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A
Mathematical Critique of Einstein's 1905 paper
by
Miles Mathis
First
written January 2002
In this
chapter I will critique Einstein's original paper of
1905—probably the most famous physics paper of all time (in
1944 a manuscript was purchased for $6 million—who knows what
it would fetch now). I have copied the 2 and 1/2 pages of
equations that lead up to gamma here, so that you can
follow along, line for line.
The first part of the article is just an introduction to
the problem, which I will not repeat here. It is perhaps better
to approach the problem fresh, anyway. It is much simpler than
history has made it.
In short, Einstein has one important
postulate to add to Newton and Galileo. It is this:
"Any
ray of light moves in the stationary system of coordinates with
the determined velocity c, whether the ray be admitted by a
stationary or moving body."
This is just a
restatement of the constancy of c, but it is a precise wording of
it, and it will come up very soon in his math. That is why I
quote it in full. It is the second of two "basic principles"
of relativity. The first principle is that systems or bodies in
"uniform translatory motion" relative to eachother have
the same laws. That is, neither one is primary: either may be
taken as "at rest." This is not so much a postulate as
it is an hypothesis (as Einstein admits in this paper) but since
I agree with the intent of both principles, it does not matter
here.
I will touch on one point before moving on.
Principle 2, quoted above, already contains a confusion of terms.
A moving body requires a second coordinate system—that is why
Einstein invents multiple systems to begin with: to explain the
motion. But the second part of principle 2 ignores this fact. He
says, "whether the ray be admitted by a stationary or moving
body." But, obviously, if the ray is admitted by a moving
body, it is not in "the stationary system of coordinates."
It is in the moving system of coordinates. A better statement of
principle 2 would have been that light is measured to travel c,
from any and all coordinate systems.
Very soon after this
Einstein introduces his two coordinate systems, which in this
paper he calls K and k, where K is the stationary system. Then he
gives us the variables, x, y, z, and t in K; and ξ, η, ζ and τ
in k. In short, given x and t, he wants to find a transformation
term that will express ξ and τ [ ξ and τ were replaced by x'
and t' in later algebraic derivations, such as the one in his
book Relativity].
The last thing we are given is a
velocity v. This, Einstein tells us, is the velocity of the
entire system k in "the direction of the increasing x of K"
But, I ask, is this the velocity of k as measured from k, or as
measured from K? Einstein does not tell us, though it is
absolutely crucial that we know. K and k each have their own
clocks and measuring rods, which are not equivalent (by the rules
of Relativity). They will therefore each measure velocity
differently. In fact, they will measure the velocity of k
differently. But Einstein does not assign v to either system.
This is Einstein's greatest error in the whole derivation of
gamma. Since v is unprimed and unGreek,
you may assume it is k as measured from K. But if this is true,
then the velocity of k as measured by k must have a value too.
Notice that Einstein never once, in this or any other derivation,
creates a variable for that v, nor does he discover an equation
which yields it. [The velocity we get in later equations—the
velocity of the current v tranformation—is the velocity of an
object moving within k. It is not the velocity of k
itself.]
Then we get the equation x' = x  vt. Einstein
says, "If we place x' = x  vt, it is clear that a point
at rest in the system k must have a system of values x', y, z
independent of time. We first define τ as a function of x', y, z
and t. To do this we have to express in equations that τ is
nothing else than the summary of the data of clocks at rest in
system k, which have been synchronized. . ."
So, he
states outright that x', y, and z are now the set of variables
"at rest in the system k." But he has already assigned
y and z to K. A point at rest relative to k is moving relative to
K, since k is moving relative to K. So "at rest in the
system k" is not equivalent to "at rest in K".
This being true, it also means that ξ, η, and ζ have just been
bumped up into a third system—the system that is moving in
k—which he has no label for. But x' is a variable with two
allegiances: he says it is now in k, but by the equation x' = x 
vt it is still connected to K. Unless he means for x and v to
also be "at rest in k", in which case we might as well
do away with K altogether—it has become superfluous. Likewise,
the variable t is also a dual citizen. Einstein has it variously
in K and k. When it is with x', it is in k. When it is with x, it
is in K. Tau (τ) is in the undefined third coordinate system,
with ξ, η, and ζ. Notice especially that Einstein has created
an x' but no t'. He has three x variables but not three t
variables.
He then says, "From the origin of system k
let a ray be emitted at the time τ_{0} along the xaxis
to x', and at the time τ_{1} be reflected thence to the
origin of the coordinates, arriving there at the time τ_{2}."
