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Galilean vs Lorentz transformation equations

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arbol
#1
Dec13-07, 03:40 PM
P: 50
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.
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Mentz114
#2
Dec14-07, 03:02 AM
PF Gold
P: 4,087
The difference between the speed of light in the stationary system S and in the moving system S' is ...
Eh ?
What are you trying to say in this post ?
p4h
#3
Dec14-07, 07:50 AM
P: 37
299,822,258m/s - 299,792,458m/s = 29,800m/s
Think he's saying that c / the speed of light isn't a constant.. o.O

Edit: That was while using the Galilean Transformation. Though I'm still not sure what his purpose for this thread was.

And as far as I can tell, his calculations is correct.

HallsofIvy
#4
Dec14-07, 09:29 AM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,689
Galilean vs Lorentz transformation equations

The speed of light relative to any coordinate system is equal to the speed of light in any other coordinate system.
arbol
#5
Dec18-07, 02:04 PM
P: 50
Quote Quote by Mentz114 View Post
Eh ?
What are you trying to say in this post ?
I am trying to contrast the values of (x,t) along the stationary system S using the Galilean and the Lorentz transformation equations. But I was wrong when I used only the motion of light from (x',t') = (0m,0s) to (x',t') = (299,792,458m,1s) along the axis of the moving system S'. I need to use the motion of the ray of light from (0m, 0s) to (299,792,458m,1s) and back to (0m,2s). I'll get back to you when I do that. Thanks.
arbol
#6
Jan4-08, 12:30 PM
P: 50
Quote Quote by Mentz114 View Post
Eh ?
What are you trying to say in this post ?
I am trying to compare or contrast the result, under the same conditions, applying the Galilean and the Lorentz transformation equations. In the following post are my findings, as I understand them. In this post I have quoted A. Einstein in his work, On The Electrodynamics of Moving Bodies.

Let S = a stationary two-dimensional space-time coordinate system, and let S' = another stationary two-dimensional space-time coordinate system. Let the x'-axis of S' lie along the x-axis of S. Let each system be provided with a rigid meter stick and a number of clocks, and let the two meter sticks, and likewise all the clocks of the two systems, be in all respects alike.
Let S' move along the x-axis of S with a constant velocity v = 29,800m/s in the direction of increasing x, and let this velocity be communicated to the x-axis, the meter stick, and the clocks of the stationary system S. To any time t of the stationary system S there then will correspond a definite position of the x'-axis of the moving system S'.
Let the length be measured from the stationary system S by means of the stationary meter stick, and also from the moving system S' by means of the meter stick moving with it; we thus obtain the coordinates x and x' respectively. Further, let the time t of the stationary system S be determined for every point thereof at which there is a clock by means of light signals (the time value is determined by an observer stationed together with the clock at every point of the x-axis, coordinating the corresponding time with a light signal, given out by every event to be timed, and reaching him through empty space); similarly let the time t' of the moving system S' be determined for every point of the moving system S' at which there is a clock at rest relatively to that system by applying the same method of light signals between every point at which the clock is located.

Galilean Transformation Equations

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) and move back to the point (x'A,t'A) = (0m,2s) along the x'-axis of the moving system S'.

According to the Galilean transformation equations,

x' = x - v*t,

which implies that

x = x' + v*t',

and

t' = t

which is equivalent to

t = t'.

By these transformation equations, the ray of light moves 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' while it moves 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. Thus, the speed c of light along the x'-axis of the moving system S' is

c = (x'b - x'a)/(t'b - t'a)

= 299,792,458m/s

while the speed c of light along the x-axis of the stationary system S is

c = (xb - xa)/(tb - ta)

= 299,822,258m/s.

The difference between the speed c of light along the stationary system S and the moving system S' is

299,822,258m/s - 299,792,458m/s = 29,800m/s.

