Anyone predisposed to dismiss this paper out of hand should be warned from the beginning that it is unlike any other critique of Relativity he or she has read. It is not a philosophical or metaphysical treatise. It is the discovery of the actual algebraic errors in Special Relativity. I follow Einstein line by line, and show precisely where the mistakes are. In this, I believe I may be the first. The most notorious critic of Einstein, Herbert Dingle, said (Nature, 1967) "I have enough mathematical insight to see that it is a waste of time to look for mathematical flaws in the theory." Few, if any, have bothered to look at the basic math in the years since, believing the equations to have been already combed by the best minds of the century, and proved beyond a reasonable doubt by field tests. In the last decade, field tests have put the equations into question again, although nearly all, if not all, of the mathematical and theoretical work has gone into making these new tests fit the equations, rather than vice versa. This paper shows that the fault lies in the transformation equations, and is easily correctable.

I rush to add that in correcting the transformation equations of Special Relativity, I am not thereby exploding Relativity in toto. I accept time dilation and length contraction as a fact. I accept the speed of light as a constant. My critique of Einstein's equations is an effort to fine tune them, not to jettison them.

Specifically, this paper was written in response to the call by the Jet Propulsion Lab for help in understanding why the equations of Relativity were yielding wrong numbers in their calculations on space satellites. To date, no explanation has been offered that explains the discrepancies, despite many replies to the call for help, and many theories published in Physical Review Letters and elsewhere. My paper is unique in that it offers new transformation equations, with which I make predictions about the numbers generated by JPL. That is, I have solved a concrete problem of applied mathematics, and I have the numbers to prove it.

I first discovered the central tenets of this paper in November of 2000, although I had been working on and off (mostly off) on the problem ever since my first reading of Einstein's book  Relativity in high school. The bulk of the paper was written before Thanksgiving of 2000, and it was in final form before the end of that year. I have spent the last two years trying in vain to get the paper published (in Physical Review Letters, Annalen der Physik, et. al.). PRL refused on length considerations-- they have a 4 page limit. AdP did not comprehend my argument. I am not sure they read it, or made any attempt to understand it. I have therefore resorted to self-publication on the web, hoping that by so doing I will be able to make proper claim to the ideas in it, should any argment arise to precedence.

I began working in earnest on the paper in late 1999, after reading an article in Newsweek reporting that the Jet Propulsion Lab was getting inaccurate numbers for its space satellites using Einstein's equations. I saw immediately that here was a concrete problem, one that went well beyond all the theoretical arguments about Relativity that I had read. What seemed to me to be necessary was a mathematical, as opposed to a theoretical, critique of Relativity-- a critique that showed specifically where the math was wrong. Corrected equations would then allow me to make real predictions about these space satellites. That is precisely what I have done in this paper. What follows is the paper almost exactly as it was presented to PRL in early 2001.

(Cover Letter)

This paper is a straightforward correction of the Lorentz equations used by Einstein in Special Relativity.  It is a correction that has never been discovered or put forward since the publication of Einstein's original paper in Annalen der Physik in 1905.  The correction is a simple algebraic critique of the primary equations Einstein presents in that paper and in the book Relativity.   This paper is not a philosophical treatise, or a discussion of theory.   It is the discovery of the actual mathematical errors made in Special Relativity by Einstein, and by Lorentz insofar as he influenced the math of Einstein.   Nearly all the math in this paper is high-school algebra, and may be understood by any educated reader.
      You may ask why a correction is necessary, considering the success of Relativity in the 20th century and the currently unassailable status of the equations.  It is necessary for several reasons.  First of all, the equations are yielding unresolvable data from space satellites.  This has been known for several decades, and was reported on most widely by Newsweek in 1999.  In addition, data from Io (Jupiter's moon) that has been an accepted part of the canon for centuries contradicts the Lorentz equations: this has never been resolved.  More recently, the binary pulsar PSR 1913+16 has presented similar data, though its significance has never been related to Special Relativity.  Other experiments and theories in the recent past have also been undermined by the Lorentz equations (and their corresponding General Relativity field equations) including calculations of redshifts, the Hubble constant, and all theories dependent on these numbers.  Likewise, data from particle accelerators is dependent upon these equations to yield useful information, and many unresolvable situations have occurred, unexplainable with the current Lorentz transformations and the field equations derived from them. 

I think you will be shocked at the simplicity of the error made by Einstein and the simplicity of the new equations.  It is not that Relativity is more difficult than we imagined.  It is that Einstein and Lorentz made it seem more complex than it really is.  The new equations and explanations are not more esoteric than those of Einstein.  They are vastly more accessible and understandable.  And they mesh much more easily with other accepted knowledge.
      One other thing must be mentioned: despite the large cosmetic changes in the equations (they look quite different-- they are much more elegant, for one thing), they provide only a small change in the actual predictions of science.  As I show in the paper, Einstein's equations have stood for so long because they mirror the correct equations in many instances.  At the speed and trajectory of a space satellite, for instance, Einstein's equations fail by only the tiniest of fractions.  Not coincidentally, this tiny failure is precisely equivalent to the observed failure.   I think you can see how powerful this makes my equations.  They correct a standing theory without destroying it, and make that theory infinitely more amenable to facts as well as to commonsense.  This at the same time that the mathematics of the theory is clarified and simplified.
      Physics has so far resisted any questioning of Special Relativity for two reasons.  One, because the specific mathematical errors could not be pointed to, incontrovertibly.  The "density" of the theory had made it very difficult to unwind.  Two, because so much of recent science depends upon the continued strength of Relativity.  Between them, the math of Relativity and Quantum Mechanics support most of the equations of contemporary physics. If Relativity should be thrown out, where would that leave us?  The corrections I offer below will, I believe, revolutionize many areas of physics; but they do not threaten to undermine the foundations of Relativity.  The phenomena of time dilation and length contraction, for instance, are made even more certain.  And the theoretical justification (or interpretation) of these phenomena is made transparent even to the layperson-- which only serves to make the theory stronger.
       This all goes to say that institutions like the Jet Propulsion Lab will breathe a great sigh of relief: they will get an answer to their decades-old problems without having to bring the walls of science down to do it.  
    

