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When I was attending UCLA (1966 - 1969) my classmates and I used to refer to difficult problems (more properly, the equations that represented those problems) as bears, seeing ourselves somewhat as Davy Crockett going into the bushes with a knife and a grin. In that metaphor we slay the bear by solving the equation. As an example of such a bear we might confront Gauss's law for the gravitational field, which law relates the flux of gravitational force through a closed surface to the amount of mass enclosed by that surface. In devising General Relativity Albert Einstein created a four-dimensional analogue of Gauss's law rendered for a non-Euclidean geometry. His resort to Riemannian geometry and its associated tensor calculus makes Einstein's Equation a steel-plated, nuclear-powered, fire-breathing Tyrannosaurus rex. Nobody goes after Godzilla with a knife and a grin and nobody has ever solved Einstein's Equation. What physicists and mathematicians present as solutions are actually well-educated guesses (nonetheless true to Reality for being guesses).
Karl Schwarzschild (1873-1916), an astronomer who was serving in the German Army on the Russian front at the time, presented the first of those guesses early in 1916. Schwarzschild gave a metric equation as his solution; that is, like Minkowski, he presented a four-dimensional analogue of the Pythagorean theorem. Schwarzschild's metric equation describes how the presence of a spherical body of uniformly distributed mass deforms Minkowski's equation (in a manner analogous to the way in which the "sum of the squares of the sides" rule would have to be modified to accommodate a right triangle drawn upon the surface of a sphere). The second of those guesses, devised in 1963 by Roy P. Kerr, describes how space and time are warped by the presence of a spherical body of uniformly distributed mass that's spinning. We also have solutions involving bodies bearing electric charge or displaying magnetic fields, but the Schwarzschild and Kerr solutions are the main ones, the ones of primary interest in the study of General Relativity.
We cannot readily use metric equations to solve problems pertaining to possible experiments or observations involving relativistic gravity. Transformation equations, like those of the Lorentz Transformation, make relativistic problem solving much easier, but to this date no one has published transformation equations that correspond to the Schwarzschild and Kerr metric equations as the Lorentz Transformation corresponds to Minkowski's metric equation. Nonetheless, physicists have been able to solve some fundamental problems in this realm. Even before he had the complete theory worked out, Einstein solved three problems that have become famous - I) that of the precession of the orbit of the planet Mercury, II) that of the gravitational Doppler shift in light, and III) that of the deflection of starlight by the gravitation of the sun.
However that may be, we can obtain transformation equations from a contemplation of the Equivalence Principle. We can then use those problems that Einstein solved and one other to test those equations to see whether they hold true to Reality. Let's start by asking What happens to a ray of light that flies straight up or straight down in a gravitational field?
Imagine that two teams of observers remain stationary relative to each other and relative to some spherically symmetric, gravitating body and that the teams occupy different altitudes in that body's gravitational field. Also, as an aid to frail human memory, let's agree to use upper-case letters to represent measurements made by members of the upper team and lower-case letters to represent measurements made by members of the lower team. Let the upper team send some light in the form of a pulse from a laser to the lower team. Upper team members describe the energy in the pulse by way of Planck's formula: they have sent N photons, each of frequency F, so the energy in the pulse equals
in which equation aitch represents Planck's constant. Members of the lower team don't want to assume that they will receive the same amount of energy that the upper team transmitted, so they will write their own equation for the energy in the pulse in the form
In writing those equations the two teams' member have made two tacit assumptions. First, they have assumed that Planck's constant has the same value for both teams. That's a good assumption because it conforms to the principle of Relativity. Fundamental constants must have the same value for all observers. Second, they have also assumed that the number of photons that the lower team receives equals the number of photons that the upper team sent. They justify making that assumption on the basis that photons carry spin, the little pieces of inherent angular momentum that all fundamental particles carry. If photons were to be created or annihilated by the vacuum, the conservation law pertaining to angular momentum would be violated and neither team will give up the conservation laws without some strongly compelling reason.
