Time dilation/ light speed-thought experiments

[NSD 09997]

If you want to measure the speed of light without regard to the Theory of Special Relativity, you don't worry about a frame of reference. You just set an experiment which measures the round trip speed of light. This takes at least two photons. One going from you to a reflector and one coming back from the reflector to you. But since you will have no way of knowing when the first photon started out, or how much delay there was for the first photon to trigger the second photon, or how to know that the photon you sent out is the one that hit the reflector, it is not possible to measure even the round trip speed of a pair of photons. Nobody ever did that and nobody would ever try to do that. Instead, we use bright flashes of light which are easy to detect and we use long distances so that we can increase our accuracy.

The Theory of Special Relativity is where a frame of reference is defined and where the one-way speed of light (which includes photons) is defined to be c in all directions within anyone frame of reference you arbitrarily choose. Everything is in that one frame of reference, no matter where it is or how fast it is moving or accelerating. All light, including photons travel at c, by definition in that one frame of reference. If you want to consider a second frame of reference, you need to use the Lorentz Transform to convert all your time, location coordinates from your first FoR to your second FoR. That's what SR is all about.

You are picking out phrases that you don't understand and talking absolute nonsense. You need to learn what the these terms mean before you can ask meaningful questions.

I explained a lot of this in post #7. Please read and study that post. It has lots of answers for you.

Sorry, I actually skipped that... and at times I really speak weird! Apology...
But I'm reading it right now...

 

[ghwellsjr]

D H (a PF Mentor) Quotes: "
If you select a frame of reference to be moving at c with respect to another frame of reference, the Lorentz Transform will result in division by zero for the Lorentz Factor (gamma) which is not allowed and so the equation relating time between the two frames is meaningless."

Should the FoRs be linked in someway for the transformation to take place?

Why did you attribute this quote to "D H"? I'm the one who made that particular statement.

But to answer your question, there are many kinds of transformation but the easiest uses the "Standard Configuation". Remember, a frame of reference is nothing more than a coordinate system with one scalar value of time and one three-dimensional point in space. The four values of the coordinates are sometimes represented like this [4,6,3,-5] where the first value is time in seconds or microseconds or years or whatever you like and the next three values are the x, y and z spatial coordinates in light-seconds or light-microseconds or light-years or whatever, respectively, just make sure that you use the same time unit in the last part of the space units if you want to make the computations simpler. Each set of values is called an "event". An event doesn't have to have anything happening or anything present at any particular time/space coordinate, it's just the name used in SR to refer to the points in the FoR.

In the Standard Configuration you make sure the two FoR's that you want to transform between have their x, y, and z axes pointing in the same directions. Then you want to specify a speed, v, along the x-axis that represents the speed difference between the two FoR's. Just make sure that absolute value of v is less than c. Finally, you line up all four values of the coordinates so that their origins coincide in time and space, [0,0,0,0] in the first FoR is the same event as [0,0,0,0] in the second FoR.

For the actual transformation, look in the wikipedia article under the heading "Lorentz transformation for frames in standard configuration". If you use v as a fraction of c, then c in the equations drops out because it is equal to 1 so just remove the two places where you divide by c squared.

 

[abbott287]

DeG, Welcome to the forum. I like the way you think.

Let's start with the bottom line--your bottom line--where you said "it has been experimentally shown that light travels at a constant speed". I'm glad you affirm that. But think about how you do an experiment to measure the speed of light. You said something very important, "Light has to come to you for it to be detected by you." So let's say you are going to measure how long it takes for light to travel from you, where you have some sophisticated electronic timing gadgetry, to a target some accurately measured distance away from you. You start the timer when the flash of light is emitted and away it goes. Can you tell when it hits the target to stop the timer? I think you will agree the answer is "no". Once the light has left your vicinity, you have no knowledge of how it is progressing. What you can do is put a reflector at the target and have the light come back to you so that you can then detect it. Isn't that what you said earlier? Now you can calculate the average speed as being twice the distance divided by the time interval, correct? Does that make sense? But you will have to agree that you are not really measuring the speed of light because it could take longer to go from you to the target than it takes to get from the target back to you. How would you know?

