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In developing a virtual-motion alternative to Einstein’s space-time warping version of General Relativity, I have assumed that we must treat two points held stationary relative to any gravitating body as if they had some relative motion between them, the amount of motion depending on their distances from the gravitating body. That assumption necessitates that observers occupying those points relate their measurements through a truncated and asymmetric version of the Lorentz Transformation, which version consists of strange manifestations of the Lorentz-Fitzgerald contraction and time dilation.
Because the virtual relative velocity between the two points comes from gravity, I want to take a look at the physics of bodies falling freely in a gravitational field, in particular where the speed of fall equals a significant fraction of the speed of light. In working out the fully relativistic Schwarzschild metric equation I made the tacit assumption that the acceleration due to gravity conforms to Newton’s law,
at all values of the radial distance. But gravity distorts radial distance, so is the assumption a good one?
We can’t answer that question through observations, natural or contrived, so we must resort to imaginary experiments. Instead of inductive analysis of measurements made of real objects, we will use deductive analysis of mathematical pictures that encode the laws of physics as we understand them, basing that analysis on the assumption that Nature conforms to the patterns of axiomatic-deductive logic.
In concept we can answer our question about Equation 1 by dropping a small body onto a super-compacted, massive body and tracking its descent with radar. The timing of the radar pulse tells us the distance to the body when the pulse strikes it and the Doppler shift of the pulse tells us how fast the dropped body moves away from us at that instant. Of course, I make the tacit assumption that we and our instruments do not move relative to the big gravitating body (as if we were using an antigravity platform to hold our position in space). Thus we obtain the data that will allow us to correlate distance fallen with speed of descent and thereby test Equation 1.
In this particular experiment we can ignore the gravitational Doppler shift. Certainly the radar pulse suffers a gravitational blue shift as it propagates from the radar to the dropped body. But the reflection of the radar pulse suffers a gravitational red shift in exactly the same proportion as it comes back to the radar. When that reflection comes back to the radar it displays only the Doppler shift due to the actual motion of the dropped body relative to the radar apparatus.
I have also tacitly assumed that, in accordance with Einstein’s second postulate, the speed of light has the same value (299, 792.458 kilometers per second) everywhere, at all times, for all observers. If a light pulse takes a little too long to go from one point to another, according to our calculation of the distance between the points from remote surveys, then we must infer, as we did in our analysis of Shapiro’s delay, that space is so deformed as to put extra distance between the points. The speed of light is a fundamental, invariant property of existence; the distance between two points is not.
Our next experiment makes explicit use of the gravitational redshift.
Imagine using a winch to lower a small body from our antigravity platform to the big gravitating body. For this task we use a perfectly massless, unbreakable, non-stretchable cord, perhaps made of polymerized unobtainium nonexistide. This use of fantasy objects (such as our antigravity platform) and fantasy materials is legitimate only if that use does not affect the outcome of the experiment. Such fantasms constitute examples of what Michael Faraday called aids to the imagination: they help us to see more clearly what’s happening in the experiment and thereby enable us to devise a proper analysis.
So the small body does work by virtue of the gravitational force acting on it and the distance it moves in the radial direction. We assert that we can convert that work into some form of storable energy by connecting the winch to a suitable generator and that we do so, storing the energy on our platform. Within the small body there exists a sample of positronium (positrons and electrons) that initially has mass m0.
As the small body descends, we must treat it as if it were moving away from us at the same speed that it would have if we had dropped it and let it fall freely. That means that its mass must increase, that the mass of the positronium in particular must increase in accordance with
in which v represents the virtual relative velocity between the positronium and our antigravity platform. But that’s wrong for a simple reason: it violates the conservation of energy theorem.
When we drop the small body and let it fall freely, it loses potential energy (relative to the big gravitating body) and gains an equal amount of kinetic energy. That kinetic energy increases the body’s mass in accordance with the requirements of relativistic dynamics. But when we lower the body in a quasi-static process and harness the work done on the winch, the body gains no additional energy. The lost potential energy ends up in storage on our platform. Thus, the small body’s mass must remain unchanged.
That statement is not true of the big gravitating body. That larger body has the effect on the small body of becoming more massive as the small body descends in its gravitational field. We can prove and verify that statement readily by imaging that we project onto the big body a beam of monochromatic light that the body absorbs. The energy in the light gets converted into mass as it’s absorbed, increasing the big body’s mass in a certain proportion. That proportion has the same value for all observers. To prove and verify that proposition, we must ask what would go wrong in Nature if that statement did not stand true to Reality.
Imagine that a very long, perfectly rigid bar connects our big gravitating body to another massive body in a direction perpendicular to the direction in which we lower our small body. There is a single point on that bar upon which we can exert a force without making the system rotate. If we have a series of markings on the bar, that balance point lies on or adjacent to one of them. Projecting light onto the big gravitating body provides the energy to increase its mass and that mass increase causes the balance point to shift closer to that big body. The new balance point will lie on or adjacent to a certain mark, the same mark for all observers. That fact necessitates that the distance that the balance point moved be a certain proportion of the length of the bar, the same proportion for all observers. And that fact necessitates that the mass of the big gravitating body change in the same proportion for all observers.