So the ray takes the place of the man in the train.
Einstein has a movement within a movement here: k is moving in K,
and the lightray is moving in k. This provides two degrees of
relativity, which Einstein is trying to calculate at once. But he
does not seem to realize that he is doing this. He doesn't have
enough variables to do it successfully, so his variables keep
sliding from K to k. He needs three sets of variables and three
coordinate systems.
The other mistake is to use a light
ray for his object moving in k. Principle 2 states that light is
measured c, whether from a moving or stationary system. This
makes light a special case. It is not like a man on a train. c,
unlike v or v', does not change with position of measurement.
This is Einstein's second greatest error in this paper.
Next
he prepares for a bit of calculus—that is why he is talking of
functions. He doesn't need calculus here, but it is the only way
he can get the reader more confused. He gives us this strange
equation:
1/2[τ(0,0,0,t) + τ(0,0,0, t + x'/(c  v) +
x'/(c + v))] = τ(x', 0,0, t + x'/(c  v))
This is a
preparatory step to differentiation, but we don't need to go
there since this equation is faulty. It is supposed to be an
expansion of
1/2(τ_{0} + τ_{2}) = τ_{1}
But the corresponding time at τ_{1} in either k
or K is not t + x'/(c  v) . First of all, this violates
Einstein's own Principle 2, which I quoted in full above. The
speed of light must be measured the same from anywhere. But
Einstein himself is subtracting v from it!
Before giving
us a partial equation for τ, supposedly from all his
differentiation, he tells us that "light is always
propagated along these axes [Y and Z], when viewed from the
stationary system, with the velocity √(c^{2}  v^{2})."
All that, as an aside, almost as a footnote. He gives us no
equations to show why light should move like that along the Y and
Z axes. Where did that square root come from? One can only
suppose he borrowed it from somewhere, most likely Lorentz. But
it just stands there, naked, with nothing leading into it and
nothing leading out. Obviously it is supposed to be the
old Pythagorean way of comparing the two lightrays in the
interferometer, that I have already critiqued. But we don't have
two light rays here. He has given us one lightray going up and
back, in the xdirection of both K and k. So his equation for Y
and Z is false—he has given us v as the velocity in the
xdirection. We have no velocity in the Y or Z direction, not
in K or k.
What is more astonishing is the fact that he
again violates Principle 2. He says, "when viewed from the
stationary system." But Principle 2 reads, "Any ray of
light moves in the stationary system of coordinates with the
determined velocity c, whether the ray be admitted by a
stationary or moving body." k is that moving body.
Finally, Einstein gives us this equation:
τ =
a[t  vx'/(c^{2}  v^{2})] where "a is a
function of v at present unknown."
Then he gives us
this
ξ = cτ
This is the equation that becomes x'
= ct' in the book Relativity. I have already shown that it
is false [in several other papers], since time dilating implies a
larger τ, not a smaller. But by substituting this false equation
into the previous one we get,
ξ = ac[t  vx'/(c^{2}
 v^{2})]
Then Einstein says that t = x'/(c  v),
which I have just shown is wrong. But he substitutes it into the
above equation.
After simplifying he gets,
ξ =
ac^{2}x'/(c^{2}  v^{2})
then he
substitutes x  vt for x', which yields,
ξ = ac^{2}(x
 vt)/(c^{2}  v^{2})
Then, as you can
see, Einstein gives us some more nonsensical equations for η and
ζ. Being in the y and z directions, they can have nothing to do
with v, which Einstein told us at the top of p. 43 was "in
the direction of the increasing x of K."
But we are
basically finished. c^{2}/(c^{2}  v^{2})
is our transformation term. Einstein calls it β here, but it has
come to be known as γ. He never had a β variable in any
equation, but it suddenly appears at the end as the thing we were
looking for all along. The variable "a" has been lost
to the dustbin in all the rewritings of these equations, with no
explanation that I have ever seen.
Einstein's final error
is a simple mathematical one. He assumes that
c^{2}/(c^{2}
 v^{2}) = 1/√(1  v ^{2} /c^{2})
When in fact it is simply
1/(1 – v^{2}/c^{2})
There is no square root!
