Mass in General Relativity - Questions, Answers, and Simple Examples of Mass in General Relativity

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 general 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.

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?

The question is somewhat ambiguous as stated. Interpreting the question as a question about the Komar mass, the answers to the questions are yes, and no, respectively. Because the pressure vessel generates a static space-time, the Komar mass exists, and can be found by 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 Komar mass of the fluid increases by more than E/c2.

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

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.

One might also ask about the answers to this question if one assumed that one were asking about the mass as it is defined in special relativity rather than the Komar mass. If one assumes that the space-time is nearly Minkowskian, the special relativistic mass exists. In this case, the answer to the first question is still yes, but the second question cannot be answered without even more data. Because the system consisting only of the gas is not an isolated system, its mass is not invariant, and thus depends on the choice of observational frame. A specific choice of observational frame (such as the rest frame of the system) must be specified in order to answer the second question. If the rest frame of the object is chosen, and special relativistic mass rather than Komar mass is assumed, the answer to the second question becomes yes. This problem illustrates some of the difficulties one faces when talking about the mass of non-isolated systems.

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 because the vessel (or the pieces of the vessel after it explodes) form an isolated system. This question does illustrate one of the limitations of the Komar formula - the Komar mass is defined only for 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, the Newtonian limit formula, or the special relativistic formula.

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 about the "charge of the universe": the charge is trivially zero."