When the projected light passes the small body, an observer on that body will see that it has a higher frequency (shorter wavelength) than an observer on the antigravity platform reports it having. The light’s properties have been changed in accordance with the gravitational Doppler shift. Unlike the ordinary Doppler shift, which involves actual relative motion, the gravitational Doppler shift involves no real motion, so only the temporal distortion of spacetime contributes to it. The total energy in a pulse of light stands in direct proportion to the frequency (in accordance with Planck’s theorem), so we can write
in which E0 represents the energy the pulse contained when it left the antigravity platform and v represents the virtual relative velocity between the small body and the platform.
That’s the energy that goes into increasing the mass of the big gravitating body, so, as seen from the small body, the mass of the big body must conform to
if the proportion in which that mass changes is to be the same for all observers. That fact means that, as experienced by the small body, the gravitational force is stronger than the observers on the antigravity platform would calculate from Equation 1. To calculate the work done by the positronium in the small body as it descends, we integrate the gravitational force acting on it over the radial distance it moves,
in which f(r)=v2 replaces the virtual specific kinetic energy of a freely falling body with the equivalent gravitational potential. The factor g0(r) represents the gravitational acceleration calculated as if the mass of the big gravitating body did not change for the small body.
At some point we stop lowering the small body and convert the positronium into a tight pulse of radiation, which we project back to our antigravity platform. Again, the radiation suffers the gravitational Doppler shift, this time the inverse of what we calculated in Equation 3; that is, we have
That’s not enough energy to recreate the positronium when the radiation comes back to the antigravity platform.
However, if we add the work that the positronium did on its descent, we should have exactly enough energy to recreate the positronium in full. Thus we know that the statement
must stand true to Reality. So now we know that we must have
That statement only stands true to mathematics if
in which K represents a constant. We know that g0(r) must involve a negative power of the radial distance so that the description of the gravitational force diminishes with increasing radial distance, as we expect. It must also involve the inverse square of the radial distance, because any other power would put a wrong coefficient into the integral of Equation 8. And we also now know that
With those data we transform Equation 8 into
In that calculation R represents the radial distance between our antigravity platform and the center of the big gravitating body. We assert that R has a sufficiently large value that its reciprocal approaches arbitrarily close to zero; thus, the square root involving it approaches unity.
So now we know that Equation 1 must take the form
We then have the associated gravitational potential from Equation 11 as
which looks like the relativistic Lagrangian. It doesn’t look anything like the classical Newtonian gravitational potential, but if r is large enough to make the second term under the radical very much smaller than one, we can use the approximation
from the binomial theorem. For our small body, then, the gravitational potential energy conforms to
Going back to Equation 3, we look again at the gravitational Doppler shift. Any light projected from the small body comes to our antigravity platform with a lowered frequency, a red shift.. There exists no actual motion between the small body and the antigravity platform, so we infer necessarily that the change in frequency comes from a distortion of the time frame encompassing the two observation points. In particular, if an apparatus on the small body measures an interval dt between two events, then observers on the antigravity platform will measure an interval dT between the same two events in accordance with
Because we assert the existence of a virtual relative velocity between the small body and the antigravity platform, we must also assert the existence of a form of the Lorentz-Fitzgerald contraction. Radial distances appear to shrink as a measuring system descends into a gravitational field. With shortened meter sticks, the measuring apparatus on the small body puts too many meters between the points where two events occur; that is, relative to the same measurement made by observers on the antigravity platform we have
Of course, that version of the Lorentz-Fitzgerald contraction is asymmetric because the gravitational field and its source give us a kind of absolute frame of reference.
If the upper-case variables represent measurements made by observers located in a relatively flat spacetime, then we can rewrite Equations 16 and 17 as
I have not put the associated transformation equations for the longitudinal and latitudinal directions with that set because measurements made in those directions don’t differ between the observers. Equations 18 thus constitute the semi-classical Schwarzschild Transformation. The associated metric equation takes the familiar form
In Equations 16 and 17 I have made use of the approximate equation
That’s just the classical Newtonian calculation of a body falling freely in a gravitational field, equating kinetic energy gained to potential energy lost. Properly, we should use the fully relativistic calculation, using Equation 13 for the gravitational potential,
That equation gives us
With that result Equations 18 and 19 become
which is the fully-relativistic Schwarzschild Transformation and metric equation. It differs slightly from the one we devised in the essay on The Fully-Relativistic Schwarzschild Metric because in that essay I used the gravitational force from Equation 1 rather than from Equation 12. Nevertheless, we now seem to have a good description of relativistic gravity and its effect on the surrounding spacetime.
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