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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby doogly » Mon Jun 28, 2010 12:32 pm UTC

No one was saying she was bad for asking this question, we were saying it is totally reasonable to find it confusing. There is a good reason why no one ever uses it! If you want to cover it as an historical topic, like the longitudinal / transverse mass, then sure, history is nice, but don't let it make your life hard.

And Blacksails' comment was extremely helpful. One should be aware that in many modern textbooks, and certainly all professional level papers, the word "mass" means "rest mass." If Secaturs was doing something normally rather helpful and comparing different treatments of SR, this inconsistency could be quite misleading.

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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby BlackSails » Mon Jun 28, 2010 12:33 pm UTC

Eebster the Great wrote:It just honestly pisses me off when people ask perfectly valid questions and rather than answer them, other people just laugh at their nonstandard terminology or otherwise beat around the bush. It strikes me as extremely pretentious to disregard a question because the person didn't ask it the way you wanted him to.


You answered it already. Now we are discussing the usefulness of the teacher asking such a question.


Secateurs: If you want a really cool (and rather difficult) SR problem, think about this: Lets say you have a submarine traveling at a relativistic speed through the water (just go with it). The submarine has the same density of water - it is neutrally buoyant and neither sinks or swims. Now the crew of the submarine sees that the water is length-contracted. Accordingly the density of the water is higher, and the submarine will become more buoyant, and float towards the surface. But from the point of view of someone at rest with the water, the submarine is contracted, and its density is higher, and it will become less buoyant and sink.

Who is right? (They obviously both cannot be, since an object cant sink and float at the same time)

Keep in mind that there is gravity here, something SR cant normally deal with. But you can instead treat it as a uniform acceleration upwards of 9.8m/s/s of the ground.

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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby Yakk » Mon Jun 28, 2010 2:29 pm UTC

BlackSails wrote:
Yakk wrote:
BlackSails wrote:That when two galaxy clusters collide, the weakly interacting dark matter should pass through undeflected, while the baryonic matter (the interstellar plasma) will slow down, and the gravitational sources of the cluser will not match the distribution of visible mass.

I'm surprised that effect is large enough.

Both dark matter and normal matter interact similarly under gravity, and at interstellar scales gravity seriously dominates.

I suppose you'll get gas/gas and gas/star interactions -- but I'd naively suspect that it would be a small effect. I guess with sensitive enough measurement, even weak effects can stand out -- but with all of that chaos, this gets hard, doesn't it?


http://arxiv.org/PS_cache/astro-ph/pdf/ ... 8407v1.pdf

It was also printed in some journal, but the arxiv is free.

Neat! I didn't know enough astronomy to know that plasma is ~10x as massive as the stars in a galaxy. Very nice experiment, using a gravitational lensing field as a noisy approximation to mass distribution then finding the peak close to where the star mass is, with some skew towards the plasma as if it took up ~10% of the mass of what was really there.

That is some nice evidence that ~90% of the mass is interacts weakly in the collision (outside of gravity) (like stars do).
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby rattusprat » Fri Jul 02, 2010 9:43 am UTC

BlackSails wrote:Secateurs: If you want a really cool (and rather difficult) SR problem, think about this: Lets say you have a submarine traveling at a relativistic speed through the water (just go with it). The submarine has the same density of water - it is neutrally buoyant and neither sinks or swims. Now the crew of the submarine sees that the water is length-contracted. Accordingly the density of the water is higher, and the submarine will become more buoyant, and float towards the surface. But from the point of view of someone at rest with the water, the submarine is contracted, and its density is higher, and it will become less buoyant and sink.
Who is right? (They obviously both cannot be, since an object cant sink and float at the same time)
Keep in mind that there is gravity here, something SR cant normally deal with. But you can instead treat it as a uniform acceleration upwards of 9.8m/s/s of the ground.


I have only studied Relativity in first year Physcis (a few years ago) and I don't recall anything about Relativistic Mass (or maybe I have just blocked it out); but would I be right in saying
Spoiler:
the relativistic mass (and density) is only different to the rest mass (and density) in the direction of relative travel, so relative density of submarine and water remain the same prependicular to direction of travel (ie the direction of gravity), so therefore submarine remains neutrally buoyant;
or would I be wrong?

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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby BlackSails » Fri Jul 02, 2010 4:24 pm UTC

You would be wrong.