P.S.(to Annalen der Physik)  The Jet Propulsion Lab published a request for assistance in Physical Review Letters several years ago, on the problem of the satellites.  They received many replies, but no solution.  None of the replies concerned Special Relativity.  I have previously sent this paper to Physical Review Letters, but they have refused to publish it due to its length.  I cannot make a suitable critique of Special Relativity in just four pages.  As your length criteria are more lenient, and as the problem directly concerns a famous theory first published in your journal, I thought it might be of interest to you.  
     
 

An Algebraic Correction
of the Transformation Equations
(the Lorentz Equations)
of Special Relativity
Abstract
 

In this paper I will show that the first equation of Special Relativity, the famous x' = x - vt,  is fatally flawed.  I will do this with simple high-school algebra. 
     I will then show that a basic substitution error by Einstein in the derivation of the term gamma (caused, in part, by the error above and in part by math borrowed from Lorentz) leads to transformation equations that are subtly flawed.  I will also show why Einstein's equations are so nearly correct, despite being arrived at by faulty math.
     Next, I will correct these errors and offer new transformation equations.  These new equations will be arrived at in a straightforward way, again with simple math.  Furthermore, I will show that Einstein's transformation equation for velocity is in fact an equation for two degrees of relativity, and that there is presently no equation for one degree of relativity.  I will offer a transformation equation for velocity with one degree of relativity.  I will derive this equation using only simple algebra-- without the use of calculus.
      I will then derive the corrected transformation equation for two degrees of relativity, for velocity.  Again, I will do this without resorting to calculus.   
       Next, I will solve the specific problem of the Jet Propulsion Lab, making a prediction for the exact amount of error in Special Relativity that leads to the faulty numbers in General Relativity. 
     Next, I will show that the present interpretation of Special Relativity-- as applying equally to objects in all relative trajectories-- is in direct conflict with other currently accepted facts, including Roemer's calculations on Jupiter's moon Io and data from the binary pulsar PSR 1913+16.   I will show the simple and inevitable resolution of this conflict.
     Next I will prove that Lorentz made the same error as Einstein, and that this error was caused by a faulty interpretation of the Michelson-Morley interferometer.  I will show that the diagram used to visualize the interferometer by Lorentz, Michelson, and every physics textbook in the 20th century is conceptually flawed.  And I will show precisely where this flaw lies, mathematically, and how it led to the equations of Special Relativity-- especially the Pythagorean component of gamma.
     Finally, I will interpret the new equations, showing how they must change our conception of the nature of Relativity, and of light, and of measurement itself. 
 

Introduction to the Problem

Only very recently has there begun to be a general acceptance, by the status quo, that Special Relativity might be subtly flawed in some way.  For most of the 20th century, of course, it was sacrosanct.  No one, in the mainstream, would have thought to question it in any way.  But now there is beginning to be an accumulation of data that does not fit Einstein's transformation equations precisely.  The data that led me to work seriously on the problem was supplied by the Jet Propulsion Lab.  For several decades, various space satellites have been found to be acting a bit strangely.  They are not where they are predicted to be, according to relativistic calculations.  The scientists who manage these craft have dismissed any number of explanations for the discrepancy, supplied to them by many of the best technicians in the field.  But still the problem is unresolved.  It has proven to be such a thorn that the JPL has even gone to the mainstream publications in the United States, begging for help.  Newsweek published a major article on it in 1999.

Einstein published his paper on Special Relativity in Annalen der Physik in 1905.  The book Relativity was published for general audiences in 1916.   It has gone through many editions, but the theory itself has not changed in the last 97 years.  Einstein made several predictions, which were confirmed by subsequent data, and the theory quickly achieved a solidity and a fame that is perhaps unmatched in history.  
     His intention was to reformulate Newton's equations for velocity to conform to the latest facts.  Light had recently been shown to have a finite and constant speed, and Einstein saw that this would affect calculations of position and velocity of measured objects.  He saw that the measurement of time would be likewise affected. 
   Light was proven to have a constant speed, regardless of the speed of the observer, by the Michelson-Morley interferometer.  The interferometer (which is diagrammed in this paper) was designed to show the velocity of the earth relative to the "ether."    It was assumed that light traveled either through, or relative to, this ether; and that therefore the velocity of the earth would have to be added or subtracted from the velocity of light.  But the interferometer found that the earth's velocity had no affect upon the measurement of the speed of light, from any direction.  This was one of the most mysterious outcomes in the history of science.  In trying to explain this null set, Heinrich Lorentz proposed a set of contractions and expansions that would offset the predicted measurements, bringing them into line with actual data.  His fudge-factor turned out to be a now famous term called gamma.
     Not accidentally, Einstein's basic transformation term is also gamma. Einstein was working independently of Lorentz, and on a different problem.  But they both used the same concepts, and the same math, and so came to the same term.  
     Einstein began his derivation by postulating two co-ordinate systems, S and S'.   S is the co-ordinate system of the observer. S' is the observed co-ordinate system. He then provides us with the basic equation x' = x - vt, which he tells us is the Galilean transformation equation from one system to the other.  This also gave him  x = x' + vt, he assumed.  He then produced the equations x = ct   and  x' = ct'  to show the distance light travels in the two coordinate systems.  He introduced gamma as the transformation term, as in the equation x' = y(x - vt)   where y is gamma.    By substituting values among these four equations, he achieved a value for y in terms of his other variables.  
     Quite simple, really.  Except that he never precisely defined his terms.   Not in the original paper.  Not in the book.  Not ever.  And no one has ever questioned these terms.  What, for example, does v stand for in the first equation?  Is it the velocity in S or S'?  One assumes it is in S, since v is not prime.  But we, the observer, are in S.  If we already know v, what are we looking for?  What I mean is, v is how the situation looks to us.  Therefore, v is already a relative velocity.  If this is true, then what does the transformation equation tell us-- what is that value of v that we get at the end?  On the other hand, if the given v is really the v in S', it should be labelled v', to be consistent.  And that begs another question.  How could we be given the v' in S'?  according to the current interpretation of relativity, we cannot know what is going on in S' without a transformation equation.  We would then need a transformation equation in order to calculate one. 
     The mysteries of Special Relativity have been considered up to now to be inherent in the problem.  We have been told that it is not comprehensible by ordinary mortals.  It is subtle and complex, and all one can do is accept the paradoxes.  That is all part of the fun, frankly.  If it were transparent, it wouldn't be deep.  This is the current wisdom, anyway.
     Unfortunately, it turns out that the confusion is Einstein's (and Lorentz's) from the beginning.  It is possible to define the terms precisely enough that all the mystery disappears.  We are then left with distressingly simple equations that almost anyone can understand.
      Over the last century there have been any number of lengthy critiques of Special Relativity.  All of these critiques, though, have been no more than philosophical attacks upon the theoretical assumptions and conclusions of Relativity as a whole.  No one has yet been able to point to the specific errors in the mathematics.  Admittedly, Einstein's explanations made this quite difficult to do.  And the spectacular successes of the theory acted as a sort of protective wall, keeping it from being seriously questioned.  Institutions like the Jet Propulsion Lab could not publicly (or even privately) question such a fortified theory, without direct mathematical evidence.   I only hope the following paper will begin the thaw. 
         
