Mystery of the Moving Car: Solving the Speed Paradox

[ghwellsjr]

Look at my second diagram above where the blue guy sends a signal at his time of 0.5 years. In the diagram, it arrives at the black guy before his clock reaches 1 year because his time is dilated in this frame. Do you agree with this or do you cling to the idea that the black guy's clock is at 1 year when the signal reaches him?

 

[choran]

Let me star with 1. Is diagram 1 indicating that the light reflection from the star will reach me, the black line traveler, at somewhat over 1 year? If so, we have to stop right there. I would say that when the light reaches the star, I will have traveled exactly halfway to the star. Are we in agreement on that? Light will bounce, and reach me in less than one year total. The time from Earth to bounce is not the same as bounce to me, because I have moved toward he star. But let's not even talk about the bounce. Let's just stop when the pulse reaches the star, OK?
Let me ask: How fast did the light travel? Well, relative to the guy on Earth who stayed behind, it traveled at c, and took one year to go 1 LY. Relative to me, the light traveled a total of 1LY minus the distance of .5 light year I traveled during the same time. Net gain for light=1/2 LY. Why is it incorrect to say that relative to me, light traveled at 1/2c from Earth to the star?

 

[Mentz114]

Let me star with 1. Is diagram 1 indicating that the light reflection from the star will reach me, the black line traveler, at somewhat over 1 year? If so, we have to stop right there. I would say that when the light reaches the star, I will have traveled exactly halfway to the star. Are we in agreement on that? Light will bounce, and reach me in less than one year total. The time from Earth to bounce is not the same as bounce to me, because I have moved toward he star. But let's not even talk about the bounce. Let's just stop when the pulse reaches the star, OK?
Let me ask: How fast did the light travel? Well, relative to the guy on Earth who stayed behind, it traveled at c, and took one year to go 1 LY. Relative to me, the light traveled a total of 1LY minus the distance of .5 light year I traveled during the same time. Net gain for light=1/2 LY. Why is it incorrect to say that relative to me, light traveled at 1/2c from Earth to the star?

(my emphasis)

I would say that the distance to the star is contracted your coordinates.

 

[Dale]

DaleSpam: In the line "You are just measuring two different things" is a valid answer for SR, but not for Newton…" If you have a sec, could you explain a bit more what you mean there? What would Newton say about the mirror box setup and the quantities being measured by the two observers.

Just what we were talking about before. In Newtonian physics the duration of a clock tick is invariant, so there is only one possible value in all frames. It is only in Einstein's relativity that the duration of a clock tick is a frame variant quantity. When you say "you are just measuring two different things" you are rejecting the Newtonian categorization (duration is the same in all frames) and accepting Einstein's (duration is different in different frames).

 

[choran]

Gotcha, Dale. That's what I thought you meant, just wanted to be sure. I can live with that. Still waiting for the headline: "Aether discovered! Scientists in an Uproar!" I checked the paper today. No news yet. LOL
What would happen though, in that rather unlikely situation? i guess SR would go, but GR, or large portions, would survive?

 

[choran]

Mentz, if so, did it contract to the point where my velocity relative to the pulse became zero in order to keep c invariant? Thanks.

 

[Mentz114]

Mentz, if so, did it contract to the point where my velocity relative to the pulse became zero in order to keep c invariant? Thanks.

Supposing I've understood the scenario correctly - there is no 'if', the distance is contracted.

It contracted just enough so that your distance/time result is c.

 

[phyti]

Well, still hard for me to internalize that one. Certain can see exactly what you are saying, it sounds good. Can't get over that at the exact moment A is measuring B's clock as slow, B at that exact moment is measuring A's clock as slow. It's not a matter of perspective--that would be easy to accept, as in the height analogy dvf gave above.
In the clock situation, we can't say "Freeze and let me back off and I'll tell you who's taller, as in the height analogy." Can't say "freeze, show your watches" and declare a winner. Not at all hard for me to see how in various real world scenarios A sees B's as slower. What is hard is the idea of symmetry, if that's the right word.