The trick is that SR is not really sufficient here - you need to account for gravity in some ad hoc way.

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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby PM 2Ring » Fri Jul 02, 2010 6:20 pm UTC

BlackSails wrote:You would be wrong.

The trick is that SR is not really sufficient here - you need to account for gravity in some ad hoc way.

Sounds messy. How do you do it? Can you use a big box full of water that's accelerating upwards at 9.81 m/sec²? I know how to derive the relativistic rocket formulae, but I've only ever tried to apply them in a vacuum.

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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby BlackSails » Fri Jul 02, 2010 6:42 pm UTC

Yeah, you just pretend the water is accelerating upwards.

The solution, which also agrees with the full GR treatment is that
Spoiler:
the submarine sees the ocean floor curve upwards at it, so it eventually hits the bottom. This obviously neglects the curvature of the Earth - with a curved earth, you just fly off into space

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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby Charlie! » Fri Jul 02, 2010 11:44 pm UTC

It's actually possible to see the solution by assuming that one exists, but that's always a little risky :D
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby Goemon » Mon Jul 05, 2010 11:36 pm UTC

BlackSails wrote:The solution, which also agrees with the full GR treatment is that


I don't get your explanation...
Spoiler:
BlackSails wrote:the submarine sees the ocean floor curve upwards at it, so it eventually hits the bottom

Why?

Does a rocket flying overhead - maintaining pace with the sub - see the surface of the water curve upwards? Does another sub at rest with respect to the water perceive the same or a different curvature? Or none?

I don't see anything about an accelerated reference system that would cause various observers (whose motion in the directions perpendicular to the acceleration are a matter of relative opinion - who's "at rest"?) to perceive any kind of curvature...
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby Charlie! » Tue Jul 06, 2010 12:27 am UTC

It "sees" it in the sense that light coming from the ocean floor curves under the influence of gravity, so the ocean floor appears to be more wrapped around the sub than you'd think.

And general relativity basically says that if you see the light ahead of you, you're going to go there, I think.
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby Goemon » Tue Jul 06, 2010 5:47 am UTC

That's just an illusion. If the sub moves, the illusion moves with it - it always looks like the floor curves up off in the distance. Certainly doesn't affect the actual depth of the sub, and the sub isn't going to crash into any illusion!

Consider, for example, two "stationary" subs at the same depth but 10,000km apart. Does each see the other buried in the rock below the ocean floor? Of course not; each sees the other well above the "curved" ocean floor. And each would expect that if they were to move toward the other (and buoyancy were not otherwise affected) they would end up side by side, not each crashing into the sea bed.
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby Eebster the Great » Tue Jul 06, 2010 7:13 pm UTC

Goemon wrote:That's just an illusion. If the sub moves, the illusion moves with it - it always looks like the floor curves up off in the distance. Certainly doesn't affect the actual depth of the sub, and the sub isn't going to crash into any illusion!

Consider, for example, two "stationary" subs at the same depth but 10,000km apart. Does each see the other buried in the rock below the ocean floor? Of course not; each sees the other well above the "curved" ocean floor. And each would expect that if they were to move toward the other (and buoyancy were not otherwise affected) they would end up side by side, not each crashing into the sea bed.


In the original example, in the sub's reference frame, the sub moves straight forward, but the ocean floor curves upward toward the front of the sub, and the sub crashes into it. In the floor's reference frame, the sub sinks and therefore takes a path that is curved downward, but the ocean floor stays flat, and the sub crashes into it. There are no "illusions" involved.

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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby Charlie! » Tue Jul 06, 2010 8:25 pm UTC

Mmh, but actually on further thought it's true that that's not how it works. The effect would be a) too small and b) not velocity-dependent. I mean, you could enhance it a bit, but... anyhow, yeah.
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby Goemon » Wed Jul 07, 2010 1:07 am UTC

Eebster the Great wrote:In the original example, in the sub's reference frame, the sub moves straight forward, but the ocean floor curves upward toward the front of the sub, and the sub crashes into it.


And that's what I don't get. I see nothing about an accelerated reference frame that would make the ocean floor curve upward. There's an illusion of curvature, but tracing the light rays backward reveals that the sea bed is in fact flat from the sub's point of view.
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby Yakk » Wed Jul 07, 2010 2:38 am UTC

So this problem doesn't exist without the presence of gravity. Thus the interaction of velocity with gravity should be responsible.