Now let the lower team convert the energy that it received from the upper team into equal quantities of matter and antimatter whose total mass equals m0 in accordance with
Next the members of the lower team want to heave that mass upward with a velocity v that's just enough to get the mass to the upper team and no higher, so they must convert some fraction of their matter-antimatter body into enough kinetic energy to increase the mass-energy of the remaining mass m0' in accordance with
Because they have not added or subtracted any energy in these processes and because Reality conserves energy, the energy of the matter-antimatter body that comes to rest at the upper team's location must then equal
We know that the mass of the heaved body will diminish as the body rises and loses energy, so the body will possess only its rest energy when it comes to rest at the upper team's location. But conservation of energy now tells us that the upper team must get back the same amount of energy that they originally transmitted, which fact I have already assumed in using the upper-case ee in Equation 5, the same ee that I used in Equation 1. But that means that we have
But now I can make the substitution from that equation and from Equation 3 to replace the masses in Equation 4 with their corresponding frequencies. I thus obtain
after dividing out the Nh on both sides of the equality sign. That result means that the frequency that the lower team receives has a higher value than does the frequency that the upper team sent.
In the next step I want to eliminate the velocity from that equation and before I do so I suggest a further simplification of our situation: I suggest that we add to our assumptions one that puts the upper team's location far enough from our gravitating body that it is at effective infinity. With that assumption in mind, I can devise an equation describing the velocity in Equation 7 by equating the kinetic energy gained by a small body of mass m falling from the upper team's location to the lower team's location to the gravitational potential energy the body loses in the fall; that is,
in which upper-case em represents the mass of the gravitating body, upper-case gee represents the Newtonian gravitational coefficient, and upper-case ar represents the lower team's distance from the center of the gravitating body as measured from the upper team's location. Making the appropriate substitution into Equation 7 gives us
So now we have described the ratio between our frequencies in terms of the static properties of our situation, obtaining an equation that expresses the fact that observers at different altitudes in a gravitational field measure different frequencies of the same pulse of light, even though those observers remain stationary relative to each other. We clearly cannot invoke the Doppler shift to account for the different frequencies, so what can we invoke?
Because the frequencies of the electromagnetic vibrations that we associate with light can be the bases for clocks, we may infer that our gravitationally mediated frequency shift comes from the gravitational analogue of time dilation. We imagine making a clock that, instead of counting the reflections of pulses of light between mirrors, as the Feynman clock does, counts time by counting the number of waves of a given radiation that pass a given point. We can use such clocks to demonstrate the gravitational deformation of time; specifically, we must use perfectly identical clocks.
Such perfectly identical clocks would have to use a specific radiation, a characteristic emission from some species of atom. We could, for example, use the 21-centimeter radiation that emanates from neutral hydrogen in space, giving our clocks a proper frequency of about 1,427.6 million cycles per second. Hydrogen maser clocks will do the trick that we need here. Hydrogen has the same properties for both teams, regardless of where they are, so the teams' clocks will be identical to each other so long as we build them to the same specifications. Now let's suppose that some radiation escapes from the lower team's clock and flies straight up and through the upper team's location. If the upper team makes the appropriate measurements on that radiation, they will discover, in accordance with Equation 9, that its frequency has a value lower than 1,427.6 mega-Hertz and, of course, that its wavelength has a value slightly greater than twenty-one centimeters. That means that if the upper team's members were to look at the lower team's clock through a telescope and compare it with their own clock, they would see the lower team's clock running slower than their clock. In fact, if two events were to occur next to the lower team's clock, the time interval t that the lower team would measure between them would correspond to the time interval T that members of the upper team would measure between those events in accordance with
We can see how that equation comes from Equation 9 if we recall that the frequency of a vibration simply equals the inverse of the period of the vibration, the time it takes the vibration to complete one cycle.
