After understanding the concepts of relativity and knowing that time and space are not absolute as Newton told us instead they follow Einstein's theory of special relativity.
Now I am going to go in depth ocean of time and space and let us figure out why these two simple known words are so difficult to understand and believe me it's really difficult and it requires deep imagination.
So, today's chapter is about simultaneity. is everything in this universe happening simultaneously. If no then why or if yes then with respect to whom.
7.1 The Relativity of Simultaneity
In order to understand the concept of simultaneity let us imagine we are calmly sitting in our spaceship which is in deep space and suddenly 2 other spaceships approach from opposite direction and pass each other as shown. They have equal velocities. Just as they pass we set off bolts of energy at points x1 and x2 simultaneously from our point of view. But are they simultaneous to the passengers in those spaceships?
Considering passenger who is in middle of spaceship he is not going to conclude that explosives were simultaneous because of the fact that during the time that light from x1 and x2 moves toward them at speed c, they move somewhat to the right because it is going to take some time for light to reach the eyes of passengers and also light is moving towards one and away from another.
So the passengers in spaceship A is going to conclude we were half way between x1 and x2 and we saw light from x2 first.
Now we can see just that change of Frame of reference changes the reality
7.2 Clock synchronization in single frame of reference
The way to synchronize two or more clocks are as follows.
We put 2 clocks and observers A and B beside each clock and measure the distance between them. Now we keep one flashbulb in the halfway of two clocks. Both the clocks are earlier synchronized and they have agreed to set their clocks at t=0 when the flash arrives. The flashbulb is then turned on and it is going to take equal time to reach both of the clocks and thus clocks are synchronized. This method can be use to synchronize any number of clocks
7.3 In the very process of synchronizing Two Clocks, a moving observer disagrees
Imagine we are sitting in train and train is moving with uniform velocity v towards left relative to two clocks A and B are being synchronized in frame of ground with the help of method discussed above. To us clocks move in right and the distance between them is contracted to
(D(✓1-v²/c²)) where D is the distance between two clocks. Now it is clear that from our frame of reference clock A intercepts the flash before clock B because A is moving towards the light source and B away from it.
This means clock A is ahed of clock B. And this proves that they are unsynchronized to us.
Now let's calculate the lag in time between two clocks from our point of view. To begin with, suppose our clocks read t=0 just as bulb fires. Between stages one and three of figure both clock A and left hand flash have moved as shown let the time they meet be t3 on our clock so light has moved a distance ct3 and clock A has moved a distance ct3, which together amount to half the distance between two clocks which is
(D/2)(✓1-v²/c²) and that is ,
ct3 +vt3=(D/2)(✓1-v²/c²)
t3=(D✓1-v²/c²)/2(c+v)
At stage four when light reaches clock B, suppose our clock reads t=t4 the right hand flash has traveled ct4 and clock B has traveled ct4. Clock B started a distance (D/2)(✓1-v²/c²) to right of flashbulb so
ct4=vt4 +(D/2)(✓1-v²/c²)
And therefore
t4= (D✓1-v²/c²)/2(c-v)
The time difference between the stages three and four is
∆t= t4-t3
∆t= Dv(1-v²/c²)/(c²-v²)
As measured by our clocks. Clock A is moving to us, so it runs slowby the factor (✓1-v²/c²) during time interval ∆t, so to us it will read time
t(A)=∆t(1-v²/c²)
= Dv(1-v²/c²)/(c²-v²)
t(A)= vD/c²
This means from our point of view clock A is set to zero first and by time clock B is set to zero, clock A has already advanced to time t(A).
In simple words moving clocks Will be out of synchronism by an amount vD/c² from our point of view.
Now I am going to go in depth ocean of time and space and let us figure out why these two simple known words are so difficult to understand and believe me it's really difficult and it requires deep imagination.
So, today's chapter is about simultaneity. is everything in this universe happening simultaneously. If no then why or if yes then with respect to whom.
7.1 The Relativity of Simultaneity
In order to understand the concept of simultaneity let us imagine we are calmly sitting in our spaceship which is in deep space and suddenly 2 other spaceships approach from opposite direction and pass each other as shown. They have equal velocities. Just as they pass we set off bolts of energy at points x1 and x2 simultaneously from our point of view. But are they simultaneous to the passengers in those spaceships?
Considering passenger who is in middle of spaceship he is not going to conclude that explosives were simultaneous because of the fact that during the time that light from x1 and x2 moves toward them at speed c, they move somewhat to the right because it is going to take some time for light to reach the eyes of passengers and also light is moving towards one and away from another.
So the passengers in spaceship A is going to conclude we were half way between x1 and x2 and we saw light from x2 first.
Now we can see just that change of Frame of reference changes the reality
7.2 Clock synchronization in single frame of reference
The way to synchronize two or more clocks are as follows.
We put 2 clocks and observers A and B beside each clock and measure the distance between them. Now we keep one flashbulb in the halfway of two clocks. Both the clocks are earlier synchronized and they have agreed to set their clocks at t=0 when the flash arrives. The flashbulb is then turned on and it is going to take equal time to reach both of the clocks and thus clocks are synchronized. This method can be use to synchronize any number of clocks
7.3 In the very process of synchronizing Two Clocks, a moving observer disagrees
Imagine we are sitting in train and train is moving with uniform velocity v towards left relative to two clocks A and B are being synchronized in frame of ground with the help of method discussed above. To us clocks move in right and the distance between them is contracted to
(D(✓1-v²/c²)) where D is the distance between two clocks. Now it is clear that from our frame of reference clock A intercepts the flash before clock B because A is moving towards the light source and B away from it.
This means clock A is ahed of clock B. And this proves that they are unsynchronized to us.
Now let's calculate the lag in time between two clocks from our point of view. To begin with, suppose our clocks read t=0 just as bulb fires. Between stages one and three of figure both clock A and left hand flash have moved as shown let the time they meet be t3 on our clock so light has moved a distance ct3 and clock A has moved a distance ct3, which together amount to half the distance between two clocks which is
(D/2)(✓1-v²/c²) and that is ,
ct3 +vt3=(D/2)(✓1-v²/c²)
t3=(D✓1-v²/c²)/2(c+v)
At stage four when light reaches clock B, suppose our clock reads t=t4 the right hand flash has traveled ct4 and clock B has traveled ct4. Clock B started a distance (D/2)(✓1-v²/c²) to right of flashbulb so
ct4=vt4 +(D/2)(✓1-v²/c²)
And therefore
t4= (D✓1-v²/c²)/2(c-v)
The time difference between the stages three and four is
∆t= t4-t3
∆t= Dv(1-v²/c²)/(c²-v²)
As measured by our clocks. Clock A is moving to us, so it runs slowby the factor (✓1-v²/c²) during time interval ∆t, so to us it will read time
t(A)=∆t(1-v²/c²)
= Dv(1-v²/c²)/(c²-v²)
t(A)= vD/c²
This means from our point of view clock A is set to zero first and by time clock B is set to zero, clock A has already advanced to time t(A).
In simple words moving clocks Will be out of synchronism by an amount vD/c² from our point of view.
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