Things float upwards in water because ... of water pressure. All you need is a difference in how water pressure works due to relativistic velocity in a gravitational field in a dense fluid, and we have a difference in buoyancy.
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby Goemon » Wed Jul 07, 2010 11:52 pm UTC

Yakk wrote:All you need is a difference in how water pressure works due to relativistic velocity in a gravitational field in a dense fluid, and we have a difference in buoyancy.


Well, if you transform the stress-energy tensor of a fluid to a coordinate system moving relative to the fluid, you find that the components of pressure perpendicular to velocity are reduced by a factor of gamma^2. Lower pressure in the vertical direction = the sub sinks.

Or, for a faintly easier explanation:
Spoiler:
For simplicity, you can replace the water pressure with a regiment of gnomes on the sea floor armed with cork guns. They fire regular volleys of corks up into the space above them to keep fish and submarines "floating": From the gnome's point of view, a moving sub (a) is shorter by a factor of gamma so is hit by fewer corks; and (b) has a weight of gamma * mg. Ergo, the moving sub sinks. From the moving sub's point of view, (a) the gnomes' firing rate is reduced by gamma, so fewer corks strike the ship; and (b) the component of the cork's momentum in the vertical direction is reduced by gamma, so each cork delivers less "lifting" power. Ergo, the moving sub sinks. Or the gnomes rise, depending on how you look at it.
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby Secateurs » Thu Jul 08, 2010 9:39 am UTC

Goemon wrote:Or, for a faintly easier explanation:
Spoiler:
For simplicity, you can replace the water pressure with a regiment of gnomes on the sea floor armed with cork guns. They fire regular volleys of corks up into the space above them to keep fish and submarines "floating": From the gnome's point of view, a moving sub (a) is shorter by a factor of gamma so is hit by fewer corks; and (b) has a weight of gamma * mg. Ergo, the moving sub sinks. From the moving sub's point of view, (a) the gnomes' firing rate is reduced by gamma, so fewer corks strike the ship; and (b) the component of the cork's momentum in the vertical direction is reduced by gamma, so each cork delivers less "lifting" power. Ergo, the moving sub sinks. Or the gnomes rise, depending on how you look at it.

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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby Charlie! » Thu Jul 08, 2010 9:51 am UTC

I hesitate to mention the nitpick that water is not made of gnomes, but it must be said. Electric fields don't fire periodically, they just hang out.

Also, the moving sub sees the gnomes as closer together, thus they are hitting it with more corks (one of those paradoxes that comes out of simultaneity being broken). So the situation still isn't resolved the same way, even when suspended via gnome force.
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby Goemon » Fri Jul 09, 2010 4:53 am UTC

Charlie! wrote:...the moving sub sees the gnomes as closer together, thus they are hitting it with more corks...


Yet another relativity of simultaneity problem... from the sub's point of view, the gnomes aren't firing in unison. They fire one at a time starting with the one furthest ahead. Some gnomes fire before the ship arrives, and others after the ship has passed. The number of hits scored is something that can be proven one way or the other; all parties have to agree on that.
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby Yakk » Fri Jul 09, 2010 2:41 pm UTC

Charlie! wrote:I hesitate to mention the nitpick that water is not made of gnomes,

Don't be ridiculous.
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby PM 2Ring » Fri Jul 09, 2010 4:26 pm UTC

Is this science, or echo-gnomics?

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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby Whelan » Fri Jul 09, 2010 4:42 pm UTC

Red light has lower energy than Violet light, doesn't this mean that redshift is robbing the light of energy, and thus violating the first law of thermodynamics?
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby thoughtfully » Fri Jul 09, 2010 5:30 pm UTC

Whelan wrote:Red light has lower energy than Violet light, doesn't this mean that redshift is robbing the light of energy, and thus violating the first law of thermodynamics?

