A Break in the
Pioneer Case

by Miles Mathis Any reader with an interest in Relativity knows that the internet is full of new solutions and corrections to Einstein’s equations, including a thousand different solutions to the Pioneer Anomaly. It is a favorite subject of amateur physicists and mathematicians. The reader is therefore likely rolling his eyes at any claim to a real solution concerning either Einstein or the Pioneer Anomaly. It must appear equivalent to claiming that there has been a solution to the Loch Ness Anomaly or the Bigfoot Anomaly.
However, most know that real physicists are working on the problem at the same time, including respected scientists at NASA and JPL. They have been for decades. So it should not be a complete surprise that something would eventually turn up. There has to be a solution. Calculations don’t just misfire for no reason. Beyond this, Einstein himself told us his theory wasn’t perfect—that no theory was perfect. What most of us in science have been hoping is that the correction could be made without bringing the walls down. We hoped for some small correction to some equation buried deep in some old paper which, once tweeked, would give us better numbers while leaving Relativity whole and healthy.
Well, that is pretty much what has happened. A simple algebraic equation from more than 100 years ago, one that is still embedded in the tensor calculus, has been found to be incorrect. Correcting it gives us a small correction to gamma, but does not threaten Relativity as a whole. In fact, the correction strengthens and extends Relativity in many ways. By clarifying the math and underlying concepts, even more uses for the transforms have been discovered.

To get right to it, the equation is x’ = ct’. This equation is part of a pair of equations that Lorentz and Einstein both used in their derivations of the transforms.

x = ct
x’ = ct’

These two equations represent the motion of light in the field of the observer and in the field of the observed object. The first equation is just a definition of velocity, and cannot be doubted. The second equation seems at first glance to be analogous to the first. How could it be in error?  It is easiest to see if you write the equations like this:

c = x/t = x’/t’

You can see that the x’s and t’s are in direct proportion. As x gets bigger, so must t. Here is the problem. According to Einstein, time dilation gives us a larger period. He says it explicitly in the book Relativity (Ch.XII, p. 37): "As judged from K, the clock is moving with the velocity v; as judged from this reference body, the time which elapses between two strokes of the clock is not one second but gamma [γ] seconds, i.e. a somewhat larger time. As a consequence, the clock goes more slowly than when at rest." This is confirmed in The Meaning of Relativity, eq. 22, where he gives us a variation of the equation in question:

Σ(Δx’v)2 – c2Δt’2 = 0

You can see that this is basically the same equation squared. But here he has made it clear that he is letting the variable t apply to periods of time, not to instants in time. This should have been obvious regardless, since instants in time could hardly be either dilated or compressed. The t variable must be a period. And a slowing of time must be a larger period. A clock ticking more slowly has fewer ticks, but it has a larger period. Just think of a pendulum. There can be no doubt on this question.
Therefore, if we know from empirical data that time is dilated and lengths contract, then we must find that x and t are in inverse proportion.

xt = x’t’
If we combine that with our first equation, we find
x = ct
xt = x’t’
x’ = ct2/t’
x’ ≠ ct’

In every case—in the 1905 paper and in all subsequent rewrites—Einstein uses this equation or an equivalent of it. In no place does he justify it beyond adding the primes to the unprimed equation. He assumes it is true simply by its form. He assumes that if the unprimed equation is true, the primed equation must also be true. But I have shown that it isn’t true.
The specific mathematical and physical reason that the equation isn’t applicable requires an analysis of the two fields and how they are operationally generated by the problem. This analysis is a bit difficult, but it can be glossed in this way: The primed field is not a physical field. It is a pseudo-field created by the incoming data only. The primed field is created by the data arriving at the observer; which is to say that x’ and t’ are located within the data, not within any real space. The primed field does not belong to the object being measured, it belongs to the data. The object being measured is far away. The data, when received, is with the observer. Therefore the primed field does not really belong to the object. The primed field is not the field that the object sees itself. This is what created the problem.

Gamma cannot be derived without this equation or its equivalent. The tensor calculus accepts all the axioms and basic equalities and proportionalities of previous math. It corrects nothing. It simply expresses the accepted transforms in a different way.
However, new transforms can easily be derived, transforms that are very close to gamma in output. The corrections also allow us to extend and strengthen Relativity by showing new uses for the transforms. The Pioneer Anomaly is explained and Relativity is logically augmented at the same time. Not only does this correction fail to damage the foundations of Relativity, it actually shows the way to solving other similar problems—ones that have been equally mystifying.

