noether's theorem, which was developed specifically to address the problem of energy conservation in general relativity, defines a conserved energy whenever one has such a time translation symmetry.

Not all systems have the required time translation symmetry, however. There is no known definition of energy in general relativity for systems which lack the required symmetry.

Types of mass in general relativity

Komar mass in a stationary space-times

Main article: Komar mass

A non-technical definition of a stationary space-time is a space-time where none of the metric coefficients $g_{\mu \nu }\,$ 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 space-times.

By definition, a stationary space-time 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 law defines the charge enclosed by a surface as the normal 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 asymptotically flat, the 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 spatial infinity, and the Bondi energy, momentum, and mass are defined by the asymptotic symmetries at 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 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:

$E=\int _{v}T_{00}dV\qquad P^{i}=\int _{V}T_{0i}dV$

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+E_binding)/c², E_binding 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.

See also

Komar mass

ADM mass

Bondi mass

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'. [1].

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 Poincare 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 extremely 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 due to the fact that 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 pseudotensors such as the Landau-Lifshitz pseudotensor.(Lanadu and Lifshitz, 1962). Pseudotensors are not gauge invariant - 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 psuedotensors also prevents any gauge-independent definition of the local energy density, as every different gauge choice results in a different local energy density.

Questions, answers, and simple examples of mass in general relativity

In special relativity, the invariant mass of a single particle is always Lorentz invariant. Can the same thing be said for the mass of a system of particles in special relativity?

Surprisingly, the answer is no. A system must either be isolated, or have zero volume, in order for its mass to be Lorentz invariant. While the density of energy momentum, the stress-energy tensor is always Lorentz covariant, the same cannot be said for the total energy-momentum. (Nakamura, 2005). Non-covariance of the energy-momentum four-vector implies non-invariance of its length, the invariant mass.

What this means in simpler language is that one must use great caution when talking about the mass of a non-isolated system. A non-isolated system is constantly exchanging energy-momentum with its surroundings. Even when the net rate of exchange of energy-momentum with the environment is zero, differences in the definition of simultaneity cause the total amount of energy-momentum contained within the system at a given instant of time to depend on the definition of simultaneity that is adopted by the observer. This causes the invariant mass of a non-isolated system to depend on one's choice of coordinates even in special relativity. Only an isolated system has a coordinate-independent mass.

Can an object move so fast that it turns into a black hole?

No. An object that is not a black hole in its rest frame will not be a black hole in any other frame. One of the characteristics of a black hole is that a black hole has an event horizon, which light cannot escape. If light can escape from an object to infinity in the object's rest frame, it can also escape to infinity in a frame in which the object is moving. The path that the light takes will be aberrated by the motion of the object, but the light will still escape to infinity[2].

If two objects have the same mass, and we heat one of them up from an external source, does the heated object gain mass? If we put both objects on a sensitive enough balance, would the heated object weigh more than the unheated object? Would the heated object have a stronger gravitational field than the unheated object?

The answer to all of the above questions is yes. The hot object has more energy, so it weighs more and has a higher mass than the cold object. It will also have a higher gravitational field to go along with its higher mass, by the equivalence principle. (Carlip 1999)

Imagine that we have a solid pressure vessel enclosing an ideal gas. We heat the gas up with an external source of energy, adding an amount of energy E to the system. Does the mass of our system increase by E/c²? Does the mass of the gas increase by E/c²?

Yes, and no, respectively. Because the pressure vessel generates a static space-time, one can utilize the concept of Komar mass to find its mass, treating the ideal gas as an ideal fluid. Using the formula for the Komar mass of a small system in a nearly Minkowskian space-time, one finds that the mass of the system in geometrized units is equal to E + ∫ 3 P dV, where E is the total energy of the system, and P is the pressure.

The integral ∫ P dV over the entire volume of the system is equal to zero, however. The contribution of the positive pressure in the fluid is exactly canceled out by the contribution of the negative pressure (tension) in the shell. This cancellation is not accidental, it is a consequence of the relativistic virial theorem (Carlip 1999).

If we restrict our region of integration to the fluid itself, however, the integral is not zero and the pressure contributes to the mass. Because the integral of the pressure is positive, we find that the mass of the fluid increases by more than E/c². Since the fluid is not an isolated system, talking about its mass may be misleading unless great care is taken. This is an example of a non-isolated system with a finite volume. Thus, as explained earlier, the mass of this system is not invariant, and depends on the choice of observational frame. The Komar formula calculates the mass of the gas in its rest frame.

The significance of the pressure terms in the Komar formula can best be understood by a thought experiment. If we assume a spherical pressure vessel, the pressure vessel itself will not contribute to the gravitational acceleration measured by an accelerometer inside the shell. The Komar mass formula tells us that the surface acceleration we measure just inside the pressure vessel, at the outer edge of the hot gas will be equal to $G\left(E+3PV\right)/r^{2}c^{2}$

where E is the total energy (including rest energy) of the hot gas

G is Newton's Gravitational constant

P is the pressure of the hot gas

V is the volume of the pressure vessel.

This surface acceleration will be higher than expected because of the pressure terms. In a fully relativistic gas, (this includes a "box of light" as a special case), the contribution of the pressure term 3 P V will be equal to the energy term E, and the acceleration at the surface will be doubled from the value for a non-relativistic gas.

The only difference between the "hot" and "cold" systems in our last question is due to the motion of the particles in the gas inside the pressure vessel. Doesn't this imply that a moving particle has "more gravity" than a stationary particle?

This remark is probably true in essence, but it is difficult to quantify.

