Mystery of the Moving Car: Solving the Speed Paradox

[choran]

First of all, Happy Thanksgiving to you all! Now, my question:

I'm on Earth viewing through a telescope. I observe a car in outer space, traveling roughly at right angles to me (i.e., across my scope's field of view, let's say left to right). That car, a 1956 Chevy (irrelevant, but my favorite car) according to its speedometer, is traveling 1/2c. It turns on its headlights. He measures the speed of light emanating from his headlights as traveling c. I measure the light traveling from his headlights as c.

Question: What do I measure the car's speed as being?

Thanks.

 

[TumblingDice]

Happy Turkey Day to you! The question requires more information. When you provide a rate of speed you need to also mention, "relative to XXX.". I mention this just to be clear about the 'speedometer' you used to provide the measurement.

For this thought experiment, if the driver is measuring his speed relative to Earth, then you measure his speed at .5 c also. But you need to be clear that everything is relative. The speedometer is meaningless without any 'road' that it's measuring against. There is no 'absolute' speed or any preferred reference frame in relativity. It's important to gain that understanding before moving forward or nothing will make sense.

 

[Psychosmurf]

First of all, Happy Thanksgiving to you all! Now, my question:

I'm on Earth viewing through a telescope. I observe a car in outer space, traveling roughly at right angles to me (i.e., across my scope's field of view, let's say left to right). That car, a 1956 Chevy (irrelevant, but my favorite car) according to its speedometer, is traveling 1/2c. It turns on its headlights. He measures the speed of light emanating from his headlights as traveling c. I measure the light traveling from his headlights as c.

Question: What do I measure the car's speed as being?

Thanks.

That depends on the velocity of whatever the car is driving on. In that case you use the velocity-addition formula. If, however, the car is out in empty space, then its speedometer would have to measure its speed as 0, not 1/2 c.

 

[choran]

OK, gotcha. Let's say it's driving on a track suspended in space. The car's speed of 1/2c is relative to that track.
To us on earth, can we say that the track appears motionless? Thanks!

 

[choran]

Re: Psycosmurt.
Can we posit that the track upon which the car is driving is motionless with respect to earth, and the car moves .5 c relative to that track, and thus relative to earth?

 

[TumblingDice]

OK, gotcha. Let's say it's driving on a track suspended in space. The car's speed of 1/2c is relative to that track.
To us on earth, can we say that the track appears motionless? Thanks!

Motionless to you would mean the road is in your inertial reference frame. The speedometer reads .5 c relative to the road (and you). You measure the car and driver's speed at .5 c also, relative to you (and the road).

 

[choran]

Tumbling Dice! One of my favorite song! LOL
Well, I think you just made my buddy very happy. I was convinced that the earthbound guy would measure the car's speed as other than .5 c, for some reason. You are right, though--we seem to be in the same reference frame.
OK, then here's my question, now that my buddy's happy: Under what conditions, keeping the facts as close to the above as possible, would I measure the car as OTHER than .5 c?
Thanks

 

[Nugatory]

Re: Psycosmurt.
Can we posit that the track upon which the car is driving is motionless with respect to earth, and the car moves .5 c relative to that track, and thus relative to earth?

We can. And in that case you on the Earth measure the car's speed to be .5c relative to you, the earth, and the track, of course.

The speed of the light from the headlights of the car is c relative to the car and relative to you. This result is consistent with the relativistic velocity addition formula $w=\frac{u+v}{1+uv}$, where $u$ is the speed of the car relative to you, $v$ is the speed of something relative to car, and $w$ is the speed of that something relative to you.

 

[TumblingDice]

...now that my buddy's happy: Under what conditions, keeping the facts as close to the above as possible, would I measure the car as OTHER than .5 c?
Thanks

Well, one way would be to all out nit-pick and bring up gravitational fields, the rotation of the Earth, or other hard core calculations that neither of you considered as premises when you created the scenario.

If I understand the spirit of the question, the only way for you to measure different than .5c would be if the track was in constant motion relative to you, to keep it simple (so it remains an inertial reference frame w/o accelerating relative to you). In that case the addition of velocities calc would be used for the speedometer and track to obtain relative to you. The speeds DO NOT just add or subtract as intuition might lead you.

Here could be a way you could concede to the first scenario, but get a chance to regain poise with a second question. Say the track is moving at .25c relative to you, and the speedometer reads .5c. How fast do you measure the car as moving? Now that's a good question to learn more!

