[Abstract]
This derivation of the Newtonian Mechanics (NM) transform equation from the equations of Special Relativity (SR) uses no assumptions other than the ‘correctness’ of the SR equations.
The derivation shows that SR velocity, time, and distance do not necessarily correspond with those of NM. However there are equivalents for them between SR and NM and these are shown.
The derivation of the NM transform equation from the equations of SR results in a transform equation that avoids ‘time dilation’, ’length contraction’, and does not involve the speed of light. NM and SR are one to one mathematical transforms of each other as shown.
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Assume the correctness of Special Relativity (SR) and that light and other electromagnetic radiation will be measured to travel the same speed c in all reference frames. Therefore the following equations of SR will be correct, and any equations mathematically derived from them are likewise correct. The transform equations of (SR) are:
1. x = 𝛾(x’ + ut’) = 𝛾(u/c)ct’ + 𝛾x’
2. ct = 𝛾ct’ + 𝛾(u/c)x’
3. x’ = 𝛾(x –ut) = –𝛾(u/c)ct + 𝛾x
4. ct’ = 𝛾ct – 𝛾(u/c)x
5. [Relativistic] Interval = I =((ct)2 – x2)1/2 =((ct’)2 – x’2)1/2 = , etc.
(as defined in Taylor and Wheeler, “Spacetime Physics” and derivable directly from Equations 1 through 4). As Eq. 5 shows, the relativistic Interval for any two events is the same in all reference frames. This Interval may henceforth be referred to as “I”.
6. 𝛾(i.e., gamma) = (1 – (u2/c2)) –(1/2), where u is the relativistic velocity between reference frames. x, t, and c are relativistic distance, time, and speed of electromagnetic radiation respectively.
By Eq. 5, the relativistic Interval=I=ct’ when x’=0; also I=ct when x=0. Therefore,
7. x = 𝛾(u/c)I when x’=0 by Eq. 1 and Eq. 5
8. ct = 𝛾I when x’=0 by Eq. 2 and Eq. 5
9. x’ = –𝛾(u/c)I when x=0 by Eq. 3 and Eq. 5
10. ct’ = 𝛾I when x=0 by Eq. 4 and Eq. 5
Eq. 7 can also be written as
11. x = 𝛾u(I/c) when x’=0
and Eq. 9 can be written as
12. x’ = –𝛾u(I/c) when x=0
Since the Interval for two events is the same in all reference frames in SR, we can henceforth refer to the (Interval/c) as “T”, using a capital letter to indicate that T is not frame specific. That is,
13. T = Interval/c = I/c
Why use T (for time) rather than some other capital letter? If the square of the Interval, I2, as defined in Eq. 5 is positive, the Interval is called “time-like” in SR (although the Interval itself is a distance) because when two events occur a time apart, there is always some actual or potential reference frame which can move sufficiently fast that in that frame both events occur at the same place (delta x or delta x’ equals zero). In that frame the two events are separated by only a time and not a distance. The relativistic (Interval/c) equals t’ exactly when delta x’=0; the relativistic (Interval/c) equals t exactly when delta x=0. Since, by SR, the Interval is the same in all reference frames for any two specific events, the Interval/c is therefore that time. All instances of one object moving relative to another fit into this category of a “time-like” Interval. (If two objects are not moving with respect to each other, then the Interval in that frame is “space-like”—the square of the Interval is negative—and is of course the same “space-like” Interval in all other reference frames moving with respect to that frame.)