Either he reduced the equation
wrong, or he has some hidden math steps here he is not sharing
with us. It may be that he thinks that those last y and z
equations will affect his transformation term β somehow. But in
his rewriting of the math in later books, he does not assume that
y and z affect calculations in the xdirection. Nor does anyone
who parrots this derivation of gamma. But if you think
about it, the only way to get a Pythagorean square root into a
linear problem in one direction, is to mess it up in just this
way—to assume, for some reason, that an xvelocity is affected
by y and z.
One final point. Einstein later changes the
math of Special Relativity. It has been said that he did this to
simplfy it for general audiences, specifically for the book
Relativity. Those who say this also point to the fact that
he changed the math to make it even more difficult after he
completed General Relativity. For, at that time, he imported the
tensor calculus used in General Relativity to express Special
Relativity. In his lectures at Princeton he used this very
difficult math. He had two derivations of Special Relativity
working after 1916: one a revamped algebraic derivation (in the
appendix of Relativity) that is very similar to the one I
showed in this paper, the other a tensor calculus
derivation.
But both changes mask the basic errors of the
original paper—they do not fix them. In the algebraic solution,
he gets rid of the Greek letters. Unfortunately he replaces ξ
with x'. But they are far from equivalent. x' is the xvariable
in a nonmoving k, as he states above. ξ is the xvariable
moving within k. It is the xvariable that belongs to the
lightray, which is moving. Nor does he fix the equation x' = ct'.
It is false whether the xvariable is ξ or x'.
The tensor
calculus derivation also takes all the givens of the paper of
1905 as still given. It fixes none of them. All it does is make
the contradictions harder to see. There is absolutely no reason
to use tensor calculus to solve the problem of Special
Relativity, as Einstein presents it and as it stands to this day.
It does not even call for regular calculus, as Einstein proved in
the appendix of his book. A problem should be solved with the
simplest math that will solve it—especially a problem of
applied math. This keeps the concepts and assumptions as near to
the surface as possible, where they may ride out the light of
day.
Now let us look at Einstein's derivation of the
equation for the addition of velocities. In his 1905 paper he did
not differentiate his x equation in order to find his relative
velocity equation, like they do now in textbooks. He simply
combined his equations algebraicly, like this:
From
earlier in the paper Einstein found: γ = gamma = 1
/√(1  v^{2}/c^{2}) τ = γ(t  vx/c^{2}) ξ
= γ(x  vt)
Now he says, if a point is moving in k, let
ξ = wτ where w "is a constant". Notice two things.
One, Einstein now has a point moving in k instead of a light ray.
He seems to have recognized his earlier mistake, and he changes
his moving object from a light ray to a point. This is really
odd. If I am right and he noticed the theoretical difference
between the light ray and the point, then why didn't he go back
and correct the first part of the derivation. On the other hand,
if I am wrong and he didn't see his mistake, why did he make the
basic change? You don't normally change your objects in the
middle of a derivation.
Two, he does not define this new
velocity variable at all, beyond saying it is a constant. This is
really extraordinary. One must look at the sentences two or three
times, to be sure that he intends w to be the velocity variable.
He does not fully define w, because he can't. If he states
outright that it is a velocity variable, then he must assign it
to one of his coordinate systems. Very soon you will see why
that would be fatal to him.
By substitution, he gets,
wτ
= γ(x  vt) wγ(t  vx/c^{2}) = γ(x  vt) wt 
wvx/c^{2} = x  vt x + wvx/c^{2} = wt + vt x(1
+ wv/c^{2}) = (w + v)t x = (
w + v )t 1 +
wv/c^{2}
Now, watch this last step very closely.
He reduces the above equation to:
V =
w + v
1 + wv/c^{2}
This is the current value for V. This
equation stands to this day. But to reduce like he did he must
assume that V = x/t We have been given that w = ξ/τ So
what does v equal? v is what x over what t?
V = x/t w
= ξ/τ v = ?/?
You may say, well maybe the x is x'.
Maybe, but what is the t? He has no third tvariable anywhere, in
this paper. And in later derivations, when he does have a t', he
has no τ. He never has three t variables.
