## Clock Rate of a Moving Clock

### Question

What is the clock rate of a moving clock (of the moving twin), compared to the clock rate of the staying home clock (of the at staying home twin).

### Background

Starting point of this exercise are two Observers A and B. Observer A will stay at home and Observer B will move at an uniform velocity v.
Observer A has a clock which will flash at the moments A1, A2, A3 etc.
Observer B his clock flashes at home at the same rate as A's clock and during the trip at the moments B1, B2, B3 etc.
A's flashes are received by B at the moments a1, a2, a3 etc. B's flashes are received by A at the moments b1, b2, b3 etc.

Based on the postulate "That all inertial obeservers are equivalent" neither A nor B can establish which one is actual moving. This requires that the recorded moments when B receives A's flashes and when A receives B's flashes should be the same. That means that the dependence (ratio) a1 to B1 should be equal b1 to A1.

### Calculation of the clock rate. part 1

B moves at 0.5 c
A's clock flashes at 2/c 4/c 6/c
In the drawing below the points A1 and A2
(in the rest of the story I will remove c)
Coordinate A1 = (x,t) = (0,2)
Coordinate A2 = (x,t) = (0,4)
B's clock flashes at the points B1,B2,B3
```   t = time
^
.                 B2
.                .
A2              a1
.             ..
b1          . .
. .       .  .
.   .   .   .
.     .    .
.   .   . .
. .      B1
A1      .
.      .
.     .
.    .
.   .
.  .
. .
..
O................  >x

Figure I

```
(Angle a1,A1 with horizontal and Angle b1,B1 with horizontal should be 45 degrees)

B will receive flash A1 at point a1
coordinate of a1 = (x,t) = (2,4)
A will receive flash B1 at point b1
coordinate of B1 = (x,t) = (x,2x)
coordinate of b1 = (x,t) = (0,3x)
x = unknow, to be calculated

In order that A receives the first flash "simultaneous" with B receiving the first flash from A (in the sense that the time of A's clock is the same as B's clock) the following equation has to be true

Ob1 / OA1 = Oa1 / OB1 = ratio
or
3x / 2 = 2V5 / xV5 = 2 / x
x² = 4/3
x = V 4/3 V = sqrt

Coordinate of B1 = (V 4/3,2V 4/3)
coordinate of b1 = (0, 3V 4/3) ratio = 3/2 V 4/3

### Calculation of the clock rate. part 2

Someone can reply and remark: the above calculation is wrong. Suppose B moves at 0.25 c to the right and A moves at 0.25 c to the left. Their relative speed is 0.5c, which is the same as the previous example.

A's clock flashes at the points A1, A2 and A3 etc
Coordinate A1 = (x,t) = (-a,4a)
Coordinate A2 = (x,t) = (-2a,8a)
B's clock flashes at the points B1,B2,B3

```                  t = time
^
.                 .                 .
.                .                .
b1              .               a1
..             .             ..
. .           .           . .
.  .         .         .  .
.   .       .       .   .
.    .     .     .    .
A2    .   .   .     B2
.      . . .      .
.       .       .
.    . . .    .
. .   .   . .
A1   .    B1          C
.   .   .
.  .  .
. . .
...
O................  >x

Figure II

```
(Angle a1,A1 with horizontal and Angle b1,B1 with horizontal should be 45 degrees)

B will receive flash A1 at point a1
coordinate of a1 = (x,t) = (x,4x)
A will receive flash B1 at point b1
coordinate of B1 = (x,t) = (a,4a)
coordinate of B2 = (x,t) = (2a,8a)
a and x = unknow

In order that A receives the first flash "simultaneous" with B receiving the first flash from A (in the sense that the time of A's clock is the same as B's clock) the following equation has to be true

Ob1 / OA1 = Oa1 / OB1 = ratio = x / a

B1C = b, a1C = 4b (b = temp variable)

A1C = a1C = 2a + b = 4b or
2a = 3b
Also a + b = x or b = x - a
2a = 3b = 3(x-a)
3x = 5a
ratio = 5/3
compare this result with calculation 1

### Remark

What the figure I shows is that neither the moments A1,B1 (when the light flashes) nor the moments a1,b1 (when the flashes are received)are simulataneous.
In figure II the moments A1,B1 and the moments a1,b1 are simultaneous.

The mathematics I have used IMO is correct, that is not so much the problem

The question is: which of the above calculations is correct ? or are the both wrong ? or are they both right, each based on different assumptions ?

IMO only a real experiment can decide.

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