Questions about relativity problem...

[leftrightleftrightleft]

Lets say you leave Earth travelling at 90% the speed of light.

One hour (relative to your watch) after you leave, you fire a laser beam back at the earth. Similarly, after 2.2942 hours on Earth, someone fires a laser beam out at you.

1) How long (relative to you) does the laser beam take to reach you?
2) How long (relative to you) does the laser beam take to reach the Earth?
3) At what point do the laser beams pass each other?

 

[lesliedellow]

Both observers would see both laser beams moving at the speed of light.

After one hour the spaceship would be 0.9 light years from Earth, and moving away from Earth at nine tenths the speed of light. Therefore both observers would say that it took 9 hours for the light beam from Earth to reach the spaceship, according to his own clock, or (applying the Lorentz Transformation) 20.6 hours according to the other observer's clock.

The situation is symmetrical until the observer in the spaceship decides he wants to turn round and come home. Then the expenditure of energy by one, and not the other, breaks the symmetry.

 

[SkyWriting]

Both observers would see both laser beams moving at the speed of light.

After one hour the spaceship would be 0.9 light years from Earth, and moving away from Earth at nine tenths the speed of light. Therefore both observers would say that it took 9 hours for the light beam from Earth to reach the spaceship, according to his own clock, or (applying the Lorentz Transformation) 27 hours according to the other observer's clock.

The situation is symmetrical until the observer in the spaceship decides he wants to turn round and come home. Then the expenditure of energy by one, and not the other, breaks the symmetry.

I've never seen "the expenditure of energy" used as a factor. Source?

 

[[serious]]

I've never seen "the expenditure of energy" used as a factor. Source?

As I understand, acceleration breaks the symmetry. The math is beyond me, but that's how it's taught in upper level physics. Well, how I was taught back in 2004.
http://en.m.wikipedia.org/wiki/Twin_paradox#Role_of_acceleration

 

[lesliedellow]

I've never seen "the expenditure of energy" used as a factor. Source?

Expenditure of energy = acceleration = curvature of spacetime.

 

[sfs]

Lessee. Time dilation factor at 0.9c is (oddly enough) 2.2942. When you have travelled for an hour by your clock, an observer in the rest frame of the earth will say 2.29 hours have elapsed by his clock. So for the earth observer, the two beams are fired at the same time, which is convenient.

Light speed is the same in all frames, so the two beams will pass each other at the midpoint between the two observers when the beams start. Where is the midpoint? From the earth frame, the ship will have travelled for 2.29 hours at 0.9c when the beams fire. 2.29 hours is 8259 seconds, times 3x10^8 m/s times 0.9 = 2.23 x 10^12 m. So the midpoint is 1.1 x 10^12 m (or .5 * .9 * 2.29 light hours).

As for timing. . . The earth dudes wait 2.29 hours before firing their laser, by their clock (assuming I'm understanding the way the problem is meant to be read). But viewed by you, their clock runs slower than yours by 2.29x, so that's 5.24 hours by your clock since you left earth before they fire. At that time you're 0.9 x 5.24 light hours away from earth, so it will require 0.9 x 5.24 hours for the light to reach you, or 4.72 hours. That's 10 hours since you left earth by your clock. (Alternatively, from the earth frame, they wait 2.29 hours to fire, at which point you're 2.06 light hours away. From their perspective their laser is only traveling at 0.1c relative to your ship, and so it takes 20.6 hours to catch up to you; 20.6 hrs + 2.29 hrs = 22.9 hrs their time, or 10 hrs by your clock.)

From your frame, you wait 1 hr to fire your laser, at which point the earth is 0.9 light hours away. Your laser travels at 0.1c relative to the earth and therefore requires 9 hrs to catch it. So your laser arrives 10 hours after liftoff by your clock.

Interesting. From the earth frame the two lasers are fired at the same time, while from your frame they arrive at the same time.

I think.

ETA: That would have taken less time if I had remembered that 5 + 4 is not equal to 11.