      

The Equation

Einstein begins his book Relativity with a famous thought experiment.  It involves a railway embankment, a train, and a man on the train.  The train moves with a constant velocity v.  Later, the man also moves, with regard to the train.  But for now we will limit ourselves to the train and the embankment.
         Let us start with an illustration.
 

          This illustration is very much like Einstein's train illustration in the book Relativity, but here the artist has tried to graph x', x, and vt.  The man is at point P.  We, the observers, are understood to be watching from the embankment in S, the co-ordinate system to the left

To go with this thought experiment, Einstein gives us this equation (p.33, Rel.),  

                                                   x' = x - vt

           In his original paper of 1905 ["On the Thermodynamics of Moving Bodies"], he gives the same equation.  But neither there nor in the book Relativity does he say where this equation comes from-- nor does he define any of the terms in the equation.  In both instances he simply pulls the equation from nowhere. 
          In the 1905 paper, the equation is completely mysterious; but in the book he gives us a small clue.  There Einstein says that this equation is the Galilean transformation for the thought problem illustrated above.  Galilean simply means "classical" or pre-Einsteinian.  If you assume an infinite c, and equal t's, then the Lorentz transformation for x reduces to this, he says.
          But let's look closely at the illustration, and see what we can find.
        
First of all, we are given that P is not moving in S'.  The man should not be moving with respect to the train at this point in the thought experiment, as I said above.  We know this for several reasons.   One, because Einstein in the book has given us v, the velocity of the train relative to the embankment.   But he has not given us a second v-- a v', say-- as the velocity of the man relative to the train.   There is only one v  in the equation above.   Two, because Relativity is now used to find the time dilation of a single observed object.  Not an object within an object (like a man moving in a train), but simply an object (the train).  Sub-atomic particles in quantum experiments are routinely found to be time-dilated, for instance, but they cannot possibly be thought of as an object within an object.  The Lorentz equations for x and t (that Einstein derives from this thought experiment) are-- or should be-- equations for only one degree of relativity.  Meaning the train relative to the observer, but not also the man relative to the train.           
          But if P is not  moving in S', then x' is a meaningless distance in this illustration.  What I mean is that it would have been simpler to place P on the y'- axis.  If P does not move relative to the x'-axis, then you just have a single velocity.  It is easiest to graph a single velocity by putting it at zero to start with.
          In this case, its Galilean equation should be simply   x = vt.   This is the equation we all learn in first-year physics.        
          But, of course, P may be placed anywhere in S' and we should be able to write a transformation equation for it.  Perhaps Einstein had a reason for not putting P on the y'-axis.  So, let's leave the man at P and see if we can make sense of that.
         
If this is supposed to be an equation for a Galilean transformation, what is x'?  What I mean is, how can there be an x prime in a Galilean transformation with one velocity?  If there is only one t and only one v, how can there be two x's?   How could a train moving with regard to a embankment, or a dot moving with regard to an eye, in a Galilean system, have an x prime?    x is its displacement with regard to the eye.  What is  x'  its displacement with regard to, in a Galilean system? 
           The truth is, a Galilean system with one velocity allows of no prime variables.
           The equation x' = x - vt  is a hybrid equation.  It is part Galilean and part Einsteinian, or relativistic.  I contend that it is a muddle.

Einstein's initial equation is supposed to be a classical equation.  It is supposed to be the classical reduction of a Relativistic equation.  But what he gives us is an equation that already assumes that every moving object carries a co-ordinate system with it.    As you can see, the equation
x' = x  -  vt  is neither classical nor relativistic.  It does not conform to any hypothetical reality in either theory.   It is not  a true classical equation, since in Galilean space it is false, even nonsensical.  Nor is it a relavistic equation.  As we know, the relativistic equation includes the term gamma.   Put simply, Einstein has derived his equations from a non-working equation.
          
