Silly question about special relativity

[paat]

I happened to listen audiobook Brian Greene - Fabric of the Cosmos. There were great explanations and nice metaphors, but I still don't get it.

Let just imagine hypothetically, that Bart is ready for launch to Mars with his nuclear-powered skateboard, next to him is radio antenna ready to broadcast Bart's favorite cartoon to the TV, which is located on Mars (and Homer sits in front of it). Bart's take-off and broadcasting start simultaneously. Would Bart make it on time? Or would he be hopelessly late? Given that his skateboard could instantaneously develop speed 299 000 kilometers per second (Near the speed of electro-magnetic wave) and distance to Mars is 150 000 000 km.

As I understand in Homer's perspective, Bart would be there almost on time. And for Bart's perspective he would be hopelessly late. But who is the person that Homer sees flying in from the window and cheering that he made it?

 

[JesseM]

I happened to listen audiobook Brian Greene - Fabric of the Cosmos. There were great explanations and nice metaphors, but I still don't get it.

Let just imagine hypothetically, that Bart is ready for launch to Mars with his nuclear-powered skateboard, next to him is radio antenna ready to broadcast Bart's favorite cartoon to the TV, which is located on Mars (and Homer sits in front of it). Bart's take-off and broadcasting start simultaneously. Would Bart make it on time? Or would he be hopelessly late? Given that his skateboard could instantaneously develop speed 299 000 kilometers per second (Near the speed of electro-magnetic wave) and distance to Mars is 150 000 000 km.

As I understand in Homer's perspective, Bart would be there almost on time. And for Bart's perspective he would be hopelessly late. But who is the person that Homer sees flying in from the window and cheering that he made it?

In Homer's frame, Bart is traveling at 299 000 km/sec while the electromagnetic wave which left Earth is traveling at 299 792.458 km/sec. So if the distance in Homer's frame is 150 000 000 km, the electromagnetic wave takes 500.346 seconds to reach him, while Bart takes 501.672 seconds to reach him. Since Bart is traveling at 0.997356645 the speed of light, in Homer's frame Bart's watch is slowed down so it's only ticking at $$\sqrt{1 - 0.997356645^2}$$ = 0.07266 the rate of Homer's watch, which means that if Bart starts his watch at the moment he fires up the skateboard, his watch will only have elapsed a time of 501.672 * 0.07266 = 36.45 seconds when he reaches Mars.

Now look at things from the perspective of Bart's frame. In Bart's frame the distance from Earth to Mars is shrunk by that same factor of 0.07266, so the distance is only 10 899 000 km. In Bart's frame he is at rest while Mars is approaching him at 299 000 km/sec and the electromagnetic wave is moving towards Mars at 299 792.458 km/sec, so we can solve 299792.458t = 10899000 - 299000t to find the time t when the wave reaches Mars in his frame, which gives a time of 18.20 seconds. And if Mars is approaching at 299000 km/sec, then it will reach him after 10899000/299000 = 36.45 seconds, so this is what his watch will read when he arrives (same as what I calculated above in Homer's frame). So in Bart's frame there was a gap of 36.45 - 18.20 = 18.25 seconds between the time the wave reached Mars and the time he did...but note that in Bart's frame, Homer's watch is slowed by a factor of 0.07266, so that in 18.25 seconds according to Bart's watch, Homer's watch only ticks forward by 18.25*0.07266 = 1.326 seconds (and if the show is playing at the normal rate in Homer's frame, the show will be slowed down by the same amount in Bart's frame, so only 1.326 seconds of the show will have passed since it started). This is consistent with what we found above in Homer's frame, where the wave arrives at a time of 500.346 seconds and Bart arrives at a time of 501.672 seconds (501.672 - 500.346 = 1.326). So both frames make all the same predictions about the different watch-readings at the moment of the wave reaching Mars and the moment of Bart reaching Mars, even though they disagree on whose watch is slowed down.