Also, the ray of light moves back from the point (x'b,t'b) = (299.792,458m,1s) to the point (x'A,t'A) = (0m,2s) along the x'-axis of the moving system S' while it moves back from the point (xb,tb) = (299,822,258m,1s) to the point (xA,tA) = (59,600m,2s) along the x-axis of the stationary system S. Thus, the speed c of light along the x'-axis of the moving system S' is

c = |x'A - x'b|/(t'A - t'b)

= |-299,792,458m|/s

= 299792,458m/s

while the speed c of light along the x-axis of the stationary system S is

c = |xA - xb|/(tA - tb)

= |59,600m - 299,822,258m|/s

= 299,762,658m/s.

The difference between the speed of light c in this case along the stationary system S and the moving system S' is

|299,762,658m/s - 299,792,458m/s| = 29,800m/s.

Along the x'-axis of the moving system S',

c = ((x'b - x'a) + |x'A - x'b|)/(t'A - t'a)

= 2*(299,792,458m)/2s

= 299,792,458m/s.

And along the x-axis of the stationary system S,

c = ((xb - xa) + |xA - xb|)/(tA - ta)

= (299,822,258m + 299,762,658m)/2s

= 599,584,916m/2s

= 299,792,458m/s.

The Lorentz Transformation Equations

Let the ray of light move from the point (x'a,t'a) = (0m,0s) to the point (x'b,t'b) = (299,792,458m,1s) and move back to the point (x'A,t'A) = (0m,2s) along the x'-axis of the moving system S'.

According to the Lorentz transformation equations,

x' = (x - v*t)/sqrt(1 - sq(v/c))

which implies that

x = x'*sqrt(1 - sq(v/c)) + v*t (1)

and

t' = (t - v*x/sq(c))/sqrt(1 - sq(v/c))

which implies that

t = t'*sqrt(1 - sq(v/c)) + v*x/sq(c) (2)

If we substitute the right side of equation (2) for t in equation (1), we get

x = (x' + v*t')/sqrt(1 - sq(v/c)) (3)

and if we substitute the right side of equation (1) for x in equation (2), we get

t = (t' + v*x'/sq(c))/sqrt(1 - sq(v/c)). (4)

By the transformation equations (3) and (4), the ray of light moves 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' while it moves 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. Thus the speed c of light along the x'-axis of the moving system S' is

c = (x'b - x'a)/(t'b - t'a)

= 299,792,458m/s

while the speed c of light along the x-axis of the stationary system S is also

c = (xb - xa)/(tb - ta)

= 299,822,259.481m/1.00009940704s

= 299,792,458m/s.

The difference between the speed c of light along the x-axis of the stationary system S and the moving system S' is

299,792,458m - 299,792,458m = 0m/s.

This means that while the ray of light moves from x' = 0m to x' = 299,792,458m during the interval of time from t' = 0s to t' = 1s along the x'-axis of the moving system S', the same ray of light moves from x = 0m to
x = 299,822,259,481m during the interval of time from t = 0s to t = 1.00009940704s. This means that

1s (of the moving system S') = 1.00009940704s (of the stationary system S), and that

299,792,458m (of the moving system S') = 299,822,259.481m (of the stationary system S).

Thus the definition of the second and the meter of the stationary system S had to be altered in order that

c = (xb - xa)/(tb - ta)

= 299,822,259.481m/1.00009940704s

= 299,792,458m/s,
in spite of the fact that our premise was that "each system be provided with a rigid meter stick and a number of clocks, and that the two meter sticks, and likewise all the clocks of the two systems, be in all respects alike." In other words, the result of the Lorentz transformation equations contradicts our given condition.