To prove this, let's go back to the illustration.  Look closely.  If P does not move,  x' is either 0, if it is understood to be the change in x relative to S'; or it is the line marked vt, if it is the change in x relative to S. 
          But let us consider the possibility that in the equation x' = x - vt, Einstein means not x(prime)  but   x(initial), as in the equation

                                      vt = x(final) - x(initial)

          This gives the equation the proper form, at any rate.  Because this form looks so familiar, every physicist in the 20th century has accepted Einstein's first equation without question.  At first glance, it does seem to fit the situation.  But Einstein's equation is subtly different from this "change in x" equation.
          This "change in x" equation is simply telling us that the man starts out some distance x from the y'-axis.  It is highly confusing for Einstein to call it x' in this problem, as I will show.  It would have been better to label it x_i -- for x(initial). 
       
I have pursued this "detail" for a reason.  Einstein derives the Lorentz transformations from this equation.  His derivation in appendix 1 in Relativity is hard to follow, and I will not describe it here.   Physics textbooks now commonly derive it in this way:
          Assume                 x' = x - vt
          Assume that the transformation from Galilean equations to Relativistic equations will be linear.  Then
                                        
Step 1:                        x' = y(x - vt)  where y is the transformation term we seek.  
                             and x = y(x' + vt')
                                        [Notice the t' in the last equation.  Einstein did not label his t in
                                          this equation t', but physics textbooks have changed
                                           the notation in order to make the equations derivable.]
      
         Now, says Einstein (following Lorentz), light travels in these coordinate systems (S and S') in this way:
Step 2:                                x = ct        and        x' = ct'

Substituting the first equations into these equations gives us:

Step 3:                               ct = y(ct' + vt') = y(c + v)t'     and
                                         ct' = y(ct - vt) = y(c - v)t  

If we substitute t' from the second equation into the first, we find that

Step 4:                              ct = y(c + v)y(c - v)(t/c) = y²(c² - v²)(t/c)  

Cancel out the t on each side and solve for y:  

Step 5:                                 y =         1/(1 - v²/c²)^1/2

This is the famous transformation term gamma. 

But notice that  x' in the equation x' = ct' cannot possibly be thought of as x_i  or  x(initial).  It is obviously the change in x during the period t'.   However, the change in x_i  during any t' will always be zero, because P is not moving in S'!   The fact is,  x_i is not a variable in S', at this point in the problem.  It is a constant.  But x' in the light equation is a variable.   It is the distance light travels according to the clock on the train.
         
          x' in the equation x' = ct' is not the same as  x_i 
         
         But Einstein substitutes one for the other.   He and every physics textbook in the 20th century have performed an illegal substitution in Step 3.    
          Therefore gamma  is arrived at by faulty math.  It is false.
 

~~~~~~~~
 

Another thing is strange here.  Everyone knows that Einstein used the Lorentz equations to find that time appeared to slow down and x appeared to get shorter.  But let's look for a moment at the two light equations above.  The light equations Lorentz and Einstein both used:

x = ct
x' = ct'

If these are true,
then   c = x/t      from the first of these equations
and     x' = xt'/t     by substitution
so           x'/x = t'/t

           This means that in these equations the apparent change in x is proportional to the apparent change in t.

           But when time slows down (in any system, or by any means of measurement), the period gets larger.  Time slowing down implies a larger  t, not a smaller t.
           That is, t should appear to get larger as x appears to get smaller.

Einstein even states this outright, in the book Relativity. He says (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 seconds, i.e. a somewhat larger time. As a consequence, the clock goes more slowly than when at rest." Again, he says "a somewhat larger time." Physicists have focused on the sentence after that, up to now. But time is not defined by the rate of the clock, not even by Einstein. Or stated more precisely, time is not measured that way. Relativity is primarily a theory of measurement, and so what is required is an operational definition of time. Not what time is as an abstract concept, but what time is as a measured quantity. Time is the length of the period, as Einstein flatly states here. A second is not a stroke of the clock. A second is the gap between strokes. It is a distance, by every operational use of that word. Time is not the strokes of the clock, it is the time between strokes of the clock. This is no semantic difference; it is a matter of the definition of time. [To consider this question further, see the links at the end of the paper-- a discussion of this definition with several scientists and mathematicians, and a paper on the operational definition of time.]
          
           x and t should be in inverse proportion!

So, we should find that
          x/x' = t'/t     t = t'x'/x    x = x't'/t
    or   xt = x't'                
And, if x = ct
then     c = x't'/t/t             
and      x'  = ct ²/t'

Only if   t = t'     does    t²/t' = t'
Einstein states that t does not equal t'

therefore  x' does not equal ct'   

Even the light equations were wrong!
 

~~~~~~~~

Let us go back for a moment to the first equation,  x' = x - vt
     Again, this is supposed to be the Galilean transformation equation for x.   Einstein says, (p. 33, Rel.)
"If in the place of law of the transmission of light we had taken as our basis the tacit assumptions of the older mechanics as to the absolute nature of times and lengths, then instead of the above we should have obtained the following equations:
   
    x' = x - vt
    y' = y
    z' = z
    t' = t

"This system of equations is often termed the 'Galilei transformation.'  The Galilei transformation can be obtained from the Lorentz transformation by substituting an infinitely large value for the velocity of light c in the latter transformation."

But this is not the case.   There is no such thing as a Galilei transformation equation.  For Galileo and Newton, no transformation was necessary for a linear problem like this.  x in S' would equal x in S.  The whole universe was a single co-ordinate system, and the train would not have been given a system of its own.   It is clear that the equation for x in a Galilean system would have looked just like the other equations (for y, z, and t).  Meaning, x' = x. Or x = vt. But not, x' = x - vt. This is one of the biggest mathematical blunders in history, sitting in the open for a century. And until now unquestioned.

The equation cannot possibly give us a reasonable value in the situation that Einstein diagrammed, with the train.  Let us say we are with Galileo, and we are looking at the train go by, and we want to calculate a velocity for the train.  How would we do this?  I think you can see that in the equation
x' = x - vt, x' is going to be zero, because x = vt.  
      And in the equation   x = x' + vt   x is always going to be 2x'.
      As a Galilean equation, the equation makes no sense.

