Ibix said:
Mass is not additive in relativity, so it's possible to have two massless light pulses that, taken together, have mass. In principle, adding a massless photon to a black hole increases the hole's mass and gives it some momentum.
Energy conservation is a complicated topic in general relativity. Since the spacetime you are describing is (probably) asymptotically flat then there are (probably) some concepts of conservation of energy available, but I don't think they'll help much in considering the detailed behaviour of your structure.
A charged black hole has a different metric to an uncharged one, but this affects the paths of uncharged particles as well as charged ones. So this should not be read as implying that photons must be charged (you could think of it as the energy of the electric field being a source of gravity which, by the way, means you need an absurdly large charge for any measurable effect). The mass of uncharged particles does not appear in the resulting maths describing their orbits.
I wonder to what extent the confusion results from the existence of different definitions of the term "mass" in general relativity from other contexts. In the highlighted language, I would use the term "mass-energy" in this context where there is ambiguity about what definition of mass is involved, rather than the unqualified term "mass", even though using the term "mass" alone would be perfectly fine in circumstances with more context to add meaning to how the term is being used.
A narrow definition of "mass" can be limited to a portion of a single energy density element (T
00) of the sixteen elements of the 4x4
stress-energy tensor of GR, while the broadest could include all elements of the stress-energy tensor of GR (and also the gravitational self-interaction effects on the left hand side of
Einstein's field equations in the Einstein tensor conventionally denoted by the uppercase letter G).
Basically, it may be helpful and clear up confusion to be more explicit about the distinction between
relativistic mass and rest mass (and other definitions of mass). As explained at this Wikipedia link (paragraph breaks added for ease of reading in the Physics Forum interface):
Relativistic mass and rest mass are both traditional concepts in physics, but the relativistic mass corresponds to the total energy.
The relativistic mass is the mass of the system as it would be measured on a scale, but in some cases (such as the box above) this fact remains true only because the system on average must be at rest to be weighed (it must have zero net momentum, which is to say, the measurement is in its
center of momentum frame).
For example, if an electron in a
cyclotron is moving in circles with a relativistic velocity, the mass of the cyclotron+electron system is increased by the relativistic mass of the electron, not by the electron's rest mass.
But the same is also true of any closed system, such as an electron-and-box, if the electron bounces at high speed inside the box. It is only the lack of total momentum in the system (the system momenta sum to zero) which allows the kinetic energy of the electron to be "weighed".
If the electron is
stopped and weighed, or the scale were somehow sent after it, it would not be moving with respect to the scale, and again the relativistic and rest masses would be the same for the single electron (and would be smaller).
In general, relativistic and rest masses are equal only in systems which have no net momentum and the system center of mass is at rest; otherwise they may be different.
The invariant mass is proportional to the value of the total energy in one reference frame, the frame where the object as a whole is at rest (as defined below in terms of center of mass). This is why the invariant mass is the same as the rest mass for single particles.
However, the invariant mass also represents the measured mass when the
center of mass is at rest for systems of many particles. This special frame where this occurs is also called the
center of momentum frame, and is defined as the
inertial frame in which the
center of mass of the object is at rest (another way of stating this is that it is the frame in which the momenta of the system's parts add to zero). For compound objects (made of many smaller objects, some of which may be moving) and sets of unbound objects (some of which may also be moving), only the center of mass of the system is required to be at rest, for the object's relativistic mass to be equal to its rest mass.
A so-called
massless particle (such as a photon, or a theoretical graviton) moves at the speed of light in every frame of reference. In this case there is no transformation that will bring the particle to rest. The total energy of such particles becomes smaller and smaller in frames which move faster and faster in the same direction.
As such, they have no rest mass, because they can never be measured in a frame where they are at rest. This property of having no rest mass is what causes these particles to be termed "massless". However, even massless particles have a relativistic mass, which varies with their observed energy in various frames of reference.
This could be the main source of confusion, even if my assertion that the dynamics of a non-charged photon around an electromagnetically neutral black hole (i.e. a
Schwarzschild or
Kerr black hole) and an
electrically charged black hole (i.e. a
Reissner-Nordström or
Kerr-Newman black hole) is incorrect. I acknowledge that I'm not particularly deeply immersed in black hole physics.
At least some of the difference between charged black holes and electrically neutral black holes is the difference between
and
which is due to the
equivalence of mass and energy, which makes the
electric field energy also contribute to the total mass, and I would be thinking of black holes of equivalent mass as black holes that have equivalent mass-energy including electrical field energy.
Incidentally, electrically charged black holes in the real world are expected to have an extremely low charge to mass ratio (which is part of what makes them hard to distinguish from electrically neutral black holes observationally). As explained in this
discussion of the Kerr-Newman metric:
[O]ne does not expect that realistic
black holes have a significant
electric charge (they are expected to have a minuscule positive charge, but only because the proton has a much larger momentum than the electron, and is thus more likely to overcome electrostatic repulsion and be carried by momentum across the horizon).
In the limit of electromagnetic charge approaching zero, a charged black hole behaves like an electromagnetically neutral black hole, and all four of the archetypical black hole types are special cases of Kerr-Newman black holes.
No astronomy observations to date have been able to definitively state that any particular black hole is charged or not charged, one way or the other.
Our most precise black hole astronomy measurements of black holes are probably those that come from the heart of the Milky Way galaxy, and those measurements just aren't precise enough yet to distinguish between various slight differences in black hole theory (not only between the four archetypical black hole types but between quantum gravity and classical gravity variants) , although they are making great process on this front.
Arguably, dual messenger photon-gravitational wave observations of neutron star-black hole mergers (which are very new with the first direct observation of any gravitational wave occurring just a decade ago in 2015), and observations of black holes "eating" nearby stars, come close, but they too are insufficiently precise at this point to make those distinctions.
