The Absurd Assertions of the SR experts - comment 1.
Mathematical discussion about The Absurd Assertions of the SR experts - comment 1.
The calculations are all done in the rest frame of the track.
The length of each moving train is 2l.
Calculation of t1, t2 and t3 for moving train to the right.
- Calculation of t1
- l = v*t1 + c *t1
- t1 = l / (c+v)
- Calculation of t2
- l + v*t2 = c *t2
- t2 = l / (c-v)
- Calculation of t3
- 2l = v*t3 + c *t3
- t3 = t2 + 2l / (c+v)
- Calculation of delta = t3-t1
- delta = t3 -t1 = l/(c-v) + 2l / (c+v) - l / (c+v)
- delta = t3 -t1 = (l * (c+v) + 2l * (c-v) - l (c-v)) / (c²-v²)
- delta = t3 -t1 = (l * (c+v) + l * (c-v)) / (c²-v²) = 2lc /(c²-v²)
Calculation of t1, t2 and t3 for moving train to the left.
- Calculation of t1
- l + v*t1 = c *t1
- t1 = l / (c-v)
- Calculation of t2
- l = v*t2 + c *t2
- t2 = l / (c+v)
- Calculation of t3
- 2l + v*t3 = c *t3
- t3 = t2 + 2l / (c-v)
- Calculation of delta = t3-t1
- delta = t3 - t1 = l/(c+v) + 2l / (c-v) - l / (c-v)
- delta = t3 - t1 = (l * (c-v) + 2l * (c+v) - l (c+v)) / (c²-v²)
- delta = t3 - t1 = (l * (c-v) + l * (c+v)) / (c²-v²) = 2lc /(c²-v²) = (2l/c)*/(1-v²/c²)
Conclusion
The above calculations are done in the track frame.
The calculations measure the difference in arrival for the Observers A' and F for two lightsignals at t1 and t3.
What the calculations show, in the train frame, is that both observers A' and F measure the same arrival time (2l/c)*/(1-v²/c²) * SQR(1-v²/c²) if they both reset their clocks at t1.
The factor SQR(1-v²/c²) is due because the moving clocks in the train frame run slower with a factor SQR(1-v²/c²).
This arrival time is equal to (2l/c)* / SQR(1-v²/c²).
Assuming length contraction then the arrival time is equal to 2* l0 / c. L0 is the length in the track frame.
This value is half the value if you sent a signal from A', which is reflected to F' back to A'. The following sketch demonstrates this.
/ / x t3 \
/ / \. \
/ / \ . \
/ / \ . \
t3 x / \ . \
/ . / \ . \
/ . / \ . \
/ . / \ . \
/ x t2 t1 x . \
/ ./ \. . \
/ . / \ . . \
/ . / \ . .\
t1 x . / \ . x t2
/ . . / \ . . \
/ . / \ . \
/ . / \ . \
/ . / t0. \
/ . / \ \
/ . / \ \
/. / \ \
t0. / \ \
A' F' F A
------> <-------
What the above sketch also shows is that the moment t1 is half ways between t3 and t0.
The Absurd Assertions of the SR experts - comment 2.
Mathematical discussion about The Absurd Assertions of the SR experts - comment 2.
Changing Length - of a moving rod with speed v
The following sketch shows a moving rod XY to the right in the rest frame of the track with has a speed v to the right
The purpose of this chapter is to calculate the length of the rod by an observer X at the end of the rod XY.
In order to calculate the length the observer will send out a light signal to the front of the rod. This signal will reach the front at point A at tA. At A the signal will be reflected. This signal will reach the observer X at point C at tC.
