Relativity puzzle

Witness · Post by **Witness** » Mon Oct 12, 2020 9:23 pm

https://i.imgur.com/Of4GgwG.png

A lab and its two equidistant mirrors (M₁ and M₂) moves to the right with speed v. There is also an outside observer "at rest".

Inside the lab nothing special happens, a flash of light goes to the mirrors and comes back at the lab simultaneously from both.

Light in vacuum has the same speed c for all observers, lab and outside.

1. Compute the time for light to get from the lab to the upper mirror (or vice versa) as seen from outside, compared to the time measured in the lab.

2. Compute the distance from the lab to the right mirror as seen from outside, compared to the distance measured in the lab (D).

The lab with its mirrors can be considered a clock. :)
Needs no more algebra than what you presumably learned when ~ fifteen; and the knowledge that speed = distance/time. Special relativity is elementary. :) :)

Witness · Post by **Witness** » Mon Oct 12, 2020 10:20 pm

↑ It's Special Relativity, not General.

Hotarubi · Post by **Hotarubi** » Mon Oct 12, 2020 11:17 pm

shemp · Post by **shemp** » Tue Oct 13, 2020 3:28 am

The answer for both is "Who gives a fuck?"

Anaxagoras · Post by **Anaxagoras** » Tue Oct 13, 2020 3:46 am

I once read a book and took a course in Special Relativity, and I would have been able to answer it then.

The two mirrors with light bouncing back and forth was exactly the same metaphor that was used to explain SR in that book.
Light must travel a greater distance when the mirrors are moving relative to the observer, because of Pythagoras, then when the mirrors are stationary.
Hence, time as measured by the person moving with the mirrors passes faster than it does for the stationary observer.

Is the second one D(1-V²/C²)?

The first question is a little harder to answer.

Is the first one something like square root of (D²+V²)/C? :notsure:

robinson · Post by **robinson** » Tue Oct 13, 2020 7:45 am

The real puzzle is when you realize the lab can be the stationary object and the observer is moving. And the math is exactly the same.

Witness · Post by **Witness** » Wed Oct 14, 2020 3:47 am

Anaxagoras wrote: ↑Tue Oct 13, 2020 3:46 am Is the second one D(1-V²/C²)?

No, something's missing, but you're on track.

Anaxagoras wrote: ↑Tue Oct 13, 2020 3:46 am The first question is a little harder to answer.

Not at all, it's much easier.

Anaxagoras wrote: ↑Tue Oct 13, 2020 3:46 am Is the first one something like square root of (D²+V²)/C? :notsure:

Can't mix distance and speed.

I'll give the solutions later, but I always found interesting that the tools needed are really elementary.

You just have to accept the mind-boggling consequences that time slows and lengths shorten. :wink:

ceptimus · Post by **ceptimus** » Wed Oct 14, 2020 5:56 am

Have a big spinning disc with the periphery moving 3/4 the speed of light. Now accelerate the whole spinning disc in a straight line, so it flies through space like a frisbee at 3/4 light speed.

A stationary observer still sees the fastest part of the disc (relative to him) going slower than light because there is a squashing effect that makes moving objects shorter along their direction of travel.

The stationary observer sees clocks on the spinning disc slow down on the 'forward' part of the disc, and speed up on the 'rearward' part.

Observers on the disc see all the clocks on the disc that are equidistant from the centre keep time with each other, but they see the clocks further away from the centre running slower than those nearer the centre of rotation.

robinson · Post by **robinson** » Wed Oct 14, 2020 8:25 pm

Mach would like to have a word with you

sparks · Post by **sparks** » Wed Oct 14, 2020 9:11 pm

Wrong but thanks for playing.

Witness · Post by **Witness** » Fri Oct 30, 2020 3:56 am

Not much success…

Here's a hint for part one:

https://i.imgur.com/tQhntaV.png

Just compute the time light takes to go along the upper red arrow and compare it to the time it takes (in the lab) to get straight up.

I'll post the solution one of these days.

robinson · Post by **robinson** » Fri Oct 30, 2020 4:47 am

shemp wrote: ↑Tue Oct 13, 2020 3:28 am The answer for both is "Who gives a fuck?"