Well, sortof.. maybe. It's complicated.
The tl;dr is that it's hard to apply those concepts to cosmological scales. Yeah, makes my brain hurt too. GR can do that to ya :)
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby eternauta3k » Fri Jul 09, 2010 8:42 pm UTC

I was just thinking about gravity bending light rays. As far as I know one could consider the Earth is following a straight line in the curved spacetime created by the Sun. Would that curvature affect a light ray in the same way? If I send a light beam from Earth in the direction the planet is moving, will the beam trace out the orbit of the earth?
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby BlackSails » Fri Jul 09, 2010 8:48 pm UTC

eternauta3k wrote:I was just thinking about gravity bending light rays. As far as I know one could consider the Earth is following a straight line in the curved spacetime created by the Sun. Would that curvature affect a light ray in the same way? If I send a light beam from Earth in the direction the planet is moving, will the beam trace out the orbit of the earth?


No, the earth has mass, light does not. Earth travels on straight lines called timelike geodesics. Light travels on different straight lines called null (or lightlike) geodesics.

There IS an orbit for light rays though, its just not the same as the earth's orbit.

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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby eternauta3k » Fri Jul 09, 2010 8:56 pm UTC

Thanks. We only had two classes about relativity this term, so we haven't really studied the implications of relativity (except basic applications of lorentz transformations). I'm looking forward to studying this in depth.
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby PM 2Ring » Fri Jul 09, 2010 8:58 pm UTC

eternauta3k wrote:I was just thinking about gravity bending light rays. As far as I know one could consider the Earth is following a straight line in the curved spacetime created by the Sun.

Yes.
eternauta3k wrote: Would that curvature affect a light ray in the same way?

Sort of.
eternauta3k wrote:If I send a light beam from Earth in the direction the planet is moving, will the beam trace out the orbit of the earth?

No, because the lightbeam is moving much faster than the orbital velocity at 1 AU from the sun. The actual spacetime curvature of our orbit is tiny, much much smaller than the space curvature.

In space, planetary orbits are pretty much confined to a plane, so we can ignore one spatial dimenion. In spacetime, our orbit looks like a helix (but bear in mind that we are using the Minkowski metric). In one year, we travel through about 3000 light-seconds worth of space while we travel through about 30000000 seconds of time.

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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby gmalivuk » Fri Jul 09, 2010 10:12 pm UTC

PM 2Ring wrote:In space, planetary orbits are pretty much confined to a plane
And that's really understating it.
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Twin paradox

Postby timonan » Sat Jul 10, 2010 1:25 am UTC

I've been wanting to get a decent grasp on relativity, because it's always been a bit over my head, so I've just started reading through About Time by Paul Davies. I've reached the section where he's talking about the Twin Paradox, and, though he does an admirable job of trying to explain it, I still can't grasp it. Maybe some of you on this site can elucidate it a bit more (for reference sake, it's pp. 59-67 of the paperback version).

The numbers are made fairly simple. Betty leaves earth in a spaceship at the start of the year 2000 to a star that is 8 light-years away (measured from earth). She travels at 240,000 km/s (80% of the speed of light). Ann stays home. Betty's acceleration throughout the trip is assumed to be instantaneous. When Betty returns from her round trip, 20 years have passed in Ann's frame of reference, while 12 years have passed for Betty, all in perfect accordance to the time dilation of .6 that takes place for Betty during her trip.

My first question is about acceleration. Davies points out that Betty's acceleration throughout the trip breaks the symmetry between the twins, which explains the time dilation. I'm curious why that is. He mentions that acceleration is not relative, but absolute. I was hoping for a more thorough explanation here. Namely, why is acceleration considered absolute and not relative, as opposed to velocity? What is it about acceleration that creates a time dilation for Betty's ship? How do we know it wasn't the earth (and I suppose the entire universe) that is accelerating away from a motionless Betty?

Later, Davies is crunching the numbers and there's a certain part I don't get. He asks the question "where is Betty located when Ann's clock reads '2007'?" From Ann's perspective, at 2007 Betty is 5.6 light-years away. But Betty's clock reads something different, 2004.2, because of the time-dilation. That makes sense. Betty then looks through a telescope, and sees that the clock on earth reads 2001.4, but she reasons that the light from Earth, measured from Ann's clock, will take 5.6 light years to get to that spot, so therefore the clock on earth actually reads 2007. That makes sense, too. But then he goes on, and I get really confused. He says that from Betty's perspective, only 4.2 years have passed, and since, from her perspective, Earth is receding at 80% the speed of light (because their velocities are relative), she observes that the time that has passed on Earth is .6 of her own, which is 2.52 years. This means that as far as Betty is concerned, the date on Earth "now" (from her perspective) is 2002.52.