In this example we see that the flight path directly away from the mass M subjects the radiation to a gravitational redshift. Because we have tied that radiation to our clocks, we must attribute that effect to a gravitation-induced dilation of time between the locations of the upper and lower teams of observers. But can the upper team really use their clock to measure events at the lower team's location? We see right away that they can do so as long as they remember to account for the time lag due to the finite speed of light. But in this case they do not actually move relative to the lower team, so they will measure the same time lag for both events and can, therefore, ignore it.
Look again at that situation and you may notice something interesting. If some radiation escapes from the upper clock and flies straight down and through the lower team's location, the lower observers will see it blueshifted relative to the radiation driving the upper clock; that is, they will measure its frequency higher than 1,427.6 mega-Hertz and its wavelength slightly shorter than twenty-one centimeters. That means that if the lower observers were to look at the upper clock through a telescope and compare it to their clock, they would see the upper clock running faster than theirs. And if the upper observers were to measure an interval T between two events occurring at their clock, then the lower observers would measure between those same two events an interval t in accordance with
But that equation shows us the inverse of Equation 10, the equation that the upper observers would use, and it shows a time contraction, not a time dilation. Now in Special Relativity we would use the same time transformation equation after merely interchanging our tees and ekses and changing the algebraic sign on the temporal offset: we would not use the inverse equation. That fact tells us something very special.
In Special Relativity two observers must use the same equation, with interchanged variables, because no one can determine an absolute state of motion to attribute to one or the other of those observers. We thus need the temporal offset in that equation to dissolve the twin paradox that would otherwise come up between two observers who each see the other's clock slowed down. In our case here, with gravity distorting time, we have a clear absolute state of motion with a well-defined central position marking the state of absolute rest, so we can use asymmetric equations to transform time and avoid creating a twin paradox from the beginning. That's fortuitous because the fact that we have no actual motion between the upper and lower teams' positions eliminates any possible temporal offset that could dissolve a twin paradox.
That analysis makes good sense: the logic that I have employed looks correct and quite appealing in its results, but do we have any proof that the effects described by their equations are actually true to Reality?
In fact, we do. Let's imagine that two observers have set up their experiments in an elevator shaft, one observer on the ground floor and the other 74 feet (22.55 meters) above him. We know that any point of a gravitationally warped inertial frame that comes to rest at our upper observer's position will do so for only an instant and then it will accelerate downward, passing through the lower observer's position at 69 feet per second (21 meters per second). We also know that such a point must have previously passed upward through the lower observer's position at that same speed. But if require that warped inertial frame to keep the full meaning of inertial frame, then we must in some way regard our two observers as moving relative to each other, with 69 feet per second between them, even though those observers have no actual velocity between them. If those observers then measure the frequency of the radiation from some specific source, they will be able to tell from the difference between their measurements, if any, whether the temporal distortions described above are a genuine feature of our Universe.
Could anyone possibly test that proposition? The proportionality factor in Equation 7 between two frames moving at 21 meters per second relative to each other equals one plus 4.907 quadrillionths. Two clocks whose rates differ by that factor would take a little over 6,457,731 years to accumulate a difference of one second in their readings. And yet in 1960 Robert V. Pound and Glen A. Rebka set up an apparatus in an elevator shaft 74 feet high and measured the effect with an accuracy of ten percent. To achieve that measurement they exploited the Mössbauer effect, in which atoms of iron-57 embedded in certain crystals emit and absorb gamma-ray photons with energies of 14.4 thousand electron-volts over a very narrow frequency range. They set up the emitting crystal at one end of the shaft, set up the absorbing crystal and its associated detector at the other end of the shaft, and used a motor to move the absorbing crystal up or down to make the classical Doppler shift compensate the expected gravitation shift of the gamma photons' frequency. They found that the speed at which the crystal had to move in order for both effects to cancel each other was, in accordance with the theory, 5.3 millimeters per hour, a pace that's not even glacial (glaciers move substantially faster).
