This question has confused me for quite some time now. I have searched it up online, and the basic answer is: 'Mass is a form of energy. When black holes die they release the amount of energy that they should. Mass is conserved.' But there's a problem with that answer. Energy is conserved, but mass isn't; it's turned into another energy store. From what I've learned in school, the amount of mass in the universe is always the same, and that's conservation of mass. But if mass is just another type of energy and can be transferred into other types of energy, mass is certainly not conserved, and the conservation of mass doesn't exist. It's like saying conservation of kinetic energy. Kinetic energy can be transferred into the electrical or thermal or other energy stores, and although the overall energy in the universe is the same, the amount of kinetic energy has changed? Am I being stupid or is my life a lie?
[Physics] Do black holes violate the conservation of mass
black-holesconservation-lawsenergyenergy-conservationmass-energy
Related Solutions
The topic of "Energy Conservation" really depends on the particular "theory", paradigm, that you're considering — and it can vary quite a lot.
A good hammer to use to hit this nail is Noether's Theorem: see, e.g., how it's applied in Classical Mechanics.
The same principle can be applied to all other theories in Physics, from Thermodynamics and Statistical Mechanics all the way up to General Relativity and Quantum Field Theory (and Gauge Theories).
Thus, the lesson to learn is that Energy is only conserved if there's translational time symmetry in the problem.
Which brings us to General Relativity: in several interesting cases in GR, it's simply impossible to properly define a "time" direction! Technically speaking, this would imply a certain global property (called "global hyperbolicity") which not all 4-dimensional spacetimes have. So, in general, Energy is not conserved in GR.
As for quantum effects, Energy is conserved in Quantum Field Theory (which is a superset of Quantum Mechanics, so to speak): although it's true that there can be fluctuations, these are bounded by the "uncertainty principle", and do not affect the application of Noether's Theorem in QFT.
So, the bottom line is that, even though energy is not conserved always, we can always understand what this non-conservation mean via Noether's Theorem. ;-)
Conservation of energy refers to systems looked from the same reference frame, it does not make sense to require that energy of the same system to be the same in different reference frames. As a consequence of time translational symmetry, energy conservation is usually true unless we drive the system externally which may break this symmetry. Similarly, momentum conservation is a consequence of space translational symmetry.
The (invariant) mass $m$ is the same in all inertial reference frames, on the other hand, energy $E$ and momentum $p$ are connected through the famous equation
\begin{equation} E^2=(pc)^2+(mc^2)^2 \end{equation} where $c$ is the speed of light. This equation is valid in any inertial reference frame, to go from one frame to another, one has to do Lorentz transformation of both energy and momentum, and it turns out the final result is that the changes in energy and momentum compensate each other and validate this equation in every frame.
For the example you gave, if there is only that ball in the universe, in reference frame A, it cannot stop by momentum conservation. If it stops, you have to exert an external force, which may explicitly change the energy of this ball even in reference frame A. Then from frame B, roughly speaking, you exert a force (you may want to work out the transformation of the force between these two frames) to the left direction and the ball gains energy, so nothing is wrong.
Best Answer
Conservation of mass is not a fundamental law of nature so the violation of it is not a problem. Mass is just a form of energy, and energy is the quantity that is supposed to be conserved. So since black holes do not violate the conservation of energy the problem you are worried about does not exist.