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arbol
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Let S' = a stationary two-dimensional space-time coordinate system, and let the x'-axis of S' lie along the x-axis of another stationary two-dimensional space-time coordinate system S. Let S' move along the x-axis of S with a constant velocity v = 29,800m/s in the direction of increasing x.
Let a ray of light move from the point (x'a,t'a) = (0m,0s) to the point (x'b,t'b) = (299,792,458m,1s) along the x'-axis of the moving system S'. Thus the speed of light in the moving system S' is
(x'b - x'a)/(t'b - t'a) = 299,792,458m/s.
Using the Galilean transformation equations
xa(x'a,t'a) = x'a + v*t'a
= 0m + (29,800m/s)*(0s)
= 0m
ta = t'a = 0s
and
xb(x'b,t'b) = x'b + v*t'b
= 299,792,458m + (29,800m/s)*(1s)
= 299,822,258m
tb = t'b = 1s,
thus the ray of light moved from the point (xa,ta) = (0m,0s) to the point (xb,tb) = (299,822,258m,1s) along the x-axis of the stationary system S, and its speed in the stationary system S is
(xb - xa)/(tb - ta) = 299,822,258m/s.
The difference between the speed of light in the stationary system S and in the moving system S' is
299,822,258m/s - 299,792,458m/s = 29,800m/s (the velocity v with which S' moves along the x-axis of the stationary system S). The Earth orbits the Sun with the approximate velocity v = 29,800m/s.
Using the Lorentz transformation equations
xa(x'a,t'a) = (x'a + v*t'a)/sqrt(1 - sq(v/c))
= (0m + (29,800m/s)*(0s))/sqrt(1 - sq((29,800m/s)/c))
= 0m
ta(x'a,t'a) = (v*x'a + sq(c)*t'a)/sq(c)/sqrt(1 - sq(v/c))
= ((29,800m/s)*(0m) + sq(c)*(0s))/sq(c)/sqrt(1 - sq((29,800m/s)/c))
= 0s
and
xb(x'b,t'b) = (x'b + v*t'b)/sqrt(1 - sq(v/c))
= (299,792,458m + (29,800m/s)*(1s))/sqrt(1 - sq((29,800m/s)/c))
= 299,822,259.481m
tb(x'b,t'b) = (v*x'b + sq(c)*t'b)/sq(c)/sqrt(1 - sq(v/c))
= ((29,800m/s)*(299,792,458m) + sq(c)*(1s))/sq(c)*sqrt(1 - sq((29,800m/s)/c))
= 1.00009940704s,
thus the ray of light moved from the point (xa,ta) = (0m,0s) to the point (xb,tb) = (299,822,259.481m,1.00009940704s) along the x-axis of the stationary system S, and its speed in the stationary system S is
(xb - xa)/(tb - ta) = 299,792,458m/s.
The difference between the speed of light in the stationary system S and in the moving system S' is
299,792,458m/s - 299,792,458m/s = 0m/s (v = 0m).
But if we begin with v = 29,800m/s in the Lorentz transformation equations, why is it that
(xb - xa)/(tb - ta) - (x'b - x'a)/(t'b - t'a) = 0m/s? Moreover, the Earth does orbit the Sun with this constant velocity v = 29,800m/s.
By the Michelson-Morley experiment we have established that
(xb - xa)/(tb - ta) - (x'b - x'a)/(t'b - t'a) = 0m/s.
The puporse for the Michelson-Morley Experiment was to determine the value of v. In other words, the value of v was not given. Given the values of the point (x'b,t'b) on the x'-axis of the moving system S', and the values of the point (xb,tb) on the x-axis of the stationary system S from the Michelson-Morley Experiment, we were not able to determine the value of v using the Galilean transformation equations because the difference between the velocity of light in the stationary system S and the velocity of light in the moving system S' was 0m/s. Thus, the Lorentz transformation equations and the two Postulates of Einstein had to be developed. We can calculate the value of v using the Lorentz transformation equations
xb(x'b,t'b) = (x'b + v*t'b)/sqrt(1 - sq(v/c))
299,822.259.481m = (299,792,458m + v*(1s))/sqrt(1 - sq(v/c))
v = 29,800m/s.
We cannot use xa(x'a,t'a) = (x'a + v*t'a)/sqrt(1 - sq(v/c)) because this equation is an identity (both sides of the equation are 0).
