Munch N. E.

Abstract. In the absence of precise notation, assumptions have been allowed to shift inappropriately in special relativity (SRT) derivations and use. For example, the use of light to relate lengths and times is an important early assumption; yet it is dropped and reassumed at will. That and other examples are described. When assumptions related to the use of light travel to relate length and time are held constant, generalized expressions for special relativity and its Doppler equations were developed using precise notation. However, no SRT solutions without conflicts could be found to those equations. Since that kind of light-use is required within the context of the 2nd principle and optical Doppler and Michelson-Morley (M-M) tests, SRT appears seriously flawed.

The nature and use of assumptions in 16 texts [1] were studied using expanded notation to track the assumptions. Seven assumptions that shift are:

2. Use or non-use of light to relate (x, x', t, t')

The shifts are best demonstrated in the context of Einstein's definition of his 2nd principle on p. 395 of [1a]: "Any ray of light moves [with constant velocity c] . . . Consequently,

"velocity [c] = light path [interval] / time interval." (1)

Important features of equation (1) are its use of intervals and its use of light in defining those intervals. Also, the 2nd principle must apply to all orientations of v and c, else it would only be useful in a sub-set of cases even though c is measured as constant in all orientations.

As a direct consequence of (1), we can write:

D x = cD t and D x'= cD t' (2a, 3a)

for light travel along the x and x' axes. The D denotes interval, primed terms are on the 'moving' frame and non-primes on a 'stationary' frame of reference. The terms D x,D x',D t,D t' there must be intervals because point-values of those terms cannot specify a velocity. Equations (2a, 3a) could equally well be written by integrating dx=cdt and dx'=cdt' between events i and j as:

(x_j-x_i)=c(t_j-t_i) and (x'_j-x'_i)=c(t'_j-t'_i) (2b,3b)

The i^th terms there are commonly set to zero as light starts as origins of the two frame pass, and only x_j,x'_j,t_j,t'_j used. But setting i^th terms to zero simply changes (2b, 3b) to:

(x_j-0)=c(t_j-0) and (x'_j-0)=c(t'_j-0) (2c,3c)

The need for intervals remains. If equations were written in that form, users would be reminded to adhere to assumptions by which the i^th terms were set to zero. Unfortunately, that is not done, assumptions are forgotten, and x_j,x'_j,t_j,t'_j are treated as unrelated point-values of position and clock-time.

When light use and intervals are abandoned, as occurs at times in all 16 derivations in [1], then a number of transformation schemes are used. One of these is 'coordinate rotation' adopted without justification in a form such as in Fig. 2 by 7 of 16 texts in [1], c.f. [2]. In the other texts, equivalent techniques with determinants, vectors or algebraic manipulations are used, also without justification. The impact of the shift away from light use is most strikingly seen in the differences between Figs 1 and 2. With that shift, the x and y values (position and clock-time) are then treated as point-values and asymmetric relationships are assumed without justifiable basis in the 2nd principle.

2. Examples of shifts from intervals to point-values:

Two examples in Chap. XII of Einstein's ref. [1b] are considered. In the first example, Einstein said:

He then substitutes those point-values instead of intervals in E-LT equations and finds two point-values of x for the same point-value of time at t=0. If x and t were intervals, this is nonsense because, as discussed above, there cannot be two different intervals of x for the same interval of t. Also, if the interval of t were zero, both intervals of x would be zero and his equations are meaningless. So, he clearly has shifted from intervals to point-values.

In the 2nd example, he said,

He substitutes point-values t_i=0 and t_j=1/g (which equal his t'_i and t'_j) at x'= 0, where g =(1—v²/c²)^—1/2. But there cannot be different intervals of t' for a single interval of x'= 0. And all intervals of t' must be zero when interval of x'=0. So there is no question he shifted to point-values and abandoned intervals and light-use and hence the context of eq. (1) of the 2nd principle.

Why did Einstein shift from intervals to point-values at this part of ref. [1b]? This is the section where he wanted to show that the E-LT can be rearranged to produce the length contraction (LC) and time dilation (TD) solutions. Resulting contradictions are discussed later.

3. Example of shift in symmetry of length and time variations:

When the 2nd principle and light-use are assumed as in eqs. (2a, 3a), we can see the symmetry of D x/D x' and D t/D t' by simply dividing (2a) by (3a), producing:

D x/D x' = D t/D t' [for constant c]

When derivations shift away from those assumptions, such as in Fig. 2, it's customary to speak of length contraction and time dilation (LC&TD) as though those asymmetric effects have always been part of SRT. They have not. Asymmetry requires:

D x/D x' = D t'/D t = f{v, c}

which is incompatible with results with light-use in Doppler or Michelson-Morley tests. The LC&TD equations are thus incorrect in those tests even though often used there.