 

[sfs]

Expenditure of energy = acceleration = curvature of spacetime.

Make the traveling twin massless, so no energy is expended. Or instead of having the outgoing twin turn around, have him meet an identical third twin who is inbound, and have them synchronize clocks. Nothing beyond special relativity is needed.

 

[sfs]

As I understand, acceleration breaks the symmetry. The math is beyond me, but that's how it's taught in upper level physics. Well, how I was taught back in 2004.
Twin paradox - Wikipedia, the free encyclopedia

Note that the Wikipedia article points out that acceleration is not needed to break the symmetry.

 

[lesliedellow]

Make the traveling twin massless, so no energy is expended. Or instead of having the outgoing twin turn around, have him meet an identical third twin who is inbound, and have them synchronize clocks. Nothing beyond special relativity is needed.

If the travelling twin was massless, he would be travelling permanently at the speed of light anyway. And if you put it down to acceleration, you have got to explain why the travelling twin is the one who is "really" accelerating.

If you try to introduce accelerating objects into Special Relativity, at first it seems to work, but pretty soon anomolies start appearing, of which the twin paradox is just the most famous.

 

[sfs]

If the travelling twin was massless, he would be travelling permanently at the speed of light anyway.

True. So calculate the stress energy tensor and take the limit as the mass approaches zero.

And if you put it down to acceleration, you have got to explain why the travelling twin is the one who is "really" accelerating.

The traveling twin is the one who is really accelerating. Attach an accelerometer to each and only one will show a non-zero reading.

If you try to introduce accelerating objects into Special Relativity, at first it seems to work, but pretty soon anomolies start appearing, of which the twin paradox is just the most famous.

Presumably you mean accelerating reference frames; SR can handle accelerating objects just fine. But I wasn't doing that: I was pointing out that the twin paradox can be resolved entirely within SR without invoking acceleration at all. (Which is not exactly news -- I was taught this in my relativity class in the early 80s.)

 

[lesliedellow]

The traveling twin is the one who is really accelerating. Attach an accelerometer to each and only one will show a non-zero reading.

You are overlooking something which bothered Einstein, Mach, and even Newton, before them. Which is that if the twins were the only two things in the universe, there would be no obvious physical grounds for nominating either of them as the one who was "really" accelerating. It doesn't matter that you can attach an accelerometer to both of them, and only one of them would register anything - you still have no explanation as to why one frame of reference is so privileged.

Presumably you mean accelerating reference frames; SR can handle accelerating objects just fine. But I wasn't doing that: I was pointing out that the twin paradox can be resolved entirely within SR without invoking acceleration at all. (Which is not exactly news -- I was taught this in my relativity class in the early 80s.)

Yes, I was referring to accerating frames, and that is precisely what you have in the twins paradox.

 

[[serious]]

You are overlooking something which bothered Einstein, Mach, and even Newton, before them. Which is that if the twins were the only two things in the universe, there would be no obvious physical grounds for nominating either of them as the one who was "really" accelerating. It doesn't matter that you can attach an accelerometer to both of them, and only one of them would register anything - you still have no explanation as to why one frame of reference is so privileged.




Yes, I was referring to accerating frames, and that is precisely what you have in the twins paradox.

I haven't heard that before. Di you have a source so I can read more about it?

 

[lesliedellow]

I haven't heard that before. Di you have a source so I can read more about it?

You could read Einstein's original paper, or a book he wrote called "Relativity: The Special and the General Theory." He makes fairly explicit reference to the current case in chapter 21.

You might also want to Google "Mach's Principle", which is not so much an answer as a statement of the problem.

 

[sfs]

You are overlooking something which bothered Einstein, Mach, and even Newton, before them. Which is that if the twins were the only two things in the universe, there would be no obvious physical grounds for nominating either of them as the one who was "really" accelerating. It doesn't matter that you can attach an accelerometer to both of them, and only one of them would register anything - you still have no explanation as to why one frame of reference is so privileged.