 

[choran]

Yes, that's the question: If the track is in motion relative to the observer on Earth at .25c, and the car is in relative motion to the track at .5 c, then the observer on Earth would measure the speed of the headlights at c, but measure the speed of the car as: ??
I assume I'd have to apply a time dilation formula of 1/square root of 1-v2/c2. I'm too old. Not worried about poise. LOL Do the math for me and I'll buy you a shot of Seagrams 7! LOL

 

[TumblingDice]

Nugatory posted the calc above when mentioning the speed of the headlamp beams. The value for c in this case is set to '1', so...
(.5 + .25) /( (1 + (.25 x .5)) =
.75 / 1.125 =

.666666 c !

If there's a math error, I get points for showing my work.

 

[choran]

OK, who do I send the Seagrams to? Thanks fellas, appreciate it! My buddy was right! OK, here's my last one, and it's for me. Don't yell, now! Remember, I'm old and not about to return to college--been there, done that, and had a fine career in another profession, now retired having a ball (you'll get there someday). BUT, I'm worried (can't enjoy my Seagrams tonight) because of Herbert Dingle (don't cringe!) and his idea that under SR, if A sees B's clock as slow, B will also see A's clock as slow. Talking straight SR now, no preferred frames. How can both be right, and what's the tiebreaker? When it's time for the showdown "Gentlemen, produce thy watches", whose watch is slow? Thanks for the help, and I admire your diligence in pursuing this most difficult of fields.
Thanks
CH

 

[Dale]

if A sees B's clock as slow, B will also see A's clock as slow. Talking straight SR now, no preferred frames. How can both be right, and what's the tiebreaker?

If both are inertial then there is no tiebreaker. B's clock runs slow in A's frame and A's clock runs slow in B's frame.

 

[Psychosmurf]

OK, who do I send the Seagrams to? Thanks fellas, appreciate it! My buddy was right! OK, here's my last one, and it's for me. Don't yell, now! Remember, I'm old and not about to return to college--been there, done that, and had a fine career in another profession, now retired having a ball (you'll get there someday). BUT, I'm worried (can't enjoy my Seagrams tonight) because of Herbert Dingle (don't cringe!) and his idea that under SR, if A sees B's clock as slow, B will also see A's clock as slow. Talking straight SR now, no preferred frames. How can both be right, and what's the tiebreaker? When it's time for the showdown "Gentlemen, produce thy watches", whose watch is slow? Thanks for the help, and I admire your diligence in pursuing this most difficult of fields.
Thanks
CH

Ask this question: Why can't they both be right? They are running two totally different experiments, and there is no reason at all a priori to think their measurements should be exactly the same. In fact, if spacetime had no geometry, their time measurements would be random (and meaningless), though there would be no reason to believe that one is any more right than the other. However, spacetime does have a geometry, but it turns out that this geometry is Minkowskian (that is, the time measurements of two inertial observers will be related by the time-dilation equations, instead of just being random) rather than Euclidean (as we like to assume). The belief that they should both be the same or that one is somehow more right than the other comes from our Euclidean prejudices and nothing more.

 

[choran]

When you say "if both are inertial", do you mean assuming both are noting in uniform relative motion relative to one another? I guess that's the answer, but a tough one to swallow! Like saying Joe is fatter than Al, but Al is fatter than Joe. Half a shot of Seargrams for Dale! LOL

 

[Nugatory]

under SR, if A sees B's clock as slow, B will also see A's clock as slow. Talking straight SR now, no preferred frames. How can both be right, and what's the tiebreaker? When it's time for the showdown "Gentlemen, produce thy watches", whose watch is slow?

Try this post: https://www.physicsforums.com/showpost.php?p=4556116&postcount=16

They're both slow and there's no paradox.

 

[dvf]

Like saying Joe is fatter than Al, but Al is fatter than Joe.

No, more like, from a distance you look smaller to me, and I look smaller to you.

 

[Dale]

When you say "if both are inertial", do you mean assuming both are noting in uniform relative motion relative to one another?

Yes.

Like saying Joe is fatter than Al, but Al is fatter than Joe.

Well, you have to be careful in expressing things in relativity. Usually, when there is some "both are fatter" statement it is because you have made a frame-variant statement without specifying the frame. If you specify the frame of every frame-variant quantity, then it becomes clear that you are not making a self-contradictory statement.

 

[choran]

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.

 

[Dale]

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.

Perhaps this will help a little. For a moment, let's consider only Newtonian physics, i.e. Galileo's relativity, not Einstein's.