As derivations progress, (2a, 3a) are re-introduced at will in the form of the light wave equations to reach the E-LT and Doppler equations where asymmetry has reverted to symmetry as illustrated in Section 5.

Consider a light wave passing over two frames K and K' containing observers P and Q respectively, as shown in Fig. 3. Frame K' is the upper one in that figure and contains two rods A'B' and A'E' of equal length at rest, and a light source at its origin A'. In Fig. 3, the frames are at rest relative to each other. A light wave starts from A' and travels along +x' to B' and along -x' to E' where it is deflected to mark points d and e on the x-axis of frame K. With no velocity, lengths A'B', A'E', Ad and Ae are all equal. Elapsed times for light travel over those paths are also equal as determined by (2a, 3a) to be light path length divided by c.

This is then repeated for K' moving at velocity v to the right in Fig. 4. That is, all in the dotted area is moving at v relative to observer P on 'stationary' frame K. A light wave again is deflected at B' and E' and marks points d and e on the x-axis. Observer P can measure lengths Ad and Ae at her leisure and see they differ from each other because frame K' has moved a distance of vD t during the elapsed time of light travel.

For precise notation we now introduce subscripts. The lengths and time intervals measured by P on K have subscript "P"; those measured by Q on K' have primes and a "Q" subscript. By (2a, 3a), the elapsed times corresponding to lengths measured by Q on his own frame K' are:

D t'_A'B',Q = A'B'_Q/c (4a)

and

D t'_A'E',Q = A'E'_Q/c (4b)

Elapsed-times seen by P on her own frame K are:

D t_Ad,P = Ad_P/c (5a)

and

D t_Ae,P = Ae_P/c (5b)

The primed values with Q subscripts seen by Q on his own frame K' are called proper values. In the eyes of Q they do not change with velocity, since no one has ever measured change in proper values on one's own frame when seen as 'moving' from another frame [3]. In fact, both P and Q agree those are the values that Q measures. Non-primed values measured by P on frame K are also proper values.

Consider a case where c and v are aligned, as shown to the right in Fig. 4 where both c and v have positive sense along the +x and +x' axes. Both P and Q observe that light-travel A'B'_Q seen by Q is less than Ad_P. Hence, by (4a,5a), D t'_A'B',Q is less than D t_Ad,P. Since light starts from A and A' at the same instant and reaches B' and d at the same instant, P says the shorter elapsed time on K' is inconsistent with the 2nd principle. If Q's elapsed time D t'_A'B',Q were correct, light would travel the distance Ad_P at speed greater than c. Hence P says A'B'_Q and D t'_A'B',Q must have contracted by the ratio Ad_P/A'B'_Q = D t_Ad,P/D t'_A'B',Q in a way that's unmeasurable by Q on his own frame. The amount of relativistic contraction seen by P is a complex issue and is discussed later..

A case where c and v are opposed is shown to the left in Fig. 4, where c is to the left along -x and -x' axes, and v is to the right along +x as seen by P. Both P and Q see that A'E'_Q is larger than Ae_P. So, A'E'_Q and its corresponding elapsed time D t_A'E',Q are said by P to be dilated.

So when the 2nd principle and light-use are assumed, we see that different relativistic results are required for differing orientations. That leads to conflicts in logic, since both contraction and dilation must then occur at the same instant on different parts of the length E'A'B'_Q on frame K'. The reader will recognize this as the train paradox, but now we see that paradox is real and cannot be explained away by lack of simultaneity since conventional clocks are not used. Only time intervals are needed and can be obtained from known lengths using (2a, 3a) when the 2nd principle and light-use are consistently assumed.

Similar conflicts are seen when light travels in a round trip from A' to B' and back to A', as in the arm parallel to v in the Michelson-Morley apparatus moving at v relative to observer P on frame K. This is same as light traveling first to the right to B' and back to the left to reach A' in Fig. 4. Now, the same length A'B'_Q must be contracted and dilated at the same instant -- a physical impossibility.

5. Example of linearity contradictions: The definition of linearity here is by Eshbach[4]:

Einstein introduced linearity in 1905 [1a]

There is no proof (experimental or otherwise) known to the author of such linearity. Nevertheless, it was assumed by authors to focus on a single root and thereby reach the LC&TD equations. Then, as derivations proceeded, light-use was re-assumed via the light wave equations until E-LT equations were reached which have dual roots as shown next.

With simple notation, E-LT along the x- and x'-axes is:

In the slightly expanded notation in (2a, 3a), this is:

(6a)

(6b)

When combined and re-written, this becomes the Inverse E-LT:

Divide the first equation by D x' and the second by D t' and substitute D x'=cD t', per (3a) and the 2nd principle:

(7)

Changing the sign of either c or v in (7) produces two roots for any value of v. When c and v are aligned (e.g., +v,+c or -v,-c) denoted by the subscript "AL", the first root of E-LT is:

(7a)

When c and v have opposite signs, denoted by subscript "OP", the second root is:

(7b)

Quite different roots indeed.