That’s the best answer

robinson · Post by **robinson** » Fri Oct 30, 2020 4:48 am

Because there is no practical or even impractical benefit to the answers

gnome · Post by **gnome** » Fri Oct 30, 2020 11:51 am

I think there are several technologies that would disagree.

robinson · Post by **robinson** » Fri Oct 30, 2020 1:29 pm

And yet you don't name them

robinson · Post by **robinson** » Fri Oct 30, 2020 1:43 pm

Just for fun, the atomic clocks in a GPS satellite encode the time in a signal so a GPS can calculate the exact location and time of the satellite, so a GPS can be accurate. To the earthbound GPS device, the satellite clock is running fast.

To the satellite, the GPS on earth is running slower. The exact figure is around 38 microseconds per day, (irony intended). From the satellite POV, everything on earth is getting behind in time by 38 microseconds per day

To the earthbound POV the satellites are getting ahead in time by 38 microseconds per day

Of course they just adjust the satellite clock to match the one down in the gravity well

This was the first time GR was used to adjust clocks so shit worked almost right

When we sent satellites to orbit other planets, then it got real

real hard

gnome · Post by **gnome** » Fri Oct 30, 2020 2:49 pm

It's clear I didn't need to name them, as you are already aware. This was my assumption.

robinson · Post by **robinson** » Fri Oct 30, 2020 9:05 pm

Maybe

But they don’t actually calculate using GR or SR

They just set the clocks twice a day

Calculating the speeding up (lower gravity) and slowing down (movement) and adjusting each clock for the variations in speed (orbits change speed), and changing gravity (earth is not uniform) isn’t done, nor is it needed

gnome · Post by **gnome** » Fri Oct 30, 2020 9:42 pm

That reminds me of what I learned when I took the insanely annoying class in Differential Equations and its cousin Calculus 2. Memorizing all these convoluted techniques for resolving integrals and derivatives, when in real life all I would ever conceivably do is approximate the result with a computer because the formulas aren't even that tidy and linear in the first place. I mean, it's good to work with some of those enough to understand what they mean, but it was just excessive and pointless. It was that sort of thing that knocked me out of my plan to get into the actuarial field--I just didn't have the ability or the motivation to memorize so many formulas and derivations. Possibly it was an effective barrier to entry just used to thin out the supply of actuaries to get them paid more. Well, it worked on me.

Calculus 3 and beyond, to Real Analysis, was a treat. I felt like I was actually grasping more genuine mathematics instead of just different ways to try algebraic substitutions to make a problem solvable.

Witness · Post by **Witness** » Fri Nov 13, 2020 3:30 am

OK, I wrote up part 1, if anyone cares (see the graph I posted):

https://i.imgur.com/0faaJpu.png

Here's a graph of the gamma:

https://i.imgur.com/8YtXQMu.png

And a small anime I made (the red circles expand at the speed of light):

https://i.imgur.com/N8wYZz3.gif

If somebody wants to tackle part 2 (length contraction) and needs a:

robinson · Post by **robinson** » Fri Nov 13, 2020 2:05 pm

robinson wrote: ↑Tue Oct 13, 2020 7:45 am The real puzzle is when you realize the lab can be the stationary object and the observer is moving. And the math is exactly the same.

Witness · Post by **Witness** » Sat Nov 14, 2020 2:06 am

robinson wrote: ↑Fri Nov 13, 2020 2:05 pm
robinson wrote: ↑Tue Oct 13, 2020 7:45 am The real puzzle is when you realize the lab can be the stationary object and the observer is moving. And the math is exactly the same.

Great! Show us…

robinson · Post by **robinson** » Sat Nov 14, 2020 2:27 am

There is no way to prove (know) if it’s the lab moving or the observer

So the math is exactly the same

Everyone moving in relation to somebody else sees the other persons clock running slower

It doesn’t matter which person you pick as the observer

If you think about it, it can’t be any other way

robinson · Post by **robinson** » Sat Nov 14, 2020 2:31 am

It’s why the speed of light is always the same for every observer, no matter how “fast” or “slow” they are moving, which of course is in relation to something else that is moving, it doesn’t matter to the observer, they are always not moving from their perspective

(Remember this is movement, not acceleration)

robinson · Post by **robinson** » Sat Nov 14, 2020 2:31 am

It’s why the speed of light is always the same for every observer, no matter how “fast” or “slow” they are moving, which of course is in relation to something else that is moving, it doesn’t matter to the observer, they are always not moving from their perspective

(Remember this is movement, not acceleration)

robinson · Post by **robinson** » Sat Nov 14, 2020 2:36 am

The light bouncing back and forth between the mirrors doesn’t know there is any motion