But even as a Lorentz equation, the equation makes no sense.  Let us look at Einstein's transformation equation for x.


x =   x' + vt  
      (1 - v²/c²)^1/2

At low values for v, x is greater than 2x'.  You may say, But if v is zero, then x = x'.  Yes, but for any velocity greater than zero, no matter how slow, x is greater than 2x'.   I do not even understand how one might go about plugging numbers into such an equation.   What if our train is going ten miles per hour?  And what if we watch it for a hour?  Does Relativity want to tell us that the train is going to appear to have gone more than twenty miles?
     The truth is, the transformation equations don't even begin to make any sense until they have been differentiated, in which case most of the difficult terms drop out.

Those with a knowledge of Special Relativity will interrupt here to point out that the transformation equation for x is only used to generate a length contraction equation, in the form
   
     L' = L[(1 - v²/c²)^1/2] 

This equation, at least, is in a sensible form.  But I must point out that length contraction along the x-axis implies a contraction of the entire x-axis.  Which is a contraction of distance.  Which should have been given us by the equation for x.  [To get to the L equation from the x equation requires more sleight of hand, which I pick apart in Appendix B, if you are interested.  Suffice it to say here that the x equation is not used by scientists, since none of them can say how it might be used.]
      I also must point out that relativistic equations are used on quanta, which have no "length."  And yet distance projections are made, such as the distance a particle will travel before breaking up. 
     And the satellites of the Jet Propulsion Lab are slowing down in ways that are subtly unpredictable by Relativity.  This is obviously a problem of distance, not of length.  Nobody at JPL cares whether the satellites are getting shorter.  They care whether the total distance traveled is getting shorter.   So the transformation equations are being misused, simply in order to make them work at all.

~~~~~~~~

Some physicists may now be shaking their heads, saying to themselves, "no, no, no."  They will say, Einstein's first equation describes a completely different situation than the one I have plotted and critiqued above.   They will say that the equation  x' = x - vt   breaks down in a wholly different way.
     The equation may be thought of in this way, they will say:

Since, in general, x = vt,    x' = x - vt   may be thought of as

       (some x) = (some x) - (some x)

It corresponds to the equation in the book  that Einstein offers:

       w = c - v

where w  is the velocity of a light ray relative to the train,
         c is the speed of light as measured from the embankment,
   and v is the speed of the train

In fact, Einstein draws the analogy directly between the two equations.  Therefore we may think of the first "some x" as taking the place of w. 

Therefore x'  is the displacement of the man relative to the train,
                vt is the displacement of the train relative to the embankment,
           so  x must be the displacement of the man relative to the embankment.

I say, this makes perfect sense except for one thing.  The notation of the variables is imprecise and confusing.  x  and  vt  appear (because of the fact that they are both unprimed) to be in the same co-ordinate system.  But they are not.  A much better notation would be the following:

             x" =  x - v't'

This tells us that we have three co-ordinate systems-- the system of the embankment, the train, and the man.  And this notation stands to remind us that the given velocity is v' : the local velocity of the train.  It is the velocity of the train as measured from the train, not as measured from the embankment.  Einstein never differentiates between the two.  He never gives us an equation to find the velocity of the train as measured from the embankment-- which would be simply v.  The v  he derives in Chapter XIII (on Fizeau) is the velocity of a moving man on the train, but it is for two degrees of relativity.  He gives us no equation (and we still have no equation) for a single degree of relativity -- the relative velocity of the train. 
     You may say, simply set the velocity of the man to zero and run the equation.  This will give the velocity of the train.  But it won't, for several reasons.  One, because the current Lorentz equation for velocity resolves to unity if you plug in zero for one of the given v's.  It tells you that your relative velocity is equal to your given velocity-- the local velocity of the train.   This is no surprise, since Einstein never differentiated between the two.   This becomes crystal clear if you set x" to zero in the last equation above.  The equation then becomes  x = v't'.     This tells us nothing.  It also does not give us an equation that can be manipulated by substitution in the way Einstein manipulated his equation.   What I mean is, x = v't'  cannot yield the term gamma. 
 

~~~~~~~~
So far I have only done a critique of the algebra of Special Relativity.  But the math more commonly used in Special Relativity is calculus.  It has taken this form:

Let us say the man at point P in the illustration above is moving.  The velocity of the man as seen from the embankment is therefore

W  =  dx/dt  =  d[y(x' + vt')]/dt'      where y is gamma

Differentiation yields the equation

W  =     v' + v    
         1 + vv'/c²

But the form of this differentiation assumes that  W =  v' + v
   where v' =  the velocity of the man relative to the train, and
             v =  the velocity of the train 

If v' = 0, then the equation resolves to W = v.   v is a given quantity, so the equation yields no information.