So now we can write
14. x = 𝛾u(I/c) = (𝛾u)T when x’=0 and
15. x’ = –𝛾u(I/c) = –(𝛾u)T when x=0
This means that ACCORDING TO SR the distance one point [i.e., x’=0 or x=0] of any reference frame [e.g., call it frame A] travels through any other reference frame [call it frame B] moving with respect to the first [frame A] is (𝛾u)T where T is the Interval/c and this Interval is the same in all reference frames for any two particular events. One point [of frame A] moves the distance (𝛾u)T [through frame B] and the other point [of frame B] moves in the opposite direction the distance MINUS (𝛾u)T [through frame A]. Since the Interval/c is the time (no distance involved) according to SR that one point of a frame moved through another frame, then it is reasonable to say that
16. x/T = 𝛾u = v (for the point x’=0 from Eq. 11)
17. –x’/T = 𝛾u = v (for the point x=0 from Eq. 12)
where v is a velocity which shows how far an object [point of one reference frame, x’=0 or x=0] has traveled through another reference frame during the time T which ACCORDING TO SR is the same in all reference frames for any two particular events. This is what we mean by velocity—the distance traveled by a point of one frame through another divided by the time of that travel. This equals
18. v = 𝛾u
Thus we now can express an SR time which is the same in all reference frames and a velocity, v=𝛾u, which enables us to state that the distance an object [one point] travels through another reference frame is vT and the distance an object [one point] at rest in the latter frame travels through the first is –vT. Notice that if T is held constant the distance moved by the point of frame A throuh frame B is the SAME as the distance moved by the point of frame B through frame A (but in the opposite direction, of course). And this is ALL according to SR.
Compare this with using t, t’, u, and 𝛾 in SR where for x’=0
1. x = 𝛾(ut’ + x’) or
x = 𝛾ut’ + 0 or
19. x=𝛾ut’
but for the same time t’, x=0 falls in the prime frame at
1. x = 𝛾(ut’ + x’) = 𝛾(u/c)ct’ + 𝛾x’ or
0 = 𝛾ut’ + 𝛾x’ or
20. x’ = –(𝛾ut’)/𝛾
meaning that at time t’, x’=0 moved the distance 𝛾ut’ through the unprimed frame while x=0 is at only the distance (1/𝛾) times that distance in the primed frame in the opposite direction. This is because the Interval is no longer the same. Obviously Eqs. 14 and 15 are much simpler to visualize for points [x’=0 or x=0] moving through each other’s reference frame than Eqs. 19 and 20, even though both methods are according to SR.
Eq. 7 and Eq. 9 are fine for when x’=0 and x=0, respectively, but what if both those values are not zero? These transform equations are incomplete. We must have equations that cover the situation where both distance values are non-zero. We need to find what function of x’ and x respectively need to be added to make the equations correct for values of x’ and x respectively not equal to zero.
When x’=0,
7. x = 𝛾(u/c)I
So the value of x when x’≠0 is the value of x when x’=0 plus an additional amount, an unknown function of x’, which we will label f(x’). Rewrite Eq. 7 as
21. x = 𝛾(u/c)Interval + f(x’)
Likewise, when x=0,
9. x’ = –𝛾(u/c)I
and the value of x’ when x≠0 is the value of x’ when x=0 plus an additional amount, an unknown function of x, which we will label h(x). Rewrite Eq. 9 as
22. x’ = –𝛾(u/c)Interval + h(x)
Our job here is to find the value of the functions f(x’) and h(x). Using Eqs. 21 and 22, we find that
23. x - f(x’) = 𝛾(u/c)Interval = h(x) - x’
or
24. x - f(x’) = h(x) - x’
f(x’) is a function only of x’ and h(x) is a function only of x. Setting x’=0, f(x’)=0 and
25. h(x)=x
Setting x=0, h(x)=0 and
26. f(x’)=x’
and thus the full **SR** Eqs. 21. and 22. become
27. x = 𝛾(u/c)Interval + x’ and
28. x’ = –𝛾(u/c)Interval + x
Of course, Equations. 27 and 28 are the same! So substituting for (𝛾u) and (I/c), Eqs. 18 & 13, we now have
29. x = vT + x’ or X = vT + X’
where T is as derived for Eq. 13.
THIS OF COURSE IS THE SAME AS THE GALLILEAN TRANSFORM EQUATION OF NEWTONIAN MECHANICS (NM)!!! However, I have derived it only and directly from the equations of SR. Eq. 29 is true at all values of v=𝛾u, Eq. 18, not just at slow speeds. It applies to all objects moving with respect to each other. v, v/c, and 𝛾(u/c) can approach infinity, in which case, u, the usual ‘velocity’ of SR, is approaching 1.0c.