. | / /
. / /
| . / /
| . C /
| / . /
| / . /
B----/---------A
| / . /
| / . /
| / . /
|/ . /
X=========Y----
- The length of the rod XY
- lxy = l0 * SQR(1-v²/c²)
- The time tA going from X to A is given by the following equation:
- lxy + v*tA = c*tA --> tA = lxy /(c-v)
- The time going from A to C is given by the following equation:
- lxy = v*t + c*t --> t = lxy / (c+v)
- The time tC going from X to C is given by the following equation:
-
tC= lxy /(c-v) + lxy/(c+v) = lxy * (2c/(c²-v²)) = (lxy*2/c)/(1-v²/c²)
or tC = l0 * SQR(1-v²/c²) * (2c/(c²-v²))
- The time tC going from X to C as measured by the moving Observer X:
-
tC = l0 * SQR(1-v²/c²) * (2c/(c²-v²)) * SQR(1-v²/c²)
tC = l0 * (1-v²/c²) * (2c/(c²-v²)) = l0 * ((c²-v²)/c²) * (2c/(c²-v²))
tC = l0 * (2c/c²) = 2* l0 /c
That means the length of the rod lxy measured by the moving Observer X = l0
Changing Length - of a moving rod with speed 0
The following sketch shows a rod ZX at rest in the rest frame of the track. The rod has a speed v to the left relative from rod XY
The purpose of this chapter is to calculate the length of the rod ZX by an observer X at the end of the rod XY.
In order to calculate the length the observer will send out a light signal to the front of the rod. This signal will reach the front at point A at tA. At A the signal will be reflected. This signal will reach the observer X at point C at tC.
| | . / . / / /
| | . / . / / /
| | C / / /
| | . / . // /
| | . / . / /
| | . / ./ /
| | . / / / . /
| . / / / .
| . | / / / ./
| . | / / / . /
| . | / / . /
| . | / / ./ /
A | B / . / /
| . | / / . //
| . | / / . /
| . | / / . //
| . |//. / /
--Z==============X---------W--Y----
- The length of the rod ZX
- lzx = l0
- The time tA going from X to A is given by the following equation:
- lzx = tA*c --> tA = lzx / c
- When the light signal reaches A the Observer X is at B
- The length lAB = lzx + v * tA = lzx *(1+v/c)
- The time going from A to C is given by the following equation:
- lAB + v*t = c*t --> t = lAB / (c-v) = lzx/c * (c+v)/(c-v)
- The time tC going from X to C is given by the following equation:
-
tC= lzx / c + lzx/c * (c+v)/(c-v)
tC= lxc/c *(1 + (c+v)/(c-v))
tC= lxc/c * 2c/(c-v) = 2 * lxc /(c-v) = 2 *l0 / (c-v)
- The time tC going from X to C as measured by the moving Observer X:
- tC = 2 * l0 /(c-v) * SQR(1-v²/c²)
- tC = 2 * l0 / c * SQR(1-v²/c²) / (1-v/c)²
- tC = 2 * l0 / c * SQR(1+v/c) / (1-v/c)
Changing Length - of a moving rod with speed 2v
The following sketch shows a moving rod XW to the right in the rest frame of the track with has a speed 2*v to the right
The purpose of this chapter is to calculate the length of the rod XW by an observer X at the end of the rod XY.
In order to calculate the length the observer will send out a light signal to the front of the rod. This signal will reach the front at point A at tA. At A the signal will be reflected. This signal will reach the observer X at point C at tC.
| | . / . / / /
| | . / . / / /
| | C / / /
| | . / . // /
| | . / . / /
| | . / ./ /
| | . / / / . /
| . B / / A
| . | / / / ./
| . | / / / . /
| . | / / . /
| . | / / ./ /
. | / / . / /
| . | / / . //
| . | / / . /
| . | / / . //
| . |//. / /
--Z--------------X=========W--Y----
- The length of the rod XW
- lxw = l0 * SQR(1-4v²/c²)
- The time tA going from X to A is given by the following equation:
- lxw + 2v*tA = c*tA --> tA = lxw /(c-2v)
- When the light signal reaches A the Observer X is at B
- The length lAB = lxw + v * tA = lxw *(1+v/(c-2v)= lxw * (c-v)/(c-2v)
- The time going from A to C is given by the following equation:
- lAB = v*t + c*t --> t = lAB / (c+v) = lxw *(c-v) /((c-2v)*(c+v))
- The time tC going from X to C is given by the following equation:
-
tC= lxw /(c-2v) + lxw * (c-v) /((c-2v)*(c+v)) = lxw/(c-2v) * (1 + (c-v)/(c+v))
or tC = lxw/(c-2v) * 2c/(c+v) = lxw* 2c /((c-2v)*(c+v))
- The time tC going from X to C as measured by the moving Observer X:
-
tC = l0 * SQR(1-4v²/c²)* 2c /((c-2v)*(c+v)) * SQR(1-v²/c²)
tC = 2*l0/c * SQR(1-4v²/c²)* 1/((1-2v/c)*(1+v/c)) * SQR(1-v²/c²)
tC = 2*l0/c * SQR(1-4v²/c²)/((1-2v/c)²* SQR(1-v²/c²)/(1+v/c)²
tC = 2*l0/c * SQR(1+2v/c)/((1-2v/c)* SQR(1-v/c)/(1+v/c)
Conclusion
If you combine the above two figures you get the following:
| | . / . / / /
| | . / . / / /
| | C / / /
| | . / . // /
| | . / . / /
| | . / ./ /
| | . / / / . /
| . B / / A
| . | / / / ./
| . | / / / . /
| . | / / . /
| . | / / ./ /
A | B / . / /
| . | / / . //
| . | / / . /
| . | / / . //
| . |//. / /
--Z--------------X=========W--Y----
The above sketch shows the following in the frame of the rod XY
- Rod XY. The length = l0
- Rod ZX. This rod moves to the left, relative from XW.