I'm so confused I don't even know how to frame the question. Betty knows that she accelerated away from the planet, so she knows that the time on earth when her clock reads 2004.2 is actually 2007, right? So does she know that her observation that the time on earth is 2002.52 is wrong, and should be 2007, despite what she observes? Or is it actually not wrong? If it's not wrong, where does it get reconciled so she'll be younger when she gets home? These numbers should be reconciled somehow, but I don't see where. Maybe on the return trip? During the accelerations?

Anyway, I still can't tie up all the loose ends. Any thoughts?

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Re: Twin paradox

Postby Tchebu » Sat Jul 10, 2010 3:15 am UTC

Long post ahead, but I couldn't find a way to explain it which I found both satisfactory and concise... my apologies.

First of all, the reason acceleration isn't relative is that you can actually feel that you're accelerating. If you were in a car without any windows to look out of, you'd still know when you're accelerating, braking or turning, because you'll be tossed around. This still holds true in relativity, and is in fact the starting point for general relativity, which identifies the forces that toss you around the car to gravitational forces, but I digress...

The fact that Ann is just chilling on Earth while Betty actually has to turn her spaceship around and in the process feels great forces similar to those you feel in a car ruins the symmetry that was present until the turn-around.

I think the easiest way to explain this would be with a space-time diagram. (See "Crappy Space-Time Diagram")
The black axes, represent the x and t coordinates of Earth. Needless to say, as time goes forward the Earth stays at x = 0.
The red axes are the space and time coordinates x' and t' in Betty's rest frame on her way away from Earth.
If you've seen space-time diagrams before skip the spoiler...

Spoiler:
If you haven't seen such diagrams before, you might be surprised that the axes get shifted toward each other, but it's fairly straightforward to see this if you stare at the formulas for the Lorentz transformations long enough.

I have omitted the ticks on the axes because paint is annoying enough as it is, but it's worth noting that the "1 second" and "1 meter" ticks on the red axes are slightly further away from the origin, than the corresponding ticks on the black axes. In fact they are found by taking the intersection point between the x' and t' axes and vertical or horizontal hyperbolas passing through the corresponding ticks on the x and t axes (this is similar, and closely related to the fact that if you were to do simply rotate axes, you'd find the red ticks, by looking for points of intersection between the axes and circles that pass through the corresponding black ticks)

The blue axes represent the coordinates x'' and t'' which are the coordinates in Betty's rest frame AFTER she turns around and heads for Earth, same principle here, but the axes are bent the other way, for obvious reasons.


Alright, so... you have probably already seen that simultaneity is a relative notion in SR. This diagram clearly demonstrates is. In Ann's frame, the lines of simultaneity are just lines of constant t, that is to say, lines parallel to the x axis. Now consider the dot that I put on the diagram. Let's suppose that that's the point where Betty turns around. So in the x,t coordinates this event has the coordinates (x,t) = (8 ly, 10 yrs). Now suppose Ann wished to know what time Betty's clock when she turned around. Well in the x', t' coordinates Betty is always at x = 0, t' = whatever time Betty's clock shows. So all we have to do is figure out the t' coordinate of that dot on the diagram. Well we can't read it off, because I didn't include ticks, but we know that the ticks on the t' axis are further from the origin than the ticks on the t axis, so whatever t' is at the time of the turn-around, it will be less than 10 years.

Now if Betty wants to know what Ann's clock was showing just before the turnaround, she will have to use HER notion of simultaneity. And Betty's lines of simultaneity are lines parallel to the x' axis (so the red line that passes through the dot is a simultaneity line). Notice that the simultaneous, according to Betty, indication of Ann's clock, is again, less than 10 years, and is in fact less than the time measured by Betty's clock at the turn-around.