Now, in Newtonian physics it makes sense to say that Al is taller than Joe without any qualifiers, this is because length is frame invariant. It doesn't matter if you do the comparison in Al's frame, Joe's frame, or the Earth's frame, the answer is the same.

However, in Newtonian physics it does not make sense to say that Al has more speed than Joe without any qualifiers, this is because speed is frame variant. In Al's frame Joe has more speed, and in Joe's frame Al has more speed, and perhaps in the Earth's frame they have the same speed.

So the statement "Al has more speed than Joe" is fundamentally meaningless because we have stated a frame-variant quantity, speed, without specifying the frame it refers to. In contrast, the statement "Al has more speed than Joe in Joe's frame" is meaningful because we have specified the reference frame referred to by the frame-variant quantity. Also, the fact that Al has more speed than Joe (in Joe's frame) does not in any way contradict the fact that Joe has more speed than Al (in Al's frame). Speed is a frame variant quantity and the two different statements are made wrt two different frames.

You can clearly see that even in Galileo's relativity there are frame variant and frame invariant quantities. You can also see that even in Galileo's relativity you have to specify the frame for frame variant quantities and you can get seeming contradictions if you drop the frame specification.

The difference between Einstein's and Galileo's relativity is not how invariant and variant quantites are treated. It is merely that the list of frame-variant quantities is larger in Einstein's relativity. So some things, such as length, duration, and simultaneity, which you could previously get away without specifying the frame, you now need to specify the frame. It isn't a big conceptual change, just a shuffling of some items from the "frame invariant" list to the "frame variant" list.

 

[choran]

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!"

 

[choran]

Oh, forgot to mention another one I just can't visualize: I shoot a single pulse of light away from me as I move at .5 c. I am it at a distant star. I continue along at .5 c. A guy who started out standing next to me, motionless with respect to me, also looks at that pulse of light move away toward the star. Now, when that pulse hits the star (let's say it was one light year from me at start), I will be .5 light years from the star when the pulse reaches the star, while my buddy will still be one light year away from the star when the pulse hits the star. Given that, it is hard to see how the beam of light travels with the same velocity relative to both of us. Yep, it's a postulate, and SR says "That's life, no exceptions" but at the end of the day, it's a tough one.

 

[choran]

"Am" should be "aim". Sorry.

 

[Mentz114]

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.

"Time is variable" is not a good way to express the idea that every clock tells the time on its own worldline. So time is not variable because this proper time shown on clocks is an invariant - all observers will agree on the time shown on a clock. This is different from Newtonian mechanics where every clock shows the same time.

 

[WannabeNewton]

I don't think there's anything that will ever get me to easily accept the notion that there is no absolute time.

See if you can get your hands on this book: https://www.amazon.com/dp/0801884225/?tag=pfamazon01-20

Read in particular chapters 5-8.

 

[choran]

OK, just ordered it. Found a used copy on Amazon--still "ouch"! LOL

 

[choran]

I like books, always have several going at once, now that I have the time. Thanks for the recommend, I will enjoy owning it. I've got several general physics texts, but most only devote a chapter or two to relativity, so this will be nice.

 

[ghwellsjr]

Oh, forgot to mention another one I just can't visualize: I shoot a single pulse of light away from me as I move at .5 c. I aim it at a distant star. I continue along at .5 c. A guy who started out standing next to me, motionless with respect to me, also looks at that pulse of light move away toward the star. Now, when that pulse hits the star (let's say it was one light year from me at start), I will be .5 light years from the star when the pulse reaches the star, while my buddy will still be one light year away from the star when the pulse hits the star. Given that, it is hard to see how the beam of light travels with the same velocity relative to both of us. Yep, it's a postulate, and SR says "That's life, no exceptions" but at the end of the day, it's a tough one.

Maybe it would help to visualize if you had some spacetime diagrams to look at. You've described everything according to the Inertial Reference Frame (IRF) of the guy who remained motionless and of the star. So here's a spacetime diagram that depicts what you described. The thick blue line represents the motionless guy, the thick red line is the star and you are shown as the thick black line. The dots represent one-year intervals of time for each object. The thin black line is the pulse of light that you shoot towards the star. The thin red line is its reflection back to you and the motionless guy:

Note that neither you nor the motionless guy can see the light arriving at the star, you both have to wait for the reflection to get back to each of you. And when you do see the reflection, you measure how long it took from the time it was emitted (at time zero) until you see the reflection, and you divide that interval by 2 and assume that the moment of reflection occurred at the half-way point of your measurement interval. This is the application of Einstein's second postulate because you are assuming that the light took the same amount of time to reach the star as it took for the reflection to reach you. Furthermore, you assume that the distance away from you that the star was at the moment of reflection was the same value in light-years.

