The Fully Relativistic Schwartzschild Solution

Of the effects usually associated with Schwartzschild's solution of Einstein's spacewarping equation - apsidal precession of planetary orbits, the gravitational redshift of light, gravitational deflection of light, and Shapiro's delay - three of them were described before Schwarzschild devised his metric equation and the fourth one could have been. But we believe that we know of one phenomenon whose description emanates directly and only from Schwartzschild's solution. One of the coefficients in the equation has the form of a fraction and the denominator contains a minus sign. When the indicated subtraction yields a zero, the coefficient "blows up"; that is, its value becomes infinite. At the place described by that infinite-valued coefficient vertical distance flattens to nothing and the river of time freezes up solid. We call the collection of all such points, forming a spherical shell around the body whose mass generates them, an event horizon (because we cannot detect any events beyond it) and we call it and the volume that it encloses a black hole.

Recall that we describe the Schwartzschild metric by the equation

(Eq'n 1)

A gravitating body of mass M satisfies the conditions for the event horizon of a black hole if the entire mass M lies below the Schwartzschild radius, the radial distance we define by

(Eq'n 2)

An observer occupying that location would measure the vertical distance between two nearby events as a nearly infinite number of meters because their measuring rod has shrunk to near zero as measured against the measuring rods that observers far from the event horizon use; that is, dr is very much greater than dR. And that lower observer would measure virtually no time elapsing between those events, even though years might pass on the upper observer's clocks.

Equation 2 gives us the formula for calculating the Schwartzschild radius of any gravitating body. In order for a body to be a black hole all of the body's mass must exist within the body's Schwartzschild radius, the radius of the event horizon (as measured from a distance, of course). Reality has made that condition difficult to satisfy, as you can discern in the fact that the sun has a Schwartzschild radius of 2.956 kilometers (the sun actually distributes its mass throughout a volume that has a radius of 690,000 kilometers) and that Earth has a Schwartzshild radius of 8.87 millimeters, about the size of a standard seedless grape.

Astrophysicists believe that the collapse of a very massive star's core when that star explodes as a supernova provides the only phenomenon that has the capability of creating a black hole. Once a black hole comes forms it may then grow by drawing in more matter. Astronomers believe that such growth of black holes provides the driving force behind quasars, the exceptionally bright objects found in galaxies far out in deep space. In those objects the light that we see comes from matter heating up by compression as it swirls into a giant black hole at the center of a young galaxy. For a short time such a central black hole can outshine its own galaxy, devouring whole stars when it has grown large enough. Eventually, though, the black hole uses up all of the matter available to it and it goes quiescent. Just such a fasting black hole is what astronomers believe floats in the center of every spiral galaxy, ours in particular. Is there any evidence to support such a belief? Professor Andrea Ghez and her coworkers in the Physics and Astronomy Department at UCLA have been using an infrared telescope over the past several years to measure the motions of a small number of stars at the center of the Milky Way, a feat only made possible at a range of 30,000 lightyears by the fact that those stars move rapidly. Doctor Ghez interprets those speeds as reflecting the motions of those stars in orbits about an object five million times as massive as our sun, an object that does not show up in any of the photographs, even though ordinary stars in that region are clearly visible in the pictures.

But for all the cosmological appeal of black holes, the theory upon which we have based our belief in them contains a subtle error: we have done something wrong in our analysis. We have developed the Schwartzschild transformation using non-relativistic assumptions and then applied the theory in a region in which those assumptions fail. In particular, we have used the classical formula for the kinetic energy of a freely falling body, thereby creating a semi-classical formulation of the theory. That fact does not invalidate the solutions that we worked out for the four problems given above because the phenomena that I described in those problems occur at distances from the sun that are much greater than the sun's Schwarzschild radius: at such distances the semi-classical Schwartzschild transformation is as valid, as an approximation to the truth, as is the classical Newtonian description of the kinetic energy of a freely falling body. However, if we want to devise an accurate description of black holes and of the phenomena that occur close to them, then we must devise equations for a fully relativistic Schwartzschild transformation.

We use the same equations that we devised for the semi-classical Schwartzschild transformation and merely modify the coefficient, the general-relativistic analogue of the Lorentz factor. In the original formulation of the theory I simply equated the kinetic energy gained by a freely falling body to the gravitational potential energy lost by that body to replace the squared velocity in the formula for the Lorentz factor. Now I must use the relativistic equivalent of that calculation, but I find that I can solve directly for the Lorentz factor. We have

(Eq'n 3)

which yields

(Eq'n 4)

Now I can make the appropriate substitution and write down the fully relativistic Schwartzschild metric

(Eq'n 5)

and its associated transformation equations

(Eq'n 6)

(Eq'n 7)

(Eq'n 8)

(Eq'n 9)

Unlike the semi-classical equations, those equations don't blow up anywhere (except at R = 0, of course) and thus offer no sign of an event horizon. If we understand General Relativity properly, then we must assert that black holes do not exist. We must then confront the fact that we don't know anything about matter at extremely high pressure. What happens inside a compact body containing the mass of five million suns? Nobody knows. At this stage I can only offer a guess: I believe that matter gets smeared out into its irreducible components, that even neutronium can't withstand the pressure, and that inside such a compact object we would find only a quark-gluon pudding. But until we can devise the fully general relativistic quantum theory of matter we must leave our speculation there.

