I'd like to draw you some spacetime diagrams that illustrate how Special Relativity handles moving observers that all measure the speed of the same flash of light even though they are moving at different speeds relative to the light source but the speed you presented will require a huge diagram so I'll do it with smaller speeds.I never understand why the speed of light is the same for all observers irrespective of their motion relative to the source of light. Now suppose I am sitting at the back of a vehicle which is travelling at the speed of 0.999999999999c and light approaches me from behind the vehicle. i.e. I am going away from the source of light while I can still see the light. Now I attach an instrument to my car for measuring the speed of light. Won't it measure 0.000000000001c.
Please help me understand why will the instrument still record 0.999999999999c and not 0.000000000001c according to the theory of relativity.
As I said in post #9, the way you measure the speed of a flash of light that is coming from behind you while you are sitting in the back of a vehicle is to start a stopwatch when you first see the flash of light coming from behind and then let it reflect off of a mirror some measured distance, I'll use six feet, in front of you and stop the timer when you see the reflection.
Here is a spacetime diagram depicting this scenario starting with you stationary with respect to the light source. You are shown as the blue line and your mirror is shown as the green line. The dots mark off one-nanosecond increments of your time. The flash of light is shown as the black line:
When the flash reaches you, you reset your stopwatch to zero. It takes six nsecs for the flash to continue on to the mirror and another six nsecs for the reflection to get back to you. You stop the stopwatch at 12 nsec. Since the light had to travel double the distance to the mirror, you calculate the speed of light to be 12 feet divided by 12 nsecs or 1 foot per nanosecond.
Now we'll repeat the measurement but this time assuming that you are traveling at 60%c with respect to the same light source and according to the same reference frame as before. Gamma, γ, at this speed for you is 1.25 meaning that the dots marking off 1-nsec intervals of time will be stretched out to 1.25 times the Coordinate Time markings. In other words, the Coordinate Time is dilated from 12 nsecs in the first diagram to 15 nsecs in this diagram. Special Relativity says that the distance to your mirror will be contracted by the reciprocal of gamma or 0.8 times what it was at rest. Since it was 6 feet at rest, the distance to your mirror now will be 4.8 feet of Coordinate Distance as you can see in the following diagram:
Many people think that you measure the same speed for light because both the distance and the time are reduced by the same factor and "cancel" each other out but the exact opposite is what is true as you can see in the diagram. And it's important to use the Coordinate Time because that is the frame in which the light travels at c. Because the mirror is moving away from the location where you started the stopwatch, it takes longer for the light to reach the mirror (although you have no awareness of this). And because you are moving towards the location of the reflection, it takes less time for the reflection to get back to you (again, you have no awareness of this). So, in fact it is a distance expansion divided by the Time Dilation factor that cancels each other out and results in the speed of light continuing to be the same. The distance expansion is shown in the diagram as the sum of 12 feet for the light to get from you to your mirror and 3 feet for the reflection to get from the mirror back to you for a total distance the light has to travel of 15 feet and it takes 15 nanoseconds resulting in a speed of 1 foot per nanosecond. It is important to realize that the distance to your mirror must contract in order for the measurement to come out the same. If it didn't, the light would have to go farther in both directions taking longer and you would get a smaller measurement for the speed of light.
But, as I said before, you have no awareness of this Time Dilation from 12 nsecs to 15 nsecs. To you, it is still 12 nsecs because that's what your stopwatch measures. And you have no awareness that in this frame, the distance to your mirror is closer to you than it was while you were at rest because any ruler that you use to measure the distance is contracted by the same amount.
Now let's repeat the measurement for you traveling a little faster, at 80%c. At this speed, gamma is 1.667 (one and two-thirds). Here is a new spacetime diagram:
As you can see, the Time Dilation has grown to 20 nsecs and your mirror is now 3.6 feet away and the distances the light has to travel have increased to 18 feet going and 2 feet returning for a total of 20 feet so the speed of light is 20 feet per 20 nsecs or 1 foot per nsec. Again, you have no awareness of these numbers. To you, it is still a total of 12 feet in 12 nsecs.
Have you noticed the trend in how the diagrams show that the mirror gets closer to you and the dots marking your 1-nsec tick marks get farther apart? Now, in an attempt to get as close to your desired speed as possible and still draw a decent diagram, I have made one more at a speed of 98%c where gamma is 5.025:
Notice that this is at a different scale from the previous diagrams. Notice that your tick marks are spaced slightly more than 5 nsecs apart and that the total time is expanded to 60.3 feet and distance to your mirror is contracted to less than 1.2 feet. The distances the light has to travel have also expanded to just under 60 feet going and just over 0.3 feet returning for a total of 60.3 feet resulting in a speed of 60.3 feet divided by 60.3 nsecs or 1 foot per nanosecond.
If we tried to make a diagram for your target speed of 0.999999999999c, gamma would be 707106.78 which would make the diagram incredibly large if you want to actually see the details. But you could do all the calculations and show that the Time Dilation, Length Contraction and distance expansion for the light paths would still result in the light speed being 1 foot per nanosecond. But again, you would have no awareness of any of this. It's no different for you than it was when you were at rest in the frame.
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