It is easy to see that since it took 2 years for the light to propagate from its point of emission to the star and to reflect back to the motionless guy, he will determine that the light reached the star 1 year after it was emitted and that the star was 1 light-year away.

Now in order for him to determine where you were at that same time (according to his rest IRF), he has to send out another pulse of light one half year after you left. This pulse reflects off of you and returns to him at his time of 1.5 years. Since it took 1 year to make the round trip, he divides that by 2 and assumes that it reached you at his time of 1 year and that you were 0.5 light-years away:

All his measurements, assumptions and calculations agree with the coordinates depicted in the IRF describing the scenario because he remains at rest in it.

You, on the other hand, come to a different conclusion because you see the reflection at your time of 1.1547 years and so you assume that since the light took the same amount of time to get to the star as it did for the reflection to get back to you, the star was 0.577 light-years away from you and the reflection occurred at your time of 0.577 years.

Also, since you arrive at the star at your time of 1.732 years, you can calculate the relative speed between you and the star by taking the distance and dividing it by the difference in the two times. That is 0.577/(1.732-0.577) = 0.577/1.155 = 0.5.

These measurements, assumptions and calculations that you make do not match the coordinates that are depicted in the motionless guy's rest IRF but we can transform all the events (the dots and the intersections of the lines) from his rest IRF to your rest IRF using a speed of 0.5 to do the transformation. Here is the resulting spacetime diagram of your rest IRF:

Now you can see that all your determinations match the coordinates of your rest IRF. And note that all the measurements made by the motionless guy are also correctly depicted in your rest IRF just like your measurements are correctly depicted in his rest IRF. Well, I didn't include his measurement of your position in your rest IRF so I'll do another spacetime diagram that includes that detail:

Hopefully, these spacetime diagrams will help you visualize how a scenario described in one rest IRF can correctly show the measurements of all observers and how transforming to the rest IRF of another observer continues to show all the measurements of all observers. In other words, it doesn't matter what IRF we use, none is preferred, not even a rest IRF of any particular observer.

One last comment: if you want to cling to the idea that nature is operating on an absolute time, then you're claiming that one IRF is preferred above all others. But clearly, since you can always transform the coordinates of any IRF to any other IRF and they all correctly show all the measurements that every observer makes, there can be no way to identify a preferred IRF based on any measurement that anyone can make, so the concept of an absolute time is useless, don't you agree?

 

[Dale]

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.

Unfortunately, that is not self consistent. The thing you have very few issues with directly implies the thing that gets you. If you accept the two postulates then the inevitable conclusion is the Lorentz transform, which contains the variable time that bothers you.

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.

There is probably a lot of truth to that. There is a group at Wash. U. (I think) that studied SR instruction and found that the biggest mental hurdle that students face is the relativity of simultaneity. They systematically studied different methods of teaching it, but I don't think that their results are often used by instructors.

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!"

This made me chuckle. "You are just measuring two different things" is a valid answer for SR, but not for Newton. You are absolutely correct, you are measuring the rate of the clock ticking in two different reference frames. In Galileo's relativity this is an invariant quantity, only in Einstein's relativity is it two different quantities. Your "what's the big deal" conceeds the whole point and accepts Einstein's categorization!

 

[choran]

Thank you both very much for the time spent!
GHWellsJr., in my example above, I thought it was a simpler issue. A and B stand together on earth. Let's say they both shoot a pulse toward the star, which by agreement, not assumption, is 1 light year away. A says "Think I'll head out toward the star". B says "Think I'll stay here". A heads off toward star at some speed, doesn't matter what it is. B stays home. (Not trying to add acceleration or anything, so let's forget about that if we can).

Now, at some point, the light pulses that A and B fired will hit the star. A, having traveled toward the star, will be closer to the star when that happens than B is, B having stayed home. So in whatever time it was that it took the light to arrive, from whoever's point of view, it seems true that the pulses moved more slowly relative to A than relative to B, for the simple reason that A moved in the same direction, and moved some finite distance? In other words, at light arrival time, A may have gone, say, 1/3 of the way to the star. So beam went 1 LY, A went 1/3 LY, SO it would seem that relative to A, the pulses traveled slower than they did relative to B.

Now, I can see that this will not be apparent when actual measurements are performed with mirrors, light speed communications, etc. But I don't see how these measurement difficulties change the underlying situation.
I must continue to cling. LOL

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.

 

[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?