Well that was the quandry that Michelson and Morley had when they were thinking about doing their experiment. Now they, along with everybody else at the time, believed that light only traveled at a constant speed relative to an ether which they thought permeated all of space. They knew that they could not measure the one-way speed of light but they assumed that it would be the same for both halves of their round-trip measurement of the speed of light as long as they were stationary in the ether.

But they also believed that the round-trip speed of light would change from the true speed as they moved through the ether and since they believed that the Earth moved through this ether it would cause the measured speed of light to change depending on how fast they were moving relative to the stationary ether.

The problem for them was that the technology of the time was not sufficiently precise enough to let them measure this very slight difference in the measured speed of light as the motion of the Earth changed during the day and during the seasons. But they figured out an ingenious work-around. They figured out that by comparing the round-trip speed of light along one axis with the round-trip speed of light at right angles to that axis, they should be able to detect very minute differences as a result of the Earth moving through the ether.

But they had a problem since they didn't know the stationary state of the ether. If both axes of their experiment happened to be at a 45 degree angle to the motion of the Earth through the ether, there would be no difference in the measured times no matter how fast they were moving through the ether. But if one axis were aligned along the motion of the Earth through the ether, they would see a positive difference and if the other axis were aligned they would see a negative difference. So to take advantage of the way these differences would show up, they made their entire apparatus capable of rotating slowly to maximize the positive and negative differences.

They were sure that during the course of a day and during the course of a year, they would be able to accurately identify the stationary state of the ether. But it always appeared that the Earth was always stationary in the ether. They actually concluded that the Earth was dragging the ether, just as you proposed, and suggested repeating the experiment at the top of a high mountain where presumably the drag would be smaller. But there were other reasons to reject their explanation and the one that held was that every non-accelerating person who measures the round-trip speed of light will get the same answer.

But how can this be if it is only when you are stationary with respect to the ether that you will get the same answer to the measured round-trip speed of light? Well, several scientists figured out that if your clocks start running slower as they travel with you through the ether and if your rulers (and everything else traveling with you) get shorter along the direction of motion through the ether, this would explain how it is possible to always measure the round-trip speed of light to be the same.

But they still had a problem: how to figure out the stationary state of the ether. Remember, it is only when stationary in the ether that the two halves of the round-trip of light will take the same time and the rulers are the correct lengths and the clocks run at the correct rate. This made for a very confusing and ambiguous science because there was no way of determining that stationary state of the ether.

What Einstein did was say, "forget about trying to find the stationary state of the ether, as long as you are not accelerating, you can assume that you are at rest in it. Your clocks will run at the correct speed, your rulers will have the correct length and most of all, the time it takes for light to travel from you to a target a fixed, measured distance away will equal the time it takes for the light to get back to you from the target."

You might recognize this last statement as Einstein's famous second postulate. It's very important in your understanding of Special Relativity to realize that Einstein was proposing something that could not and can not be measured, the time it takes for light to get from point A to point B. He is stating without any proof that it travels at c in anyone direction. Now you should not take this as a true statement in and of itself, it is only true within the context of the Theory of Special Relativity. It's what makes the Theory useful.

But you must also realize that it is true for only one assumed rest state of the ether at a time. But we don't call it that or refer in any way to ether, rather we call it a Frame of Reference.