- tC = (2*l0/C) * SQR(1+v/c) / (1-v/c)
- Rod XW. This rod moves to the right, relative from XW.
- tC = (2*l0/C) * SQR(1+2v/c)/((1-2v/c)* SQR(1-v/c)/(1+v/c)
Because the two arrival times tC are different, the length of the rods ZX and XY are also different.
Results and Comments
For c=300000 km/sec l0= 100000km and v=30000km/sec the results are:
v=v v=0 v=2v
xy zx xw
tC .6700252 .7407407 .742269 rest frame
tC .6666666 .7370277 .738548 moving observer
The bottom line shows the time that the light signal reaches the moving observer at X for resp. v=v, v=0 and v=2v. The times are different. This implies that for the moving observer X the length of the moving rods are different.
Let us study the length of a moving rod XY with speed v in more detail.
- First we have a rod with a length l0. In order to measure its length we sent out a light signal at X which is reflected at Y.
- The reflection time tC at X is 2*l0 /c.
- Next we move the rod with a speed v to the right. Again we measure time at X in rest frame.
- We should expect 2*l0 /(c-v)
- However we measure 2 * l0 / (c-v)* SQR(1-v²/c²)
That means the moving rod is contracted
- Next we have a moving observer with speed v, measured in the rest frame. This moving observer sends a lightsignal and measures the reflection time in the rest frame. What does he or she measures:
- a) tC = 2*l0 /(c-v)
- b) or tC = 2 * l0 / (c-v)* SQR(1-v²/c²) i.e. length contraction.
- tc = (2 * l0 / c)* SQR(1+v/c) / SQR(1-v/c)
-
IMO it is a i.e. no length contraction.
- Next we have a moving observer with speed v, measured in the rest frame. This moving observer sends a lightsignal and measures the reflection time but now in her own reference frame i.e. with a moving clock. What does he or she measures:
- a) tC = 2*l0 /(c-v) * SQR(1-v²/c²) or
- b) tC = 2 * l0 / (c-v) * SQR(1-v²/c²) * SQR(1-v²/c²) i.e. length contraction.
- tc = (2 * l0 / c) (1+v/c)
-
IMO again it is a i.e. no length contraction.
- If a moving observer wants to measure the length of a moving rod with the clock in the rest frame again we have two options. What does he or she measures:
- a) tC = (2*l0/c)/(1-v²/c²)
- b) or tC = (2*l0/c)/(1-v²/c²) * SQR(1-v²/c²) i.e. length contraction.
- tc = (2 * l0 / c) /SQR(1-v²/c²)
-
IMO again it is b i.e. length contraction.
- If a moving observer wants to measure the length of a moving rod with a moving clock i.e. in the frame of the moving observer this becomes:
- a) tC = (2*l0/c)/(1-v²/c²)* SQR(1-v²/c²)
- b) or tC = (2*l0/c)/(1-v²/c²) * SQR(1-v²/c²) * SQR(1-v²/c²) i.e. length contraction.