Eq. 29 says nothing about light. v is simply the ratio of distance traveled through a reference frame by a point of another frame during the frame-independent SR (Interval/c)=T (Eq. 13) which, by definition is the time on a clock not moving in its reference frame (i.e., delta(X)=0 or delta(X’)=0). v therefore has nothing to do with the speed of light. (I/c)=(ct’)/c=t’ when delta(X’)=0 and (I/c)=(ct)/c=t when delta(X)=0.
Notice also that if T=0, X=X’. There is no length ‘contraction’. X is the sum of how far the origin X’=0 traveled from X=0 during the frame independent time function I/c=T, plus the value of X’. For a given value of vT, changes in X equal exactly changes in X’ and these changes in X caused by changes in X’ are not affected by speed. (In contrast, when t or t’ are used instead of T, offsets in x’ or x affect x or x’ by the factor 𝛾 where 𝛾 is a function of speed. See Eq. 1 and Eq. 3.)
SUMMARY
All of this has been about objects (and the reference frames in which they are each ‘at rest’) moving relative to each other, which means that a point of one reference frame changes location in the other frame linearly with succeeding time. In SR this time is usually characterized by different values in the two reference frames (t and t’) but I have shown that this time can be characterized, ACCORDING TO THE RULES OF SR, as a single value for time in all reference frames, namely the SR (I/c)=T which is the same in all reference frames for any two given events. This is the same as the time on a non-moving clock read in its reference frame. It is also the time of everyday Newtonian Mechanics (NM) and in SR is called ‘proper time’. Furthermore, the ratio of the distance one point moves through the other frame to this frame-independent T=(I/c) is what we normally mean by velocity and equals v=𝛾u where 𝛾 is the “gamma” of SR defined in Eq. 6 and u is what is usually called velocity in SR. Since 𝛾 (gamma) is dimensionless, 𝛾u is equally valid to use as a velocity and this 𝛾u=v is the velocity of NM.
Thus Eq. 29, derived DIRECTLY and ONLY from SR, shows that the equations of NM are just as good as the usual ones of SR (in fact better and simpler) to describe the relationship of objects moving with respect to each other even at extreme speeds if we recognize that the NM time T in Eq. 29 equals
[((ct’)2 – x’2)1/2]/c = [((ct)2 – x2)1/2]/c = (I/c) = T
and that the NM speed v in Eq. 18 equals 𝛾u where u is the relativistic ‘velocity’ of SR.
RECAP
Look at the Equations 1 through 6 of SR. I won’t repeat them here.
7. x = 𝛾(u/c)Interval when x’=0
9. x’ = –𝛾(u/c)Interval when x=0
Define NM time T, constant in all reference frames, in terms of the SR Interval:
13. T = Interval/c
Define v, the velocity of NM, in terms of SR ‘velocity’ u:
16. x/T = 𝛾u = v (for the point x’=0 from Eqs. 7, 13, and 18)
17. –x’/T = 𝛾u = v (for the point x=0 from the Eqs. 9, 13, and 18)
Therefore
14. x = 𝛾u(I/c) = vT when x’=0 and
15. x’ = –𝛾u(I/c) = –vT when x=0
Eqs. 14 and 15 (and 7 and 9) are fine when x’=0 or x=0 but what if neither is zero? We must find f(X’) and h(X) in the following equations:
21. X = 𝛾(u/c)Interval + f(X’) = vT + f(X’) from Eq. 7
22. X’ = –𝛾(u/c)Interval + h(X) = –vT + h(X) from Eq. 9
Set X’=0
X = 𝛾(u/c)Interval + 0 = vT + 0
0 = –𝛾(u/c)Interval + h(X) = –vT + h(X)
Adding,
25. h(X) = X
Set X=0
0 = 𝛾(u/c)Interval + f(X’) = vT + f(X’)
X’ = –𝛾(u/c)Interval + 0 = –vT + 0
Again adding,
26. f(X’) = X’
Both Equations 21 and 22 then become
29. X = vT + X’
the Gallilean transform equation of NM, DERIVED FROM SR.
How to derive Newtonian Mechanics Directly
from Only the Equations of Special Relativity
Copyright August 1997, May 2006
Dr. Sherwood Kaip
El Paso, TX
This material may be reproduced if author attribution is given.