In our universe, inertial frames are different from noninertial frames. Whether that's due to the presence of the rest of the matter in the universe, as Mach speculated, or not, in our universe the accelerometers do indeed distinguish the two twins.

Yes, I was referring to accerating frames, and that is precisely what you have in the twins paradox.

The point of introducing the third twin is that you still have the same paradox (and the same resolution) even if you formulate the scenario with zero acceleration.

 

[lesliedellow]

In our universe, inertial frames are different from noninertial frames. Whether that's due to the presence of the rest of the matter in the universe, as Mach speculated, or not, in our universe the accelerometers do indeed distinguish the two twins.

Where are these inertial frames you speak of? Is the Earth an inertial frame? Well, not really; it is accelerating round the Sun. Is the Sun an inertial frame? Well, not really: it is orbiting the galaxy. Is the galaxy an Inertial frame? Well not really: it is accelerating towards the Andromeda galaxy.

Inertial frames only exist as approximations in an essentially curved spacetime.

 

[sfs]

Where are these inertial frames you speak of? Is the Earth an inertial frame? Well, not really; it is accelerating round the Sun. Is the Sun an inertial frame? Well, not really: it is orbiting the galaxy. Is the galaxy an Inertial frame? Well not really: it is accelerating towards the Andromeda galaxy.

Inertial frames only exist as approximations in an essentially curved spacetime.

Yeah, I know. They're spectacularly useful approximations that form the basis of Special Relativity, which seems to be the subject of this thread. Your point?

But if you really think a different approach would be more useful, work through your original suggestion: calculate the curvature caused by the expenditure of energy in turning the traveller around and show that it gives the correct answer to the twin paradox.

 

[mzungu]

γλυκύ δ᾽ἀπείρῳ πόλεμος.

I love your siggy but would it not be better to quote it in full!

γλυκύ δ᾽ἀπείρῳ πόλεμος.
πεπειραμένων δέ τις ταρβεῖ προσιόντα νιν καρδία περισσῶς.
- Πίνδαρος

War is sweet to those who have no experience of it,
but the experienced man trembles exceedingly at heart on its approach.
- Pindar

 

[lesliedellow]

Yeah, I know. They're spectacularly useful approximations that form the basis of Special Relativity, which seems to be the subject of this thread. Your point?

My point is that there are no frames of reference which can be regarded as especially privileged, and therefore none which can be identified as being the one which is "really" undergoing acceleration. At least not without first identifying a physical reason why that frame should be so identified.

Once you have done that, you can hang an accelerometer off of it, and what you observe will make perfect sense.

 

[sfs]

My point is that there are no frames of reference which can be regarded as especially privileged, and therefore none which can be identified as being the one which is "really" undergoing acceleration. At least not without first identifying a physical reason why that frame should be so identified.

Once you have done that, you can hang an accelerometer off of it, and what you observe will make perfect sense.

And my point is that you're making a philosophical assertion rather than a scientific one here. It is simply a fact that we can identify which frame of reference is (to a very good approximation) an inertial one and which isn't, even without knowing the underlying reason for the existence of inertia. And if we're doing calculations using SR (which physicists do routinely), that's the framework we're working in. It's quite possible that some more general framework would provide a better-grounded, more satisfying explanation for all relativistic phenomena, including the twin paradox, but there's nothing special about the paradox that demands a more fundamental framework, any more than anything else in SR.

I would kind of like to know what the point of the original question was. Was it a quiz? If so, did anyone pass it? Was it a homework problem?

 

[sfs]

I love your siggy but would it not be better to quote it in full!

γλυκύ δ᾽ἀπείρῳ πόλεμος.
πεπειραμένων δέ τις ταρβεῖ προσιόντα νιν καρδία περισσῶς.
- Πίνδαρος

War is sweet to those who have no experience of it,
but the experienced man trembles exceedingly at heart on its approach.
- Pindar

I prefer pithy: that way I get the maximum density of pretension out of the quotation. Also, I've never seen the full fragment before. (I've never seen the word "νιν" before, either, and it kind of threw me.)