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I | 300m | 300m | 300m
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A 400m B 500m C
A, B, C are passing each other in the horizontal x direction and are momentarily at location A. Each has a light source at rest with respect to itself. At that moment when they are together, they all turn on their lights. After a short time, the light from each has traveled at speed c from each in all directions for time t = 1 musec. I=ct=300m. Now B is 400m from A, C is 900m from A, and C is 500m from B. This is the physics. There is nothing else. Nothing about energy, momentum, etc. By Special Relativity (SR):
I2 = (ct)2 - x2 = (ct')2 - x' 2 = (ct'')2 - x'' 2 = . . .
(ct)2 = I2 + x2; (ct')2 = I2 + x' 2 ; (ct'')2 = I2 + x'' 2 ; . . .
u/c = x/ct; u'/c = x'/ct'; u''/c = x''/ct''; . . .
g' = (1/(1 - (u'/c)2))1/2 = ( (ct' 2) / ((ct' 2) - (ct' 2)(u'/c)2) )1/2 =
((ct' 2) / ((ct' 2) - x' 2) )1/2 = ((ct' 2)/I 2)1/2 = (ct’)/ I
x' = g(x + ut) = g(u/c)ct + gx
Since A has its light source at rest in its reference frame, x = 0 when comparing another reference frame moving with respect to A. Same applies to B and C for their light sources. Then the preceding equation becomes
x' = g(u/c) I
For A with respect to B,
xAB = 400m, I = 300m, tA= I/c = 1 musec, (ct)B = 500m, tB = (5/3) musec, u/c = 400/500, g = 500/300, xAB = g(u/c) I = (500/300) (400/500) 300 = 400m as illustrated below.
| \ ct'
I | \ I2 = (ct')2 - x'2 ; g = (ct')/ I; (u'/c) = x'/(ct'); x' = g(u/c)I
For A with respect to C,
xAC = 900m, I = 300m, tA = 1 musec, (ct)C = 948.683298m, tC = 3.162228 musec, u/c = 0.948683, g = 3.162278, xAC = g(u/c) I = 3.162278 • 0.948683 • 300 = 900m
For B with respect to C,
xBC = 500m, I = 300m, tB = 1 musec, (ct)C = 583.095189m, tC = 1.943651 musec, u/c = 0.857492, g = 1.943651, xBC = g(u/c) I = 1.943651 • 0.857492 • 300 = 500m
For B with respect to A,
xBA = -400m, I = 300m, tB = 1 musec, (ct)A = 500m, tA = (5/3) musec, u/c = (-400/500), g = (500/300), xBA = g(u/c) I = (500/300) (- 400/500) 300 = - 400m
For C with respect to B,
xCB = -500m, I = 300m, tC = 1 musec, (ct)B = 583.095189m, tB = 1.943651 musec, u/c = - 0.857492, g = 1.943651, xCB = g(u/c) I = 1.943651 • - 0.857492 • 300 = - 500m
For C with respect to A,
xCA = -900m, I = 300m, tC = 1 musec, (ct)A= 948.683298m, tA = 3.162228 musec, u/c = - 0.948683, g = 3.162278, xCA = g(u/c) I = 3.162278 • - 0.948683 • 300 = - 900m
The physics here is only about objects and light ‘wavefronts’ which have changed their location over time. That’s all. There is nothing about collisions, momentum changes, energy, etc. The quantities are location and time. Additionally, changes of location are called distances and changes of location (distances) during a given time are velocity (or speed). So we have distances, times, and the ratio of distance to time called speed or velocity. The above showed how these distances, times, and velocity are related in SR.
Notice that in “For A wrt to B” tA=1.0 musec and tB=(5/3) musec while in “For B wrt to A” tA=(5/3) musec and tB=1.0 musec. Each object has several different ‘times’ and if we had considered more objects traveling different distances we could have many more ‘times’.
There are two key equations in SR which are very similar to equivalent equations which can be derived for Newtonian Mechanics (NM).
The SR versions are
I2 = (ct')2 - x' 2
g = (ct’/I)
The NM versions are
I2 = (c’t)2 - x' 2
g = (c’t/I)
The quantity (ct’) equals exactly the quantity (c’t). Therefore, I2, x’2, and g will have the same values in both SR and NM.
However, t does not equal t’ and c does not equal c’ and this is where SR and NM differ.
In both SR and NM I equals the distance the light has traveled from its source, possibly spherically, at speed c for time t, i.e., I=ct. So t=I/c. This makes the light essentially a timing signal. The light was emitted from a source not moving in its reference frame. It moved a certain distance I at speed c during time t in all directions in that frame. From the point of view of another reference frame moving with respect to that reference frame, the light moving perpendicularly (call it vertically) moved not only the ‘vertical’ distance I but also the perpendicular distance x’ for a total distance of
H = (I2 + x’2)1/2
In NM, this obviously increased distance, H during the time t would be called something like c’, so the distance H would be
H = (c’t)
I2 = (c’t)2 - x' 2
the NM version of I.
SR insists (postulates, assumes, etc.) that the speed of the light be the same in both reference frames, i.e., that the ratio H/t be the same as I/t. Since H is greater than I, this is impossible physically. However, you can mathematically arbitrarily increase t (and call it t’ to differentiate it from t). But now this is mathematics, not reality. This is why the SR version is
I2 = (ct’)2 - x' 2
I2 = (c’t)2 - x' 2
It also accounts for the two different forms of the gamma function (g), although the numeric value is the same in both SR and NM, c’t=ct’. Of course, c≠c’ and t≠t’.
Notice that this mathematical ledgerdemain can be applied to any ‘timing signal’ of a known speed in its reference frame, whatever that speed is. I.e., you could have a Special Relativity where the timing signal travels at speed c=300 m/sec instead of 300 million m/sec and is assumed to also travel that same speed in all reference frames moving with respect to it.
Notice also that although light from a given source (just like anything else) is obviously traveling a different speed in reference frames moving with respect to the source’s, it is assumed (along with respect to everything else according to SR) to be moving at speed c from its source. This was known as the Ritz theory.
x' = g(u/c) I; (x’/ I) = g(u/c)
shows that x’, the distance the object travels, can be many times the distance I, that the light travels, even approaching infinity as u approaches c. All the above objects traveled a greater distance than the three lights did between the time they were together and when their lights had traveled 300m for 1 musec at speed c.
u’/c = x’/ct’ = x’/(x’ 2 + I2)1/2
easily shows why “nothing can go faster than light” in SR. The denominator will always be bigger so u'/c will always be <1.0.
In NM, (x’/I)=(x’/ct)=(v/c). In SR, (x’/I)=g(u/c)=(ct’/ct)(x’/ct’)=(x’/ct)=v/c which is why SR math can do these time-distance problems although based on erroneous (or wierdly altered, if you prefer) concepts of time and velocity. This wierd math is carried over into more complex topics such as momentum and kinetic energy and seems to work somewhat because of its ties to NM as shown here.