Nor does it ever travel farther, no matter what the relative motion

It helps to realize the stationary lab with the two mirrors might be moving at a fraction of the speed of light, from some observers POV

But it’s relative motion

The lab and the mirrors are not actually moving at all, according to them

robinson · Post by **robinson** » Sat Nov 14, 2020 2:36 am

The light bouncing back and forth between the mirrors doesn’t know there is any motion

Nor does it ever travel farther, no matter what the relative motion

It helps to realize the stationary lab with the two mirrors might be moving at a fraction of the speed of light, from some observers POV

But it’s relative motion

The lab and the mirrors are not actually moving at all, according to the mirrors and the light between them

Witness · Post by **Witness** » Sat Nov 14, 2020 2:43 am

robinson wrote: ↑Sat Nov 14, 2020 2:27 am There is no way to prove (know) if it’s the lab moving or the observer

So the math is exactly the same

I don't disagree. Just show us that "the math is exactly the same", without hand waving. I. e. we need proof.

Witness · Post by **Witness** » Sat Nov 14, 2020 2:49 am

robinson wrote: ↑Sat Nov 14, 2020 2:31 am It’s why the speed of light is always the same for every observer, no matter how “fast” or “slow” they are moving, which of course is in relation to something else that is moving, it doesn’t matter to the observer, they are always not moving from their perspective

You have your implications backwards: it is because all experiments showed the constancy of the speed of light (in a vacuum) that the ether theory was ditched and special relativity introduced.

robinson wrote: ↑Sat Nov 14, 2020 2:31 am (Remember this is movement, not acceleration)

Are you implying that an accelerated observer measures a different light speed?

Witness · Post by **Witness** » Sat Nov 14, 2020 2:51 am

I'm sure you could, with some effort, even triple post. :mrgreen:

robinson · Post by **robinson** » Sat Nov 14, 2020 11:08 am

Only losers triple post lol

robinson · Post by **robinson** » Sat Nov 14, 2020 2:22 pm

Not if you are doing the maths

Witness · Post by **Witness** » Sun Nov 15, 2020 3:18 am

robinson wrote: ↑Sat Nov 14, 2020 2:22 pm Not if you are doing the maths

Still waiting… :mrgreen:

robinson · Post by **robinson** » Sun Nov 15, 2020 3:33 am

The math is exactly the same

https://www.pitt.edu/~jdnorton/teaching ... index.html

What happens depends on the observer, the unmoving observer in the thought experiment sees light traveling farther, but if you think about it, the lab can be stationary and the observer is moving

It has to be that way

The “moving” clock is not moving to the observer in the lab

Time is perfect and the light doesn’t travel any farther, even as the moving observer sees it does

robinson · Post by **robinson** » Sun Nov 15, 2020 3:34 am

Which is why it’s so puzzling

robinson · Post by **robinson** » Sun Nov 15, 2020 3:38 am

It helps to realize no matter what the movement we ascribe to the lab and the observer, they are both moving in respect to something else

To an observer “not moving” at the center of the galaxy, they are both moving really really fast, which is why that observer sees both of them with slower clocks

Then there is the motion around the sun, and the entire frame is moving toward another galaxy, and that frame is rushing towards a great attractor

So the math gets more complicated

But to the lab, nothing is moving in the lab, except photons, at the speed of light

Witness · Post by **Witness** » Sun Nov 15, 2020 5:04 am

robinson wrote: ↑Sun Nov 15, 2020 3:33 am The math is exactly the same

https://www.pitt.edu/~jdnorton/teaching ... index.html

Zero math there. Still waiting… :mrgreen:

robinson · Post by **robinson** » Sun Nov 15, 2020 8:43 am

shemp wrote: ↑Tue Oct 13, 2020 3:28 am "Who gives a fuck?"

:D :D

Witness · Post by **Witness** » Mon Nov 16, 2020 1:01 am

robinson wrote: ↑Sun Nov 15, 2020 8:43 am
shemp wrote: ↑Tue Oct 13, 2020 3:28 am "Who gives a fuck?"
:D :D

Begs the question: why did you show up to comment in this thread then?

Still got nothing about "exactly the same math", eh? Quelle surprise.

robinson · Post by **robinson** » Mon Nov 16, 2020 4:22 am

I showed up? That's not how this works, that's not how any of this works