So far this is all perfectly symmetric. Both observers see the other's clock run slower, by exactly the same factor, in perfect accordance with time dilation. In fact I could redraw that diagram, by making the x' and t' axes orthogonal to each other while bending the x and t axes instead (in the opposite direction of course) and do the same analysis. You might be asking "so which one is ACTUALLY going slower" if so, see spoiler.
Spoiler:
At a first glance it may seem like the observers might disagree on the physics that might be happening around them, leading to some contradictions or something, because observers disagree on time between events, or perhaps even their order. This doesn't happen however, because although the notion of simultaneity is relative (which is what leads to this disagreement as illustrated above), what's important for the physics is that two events which are outside each other's light cone stay that way in all inertial frames, and that does indeed hold, so while the events may not happen in the exact same sequence for all observers, the causal relations between them are preserved. Whether the observers see two spacelike separated events as being simultaneous to the same ticks of their clocks is irrelevant to the causal structure of the universe. Also note that, while each of them might see the other observer's clock tick slower, Ann is fully equipped to understand that Betty sees the same effect, and can even calculate what time on HER clock Betty is seeing and vice versa.


Now suddenly Betty turns around, and what that does, is change her rest-frame coordinates from the x',t' coordinates to the x'', t'' coordinates, which are angled in the opposite direction. In particular what this does is change her simultaneity lines from the red line on the diagram to the blue line on the diagram, parallel to the x'' axis. Now let Betty ask what time Ann's clock is showing, simultaneously with the moment just after the turn-around. Well that would be value of t, at the intersection between the t axis and the blue simultaneity line. Notice that now that value of Ann's time is actually more than 10 years, by exactly the amount by which it was smaller than 10 years just before the turn-around. In other words, Betty sees Ann suddenly age by several years. However Ann's simultaneity lines are completely unaffected, so she doesn't end up seeing anything special happen to Betty. To Ann, Betty's clock is simply behind hers, as it should be, and now Betty is moving towards her instead of away, still with a slower clock. This is what is meant by acceleration breaking the symmetry between Ann and Betty. For the rest of the return journey the symmetry is restored, and both observers see the other's clock tick slower as usual, but the extra time that Ann picked up on Betty during the turn-around stays there as an objective fact of life that both observers can agree on upon meeting back on Earth.

Obviously in a real setting, the turn around won't be instantaneous, so Betty's simultaneity lines will gradually change from the red lines to the blue ones, and Ann will seem to simply age really fast, rather than instantly age by several years.

Also, as I said before, while turning around, Betty will feel very strong forces as her spaceship brakes and then accelerates again to cruising speed. In general relativity such forces are said to be equivalent to gravity, so in fact what Betty experiences can be seen as being temporarily placed in a strong gravitational field. One of the consequences of being in a strong gravity field, according to GR, is that your clocks slow down. Which gives the exact same result as what we had before: Betty's clock, and thus aging slows down, compared to Ann, during the turn around, so Betty sees Ann age faster than herself. In fact this twin paradox is one way to derive this GR effect.
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby thoughtfully » Sat Jul 10, 2010 11:45 am UTC

Back in the 80's, Caltech produced an instructional video series that tracks pretty well with the first year or two of an undergraduate physics curriculum. The production values are excellent, and it's very accessible. Lessons 42 and 43 cover Special Relativity, and might help to clear up lingering confusions, while being entertaining to boot. The twin paradox is treated at the end of lesson 43.
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby PM 2Ring » Tue Jul 13, 2010 10:56 am UTC

BlackSails wrote:If you want a really cool (and rather difficult) SR problem, think about this: Lets say you have a submarine traveling at a relativistic speed through the water (just go with it). The submarine has the same density of water - it is neutrally buoyant and neither sinks or swims. Now the crew of the submarine sees that the water is length-contracted. Accordingly the density of the water is higher, and the submarine will become more buoyant, and float towards the surface. But from the point of view of someone at rest with the water, the submarine is contracted, and its density is higher, and it will become less buoyant and sink.

Who is right? (They obviously both cannot be, since an object cant sink and float at the same time)

Keep in mind that there is gravity here, something SR cant normally deal with. But you can instead treat it as a uniform acceleration upwards of 9.8m/s/s of the ground.

From another forum:
KJW wrote:I believe I have the correct answer (though I probably should perform a written check):

I believe the submarine will stay buoyant. Having the confidence that relativity is consistent, one really only needs to examine the situation in one frame-of-reference, choosing that frame-of-reference according to the ease of solving the problem, then correctly solve the problem in that frame-of-reference. I choose the frame-of-reference in which the submarine is at rest and the water is in motion. If we assume that only buoyancy is to be considered, then we only need to consider the vertical forces on the submarine. Because only the component in the direction of motion and the time component of the energy-momentum 4-vector of the water are altered by the motion of the water relative to the submarine, the vertical component (which is orthogonal to both), will be unaltered. Therefore, the vertical component of the forces will be unaltered and the submarine will remain buoyant.