So now let's apply what we have learned to your scenario. We will use the Frame of Reference in which both observers start at rest and the first one, A, remains at rest. The second observer, B, starts off at 0.6c in some direction. After 10 seconds, A emits a flash of light in B's direction when B is 6 light-seconds away from A. After another 10 seconds, B will be 12 light-seconds away from A and the light will be 10 light-seconds away from A so the light will not have yet reached B. So far so good. But in between these two statements you say:

Now if the light wave is traveling at the same speed relative to both of them, it should reach the second observer in just 6 seconds (he’s only 6 light seconds away).​

I think what you are saying here is that relative to A, the light must be traveling at 1.6c in order for it to be traveling at c relative to B. And after the second 10-second interval, the light will have traveled 16 seconds and therefore will have passed B because he has only traveled 12 light-seconds.

So now we have a contradiction in that after 20 seconds, the light has passed B from B's point of view but it has not passed B from A's point of view, is this what you are saying?

The problem with your thinking is that you are jumping between Frames of Reference without properly applying the Lorentz Transform (look it up in wikipedia). You need to describe the entire scenario all within one FoR, which is what you did but if you want to see what things look like in a different FoR, you need to correctly apply the Lorentz Transform to obtain a new set of coordinates to your scenario.

So in your original FoR, you had three events for A. We will describe these with a time coordinate followed by an x-dimension coordinate leaving off y- and z- since they are always zero. So the three events are:

[0,0] (start of scenario with A at location 0)
[10,0] (start of light flash at time 10 and location 0)
[20,0] (end of scenario, 10 seconds later with A still at 0)

You also had three events for B:

[0,0] (start of scenario with B at location 0)
[10,6] (when light flashes at time 10, B is at 6)
[20,12] (end of scenario, 10 more seconds later with B at 12)

And two events for the light:

[10,0] (start of light flash at time 10 and location 0)
[20,10] (end of scenario, 10 seconds later with the light at 10)

Now when we transform all these events into the FoR in which B is at rest, we get a new set of coordinates:

First for A:

[0,0] (start of scenario)
[12.5,-7.5] (start of light flash)
[25,-15] (end of scenario)

Three events for B:

[0,0] (start of scenario)
[8,0] (start of light flash)
[16,0] (end of scenario)

Two events for the light:

[12.5,-7.5] (start of light flash)
[17.5,-2.5] (end of scenario)

The first thing we want to notice is that since B is at rest in this new FoR, his x coordinates are alway 0. Next we can note that what took 20 seconds for the whole scenario to last now takes 25 seconds for A and 16 seconds for B. Before, in A's FoR, B was traveling at 12/20 = 0.6c, now in B's FoR, A is traveling at -15/25 = -0.6c. And we can see that the light is traveling from -7.5 light-seconds to -2.5 light-seconds so it never reaches B just like in A's FoR.

And finally, it is easy to see how the speed of light in B's FoR is c. The distance the light has traveled is -2.5 - (-7.5) = 5 light-seconds and the time in which this happened is 17.5 - 12.5 = 5 seconds. The speed of light is the distance divided by the time which is 5/5 = 1 light-second per second.

Now this may be very unsatisfying to you because as you pointed out at the beginning, "Light has to come to you for it to be detected by you" and since the light never got to B, how's he supposed to observe it traveling at c? Well you could let the scenario go on a little bit longer and the light will reach B but no matter how long you let it go on, unless you put a reflector in your scenario at some positive distance away from B and let the light reflect back to him, your statement will not be affirmed.

I think a more interesting scenario is to consider your same two observers but let the light flash at time zero when they are all together, then have a pair of reflectors, say 10 light-seconds away from each of them, one stationary with respect to A and one stationary with respect to B and watch how the round trip speed of the one light flash can be measured by both of them to be c. Note that you will have to pretend that A's reflector is semi-transparent in order for the same light flash to be both reflected and continue on to B's reflector. You can do this in the FoR in which A is stationary and then again in the FoR in which B is stationary and you will see how they both can measure the round-trip speed of light to be c, how they can both assume the one-way speed of light is c, how they each observe the other one to be length contracted and how they each can see the other one experiencing time dilation.

What a fantastic and clear headed post. Thanks for taking the time to write it all out. :)