Einstein's equation for velocity tells us how fast the man appears to us to be moving, if the man is moving in the train.  But if the man is not moving with regard to the train, the equation tells us nothing about the apparent velocity of the both the train and the man relative to the embankment.   No one has seemed to notice that the train has a relative velocity of its own.   Or, if you take the given v as the velocity of the train as seen from the embankment, then no one has noticed that the train will have a local velocity that is different from this observed velocity.  
       Look again at the beginning of this calculus problem as I have stated it here.  I have stated it as Einstein and the current textbooks state it:  "Let us say that the man... is moving."   Notice that there is no distinction in this sentence between 1) the man moving because he is moving with regard to the train, or 2) the man moving simply because he is seated in the train, and the train is moving. 
       By differentiating an equation of this form, Einstein has arrived at a velocity that is in fact relative to two degrees.  That is, the man relative to the train, and the train relative to the embankment.  The current transformation equations do not derive a value for the relative velocity of the train.  Einstein and all the physicists of the 20th century have not even noticed that this value is necessary-- that it is, in fact, the value we were seeking in the first place.  Nor have they noticed that physics has ended up conflating, or substituting, one value for the other.  This confusion of terms has never even been noticed, much less resolved.
 

~~~~~~~~

In glossing the calculus of Special Relativity in the section above, I said that Einstein's equation for velocity gives us a number as long as the man is moving with regard to the train.  What I did not say is that it gives the wrong number for that as well.  It is wrong not only for the substitution and conceptual mistakes I have already outlined, but also for the following reason.
   
The main feature of the Lorentz equations is the term y, which I have let stand for the Greek letter gamma (since this program does not allow for Greek fonts).  As I said above, Lorentz and Einstein calculated gamma to be

                                                  y = 1/(1 - v²/c²)^1/2

Where did they get this?  Lorentz arrrived at gamma first, and his thinking was not precisely the thinking of the substitution equations I have listed the steps for above.   It is obvious from its form that gamma comes from applying the Pythagorean theorem to something.  But what?  Lorentz intially came up with his equations to answer the findings of the Michelson-Morley interferometer experiment.  This was before Einstein proposed the theory of Special Relativity.  I break down the interferometer experiment in appendix A, but a simpified illustration here will show where the Pythagorean theorem comes from.  This illustration is directly from a college physics textbook, in the chapter on Relativity.



We have already seen two algebraic errors by Einstein in the invention and derivation of the Lorentz equations.  The third-- Lorentz' use of the Pythagorean theorem in deriving his original equations-- arises from the problem illustrated above.
       What we find in the illustration is a spaceship with a light projector inside. The spaceship at the top is the S' system, and it illustrates the path of the lightray as seen from inside the spaceship (a). Below is illustrated how the path of the lightray would look from the outside (b), to a stationary observer on the earth. This is the S system, obviously.
       We are told that the observer on earth would observe the process in (b) as it is illustrated.   But notice that in (b) the lightray is moving tangentially to the observer on earth.  I hope is is obvious that our observer cannot see this lightray.  No one can see a tangential lightray!  We only see lightrays that come into our eyes.  Every lightray we see is coming directly toward us.  We have no knowledge of lightrays moving away from us or moving tangentially or even just missing us.  To have information about the situation on this spaceship, we must be sent a signal from the ship directly toward us.  In this illustration, the editors of the book are performing equations on imaginary light paths.  Not observed paths, but abstractions.  This is a grievous conceptual error.
      In the (a) part of the illustration, time is being measured by observation.  In the (b) part, time is measured by the imagination.  Or, to put it another way, in (a) the local observer is collecting real data.  Lightrays are entering the "receiver."  In (b) the observer is not basing his equations on collected data.  He is not even collecting any data.  There are no lightrays coming toward him.  In reality he would not be seeing anything.  The spaceship would pass him by, unknown.  He is making assumptions.  He is assuming that if  he could see the same lightray (a) sees, it would be traveling in this manner.  But this is not observation, much less measurement.  It is simply bad methodology, and bad math.
    
The Michelson Morley interferometer was invented to to test the situation described above.  You can see how the Pythagorean theorem would be used to calculate the distance light travelled in (b) given the distances D and L.  D and L are the sides of the triangle and the path of the lightray in (b) is the hypotenuse.  The Lorentz equations, applied to the interferometer, work in exactly the same way.   The Lorentz transformations take us mathematically from (a) to (b).
     This would be fine if the light ray appeared from the earth to travel that path or that distance.  But, as I said, the little man waving does not observe that hypotenuse.  It absolutely cannot be part of his data!
     One of the outcomes of Einstein's relativity is that all events are local.  That is, all measurements (of time, distance, etc.) are good only for the measurer.  Another measurer in another place will get different measurements.  And yet, by applying the Pythagorean theorem to this situation, the authors of the textbook are attempting a non-local measurement.  They are taking information obtained in a local field [specifically, the distance D, obtained by local measurement in (a)] and transferring it into a non-local reference field [the field in (b)].  This is not allowed, by the very theory they are trying to prove.  In this way, the argument is circular.  In order to prove that all events are local, and that time and distance are relative, they assume that quantities can be transferred from one system to another, and that D and L in (a) are the same as D and L in (b).  But quantities like D are transferable only if t and x are equivalent in both fields.  Besides, D is a local measurement of the co-ordinate system (a), while L is an observed distance in (b), and yet they are treated exactly the same.  No transformation equations are done on either one before they are plugged into the same right triangle! 
       I say "they," but it is not just the authors of this textbook or the artists of this diagram.  Lorentz and Einstein do the same thing.  Every illustration or conceptual analysis of this problem I have ever seen makes this same error.  The Lorentz equations came from precisely this sort of diagram, and the Michelson-Morley experiment accepts it as a given.  It is the very reason that the Lorentz equations have the form they do.  If this diagram had not been the accepted view at the time of the Michelson-Morley experiment, the Lorentz equations would not have had the form of the Pythagorean theorem.  As I will show, the true equations for simple time dilation have no Pythagorean component at all.  [The equation for an object moving at an angle to an observer will use plane triangle trigonometry, but not the Pythagorean theorem].*  

*To read more about Michelson-Morley, or to see a diagram of the interferometer-- and see its equivalence to the diagram above-- see appendix A.  

 