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Appendix: The Law of Gravity

In deriving the Schwartzschild transformation, in both its semi-classical form and its fully relativistic form, I have assumed that Newton's equation accurately describes gravity around a massive body at all radii down to zero. Can I truly maintain that the law of gravity requires no modification at the extreme conditions to which we want to apply it? Surely, at the very least, Relativity itself necessitates some modification to keep Newton's law true to Reality.

Let's imagine that we have a body of very large mass M and that we have established two experimental stations, one directly above the other, at widely different distances from its center. Observers on those stations could now perform the following experiment: the upper experimenters drop a small body of mass m, letting fall freely to the lower station; the lower experimenters catch the body in a device that harnesses and stores all of its kinetic energy; then the lower observers put the body into a massless bucket hanging from a massless cable and use the harnessed energy to drive a winch that lifts the body back to the upper station. We know that the kinetic energy that the body gains as it falls comes from the potential of the gravitational field and we know that the work that the winch does upon the body goes into the potential of the gravitational field. We know that Reality upholds conservation of energy, so we know that the gravitational potential affecting the freely falling body precisely equals the gravitational potential affecting the slowly ascending body. Thus, we know that Relativity does not alter Newton's law of gravity: we expect to find no velocity-dependent effects altering our mathematical description of the gravitational field.

But now we may ask whether gravity has some inherent difference from Newton's law. At very small distances from the center of a gravitating mass does the force of gravity differ from that famous inverse-square law?

We know that if gravity does indeed differ from Newton's description, it can only do so at distances smaller than about one centimeter. Physicists have used massive bodies separated by distances of a few centimeters in sensitive torsion balances to determine the value of the proportionality constant in Newton's formula. Astronomers have used that value in describing the orbits of the planets and the lesser bodies of the solar system and NASA's astrogators have used it in prescribing the trajectories of their space probes. From the fact that most space probes successfully reach their intended destinations we may infer that the law of gravity applicable over cosmic distances, described by Newton's formula, differs in no discernible way from the law of gravity that operates over small distances. But Newton's law may still suffer some alteration over ultrasmall distances, though such an alteration would have no effect on astronomical phenomena: we can still expect, as I deduced above, not to find any black holes, any objects displaying event horizons.

Nonetheless, I would still like to deduce some things that we can say about any possible alterations to Newton's law. If the structure of Reality necessitates that we make such an alteration in our description of gravity, then mathematically we must either add a term to Newton's formula or multiply that formula by a new factor. Such an added term must diminish to effective zero at distances on the order of one centimeter from the gravitating body's center of mass and the new factor must approach a value of one at those same distances. We can simplify this discussion by exploiting the fact that we can represent a formula with an added term by the same formula multiplied by a factor that comprises one plus or minus some other term. If we take that approach, then we can say with certainty that we must correctly write Newton's law of gravity multiplied by the factor

(Eq'n A.1)

in which formula r_{g+} represents
a distance from the body's center of mass that serves as a scale factor. We gain
four different possibilities from the plus-or-minus signs in that formula, one
of which we can dismiss immediately as false to Reality.

If gravity were to conform to the version
of Newton's law that comes from multiplication of the classical form by F_{N}
with two minus signs, then we have the possibility that at some distance from
the center of mass the term f(r_{g+}/r) will equal one and the resulting
division by zero will represent an infinite force. But the application of force
yields linear momentum and the application of infinite force, for so much as an
infinitesimal instant of time, would yield infinite linear momentum. That yield
would violate the finite-value theorem, so we must infer that we cannot have an
infinite force and we must infer further that F_{N} as described above
cannot have two minus signs.

We can also infer that the scale distance
r_{g+} cannot have any relation to the mass of the gravitating body from
which its gravitational field emanates. If Reality did not enforce that
proposition, then bodies of different masses would exert different forces upon
each other, again violating the law of conservation of linear momentum.

But now I cannot infer any further restrictions
on F_{N} that would necessitate making f(r_{g+}/r) equal to
zero, thereby confirming Newton's formula as the correct description of the law
of gravity. I won't have the means to achieve that final inference until I
deduce enough of the quantum theory of matter to deduce the law of gravity ab
initio. For now, then, I must rely on the empirical data to which I alluded
above to state that Newton's law requires no modification for the purpose of our
relativistic study. I don't like doing that because empirical data do not
distinguish between the contingent and the necessary. In creating the Map of
Physics I want to create an axiomatic-deductive structure of necessity upon
which we can display the empirical-inductive works of modern physics. However,
in these early stages of the weaving of our logical web we must occasionally
fall back on the empirical data to bridge gaps in the theoretical structure
until we can fill those gaps with solid logic. So, onward!

habg