In its inception, the math of Special Relativity was algebra. Except for one step, Einstein's 1905 derivations¹ of both gamma and the addition-of-velocity equation were algebraic. Even this one step of calculus was unnecessary, as Einstein proved in the appendix to his book Relativity, where he did without it. This is not surprising, since the problem concerns linear uniform motion.

Einstein's derivations stand to this day. Gamma and the addition-of-velocity equation have never been corrected. They are considered uncorrectable. They underlie the equations of four-dimensional space (Minkowski) and the field equations of General Relativity. The tensor calculus takes them as given. However, in recent decades there have been a number of discrepancies found in the use of the equations on data from both accelerators and space satellites. Physical Review Letters published several papers on the satellite problem a few years ago. It was never solved to everyone's satisfaction, using the mechanical analyses offered. I became convinced at that time, and remain convinced, that the problem is in the basic equations. So I have returned to the algebraic derivations of nearly a century ago. I am now in a position to prove that although his final equations are close enough for much prediction, they are not correct in form. He ignores one very important step, and this step completely compromises the math. Nor was this step uncovered in later emendations. All current derivations yield equations for two degrees of relativity. First-degree relativity is ignored. This paper is my announcement of the discovery of First-Degree Relativity.

Relativity is caused by motion. An object in motion relative to a second object no longer shares the co-ordinate system of that object. We must therefore create two systems to explain them. Specifically, the length and time variables will differ, and at least one transformation equation will be required to go from one to the other. The transformation equation(s) must include the speed of light, since the finite speed of light is what makes them necessary in the first place. If c were infinite, then all space would be one co-ordinate system, as with Galileo. This is Einstein's set-up, which I fully accept. It implies that clocks and measuring rods will not match up across systems. The result is length contraction and time-dilation, which I also do not question.

Einstein gives us two systems, A and B, say. A is at rest, B is moving. Another way to state this is that observations are made from A, which is at rest relative to itself; and B is moving relative to A. Then we are given a constant velocity, v, that is B relative to A (v is linear, along the x-axis). We are also given x and t in A, and x' and t' in B. We seek the transform between them. This is all we are given involving A relative to B, but the mistake has already been made.

What is wrong is that Einstein failed to assign the given v to either A or B. If the clocks and measuring rods in A are different from B, then A and B will measure velocity differently. That is, they must get different numbers for the velocity of B. But Einstein did not notice this. He did not notice that v, as given, is already a relative velocity. No one else has noticed this in 98 years. In this problem, we should have the velocity of B relative to A, measured from A; and the velocity of B relative to A, measured from B. One motion, two different numbers. This is what I call First-Degree Relativity. We have a v and a v' now, and we need a transformation equation from one to the other. How can we get this?

First, let me clarify the situation. I know that some will say here that B has no velocity measured from B. Velocity is a relative term, one that requires a background against which to measure. This is true. B has no velocity relative to B. But B does have a velocity relative to A, and B can easily measure that velocity itself. What is interesting is that A and B have different operations for measuring this velocity, and it is the difference in operations that yields the different numbers.

If you are in a car, and you want your velocity relative to the ground, it is a simple calculation. You only need mile markers and a watch. You count the mile markers as you pass them, from a negligible distance. In other words, you are at mile-marker 5 when you see or read or tabulate "5". Therefore your x and t are both local numbers. They are unaffected by the speed of light. But if a police officer wants to calculate your speed, his operation is different. He must have one given to start with. He must have either the time or the distance as a given. If the distance is given, then he can use incoming data to calculate time; if time is given then he can use incoming data o calculate distance. All his incoming data is in the form of electromagnetic waves, or light. This must affect his calculations, according to the rules of relativity. In real life (before the age of speed-guns) an officer would use his own watch and the mile markers, just like you. But he would not be at mile-marker 5 when he saw it. He would be x-distance away. Obviously, if you are going very fast, or he is a long way away, this begins to affect his calculation of your velocity. There is a lag as the light travels from you to him.

There are really two events: you passing mile-marker 5 and him seeing you pass it. There is a distance x between the two events, and the difference in time is the time it takes light to travel x. Therefore the ticks on your clock will be late, read by him. And they will be later and later, the farther away you get, by this simple equation.

The t-variables are understood to be the periods, or the changes in t. Likewise, the x-variables are changes in x. t' is the period of your clock. t is the period of your clock seen by the officer. Notice that t is not the period of the officer's clock. It is the period of your clock, seen by him. Using this logic, we can derive x and v equations for First-Degree Relativity.

The warning v ≠ xt is just a reminder that the officer must have a given to work from. He must be given an x or a t, in which case that variable is not unprimed. Unprimed variables are relative variables here, by definition. They must be calculated from givens; they cannot both be calculated at once. As proof of this, imagine the officer spots you at night. Let us say that you left your turn blinker on. This is all he can see. He cannot see the mile markers, he has no speed gun. Can he calculate your speed? No, not even if he knows that the period of your blinker is constant. Only if the period of your blinker happens to be one second, and he knows it, can he calculate your speed—because then t' is given.

If that is not enough, think of Michelson measuring the speed of light. What if he did not know the distance from Mt. Wilson to Mt. Baldy in advance. Could he calculate c? No. He needs x to calculate c, or c to calculate x. He cannot find two variables from one experiment. In the calculation of relative velocity, as with the calculation of c, given variables cannot be relative variables without a reductio ad absurdum.