New Transformation Equations

Now let us derive new equations, correcting the mistakes we have uncovered.  Since the end result of the transformation equations has always been the ability to derive a relative velocity from a local velocity (or other known quantities), we should ask, what do these terms mean?  What is a local velocity and what is a relative velocity?  It turns out that these definitions are strictly practical.  That is, these velocities are determined by how we measure them.  Historically we have always measured velocity by one of two methods:
         1) We measure our own velocity by using a clock and by measuring our change in x relative to a known background.  As an example, if we were driving in a car (but did not have a built-in speedometer) we would have to make use of mile markers.  We would take note of the markers as we passed them; and then, using our on-board clock, we would calculate  the velocity.  Please notice that in this case we see the markers from a negligible distance.  The speed of light does not affect our calculation, because we are at mile marker x when we see mile marker x. 
        2) We measure the velocity of an object at some distance.  This measurement is arrived at in a completely different way than the first one.  Usually we are given x, as in the first problem.  We know  x  because we have already marked it off, or we have it as an accepted number from previous experiments.  But  t  is different.  We use our own clock, it is true.  But, because the object is at a distance, and because light has a finite speed, we do not see the object at the same time that the object sees itself.
          To make this clearer, imagine that the object is a blinking light.  In this case, there are actually two events.  The object blinking, and our receipt of the blink.  These two events take place x distance apart, and the gap in time is the time it takes for light to travel x. 
            Let us make up our own thought problem to illustrate this.

Thought problem one:

Apparatus:
           1) A blinker that blinks at a rate of one blink per second.
           2) A tunnel marked off with lines, like a ruler, to indicate distance.  
           3) An eye, with a clock that ticks at a rate of one per second, at the beginning of the tunnel.



Experiment:
           The blinker and the eye begin at rest, next to eachother.  Their blinks and ticks are exactly synchronous.  The blinker then takes off and goes through the tunnel at a constant velocity.  It measures its own velocity based on the number of marks it passes for each blink.  It reads the marks from a negligible distance.  That is, it reads the marks as it passes them .
           The eye also measures the velocity of the blinker.  It measures the velocity of the blinker relative to its own clock.  It measures by seeing the blinks, which are blinks of visible light.  The eye is given x'.  It has walked off the distance in a previous experiment (or you may want assume the eye is the one who painted the lines on the tunnel).
            The blinker is set on a course directly away from the eye.  Assume that it reaches v' instantaneously.

Question:
            Will the eye and the blinker measure the same velocity?
            If not, how can the velocity measured by the eye be known given the velocity as measured by the blinker itself (and vice versa)?

Answer:
            Let t' = the period of each clock, from its own vicinity.  This is the period measured when the two clocks are side by side at the beginning.  Notice that the blinker is a clock.  Each blink is a tick of the clock.
            x' = distance blinker has gone relative to tunnel marks, according to its own visual measurements.
            v '= velocity blinker is going, by it's own calculation.
     Let  t = period that the eye sees blinks from blinker.  This gives us the apparent period.
            v = velocity eye calculates blinker to be going, based on visual evidence.
                          This is the apparent velocity.
           
            If you are with the blinker, then you will measure your own velocity like this

                                                           v' = x'/t'

           Let us say that your first blink is at the 1km mark.  Your second at the 2km mark, and so on.
           Obviously, your v' = 1km/s

           What then is v, the velocity of the blinker as measured by the eye?
   
             To discover this, we must first find  T₁.  That is, when does the eye receive the first blink, according to its clock?

                       t = period
                       T = time
                                                   
Well, @ T₁' = 1s, 
             x' = 1km, so the light must travel back to the eye 1km.  It takes the light 1km/c  to do this.  So we would expect the eye to receive blink #1 at
   
                                       T₁ = T₁' + (x'/c) = 1.000003s
And
           @ receipt of second blink, T₂ = 2.000006.
           @ receipt of third blink, T₃ = 3.00001.
                and so on.

            So, for a simple blinker, the general equation would be

                                        T_n = T _n ' + (x_n' /c)   
                                                   
                                          t  =  T₂   -  T₁
                                          t  =  t' + (change in x'/c)

A blinker with a period of 1s and a local velocity of 1km/s will appear to have period of 1.000003s.
This period will be stable.

Now let us calculate the apparent velocity.

                            v = x'/t
                              = x'/[t' + (x'/c)]
                              = .999996km/s
  
You may say, "Wait, why did you use x' in that equation?  And why did you assume x' = 1km  when you said that the light must go 1km to get back to the eye, in the time equation?  You can't assume these things!  Relativity tells us that the clock will slow down and that x will shrink.    x should be less than x'."
            I am not assuming x' is the distance to use in the equation for apparent velocity.  I am given it.  The velocity of an observed object is either the given distance divided by the apparent time or the apparent distance divided by the given time.  These are the only possible calculations for an observed velocity.
           In the present case, v = x'/t     or   v = x/t'     but not     v = x/t
           The same goes for the light ray traveling back to the eye, in the time equation.  x' is simply a given here, just as c is a given.  Without them, any equations-- mine or Einstein's-- would be useless. 
           If I was not given x' (or v' and t', which is the same thing), there is no way I could know it or calculate it.  And there is no way I could calculate v.
           Think of it this way:  A train passes at night.  We don't know the velocity, and we can't see the mile markers.  All we can see is a pulse clock on the train.  Can we know its velocity relative to us?   No.  The Lorentz transformations, as used up to now, can tell us nothing.  We must be given a local velocity v', or we must know x'.  The apparent velocity of the pulse clock is determined by its period and its speed.  That is, it could be ticking slowly and going slowly, or ticking faster and going faster: in both cases it would look the same.
           It is true, though, that x will look shorter to the observer, as Einstein said.  But this x is not x'.  Nor is it the x used in the apparent velocity equation, as I have shown.  That x is given as x'.  What we are seeking for x  here is the apparent distance.        
            It is calculated like this:

                                    apparent x = (apparent v)  X   t'

         If you are still unclear on why I used  t'  instead of  t, think of it this way.  What we want is to multiply the apparent velocity v by the time on our clock, right?   We want to know what x is at T₁ , and T₂ , and so on, on our own clock.  That is what it means to measure by your own clock.  If you know a runner's speed, and want to calculate how far he runs in a time interval, you would not check where he was as your watch ticked 1.000003, would you?  You calculate using your standard time interval, your own second hand. 