 

[paat]

Thank you very much, it was most helpful. Although still it is quite a brain massage ;)

 

[genneth]

In Homer's frame, Bart is traveling at 299 000 km/sec while the electromagnetic wave which left Earth is traveling at 299 792.458 km/sec. So if the distance in Homer's frame is 150 000 000 km, the electromagnetic wave takes 500.346 seconds to reach him, while Bart takes 501.672 seconds to reach him. Since Bart is traveling at 0.997356645 the speed of light, in Homer's frame Bart's watch is slowed down so it's only ticking at $$\sqrt{1 - 0.997356645^2}$$ = 0.07266 the rate of Homer's watch, which means that if Bart starts his watch at the moment he fires up the skateboard, his watch will only have elapsed a time of 501.672 * 0.07266 = 36.45 seconds when he reaches Mars.

Now look at things from the perspective of Bart's frame. In Bart's frame the distance from Earth to Mars is shrunk by that same factor of 0.07266, so the distance is only 10 899 000 km. In Bart's frame he is at rest while Mars is approaching him at 299 000 km/sec and the electromagnetic wave is moving towards Mars at 299 792.458 km/sec, so we can solve 299792.458t = 10899000 - 299000t to find the time t when the wave reaches Mars in his frame, which gives a time of 18.20 seconds. And if Mars is approaching at 299000 km/sec, then it will reach him after 10899000/299000 = 36.45 seconds, so this is what his watch will read when he arrives (same as what I calculated above in Homer's frame). So in Bart's frame there was a gap of 36.45 - 18.20 = 18.25 seconds between the time the wave reached Mars and the time he did...but note that in Bart's frame, Homer's watch is slowed by a factor of 0.07266, so that in 18.25 seconds according to Bart's watch, Homer's watch only ticks forward by 18.25*0.07266 = 1.326 seconds (and if the show is playing at the normal rate in Homer's frame, the show will be slowed down by the same amount in Bart's frame, so only 1.326 seconds of the show will have passed since it started). This is consistent with what we found above in Homer's frame, where the wave arrives at a time of 500.346 seconds and Bart arrives at a time of 501.672 seconds (501.672 - 500.346 = 1.326). So both frames make all the same predictions about the different watch-readings at the moment of the wave reaching Mars and the moment of Bart reaching Mars, even though they disagree on whose watch is slowed down.

Times like this I wish we could vote up certain posts. I'm certainly going to try and remember to point people at this precise example every time this question comes up.

 

[robphy]

Times like this I wish we could vote up certain posts. I'm certainly going to try and remember to point people at this precise example every time this question comes up.

You can maintain an entry of favorite posts in your blog.
Actually a nice feature for PF would be to make it easier to save such posts to a blog entry of local bookmarks with a click of a link.

 

[neutrino]

Times like this I wish we could vote up certain posts. I'm certainly going to try and remember to point people at this precise example every time this question comes up.

Ditto. It was almost like your favourite superhero beating up the Paradoxman.

Great presentation, JesseM! I loved reading that post.

 

[neutrino]

Times like this I wish we could vote up certain posts.

You can maintain an entry of favorite posts in your blog.
Actually a nice feature for PF would be to make it easier to save such posts to a blog entry of local bookmarks with a click of a link.

We could have something similar to a Report button, but this will be only to report the good stuff. Maybe if a posts gets a certain amount of reports, it goes into a hall of fame.

 

[robphy]

We could have something similar to a Report button, but this will be only to report the good stuff. Maybe if a posts gets a certain amount of reports, it goes into a hall of fame.

Yeah... maybe something like in Amazon's contributed reviews "48 of 53 people found the following review helpful".

Maybe this and more discussion along these lines should continue in Forum Feedback.

 

[jacksnap]

Cheers JesseM, I actually read that post and understood it, which is amazing for me.
Very clear and a good example. (I tend to just read here, but that post I thought deserved a thank you.)

One question, after the signal and Bart have set off, will Bart not see the signal traveling away from him at the speed of light (299 792.458 km/sec) so should get there sooner in that respect. this is the part I don't quite grasp that the speed of light is constant in anyones frame.

PS recommended posts, or thread votes would be a great addition to this board.