We can follow the same argument if v = -29,800m/s, or S moves with v = 29,800m/s 0r -29,800m/s.
Let a ray of light move from the point (x'a,t'a) = (0m,0s) to the point (x'b,t'b) = (299,792,458m,1s) along the x'-axis of the moving system S'. Thus the speed of light in the moving system S' is
(x'b - x'a)/(t'b - t'a) = 299,792,458m/s.
Using the Galilean transformation equations
xa(x'a,t'a) = x'a + v*t'a
= 0m + (29,800m/s)*(0s)
= 0m
ta = t'a = 0s
and
xb(x'b,t'b) = x'b + v*t'b
= 299,792,458m + (29,800m/s)*(1s)
= 299,822,258m
tb = t'b = 1s,
thus the ray of light moved from the point (xa,ta) = (0m,0s) to the point (xb,tb) = (299,822,258m,1s) along the x-axis of the stationary system S, and its speed in the stationary system S is
(xb - xa)/(tb - ta) = 299,822,258m/s.
The difference between the speed of light in the stationary system S and in the moving system S' is
299,822,258m/s - 299,792,458m/s = 29,800m/s (the velocity v with which S' moves along the x-axis of the stationary system S). The Earth orbits the Sun with the approximate velocity v = 29,800m/s.
Using the Lorentz transformation equations
xa(x'a,t'a) = (x'a + v*t'a)/sqrt(1 - sq(v/c))
= (0m + (29,800m/s)*(0s))/sqrt(1 - sq((29,800m/s)/c))
= 0m
ta(x'a,t'a) = (v*x'a + sq(c)*t'a)/sq(c)/sqrt(1 - sq(v/c))
= ((29,800m/s)*(0m) + sq(c)*(0s))/sq(c)/sqrt(1 - sq((29,800m/s)/c))
= 0s
and
xb(x'b,t'b) = (x'b + v*t'b)/sqrt(1 - sq(v/c))
= (299,792,458m + (29,800m/s)*(1s))/sqrt(1 - sq((29,800m/s)/c))
= 299,822,259.481m
tb(x'b,t'b) = (v*x'b + sq(c)*t'b)/sq(c)/sqrt(1 - sq(v/c))
= ((29,800m/s)*(299,792,458m) + sq(c)*(1s))/sq(c)*sqrt(1 - sq((29,800m/s)/c))
= 1.00009940704s,
thus the ray of light moved from the point (xa,ta) = (0m,0s) to the point (xb,tb) = (299,822,259.481m,1.00009940704s) along the x-axis of the stationary system S, and its speed in the stationary system S is
(xb - xa)/(tb - ta) = 299,792,458m/s.
The difference between the speed of light in the stationary system S and in the moving system S' is
299,792,458m/s - 299,792,458m/s = 0m/s (v = 0m).
But if we begin with v = 29,800m/s in the Lorentz transformation equations, why is it that
(xb - xa)/(tb - ta) - (x'b - x'a)/(t'b - t'a) = 0m/s? Moreover, the Earth does orbit the Sun with this constant velocity v = 29,800m/s.
By the Michelson-Morley experiment we have established that
(xb - xa)/(tb - ta) - (x'b - x'a)/(t'b - t'a) = 0m/s.
The puporse for the Michelson-Morley Experiment was to determine the value of v. In other words, the value of v was not given. Given the values of the point (x'b,t'b) on the x'-axis of the moving system S', and the values of the point (xb,tb) on the x-axis of the stationary system S from the Michelson-Morley Experiment, we were not able to determine the value of v using the Galilean transformation equations because the difference between the velocity of light in the stationary system S and the velocity of light in the moving system S' was 0m/s. Thus, the Lorentz transformation equations and the two Postulates of Einstein had to be developed. We can calculate the value of v using the Lorentz transformation equations
xb(x'b,t'b) = (x'b + v*t'b)/sqrt(1 - sq(v/c))
299,822.259.481m = (299,792,458m + v*(1s))/sqrt(1 - sq(v/c))
v = 29,800m/s.
We cannot use xa(x'a,t'a) = (x'a + v*t'a)/sqrt(1 - sq(v/c)) because this equation is an identity (both sides of the equation are 0).
We can follow the same argument if v = -29,800m/s, or S moves with v = 29,800m/s 0r -29,800m/s.
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