       You may say, "But you have defined  t  as the time for the eye, and  t'  the time for the blinker.  Now you want to switch."  No.  I never defined  t  as the time for the eye.   I calculated  t  to be the apparent period of the blinker, as measured by the eye.   This does not mean that the eye's clock is ticking every 1.000003 seconds.  It means, of course, that the blinker's clock looks like it is ticking every 1.000003 seconds, from the eye.  But the eye's clock is ticking at a normal interval, for the eye; just as the blinker's clock is ticking at a normal interval, for the blinker.  This normal interval-- the rate a clock goes as seen from its own vicinity-- I have defined as t'.  
       Notice that if the eye's clock had a period of t, then it would not see the blinker's clock as slow.  It see's the blinker's clock as having a period of t, right?  If the eye's clock also had a period of t, there would be no difference.  The blinker's clock is slow, relative to the eye's clock, which therefore is not slow.  Very simple. 
        
            So, @ T' = 1,          
                       v  = .999996km/s.
      
            And     x = .999996km/s = .999996km.
                                       1s
            
            This is just what we would expect. 
             t has apparently slowed down.  And  x has apparently shrunk.  That much is consistent with Einstein, at least.
            But you can see that we have had to be very careful about our t's and x's and v's.  You cannot just substitute an x or a t into an equation because it looks similar to another x or t.    You must think about what is really happening.
                             
So, to sum up:
            The blinker's period will appear to slow down, but the period will remain stable (it will not continue to slow down further the farther away it gets).
            Therefore, the velocity will also appear to be slow.  If the blinker sends you a message telling you that its v' is 1km/s, then it will have appeared to slow down relative to that.
            If the blinker has a length along the x-axis, then the blinker will be calculated to appear shorter, because there is an apparent contraction along the x-axis.  If you measured the blinker when it was at rest next to you, then your calculation will be short relative to that.

We have found that  x = vt'
                         and  v = x'/t
                          so,   x/t'  = x'/t
                         and  xt = x't'   just as Relativity predicted.

x and t are inversely proportional.  As t appears to get larger, x appears to get smaller

By substituting quantities we can now easily derive the direct transformation equations, and calculate v  from v'  or  x  from  x' and v':                            
                     
                         v =  x'/t  =  x/t'
                         t = t'  +  (x'/c)
                           = t' + (v't'/c)
                           = t' (1 + v'/c)
                         v = x'/[t'(1 + v'/c)]
                         x' = v't'              
                          
                            v =      v'   
                                  1  + (v'/c)    
                           v' =     v   
                                  1 - (v/c)     
                           x  =        x'         
                                    1 + (v'/c)         
                 

Now that we have our new equations, I predict this complaint: "You are assuming that t and x are absolute, before you even start.  Your marked off tunnel is an absolute system of coordinates,  and your t's even coincide.  They may appear to be .000003s apart, but they are really the same.   Relativity does not rely on these assumptions.  It transcends them."
          My answer is that I have done precisely what Einstein did.   I started with given quantities and derived unknowns from them by discovering the proper equations.  You are calling my givens "absolutes," but the terminology is meaningless.  They are not absolutes, they are accepted values.  Einstein does not derive his relative velocities from thin air.   In his transformation equations, you must have certain information to start with.  I claim that my givens are no more absolute than his.  My givens are exactly the same as his.  If my elucidation of his process makes you think that these givens are absolutes, then I can force you to admit that Einstein's givens are also absolutes.
       
In fact I will do so now.  Einstein says (p.18, Rel.) that the train has a given velocity v.  The implication is that this velocity is a local velocity.  It must be the velocity of the train, as measured from the train.  He cannot mean the velocity of the train as measured from the embankment.  For if he meant as measured from the embankment, then we would already be given a relative velocity, and we would not need fancy transformation equations to find it. 
         Einstein then shows, correctly, that the train's clock will look slow to the eye on the embankment and that the train's measuring rods will look short.  And he presents the Lorentz equations for t and x.    But then he never uses his relative x and t, that he derives by these equations, to calculate a relative v.  Isn't this strange?  He never concludes that there is a relative v that is different than the v given in the problem.  He derives two t's and two x's, but never derives the second v, the relative v.
          Later, he derives the Lorentz transformation for velocity, using his t and x equations (Ch.8, on Fizeau).  But this is for the addition of velocities.  It is for the situation in which the man on the train is moving relative to the train and the train is moving relative to the embankment (eye).  You have two relative velocities and you want to find the third.   This equation gives us a velocity of two degrees of relativity.
          If Einstein had derived an apparent velocity for just the train relative to the embankment, then it would have been clear that something was wrong.  Once you have two v's, a relative v and a given v (the v in the illustration-- my v'), someone might ask, what was the given v? 
          In other words, Einstein derived a Lorentz equation that gave him a relative x.  With this relative x, he can obviously calculate a relative v.  He has two t's and two x's.  He will certainly have two v's.  The relative v from this transformation equation would be the v of train relative to the embankment.  What then was v'? 
         Or, to state it another way, What was the v in equation x' = x - vt?
         It is the v of the train relative to the tracks, as measured from the train!  Einstein must be given this velocity before he ever starts:  relative v is  dependent  on v'.    So Einstein must have his railroad track already marked off before he can calculate his relative x and t! 
         He even admits this.  At the top of the same page (p. 18) he says, "Of course we must refer the process of the propagation of light (and indeed every other process) to a rigid reference body (co-ordinate system)."   [his parentheses]
          His problem implies the existence of a pre-existing system, like my tunnel.  But this system remains hidden throughout the problem.  Regardless, this system-- whether his or mine-- is not an "absolute."  It is not an absolute in the sense of contradicting the relativity of measurement.  It is a given, a postulate that allows for the calculation of unknowns.
         You may say, "Yes, but there are two co-ordinate systems (S and S').  Distance in one will not be the same distance in the other.  x does not equal x'.  You cannot just transfer x' into your equation-- as you did when the blink was traveling from the blinker to the eye-- as if you already knew the distance."
         Einstein did.