A full derivation is published elsewhere, but I will provide the highlights of a simple algebraic derivation here. Correcting the variable assignments leads us first to the equation

Δt = Δt’ + Δx’/c

This equation is arrived at by imagining an object starting at rest next to an observer, then traveling directly away from that observer at a constant velocity for one second. At the end of that second the object fires a laser at the observer, and the receipt of the laser is the observational data.
In the corrected derivation, it is found that there must be a v and a v’ from the very beginning. Einstein assigns only a v (the velocity of the object relative to the observer as measured by the observer), but v’ is also assignable as the velocity of the object relative to the observer as measured by the object. Note that this is not the same as saying the velocity of the object relative to the object, which is of course absurd. The object can measure its own velocity, Einstein was just never interested in this variable. We find that

Δt/Δt' = 1 + (v’/c)
Δx'/Δx = 1 + (v'/c)
v = v'/[1 + (v'/c)]
v’ = v/[1 - (v/c)]
1 + (v'/c) = 1/[1 - (v/c)]

These equations may be labeled first-degree Relativity, and they will have their uses, since they allow direct velocity transforms that were unavailable with Einstein’s math. Notice that they mirror the current frequency transforms of light:

f = f’[1 + v/c]

This is not a coincidence.

Using these equations it is quite simple to derive an addition of velocities equation that mirrors the one that Einstein derived. After the correction, it turns out to be

v = v’ + v’’ - (2v’v’’/c)
1 – v’v’’/c2

Or, you can now plug in local velocities (velocities of spacecraft as measured by onboard computers, for instance, instead of velocities of spacecraft measured from earth) and this is the transform:

v =    w + w'
1 + [(w + w')/c ]

You can see that this is very close to the current equation, except that we now must add the velocities in the denominator rather than multiply them. Although the second equation looks closer in form, the first equation is actually closer in output and theory.

What about gamma? It is found that gamma cannot be replaced by a single equation, since Einstein’s mistake led to a slight oversimplification of the problem. Different situations and trajectories yield different transforms. We can no longer treat all motion in all direction as equivalent in Relativity. Einstein found that time was always dilated in SR, but this turns out to be false. With approaching objects it is compressed. The period is smaller. This is quite easy to show with the binary pulsar PSR 1913+16 and all other objects that may be thought of as “clocks in the sky”. Any periodic motion may be called a clock, and a pulsar may be thought of as a pulse wristwatch at a great distance with a very fast tick. It is known empirically that the period of PSR 1913+16 slows down when it is moving away from us and speeds up when it is moving toward us, which by itself must prove my point. Relativity is, in one sense, just the Doppler Effect applied to clocks. Mountains of other evidence also exists, going all the way back to Rřmer in the 1670’s and his measurements of the eclipses of Io. These eclipses were another “clock in the sky,” since they were strictly periodic. He also saw shorter periods when Io was moving toward the earth.
This does not affect the Pioneer problem, since these satellites are never moving toward the earth. But they are often moving at an angle rather than directly away, and the transforms are shown to be slightly more complex in that situation. Any perpendicular velocity component over any dt does not count toward the transform or the Relativity.
For some uses, gamma will be replaced by the simple transform above: 1/[1 – (v/c)]. In other uses it will be replaced by

t/τ’ = c2 - v’v’’
(c – v’)(c – v’’)

In energy equations it will be replaced by

ET = moc2{1 + [(v2 + cv)/(2c2 – 4cv)]}    or
ET = moc2[1 + (v’/2c)]
[1 – (v’2/c2)]

This last equation shows why gamma works so well in accelerators, despite being incorrect. The extensions of Relativity allowed by the new corrections allows us to show that it is v’ that is limited at c. The variable v is actually limited at c/2, so we must read the new equations accordingly.

Of course the correction implies many other tweeks here and there, whenever Relativity is used to solve a problem. But in no problem that I am familiar with do the corrections show or imply any ultimate failures of Relativity. To the contrary. The new transforms put Relativity on an altogether firmer footing. Besides our new variable v’ opening up a host of new applications for Relativity, the new math also vastly simplifies all the derivations and concepts. Relativity will finally become transparent to everyone.

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