- tc = 2 * l0 / c
-
The last line implies that a moving observer with a moving clock does not measure length contraction.
In order to study the length of a moving rod three things are important:
- First we have a rod Lx at rest.
- The reflection time in this case is: 2 * Lx / C
- Secondly we have a rod Lx moving to the right with a speed v.
- The reflection time for an Observer at rest is: tB = 2 * tA = 2 * Lx / (c-v)
- Third we have a rod Lx at rest and a moving Observer to the right.
- The reflection time is : 2 * Lx + v * tA = c * tA
- tA = 2 * Lx / (c-v)
B / | A
| . / | . /
| . / | . /
| . / | . /
| . / | . /
A . | . /
| . / | . /
| . / .|. /
| . / . | . /
| . / . | . /
--X---------Y Y---------X---------Y
Observer X at Rest Rod XY at Rest
Moving rod to right Moving Observer Y to right
Case 2 Case 3
The arrival times of the signals in case 2 and 3 seem to be identical, but are they realy ?
- In case 2 there is length contraction.
- That means what we should measure is 2 * L0 * SQR(1-v²/c²) / (c-v)
- In case 3 there is time dilation.
- That means what we should measure is 2 * L0 / (c-v) * SQR(1-v²/c²)
That means case 2 and 3 are the same.
Comments 2
- Start with a rod at rest with length l.
Reflection time in rest frame t1 = 2*l/c
- Move a rod with length l' and speed v.
Reflection time in rest frame t2 = 2*l'/(c-v)
Length contraction gives l' = l * SQR(1-v²/c²)
Reflection time in rest frame t2 = 2*l*SQR(1-v²/c²)/(c-v)
-
Reflection time in rest frame t2 = (2*l/c) * c * SQR(1-v²/c²)/(c-v)
Reflection time in rest frame t2(as f of t1) = t1 * c * SQR(1-v²/c²)/(c-v)
Reflection time in rest frame t2(as f of t1)= t1 * SQR((c+v)/(c-v))
- Move a rod with length l' and speed v.
Reflection time in moving frame t3 = l'/(c-v)+ l'/(c+v)
Reflection time in moving frame t3 = 2*l'*c /(c²-v²)
This gives 2*l'= t3 * (c²-v²)/c
- Length contraction gives l' = l * SQR(1-v²/c²)
Clock in moving frame runs slower with factor SQR(1-v²/c²)
Reflection time in moving frame t3 = 2*l*SQR(1-v²/c²) * c * SQR(1-v²/c²)/(c²-v²)
Reflection time in moving frame t3 = 2*l*(1-v²/c²) * c /(c²-v²)
Reflection time in moving frame t3 = 2*l/c
- Consider rod with length l" and speed v to the left in moving frame v to the right.
This rod has a speed 0 in rest frame.
Reflection time in moving frame t4 = 2 * l"/(c-v)
- When length contraction is involved then l"=l'* SQR(1-v²/c²)
Reflection time in moving frame t4 = 2 * l'* SQR(1-v²/c²) /(c-v)
Reflection time in moving frame t4 = (2 * l' /c) * SQR (c+v)/(c-v)
Reflection time in moving frame t4 = t3 * (c²-v²)/c² * SQR (c+v)/(c-v)
- When length expansion is involved then l"=l'/ SQR(1-v²/c²)
Reflection time in moving frame t4 = 2 * l'/(SQR(1-v²/c²) *(c-v))
Reflection time in moving frame t4 = t3 * ((c²-v²)/c )/(SQR(1-v²/c²) *(c-v))
Reflection time in moving frame t4 = t3 * ((c+v)/c )/SQR(1-v²/c²)
Reflection time in moving frame t4 = t3 * SQR (c+v)/(c-v)
Item 3 shows what we measure for a rod which moves with a speed v relative in a rest frame, including length contraction.
Item 7 shows what we measure for a rod which moves with a speed v relative in a rest frame, including length contraction.
Those formulas should be identical, but they are not.
Item 8 shows what we measure for a rod which moves with a speed v relative in a rest frame, including length expansion.
Comparing item 3 with 8 the formulas should not be identical, but they are
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Created: 9 February 2001
Modified: 18 February 2001
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