Mr. C attacks some generally accepted notions about black holes. It appears that the introduction of test particles is inadmissible to him. A test particle, freely falling in a gravitational field, feels no change in energy and momentum, and mathematically, we describe this situation in terms of comoving coordinate frames. This does not fit in C’s analysis, so, test particles are forbidden. A test particle is an object with almost no mass and almost no size, such as the space ship Cassini orbiting Saturn. C calls the use of almost“poetry”, but in fact this is a notion that can be defined in all mathematical rigor, as we learn in our math courses. C is “self taught”, so he had no math courses and so does not know what almost means here, in terms of carefully chosen limiting procedures.
Mr. C. adds more claims to this: In our modern notation, a radial coordinate r is used to describe the Schwarzschild solution, the prototype of a black hole. “That’s not a radial distance!”, he shouts. “To get the radial distance you have to integrate the square root of the radial component grr of the metric!!” Now that happens to be right, but a non-issue; in practice we use r just because it is a more convenient coordinate, and every astrophysicist knows that an accurate calculation of the radial distance, if needed, would be obtained by doing exactly that integral. “r is defined by the inverse of the Gaussian curvature”, C continues, but this happens to be true only for the spherically symmetric case. For the Kerr and Kerr-Newman metric, this is no longer true. Moreover, the Gaussian curvature is not locally measurable so a bad definition indeed for a radial coordinate. And why should one need such a definition? We have invariance under coordinate transformations. If so desired, we can use any coordinate we like. The Kruskal-Szekeres coordinates are an example. The Finkelstein coordinates another. Look at the many different ways one can map the surface of the Earth on a flat surface. Is one mapping more fundamental than another?
“The horizon is a real singularity because at that spot the metric signature switches from (+,-,-,-) to (-,+,-,-)”, C continues. This is wrong. The switch takes place when the usual Schwarzschild coordinates are used, but does not imply any singularity. The switch disappears in coordinates that are regular at the horizon, such as the Kruskal-Szekeres coordinates. That’s why there is no physical singularity at the horizon.
But where does the black hole mass come from? Where is the source of this mass? R μν = 0 seems to imply that there is no matter at all, and yet the thing has mass! Here, both L and C suffer from the misconception that a gravitational field cannot have a mass of its own. But Einstein’s equations are non-linear, and this is why gravitational fields can be the source of additional amount of gravity, so that a gravitational field can support itself. In particle theories, similar things can happen if fields obey non-linear equations, we call these solutions “solitons”. A black hole looks like a soliton, but actually it is a bit more complicated than that.
The truth is that gravitational energy plus material energy together obey the energy conservation law. We can understand this just as we have explained it for gravitational waves. And now there is a thing that L and C fail to grasp: a black hole can be seen to be formed when matter implodes. Start with a regular, spherically symmetric (or approximately spherically symmetric) configuration of matter, such as a heavy star or a star cluster. Assume that it obeys an equation of state. If, according to this equation of state, the pressure stays sufficiently low, one can calculate that this ball of matter will contract under its own weight. The calculation is not hard and has been carried out many times; indeed, it is a useful exercise for students. According to Einstein’s equations, the contraction continues until the pressure is sufficiently high to stop any further contraction. If that pressure is not high enough, the contraction continues and the result is well-known: a black hole forms. Matter travels onwards to the singularity at r = 0, and becomes invisible to the outside observer. All this is elementary exercise, and not in doubt by any serious researcher. However, one does see that the Schwarzschild solution (or its Kerr or Kerr-Newman generalization) emerges only partly: it is the solution in the forward time direction, but the part corresponding to a horizon in the past is actually modified by the contracting ball of matter. All this is well-known. An observer cannot look that far towards the past, so he will interpret the resulting metric as an accurate realization of the Schwarzschild metric. And its mass? The mass is dictated by energy conservation. What used to be the mass of a contracting star is turned into mass of a “ball of pure gravity”. I actually don’t care much about the particular language one should use here; for all practical purposes the best description is that a black hole has formed.
But has it really? Isn’t it so that the collapsing star hangs out forever at the horizon? Well, in terms of the Schwarzschild coordinates, this is formally true! The Schwarzschild solution is the asymptotic limit of the solution in the forward time direction. At finite times, the region behind the horizon does not exist. However, for this analysis, one can better use the Eddington-Finkelstein coordinates, where one does notice that the future part of the horizon does exist. This discussion is compounded a bit because the construction of the maximal extension of spacetime is subtle, and it is certainly not understood by C. Think of a map of the North Pole of the Earth, where it could be that coordinates were chosen such that they cannot be extended across the equator. Formally, the equator is then a horizon. But nobody who’s walking on the equator has any trouble with that.
These self proclaimed scientists in turn blame me of “not understanding functional analysis”. Indeed, L maintains that there is a difference between a mathematical calculation and its physical interpretation, which I do not understand. He makes a big point about Einstein’s “equivalence principle” being different from the “correspondence principle”, and everyone, like me, who says that they in essence amount to being the same thing, if you want physical reality to be described by mathematical models, doesn’t understand a thing or two. True. Nonsensical statements I often do not understand. What I do understand is that both ways of phrasing this principle require that one focuses on infinitesimally tiny space-time volume elements.
PS: For the curious, Mr L is C.Y.Lo: Chief Scientist and bottle-washer at a non-existent research institute.