The concept (still talking SR only) that if A and B are moving relative to one another, A has just as much right to consider himself as stationary as B does, and vice versa, leading to what I just can't swallow. Rocket leaves earth--universe did not get up and move away from rocket. Earth did not pick up and go--the rocket left.
Muon example--Rocky Mountains did not rush up to meet the muon, no matter how much a conceited muon might
believe that. lol Rocky mountains did not become foreshortened, etc. Muon decayed at speed. Muon might measure my clock as slow, but the muon, I believe, would be wrong. Oh, well, back to the same old problem.
I think one has to go through physics boot camp, have a D.I. for several years, and then this stuff is internalized.
Otherwise, no shot. Thanks again, buddy.

refer to drawing:
A and B are moving at random speeds less than c. The hyperbolic curves cross each path indicating 1 time unit intervals. Each clock emits signals (blue) at 1 tu intervals. Begin at t = 1.
While diverging, the detection interval is greater than the emission interval, for A and B.
Begin at t = 1, run film backward or rotate drawing 180º.
While converging, the detection interval is less than the emission interval, for A and B.
As ghwells demonstrated, A and B are NOT observing time dilation, but doppler shifts, i.e. the relative frequency of signals between moving objects (and clocks are just frequencies). The clocks have to be compared simultaneously, and in close proximity, to determine their time dilation.

Can't get over that at the exact moment A is measuring B's clock as slow, B at that exact moment is measuring A's clock as slow.

As the space-time drawing shows, A and B are observing historical events at different times. If each sent a signal to the other at t1, it would return at t2. The red curves show each would measure the same round trip time, and observe the other clock at t = 1, i.e. symmetrical results.

The remainder of your post requires a refinement in definitions of propagation speed of
light, measured speed of light, events independent of observers, and perception of events by observers.
https://www.physicsforums.com/attachments/64425

 

[phyti]

That is a very clear explanation, I for some reason I have no problem with agreeing. When it is put that way, it seems (is?) quite uncontroversial and sensible. If I look at SR as simply being an issue of how to measure velocity given the relativity postulate and the light invariance postulate, I have very few issues. The idea that time is variable and observer dependent is what gets me. I still see these two things as different. I don't think there's anything that will ever get me to easily accept the notion that there is no absolute time. I know this probably makes no sense, but that's my difficulty. Maybe it's how SR is taught. I don't see time slowing, I see and can accept how its measure, if we use light, will vary.

Maybe it's also an issue of some of the basic examples, like the one we always see of the parallel mirrors on the train, where the light bounces straight up and down for the guy on the train, but traces a longer path (the hypotenuses of two triangles) for the ground-based observer. My reaction is simply to yell "Hey, what's the big deal, you are just measuring two different things, of course you will get different answers!"

You have the right idea. The light clock is the simplest example to demonstrate time dilation. In this clock, light moves vertically to a mirror and returns to a detector. After accumulating k signals, a tick (visual/audible) is produced. The signal is resolved into a horizontal x component which compensates for the speed of the clock and a y component which becomes the working part of the clock. Since c is constant, the y component must move at <c, i.e. the clock runs slower. Since the observer is a composite material object, moving with the clock, he also runs slower, and therefore does not detect any change in his clock.
Each observers clock uses an independent light source, and functions at a rate dependent on observer speed (relative to light, v/c).

 

[choran]

Thank you all for the responses. May I ask if you all concur with the post of Mentz114 re: distance contraction?
Having a bit of trouble with that one. Thanks again.

 

[Nugatory]

Thank you all for the responses. May I ask if you all concur with the post of Mentz114 re: distance contraction?
Having a bit of trouble with that one. Thanks again.

You asked "Mentz, if so, did it contract to the point where my velocity relative to the pulse became zero in order to keep c invariant?"

Mentz answered "It contracted just enough so that your distance/time result is c."

He's right, which is to say that it did not contract to to the point where your velocity relative to the pulse became zero. It contracted to the point where your velocity relative to the pulse was exactly c.

You calculate this by taking the total distance the flash traveled (out and halfway back) and dividing by the total time it was in flight (speed equals distance covered divided by time taken, right?) using your notion of time and distance.