The dual results of E-LT can be confirmed in another way. When D x is wavelength of light and D t is period of those light waves, equation (7) is identical to Einstein's Doppler equations and written in that form in some texts, c.f., [5]. Doppler results are well known to vary with orientation of c and v, with one root for an approaching source (blue-shift) and a second root for a receding source (red-shift). Those dual results contradict the type of linearity assumed in SRT derivations.

6. Kinematics vs. kinetics: The assumed kinematics, by definition, ignores effects of forces such as gravity -- a situation found nowhere in this universe. If photons have mass [12], SRT is not applicable anywhere in the known universe.

Together, these shifts provide ample evidence to reject SRT since the assumed type of light-use is required in applications such as Michelson-Morley and optical Doppler tests.

7. Algebraic Relationships We first define two new terms "G _AL" and

"G _OP":

(8a, 8b)

Using simple algebra, the expression g =(1-v²/c²)^-1/2 can be re-written as the product of those terms:

Einstein's Lorentz transformation in (7a, 7b) can also be expressed in terms of those factors as:

It is seen that E-LT_OP is the inverse of E-LT_AL. So, reversing either c or v in one produces the other.

Einstein's Doppler equation [6] for an observer on the 'stationary' frame is:

for an emitter at frequency n approaching the observer. This is the same as E-LT_AL. For a receding source, Doppler frequency ratio is G _OP/G _AL which is E-LT_OP. So, E-LT and Einstein's Doppler equations are identical when light-use is assumed.

Using algebra only, the product of g and E-LT_AL is seen to be the corresponding G ²:

(9a)

(9b)

These relationships will be used in Section 11 and other ways [7]. The G ² terms will be seen later to be the Newtonian solutions to SRT and Doppler equations.

8. View of Fig, 4 using expanded notation: We look now in more detail at the case where v and c are aligned in the same direction, as shown to the right in Fig. 4. The light-wave travel Ad_P seen by P on her own frame K is:

Ad_P = cD t_Ad,P = vD t_Ad,P + A'B'_P

or

(c - v)D t_Ad,P = A'B'_P (10a)

where D t_Ad,P is the time of light-travel over Ad_P. For a consistent view by P from frame K, relativistic length A'B'_P is used rather than the proper length A'B'_Q seen by Q. Note that A'B'_P is quite different than A'B'_Q -- a difference that is unseen with imprecise notation. As in Section 4 for +c, +v orientation, P observes that light-travel A'B'_P is less than Ad_P, and hence that D t'_A'B',P is less than D t_Ad,P [8]. P says this is inconsistent with the 2nd principle and hence that A'B'_P and D t'_A'B',P must have contracted in some way.

Where c and v are opposed as shown to the left in Fig. 4, in the view of P, A'E'_P is larger than Ae_P and, together with its corresponding elapsed-time D t_A'E',P, are said by P to be dilated, i.e., enlarged. Equation (10a) becomes:

Ae_P = cD t_Ae,P = A'E'_P - vD t_Ae,P

or

(c + v)D t_Ae,P = A'E'_P (10b)

where D t_Ae,P is the time of light-travel over Ae_P.

9. General equations for SRT and Doppler: For cases where c & v are aligned (e.g. +v, +c), dividing (4a) by (10a) produces:

(11a)

Using (8a, 2a, 3a) and rearranging:

(11b)

For cases where c and v are opposed (e.g., -c, +v), dividing (4b) by (10b) produces:

(11c)

(11d)

Equations (11a) and (11c) are each comprised of three ratios which are assigned names as follows:

The generalized expression for SRT equations can now be written simply:

G ² = (D-ratio)(R-ratio) (12a)

or

G ²/(D-ratio) = (R-ratio) (12b)

The relative proportions of the D-ratio and R-ratio vary for differing assumptions, but their product must always be G ² as long as light use and 2nd principle are assumed.

n '_Q = c/l '_Q = c/A'B'_Q = 1/D t'_Q = 1/T '_Q (13a)

and is the same (in the eyes of Q) regardless of magnitude or sense of v seen by P. The reduced wavelength l ' seen by P when the source is approaching P (i.e., +c, +v) is shown to the right in Fig. 5, and l ' for receding source (-c,+v) to the left. The frequency seen by P when v and c are aligned is:

n '_P = c/l '_P = c/A'B'_P = 1/D t'_P = 1/T '_P (13b)

where A'B'_P is defined in (10a). When v and c are opposed, frequency n '_P is:

n '_P = c/l '_P = c/A'E'_P = 1/D t'_P = 1/T '_P (13c)

When the 2nd principle and light-use apply, these equations can be combined with (11a, 11c) to produce generalized equations for Doppler frequency ratio in the form:

n '_P/n '_Q = G ²/(D-ratio) = (R-ratio)

For c & v aligned, that is, an approaching source:

(14a)

For c & v opposed, that is, a receding source:

(14b)

These equations are more precise than other known SRT equations when light is used.