But this is all bonus information. You do not need to accept my calculations to see that First-Degree Relativity exists, and that Einstein did not include it in his conception. Here is his equation for velocity.

In it, v is the given I talked about above; w is the given velocity of an object moving in B. In the paper of 1905, it was a lightray in the first section of the paper and a moving point in the second section. In the book Relativity, it was a man moving inside a moving train. V is the velocity of the point or man as measured from A, the ground. So you can easily see that it is two degrees of relativity—the man relative to the train and the train relative to the embankment, for instance. Or the point moving relative to B, and B moving relative to A. I don't think anyone can deny that we have a compound velocity here: that is why we have three velocity variables.

But that begs the question, where is Einstein's velocity equation for one degree of relativity? How do we transform the velocity of a quantum, for instance? A quantum is not an object within an object. It is not analogous to a man on a train. If it is time-dilated, it should be dilated to us directly. The machine, whether it be linear accelerator or synchrotron, is not moving relative to the ground, like a train. Likewise with space satellites. They should be relative to us in only one-degree, like the car to the officer. You may say, what about the speed of the earth? Well, our incoming data is the speed of the satellite relative to the earth. We never measure the satellite relative to the ether, since we have not had the concept of an ether for a century. So the speed of the earth is not involved. Does the officer include the speed of the earth in orbit in his calculations? The situation is precisely analogous, since we are on the earth when we measure the satellite. The discrepancy in our predictions and our measurements on satellites involves first-degree relativity. It is the discrepancy between our local calibrations on the satellite when it was in port and still a local object, and its data transmitted to us from space.

But this is also bonus information. Whether you accept it or not has no bearing on the fact that Einstein has no velocity transformation from v to v'. He can't: he has no velocity variable that could possibly be assigned to v as measured by B. This must affect his later equations, no matter what interpretation is given to them. He runs into the same problem with the variable w. He does not assign it either to the measurement by the man or by the train. It is undefined, to this day.

If you say the given v is the dilated time of the train, then we are given a relative velocity. Obviously a reductio. If you say the given v is the non-dilated time of the train, then you have just admitted the existence of local time.

But now to gamma. I have shown that gamma does not apply to first-degree dilation. Does it apply to second-degree? In first-degree, we had t = t'c/(c - v), where c/(c - v) was our transformation term. To put it in the form of gamma, it is 1/[1 - (v/c)]. Gamma currently is 1/√(1 - (v²/ c²). Also interesting to note is that Einstein found gamma to be, in his next to last step of 1905-- c²/(c² -v²). One would think that I am on course, since my term seems to be halfway there, as it were. Unfortunately it is not that simple. We now we need more variables, since we have only done the train, but not the man. We need a third set of variables. And a fourth. And a fifth. Also, we have three co-ordinate systems. To be consistent, the man must have one of his own, C.

t’’ = the time of the man, measured by the train

t’’’= the time of the man, measured by the man

x" = the distance of the man, measured by the train

x''' = the distance of the man, measured by the man.

w = the velocity of the man relative to the train, measured from the train.

w' = the velocity of the man relative to the train, measured by the man

w is equivalent to Einstein's w, but he had none of these other variables. We want to find V, the velocity of the man relative to the ground, measured by the ground. To do this we must find t''''—the time of the man as seen from the ground. First notice that t'''' ≠ t. t is dilated once. t'''' is dilated twice. Also, t' ≠ t". The time of the man as seen from the train is not the same as the train's own time. t'' is dilated. t' is not.

How then do we solve? None of the variables from the first part are in the second part: there is no possibility of substitution. The only way to solve is to notice that t'''' = t''' + x"'/c + x'/c. The dilations are additive when the velocities are in a line. This is because the dilations are gaps caused by the distance between the observer and the observed, and these distances are additive as long as they are taken from local fields. Some will complain that Einstein does not do this, but he does. To find x, you must be given x' in Einstein's equations. x' is defined by him as at rest in B.² x' is a distance measured in its own system: that is a local field, by my definition.

t'''' = t''' + wt''/c + vt/c

Putting all the t's on the right side in terms of t''' will give us a transformation equation of two degrees of relativity.

t'' = t''' + x'''/c (by analogy to our very first equation) = t'''c/(c - w)

To go from t to t''', we use t = t'c/(c - v) and let t' = t'''. The time of the train measured by the train is the same as the time of the man measured by the man: since they are both looking at their own clocks from a negligible distance, c does not enter into the measurement.

That is our transformation term for t for two degrees of relativity. Nothing like gamma, since of course it must include both w and v. With this corrected term, we find that

This equation yields numbers that are close to Einstein's equation in most situations, but my equation finds more slowing at speeds nearer c, or at great distances. I believe this will solve the Jet Propulsion Lab's satellite problem, once it is correctly plugged into the field equations. My first-degree transforms are also very useful in experimental situations, as I have shown. In addition, my equations for x and t for both degrees may be used without any further tinkering. Einstein's equations for x and t have always been problematic.

Now that gamma and V are corrected, we may correct all the equations that rely on them, including mass increase and the field equations. Obviously this must wait until my next letter, or until the publication of my entire paper.

1"On the Electrodynamics of Moving Bodies", Annalen der Physik, 17, 1905.

2"On the Electrodynamics of Moving Bodies", Annalen der Physik, 17, 1905, p. 8.