The other guy does the same thing using his notion of time and distance. He gets the same result c, and he explains this by pointing out that you were dividing a contracted (smaller) travel distance by a shorter time interval.

 

[choran]

Thanks, Nug: In my example, light went 1 LY. I traveled 1/2 LY behind it, in same direction. No return beam, no additional facts. Are you saying "You didn't really travel 1/2LY" or are you saying "The light pulse didn't really travel 1 LY" or ? How far did I go, and at what velocity?

 

[Nugatory]

Thanks, Nug: In my example, light went 1 LY. I traveled 1/2 LY behind it, in same direction. No return beam, no additional facts. Are you saying "You didn't really travel 1/2LY" or are you saying "The light pulse didn't really travel 1 LY" or ? How far did I go, and at what velocity?

I'm sorry, I was describing the situation in one of the space-time diagrams where the light is reflected back from the destination planet so you meet it again mid-journey.

However, the same general logic applies for the one-way case you're asking about. Take the distance that the light travels according to you, divide by the time it took in flight, and you'll get c.

When figuring the distance you have to account for length contraction (for you, the distance between the two planets is contracted down to .886 light-years) and the motion of the destination planet towards you (it's approaching you at .5c, the light beam is moving away from you at c, net-net the light beam only has to cover half the contracted distance). You also have to account for time dilation, and when you do, you conclude that the light flash traveled .443 light-years in .443 years, so its speed was c. It takes both time dilation and length contraction to make it come out right.

Meanwhile, the guy at rest with respect to the two planets saw the light flash travel one light-year in one one year, so he also calculates a speed of c.

 

[choran]

OK, thanks Nagatory, I appreciate it.

 

[ghwellsjr]

Let me star with 1.

Ok, here's the first diagram:

Is diagram 1 indicating that the light reflection from the star will reach me, the black line traveler, at somewhat over 1 year?

Yes, both the Proper Time for you, the black traveler, and the Coordinate Time are somewhat over 1 year when the reflection gets back to you.

If so, we have to stop right there. I would say that when the light reaches the star, I will have traveled exactly halfway to the star. Are we in agreement on that?

Yes, but only in the Inertial Reference Frame (IRF) depicted in diagram 1 which is the rest frame of the motionless guy and the star. Any time you discuss "when" something happened remotely, you must specify the IRF in which it applies. Here is your rest frame during the trip:

In your rest frame, you are not halfway between the motionless guy and the star when the light reaches the star. It looks like the star is twice as far away from you as the guy on Earth is from you.

Light will bounce, and reach me in less than one year total.

You're going to have to show me how that happens. Can you draw a diagram to depict this?

The time from Earth to bounce is not the same as bounce to me, because I have moved toward he star.

That's true in the Earth's rest frame but the two times are equal in your rest frame. But then it's the star that is moving, not you.

But let's not even talk about the bounce. Let's just stop when the pulse reaches the star, OK?

OK, and we'll look at what happens in both of the above rest frames.

Let me ask: How fast did the light travel?

You stated in post #21 that you have very little issues with either of Einstein's two postulates. The second one states that light travels at c in any IRF so that is your answer.

Well, relative to the guy on Earth who stayed behind, it traveled at c, and took one year to go 1 LY. Relative to me, the light traveled a total of 1LY minus the distance of .5 light year I traveled during the same time. Net gain for light=1/2 LY. Why is it incorrect to say that relative to me, light traveled at 1/2c from Earth to the star?

It's wrong because you're combining statements about two different IRF's. You have to stick with one at a time. You can't use some parameters "relative to the guy on earth" and some parameters "relative to me" and do arithmetic on them and expect to come up with a valid conclusion. This is the source of virtually all of the so-called paradoxes in Special Relativity.

In the Earth's IRF as depicted in the first diagram, it is true that the light took 1 LY to reach the star. But relative to you as depicted in the second diagram, the light took 0.577 LY to reach the star.

The Lorentz Transformation is a result of Einstein's two postulates that you said you have very little issue with and that is how the second diagram was produced from the first diagram. So if you don't like the conclusion, then you have big issues with Einstein's two postulates. So which is it?