Unfortunately, it is not clear how to measure the "gravitational field" of a single relativistically moving object. It is clear that it is possible to view gravity as a force when one has a stationary metric - but the metric associated with a moving mass is not stationary.

While definitional and measurement issues constrain our ability to quantify the gravitational field of a moving mass, one can measure and quantify the effect of motion on tidal gravitational forces. When one does so, one finds that the tidal gravity of a moving mass is not spherically symmetrical - it is stronger in some directions than others. One can also say that, averaged over all directions, the tidal gravity increases when an object moves.

Some authors have used the total velocity imparted by a "flyby" rather than tidal forces to gain an indirect measure of the increase in gravitational "effective mass" of relativistically moving objects (Olson & Guarino 1985)

While there is unfortunately no single definitive way to interpret the space-time curvature caused by a moving mass as a Newtonian force, one can definitely say that the motion of the molecules in a hot object increases the mass of that object.

Note that in General Relativity, gravity is caused not by mass, but by the stress-energy tensor. Thus, saying that a moving particle has "more gravity" does not imply that the particle has "more mass". It only implies that the moving particle has "more energy".

Suppose the pressure vessel in our previous question fails, and the system explodes - does its mass change?

The mass of the system doesn't change. This illustrates one of the limitations of the Komar formula - it only applies to stationary systems. If one applies the Komar formula to this non-static non-stationary system, one gets the incorrect result that the mass of the system changes. The pressure and density of the gas remains constant for a short time after the failure, while the tension in the pressure vessel disappears immediately when the pressure vessel fails. One cannot correctly apply the Komar formula in this case, however - one needs to apply a different formula, such as the ADM mass formula, or the Newtonian limit formula.

Does the Komar mass formula imply that the tension in a rotating disk reduces its mass?

One must be careful to use the correct definition of "tension" when applying the Komar mass formula. In mechanics and engineering, the stress-energy tensor is defined in the co-moving frame. In general relativity, one defines the stress-energy tensor in a spatial frame, thus one does not include convective terms in the stress-energy tensor [3]. This fundamental difference means that a rotating disk does not have the stresses associated with it that one would expect from the engineering usage of the term.

For example, consider the following stress energy tensor $T^{{\hat {a}}{\hat {b}}}$ in a cylindrical frame field: $\left({\hat {t}},{\hat {r}},{\hat {\theta }},{\hat {z}}\right)$

${\begin{pmatrix}E(r)&0&P(r)&0\\0&0&0&0\\P(r)&0&0&0\\0&0&0&0\end{pmatrix}}$

It represents a flow of circulating material with angular momentum (i.e. a rotating disk), which satisfies the continuity equations $\nabla _{a}T^{ab}=0$, but does not have any "pressure" or "stress" terms.

This is not the only stress-energy tensor possible for a rotating disk, but it illustrates the important difference between the engineering definition of stress and the definition used in General Relativity and why care must be used.

What is the mass of the universe? What is the mass of the observable universe? Does a closed universe have a mass?

None of the above questions have answers. We know the density of the universe (at least in our local area), but we can only speculate on the extent of the universe, making it impossible for us to give a definitive answer for the mass of the universe. We cannot answer the second question, either. Since the observable universe isn't asymptotically flat, nor is it stationary, and since it may not be an isolated system, none of our definitions of mass in General Relativity apply, and there is no way to calculate the mass of the observable universe. The answer to the third question is also no : the following quote from (Misner, et al, pg 457) explains why:

"There is no such thing as the energy (or angular momentum, or charge) of a closed universe, according to general relativity, and this for a simple reason. To weigh something one needs a platform on which to stand to do the weighing ...

"To determine the electric charge of a body, one surrounds it by a large sphere, evaluates the electric field normal to the surface at each point on this sphere, integrates over the sphere, and applies the theorem of Gauss. But within any closed model universe with the topology of a 3-sphere, a Gaussian 2-sphere that is expanded widely enough from one point finds itself collapsing to nothingness at the antipodal point. Also collapsed to nothingness is the attempt to acquire useful information bout the "charge of the universe": the charge is trivially zero."

References

• "E. Noether's Discovery of the Deep Connection Between Symmetries and Conservation Laws".

• Komar A (1959). "Covariant Conservation Laws in General Relativity". Phys. Rev. 113: 934–936.

• Arnowitt R, Deser S, and Misner C W, (1960) Phys. Rev. 117, 1695

• Bondi H, van de Burg M G J, and Metzner A W K, Proc. R. Soc. London Ser. A 269:21-52 Gravitational waves in General Relativity. VII. Waves from axi-symmetric isolated systems (1962)

• Landau L D and Lifshitz E M (1962) The Classical Theory of Fields

• Held (1980). General Relativity and Gravitation, One Hundred Years After the Birth of Einstein, Vol 2. Plenum Press. ISBN 0-306-40266-1.

• Wald, Robert M (1984). General Relativity. University of Chicago Press. ISBN 0-226-87033-2.

• Nakumura (2005). "Covariant thermodynamics of an object with finite volume".

• Carlip (1999). "Kinetic Energy and the Equivalence Principle".

• ""If you go too fast, do you turn into a black hole?"".

• Misner, Thorne, Wheeler (1973). Gravitation. W H Freeman and Company. ISBN 0-7167-0344-0.{{cite book}}: CS1 maint: multiple names: authors list (link)

• Olson, D.W. (1985). ""Measuring the active gravitational mass of a moving object"". American Journal of Physics. doi:10.1119/1.14280. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

External links

• "Is energy conserved in General Relativity?