What we
need to solve for an addition of velocities, amazingly enough, is
four tvariables and five x variables. t_{0}
= the time of K from K t = the time of the point as seen from
K t' = the time of k as seen from K t'' = the time of k as
seen from k τ = the time of the point as seen from k. τ'
= the time of the point as seen from the point but t" =
τ' = t_{0} We need five xvariables x =
displacement of the point, as measured by K x' = displacement
of k, as measured by K x" = displacement of k as measured
by k. ξ = displacement of the point, as measured by k ξ'
= displacement of the point as measured by the point
Einstein
says that v is the velocity of k relative to K. w is the
velocity of a point relative to k. V is the velocity of that
point relative to K. but to solve we also need, w' =
velocity of the point measured by the point v' = velocity of k
as measured by k
w' = ξ'/t _{0} v' = x"/t_{0} τ
= t_{0} + (ξ'/c) w = ξ'/[t_{0} + (ξ'/c)] =
w'/(1 + w'/c) w' = w/(1  w/c) v = x"/t' t' = t_{0}
+ (x"/c) v = x"/[t_{0} + (x"/c)] =
v'/(1 + v'/c) v' = v/[1  (v/c)] t = t_{0} + (ξ' +
x")/c V = (ξ' + x")/t = (ξ' + x")/[t_{0}
+ (ξ' + x")/c] Multiply the last equation by
1/t_{0}//1/t_{0} {Eq. 5} V
= v' + w'
1 + [(v' + w')/c]
V =
v
+ w
1  (v/c)
1  (w/c)
v
+
w
1 + 1  (v/c)
1 
(w/c)
c
=
v + w  (2vw/c)
1  v/c  w/c + vw/c^{2} + [v +
w  (2vw/c)]/c
=
v + w  (2vw/c)
1  vw/c^{2}
Just to be sure that gamma does not apply to
the transformation of two degrees for t, let us find τ' in terms
of v and t, like Einstein did. t = t_{0} + (ξ' +
x")/c w' = ξ'/t_{0} v' = x"/t_{0} t
= t_{0} + (w't_{0}+ v't_{0})/c w' =
w/[1 – (w/c)] and v' = v/[1 – (v/c)] t_{0} = τ' t
= τ' + wτ'/[c(1  w/c)] + vτ'/[c(1  v/c]) t = τ'{1 +
[w/c(1  w/c)] + v/c(1  v/c)} t/τ' = (c  v)(c + v)
+ w/(c  v)
+ v/(c  w)
(c  v)(c  w)
(c  v)(c  w)
(c  v)(c  w)
= c^{2}  wv
=
1  wv/c^{2}
(c  w)(c  v)
(1  w/c)(1  v/c)
Similar,
but not gamma. Not surprising, since gamma only has
one velocity variable. Notice that in Einstein's derivation of
gamma, regarding x and t, he already had two velocities.
His setup for the addition of velocity section is exactly the
same as his setup for x and t, in the first section. The only
difference is he had a light ray moving—as his second
velocity—in the first part, and a point in the second part. But
in both sections he is seeking equations for two degrees of
relativity.
Now, you may say, why not use "equation
5", in the section on V above? It looks very much like
Einstein's equation, except that we are adding the velocities in
the denominator rather than multiplying them. At most speeds this
would only be a small correction to Einstein and would seem to
imply that his math was not that far off.
We can't use
that equation for one very important reason. The velocity
variables don't match Einstein's. Mine are prime, his were not.
Mine are the local velocities of k and the point. His are
actually undefined, so it is impossible to say what he meant for
them to stand for. But I assume he intended them to be the
measured velocities of the point and k, from k and K,
respectively. I assume this because he never provides us with an
equation to discover the velocity of k relative to K, or the
point relative to k. He doesn't derive these equations because he
thought those were his givens. But now that it turns out that his
final equation suggests that his givens were local velocities,
measured by the moving systems themselves. And this means that
the velocity of k relative to K and the velocity of the point
relative to k are still unknowns. Einstein has derived no
equations for firstdegree relativity!
The other reason
not to use equation 5 is that in most real situations we will not
be given the local velocities. In using the relativity equations
on quanta, for instance, the givens are not local velocities. We
would be given relative, or measuredfromadistance, numbers to
begin with, and would need an equation to determine the addition
of these numbers. The famous experiment of Fizeau (explained by
Einstein) is another example. We are given the speed of the
liquid. But this is our determination of the speed of the liquid,
not the liquid's. The given is not a local measurement of the
system.
Please notice that my new equation for the
addition of velocities gives us numbers that are very close to
Einstein's in most situations. It differs from his in having
another easily comprehensible term in the numerator and a minus
sign instead of a plus sign in the denominator. But it may be
used with confidence, since it has been derived from a thoroughly
analyzed situation, as above, from five different coordinate
systems.
The true transformation equation for velocity
of one degree of relativity is the one I used above, in my
derivation of V.
v =
v'
1 + v'/c
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