### 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.

1. Calculation of t1
• l = v*t1 + c *t1
• t1 = l / (c+v)
2. Calculation of t2
• l + v*t2 = c *t2
• t2 = l / (c-v)
3. Calculation of t3
• 2l = v*t3 + c *t3
• t3 = t2 + 2l / (c+v)
4. 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.

1. Calculation of t1
• l + v*t1 = c *t1
• t1 = l / (c-v)
2. Calculation of t2
• l = v*t2 + c *t2
• t2 = l / (c+v)
3. Calculation of t3
• 2l + v*t3 = c *t3
• t3 = t2 + 2l / (c-v)
4. 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.

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.

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.

1. 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.
2. 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
3. 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.
4. 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.
5. 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.
6. 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:

1. First we have a rod Lx at rest.
The reflection time in this case is: 2 * Lx / C
2. 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)
3. 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.

Reflection time in rest frame t1 = 2*l/c
2. 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)
3. 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))
4. 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
5. 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
6. 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)
7. 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)
8. 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