Before leaving Doppler, let's look at results if non-relativistic Newtonian physics were assumed. Variable light speeds are thereby possible and D t'_A'B',Q would equal D t_Ad,P or D t_Ae,P and (14a,14b) then reduce to classical Newtonian solutions [10]:

n '_P/n '_Q = c/(c-v) or n '_P/n '_Q = c/(c+v)

as expected. These are also G ²_AL and G ²_OP.

11. Three attempts to solve the general SRT and Doppler equations: We now test to see if a viable solution of (14a, 14b) can be reached using three possible relationships of the D-ratio and R-ratio:

• "D-ratio = R-ratio = G "

• "D-ratio = g , R-ratio = E-LT "

• "D-ratio = E-LT, R-ratio = g "

In this we assume the D-ratio and the R-ratio are equal because there are no known reasons why relativistic effects should differ with magnitude of length or elapsed time. That is, there's no logical reason why the longer length Ad_P would contract in a different ratio to reach A'B'_Q than the shorter length A'B'_Q contracts to reach A'B'_P. Since the product of those two ratios is G ², each ratio equals G . For c & v aligned, (14a) and (11b) produce:

(15a)

and for c & v opposed, (14b) and (11d) produce:

(15b)

The above equations could also have been reached using this same logic with the early equations [11] in Einstein's 1905 derivation. Then, no further kinematic SRT or Doppler equations would have been needed. But, that would have left the fatal flaw of a need for simultaneous contraction and dilation of the moving length [9]. A 2nd flaw is that this Doppler ratio (n '_P/n '_Q) is equal to either G _AL or G _OP which contradicts Einstein's Doppler equations shown in Section 7 to be a ratio of G _AL to G _OP. Assuming that his Doppler equations match actual Doppler data, we can reject this solution on both counts.

It was shown in (9a, 9b) that the product of g and E-LT equals G ² so g could be the one of the D- and R- ratios and E-LT the other. But in which order? This test assumes the above listed order; the next test assumes the reverse order. For aligned c & v, the D-ratio is:

D t_Ad,P/D t'_A'B',Q = Ad_P/A'B'_Q = g

and the R-ratio, from (14a, 7a) is:

A'B'_Q/A'B'_P = E-LT_AL = g (1+v/c) = n '_P/n '_Q

For c & v opposed, the D-ratio is:

D t_Ae,P/D t'_A'E',Q = Ae_P/A'E'_Q = g

and the R-ratio, from (14b, 7b) is:

A'E'_Q/A'E'_P = E-LT_OP = g (1-v/c) = n '_P/n '_Q

At first, this seems an attractive solution, since both length contraction and time dilation (LC&TD) and E-LT equations are included, and the R-ratio which is also the Doppler ratio is equal to E-LT -- the same solution reached by Einstein. But the fixed value of g for the D-ratio for all orientations of v and c contradicts the presumed universality of the 2nd principle. The unchanging gamma assumes that Ad_p = Ae_P which is easily refuted by inspection of Fig. 4. So this solution and its equations above can be rejected.

The reversal of assignments of g and E-LT to the D- and R- ratios in (13a, 13b) produces results as follows. For aligned c & v, the D-ratio is:

D t_Ad,P/D t'_A'B',Q = E-LT_AL = g (1+v/c)

and the R-ratio (which is also Doppler) is:

A'B'_Q/A'B'_P = D t'_A'B',Q/D t'_A'B',P = g = n '_P/n '_Q

and for c & v opposed, the D-ratio is:

D t_Ae,P/D t'_A'E',Q = E-LT_OP = g (1-v/c)

and the R-ratio is:

A'E'_Q/A'E'_P = D t'_A'E',Q/D t'_A'E',P = g = n '_P/n '_Q

The above equations for R-ratio fail to provide the required changes in Doppler shift with orientation of c & v. That is because g purposely ignores such orientation requirements. Since Doppler frequencies clearly must shift with orientation, these equations can also be rejected.

Additional details and understanding of all three of the above tested solutions can be gained from the numerical examples in Table 1.

All tested alternative solutions have serious conflicts when the travel of light is consistently used to relate lengths and times. Since that use of light is required within the context of the 2nd principle and in optical Doppler tests and in Michelson-Morley tests, those flaws indicate that SRT itself is seriously flawed. Past unawareness of such flaws is attributed to inadequate control of assumptions and imprecise notation. A remedy seems to lie in the use of more precise notation, and perhaps developing a better understanding of the experimental data which underlie SRT.