Also, the ray of light moves back from the point (x'b,t'b) = (299.792,458m,1s) to the point (x'A,t'A) = (0m,2s) along the x'-axis of the moving system S' while it moves back from the point (xb,tb) = (299,822,259.481m,1.00009940704s) to the point (xA,tA) = (59,600.0002945m,2.00000000988s) along the x-axis of the stationary system S. Thus, the speed c of light along the x'-axis of the moving system S' is

c = |x'A - x'b|/(t'A - t'b)

= |-299,792,458m|/s

= 299792,458m/s

while the speed c of light along the x-axis of the stationary system S is again

c = |xA - xb|/(tA - tb)

= |59,600.0002945m - 299,822,259.481m|/(2.00000000988s - 1.00009940704s)

= 299,762,659.481m/0.9999006084s

= 299,792,458m/s.

The difference between the speed of light c in this case along the stationary system S and the moving system S' is also

299,792,458m/s - 299,792,458m/s = 0m/s.

This means again that while the ray of light moves from x' = 299,792,458m to x' = 0m during the interval of time from t' = 1s to t' = 2s along the x'-axis of the moving system S', the same ray of light moves from
x = 299,822,259.481m to x = 59,600.0002945m during the interval of time from t = 1.00009940704s to
t = 2.00000000988s. This means that

1s (of the moving system S') = 0.99990060284s (of the stationary system S), and that

299,792,458m (of the moving system S') = 299,762,659.481m (of the stationary system S).

Thus the definition of the second and the meter of the stationary system S had to be altered again in order that

c = |xA xb|/(tA - tb)

= 299,762,659.481m/0.9999006084s

= 299,792,458m/s,
in spite again of the fact that our premise was that "each system be provided with a rigid meter stick and a number of clocks, and that the two meter sticks, and likewise all the clocks of the two systems, be in all respects alike." In other words, the result of the Lorentz transformation equations once again contradicts our given condition.

Along the x'-axis of the moving system S',

c = ((x'b - x'a) + |x'A - x'b|)/(t'A - t'a)

= 2*(299,792,458m)/2s

= 299,792,458m/s.

And along the x-axis of the stationary system S,


c = ((xb - xa) + |xA - xb|)/(tA - ta)

= (299,822,259.481m + 299,762,659.481m)/2.00000000988s

= 599,584,916m/2.00000000988s

= 299,792,458m/s.

In conclusion, by the Lorentz transformation equations, any ray of light moves in the stationary system S with the determined velocity c, whether the ray be emitted by the stationary system S or the moving system S', if and only if the definition of the second and the meter of the stationary system S is altered, contradicting the given condition that the definition of the second and the meter of the stationary system S is in all respects alike to the definition of the second and the meter of the moving system S.
Mentz114
#7
Jan4-08, 04:33 PM
PF Gold
P: 4,087
Hi,
if you are using one numerical example to make your point, I have to tell you I won't even read it. To demonstrate something in general you must use algebra, not numeric calculations which are always subject to imprecision.

M
arbol
#8
Jan9-08, 01:49 PM
P: 50
S = a stationary two-dimensional space-time coordinate system, and S' = another stationary two-dimensional space-time coordinate system. The x'-axis of S' lies along the x-axis of S, and each system is provided with a rigid meter stick and a number of clocks, and the two meter sticks, and likewise all the clocks of the two systems, are in all respects alike. Moreover, S' moves along the x-axis of S with a constant velocity v = 29,800m/s in the direction of increasing x, and to any time t of the stationary system S there then will correspond a definite position of the x'-axis of the moving system S'.

The points A'(0m,0s), D'(299,792,458m,0s), and E'(299,822,258m,0s) on the x'-axis of the moving system S' are transformed to the points A(0m,0s), D(299,792,458m,0s), and E(299,822,258m,0s) respectively on the x-axis of the stationary system S by the Galilean transformation equations

x = x' + v*t'

and

t = t'.