But gravitation distorts dimensions other than time. We know that in Special Relativity time dilation entails the Lorentz-Fitzgerald contraction of space. A similar distortion of distance occurs in a gravitational field, even though we might expect the effect to require actual relative motion between two observers and not merely virtual relative motion.
We can understand the distortion of space as a logical consequence of the distortion of time. Look again at our derivation of Equation 10 and note that the upper and lower observers have taken a gravitational redshift of radiation emanating from the lower team's location as indicative of a time dilation. In the same way they can take the lengthening of the waves of that radiation as being indicative of a contraction of lengths measured in the radial direction of our system. Thus, for example, if the lower observers were to measure a differential distance dr between two points separated from each other in the vertical direction at their location, then the upper observers would have to infer an equivalent differential distance dR between those same points in accordance with
which we might take to represent the gravitational analogue of the Lorentz-Fitzgerald contraction. By the same reasoning and because the lower observers you see the upper observers' radiation blueshifted, if the upper observers were to measure a differential distance dR between two points at their location, then the lower observers would infer the equivalent distance in their frame to be dr in accordance with
which is like a Lorentz-Fitzgerald dilation.
We now have only two more dimensions to consider. Because we have used a spherically symmetric body to generate our gravitational field, we must use spherical coordinates for the most efficient algebraic descriptions of the system. Having considered the radial dimension, we now have the longitudinal and the latitudinal dimensions to consider.
Distances measured purely in the longitudinal and latitudinal directions connect points that all lie at the same altitude in our gravitational field and we have found so far that only a difference in altitude in that field produces the distortions of space and time. Further, the longitudinal and latitudinal directions are perpendicular to the radial direction, which is the direction in which virtual motion occurs in our analysis, so we should suspect that those directions play a role in this case analogous to the role of the directions perpendicular to relative motion in Special Relativity; that is, we should expect that observers measuring distances in those directions will get the same numbers of meters (or feet or barleycorns or whatever units they use).
We must apply one caution to that analysis. We use angles as the fundamental coordinates of the longitudinal and latitudinal directions. Actual distance in those dimensions thus comes from the product of those angles (measured in radians) and the radial distance of the endpoints of the interval from the center of the coordinate grid, from the center of the gravitating body in our case. That means that we would measure the same number of kilometers in a given longitudinal/latitudinal interval but we would measure different angles between the same points. That fact has a rather interesting effect on orbits of bodies revolving about the gravitating body, as we shall see in another essay.
Now let's suppress the longitudinal and latitudinal dimensions temporarily, just as we suppress the perpendicular directions in Special Relativity when we don't need them, and look at something we can do with Equations 11 and 13. Those two equations comprise transformation equations analogous to the equations of the Lorentz Transformation, so it seems reasonable to apply them to Minkowski's Theorem; that is, we square both equations, multiply the squared time equation by the square of lightspeed, and subtract the result from the squared space equation. Using differential spatial and temporal intervals, we get
We thus have, by deduction, the metric equation that Karl Schwarzschild devised in 1916 as a solution of Einstein's Equation for the space and time around a spherically symmetric body of mass M. So Equations 11 and 13 comprise what we should rightly call the Schwarzschild Transformation and Equations 10 and 12 comprise the inverse Schwarzschild Transformation.
For convenience I list here all four equations of the Schwarzschild Transformation in their differential form (simply because that's the form that should be used in a highly curved coordinate system like polar coordinates). In the longitudinal and latitudinal equations I have used the usual expressions of distance measured in those dimensions. In accordance with convention I use the Greek letter phi to represent angular displacement in the latitudinal direction, measured from minus ninety degrees at the south pole of the grid to plus ninety degrees at the north pole, and the Greek letter theta to represent the longitudinal displacement of a point from some arbitrarily defined prime meridian of the grid, usually considered to be increasing in the direction called "east" (counterclockwise as seen from a point above the north pole of the grid). We thus have:
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