 

[JesseM]

One question, after the signal and Bart have set off, will Bart not see the signal traveling away from him at the speed of light (299 792.458 km/sec) so should get there sooner in that respect. this is the part I don't quite grasp that the speed of light is constant in anyones frame.

Yes, that was included in the calculation, in this part:

In Bart's frame the distance from Earth to Mars is shrunk by that same factor of 0.07266, so the distance is only 10 899 000 km. In Bart's frame he is at rest while Mars is approaching him at 299 000 km/sec and the electromagnetic wave is moving towards Mars at 299 792.458 km/sec, so we can solve 299792.458t = 10899000 - 299000t to find the time t when the wave reaches Mars in his frame, which gives a time of 18.20 seconds.

To elaborate on this a little, note that in a time of 18.201632 seconds in Bart's frame (a slightly more exact version of what you get from the equation above), the signal has moved away from Bart a distance of (299792.458 km/sec)*(18.201632 sec) = 5456712 km. And Mars, which in Bart's frame is approaching Bart at 299000 km/sec, has gotten closer by (299000 km/sec)*(18.201632 sec) = 5442288 km, so since Mars was originally 10899000 km away when the signal was emitted, after 18.201632 seconds it's now only 10899000 - 5442288 = 5456712 km away, exactly the same distance as the signal, showing that this is the time the signal should reach Mars in Bart's frame.

Basically, if you look at the equation I used:

299792.458t = 10899000 - 299000t

...the left side is telling you the distance of the signal from Bart as a function of time t, while the right side is telling you the distance of Mars from Bart as a function of time, so setting them equal and solving for t tells you the time when the signal reaches Mars in Bart's frame.

 

[O Great One]

So in Bart's frame there was a gap of 36.45 - 18.20 = 18.25 seconds between the time the wave reached Mars and the time he did...(and if the show is playing at the normal rate in Homer's frame, the show will be slowed down by the same amount in Bart's frame, so only 1.326 seconds of the show will have passed since it started)

But, the show playing slowly implies that the electromagnetic wave isn't traveling at the speed of light relative to Bart doesn't it? Because he is 'catching up' to it so that only 1.326 seconds of the show has passed?

 

[JesseM]

But, the show playing slowly implies that the electromagnetic wave isn't traveling at the speed of light relative to Bart doesn't it? Because he is 'catching up' to it so that only 1.326 seconds of the show has passed?

This isn't a matter of the speed of any individual signal, but of the rate that successive frames of the image are being sent from Earth using successive signals (and here I'm using the word 'frames' in the sense of the frames of a movie/TV image here, not in the sense of a reference frame in relativity...unfortunate terminology for this example!) In the reference frame of Earth and Mars they're being sent at the normal rate for TV, around 30 frames per second which translates to a new image being sent out every 0.03 seconds, but in Bart's reference frame the transmitter is slowed down, so it's only sending out a new image about once every 0.03/0.07266 = 0.4 seconds (less than 3 frames per second). The signals carrying each successive image are still moving at the speed of light in Bart's reference frame, it's just that the transmitter is sending out new images less often.

 

[O Great One]

But one could have the same scenario with the electromagnetic wave already generated some time in the past and once it reaches Bart on Earth, Bart takes off etc. In this case, Bart was at rest in the Earth-Mars frame when the electromagnetic wave was generated.

 

[JesseM]

But one could have the same scenario with the electromagnetic wave already generated some time in the past and once it reaches Bart on Earth, Bart takes off etc. In this case, Bart was at rest in the Earth-Mars frame when the electromagnetic wave was generated.

Are you talking about using a non-inertial frame where Bart is at rest both before and after he accelerates? In non-inertial frames the usual equations of SR don't work, and the coordinate velocity of light waves is not necessarily constant. It's better to just analyze this problem from two different inertial frames--the frame where Earth and Mars are at rest, and the frame where Bart is at rest after he accelerates (once you decide to use this frame, you're free to use it to analyze what's going on in the time period before Bart accelerates too--don't let the fact that I called it 'Bart's rest frame' mislead you, an inertial frame always represents the measurements of a hypothetical observer who moves at constant velocity forever, it isn't tied to the movements of any real physical object which can change velocity).