A ray of light moves from the point A'(0m,0s) to the point D'(299,792,458m,1s) along the x'-axis of the moving system S' while the point D'(299,792,458m,0s) on the x'-axis of the moving system S' itself moves from the point D(299,792,458m,0s) to the point E(299,822,258m,1s) along the x-axis of the stationary system S by the Galilean transformation equations. The ray of light takes the duration t'd - 0s = 1s to move through a distance x'd - x'a = 299,792,458m with the speed c = 299,792,458m/s along the x'-axis of the moving system S', and simultaneously during that same period te - 0s = 1s, the point D'(299,792,458m,0s) on the x'-axis of the moving system S' itself moves through a distance xe - xd = 29,800m with the velocity v = 29,800m/s along the x-axis of the stationary system S. At the time t'd = te = 1s, the ray of light is located at the point D'(299,792,458m,1s) on the x'-axis of the moving system S' (and the point D'(299,792,458m,1s) itself is located at the point E(299,822,258m,1s) on the x-axis of the stationary system S).
arbol
#9
Jan15-08, 01:31 PM
P: 50
Quote Quote by Mentz114 View Post
Hi,
if you are using one numerical example to make your point, I have to tell you I won't even read it. To demonstrate something in general you must use algebra, not numeric calculations which are always subject to imprecision.

M
Thank you for your feedback. In the following version I am using algebra.

S = a stationary two-dimensional space-time coordinate system, and S' = another stationary two-dimensional space-time coordinate system. The x'-axis of S' lies along the x-axis of S, and each system is provided with a rigid meter stick and a number of clocks, and the two meter sticks, and likewise all the clocks of the two systems, are in all respects alike. Moreover, S' moves along the x-axis of S with a constant velocity v in the direction of increasing x, and to any time t of the stationary system S there then will correspond a definite position of the x'-axis of the moving system S'.

By the Galilean transformation equations


x = x' + v*t' (1)


and


t = t' (2),


the points (x'd, t'D), (x'e, t'E), and (x'd, t'd) on the x'-axis of the moving system S' are transformed to the points
(xd, tD), (xe, tE), and (xe, te) on the x-axis of the stationary system S respectively.

xd = x'd,


tD = t'D = tE = t'E = 0s,


xe = x'e = x'd + v*t'd, and


te = t'd.


A ray of light moves from the point (x'a, t'a) = (0m, 0s) to the point (x'd, t'd) along the x'-axis of the moving system S' while the point (x'd, t'D) on the x'-axis of the moving system S' moves from the point (xd, tD) to the point (xe, te) along the x-axis of the stationary system S.

If a particle moves along an axis from the point (xa, ta) to the point (xd, tD + t), then the interval of time that the particle takes to move from one point to the other is t = (tD + t) - tD.

The duration of the ray of light to move from the point (x'a, t'a) to the point (x'd, t'd) along the x'-axis of the moving system S' is
t'd - t'D, where t'd = t'D + t', and the duration of the point (x'd, t'D) on the x'-axis of the moving system S' to move from the point (xd, tD) to the point (xe, te) along the x-axis of the stationary system S is te - tE, where te = tE + t.

The ray of light moves through the distance x'd - x'a along the x'-axis of the moving system S' during the interval of time t', and the point (x'd, t'D) on the x'-axis of the moving system S' moves through the distance xe - xd along the x-axis of the stationary system S during the interval of time t.

The unit of time in the moving system S' is the second, which is the duration of 9,192,631,770 vibrations of the radiation (of a specified wavelength) of a cesium atom.

From equation (1),


x = c*t = x' +v*t' (3).

If we keep c = 299,792,458m/s constant, equation (3) implies that

t = ((x' + v*t')/c)s.

And from equatuon (2) we can infer that

in the stationary system S, (t/t')s = the duration of 9,192,631,770 vibrations of the radiation (of the same wavelength) of the same cesium atom.

The unit of length in the moving system S' is the meter, which is c*((1/299,792,458)s)m.

Thus from equation (1), we can also infer that

((x' + v*t')/x')m = c*((1/299,792,458)s)m.

In conclusion, if we keep c constant, the two meter sticks, and likewise all the clocks of the two systems, are not in all respects alike.


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