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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby BlackSails » Tue Jul 13, 2010 2:54 pm UTC

That analysis is right for a situation where we neglect gravity. But here on Earth, we live in a schwarzschild metric! No minkowski metrics in my house!

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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby Charlie! » Tue Jul 13, 2010 3:32 pm UTC

Yeah, KJW from "another forum" is incorrect. Like I said, you can find an answer by assuming one exists, but that's dangerous. You not only are assuming that a solution exists, but you have to assume that one of the scenarios is truly "well-behaved," and be able to pick that one or the whole thing doesn't work. His choice of the sub's frame seems to be that last kind of mistake. Also, he doesn't account for the density increase.
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby PM 2Ring » Tue Jul 13, 2010 3:42 pm UTC

Charlie! wrote:Yeah, KJW from "another forum" is incorrect. Like I said, you can find an answer by assuming one exists, but that's dangerous. You not only are assuming that a solution exists, but you have to assume that one of the scenarios is truly "well-behaved," and be able to pick that one or the whole thing doesn't work. His choice of the sub's frame seems to be that last kind of mistake. Also, he doesn't account for the density increase.

I also wondered about his neglect of the density issue. OTOH, he does seem to know his stuff when it comes to GR, but (as mentioned at the start of his post) he was just doing this off the top of his head, with no written analysis. And I'm not sure how much of this thread he's read; he may've just read the initial problem description.

I've been trying to encourage him to register here, but he's pretty busy. I've given him a link to this thread, and will post any updates from him, if he doesn't do so himself.

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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby Goemon » Wed Jul 14, 2010 1:10 am UTC

KJW wrote:...Because only the component in the direction of motion and the time component of the energy-momentum 4-vector of the water are altered by the motion of the water relative to the submarine, the vertical component (which is orthogonal to both), will be unaltered. Therefore, the vertical component of the forces will be unaltered and the submarine will remain buoyant..


The vertical component of a water molecule's 4-velocity = dz/dtau is unaffected. But the velocity of the water physically measured and felt by the submariners in motion isn't dz/dtau, it's dz/dt = (dz/dtau) / (dt/dtau). The velocity (and momentum) are reduced by a factor of dt/dtau = gamma in the vertical direction.
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby PM 2Ring » Thu Jul 15, 2010 5:39 pm UTC

KJW wrote:According to Goemon, one should use dz/dt instead of dz/d[tau], leading to a reduced z-component of the velocity by a factor of [gamma]. But, using dz/dt instead of dz/d[tau] also requires that one use the relativistic mass for the momentum, increasing the mass by a factor of 1/[gamma]. Therefore, the z-component of the momentum is unaltered whether one uses dz/dt or dz/d[tau].

I think length contraction is a red herring here, and too easy to misapply.

Unlike the twin clock paradox, this situation seems quite symmetric, which suggests to me that neither sinking nor floating is correct and that the submarine instead remains buoyant.

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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby Bowshewicz » Sat Jul 17, 2010 11:47 am UTC

I have a question about length contraction, similar to the ladder paradox, but a bit different. First, to make sure I understand correctly so far:

If a 12-inch ruler were to move past the chair I'm sitting in, and it were traveling at 0.86c, it would appear to me that the ruler were only 6 inches.
Length contraction works even for two separate objects, so if there were two billiard balls that were 12 inches apart moving at the same speed, I would perceive them to be only 6 inches apart.

What has me stumped is a situation where there are two billiard balls outside my room, one right against the outer wall, and one on the other side half a room length away (So the distance between them is one and a half room lengths). If these billiard balls were to move together at 0.86c until the second ball is now against the outer wall, it would appear to me that the distance between them is only 75% the length of the room.

This works out fine in the reference frame of the billiard balls, as my room is contracted to be smaller, but in my reference frame, it would seem that one of them would have to appear in the room, when this clearly never happens. Any attempt I make to apply relative simultaneity has the distance between the balls changing, which I don't think is right, either.

Would someone explain what happens in this situation? I'm very confused about it.

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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby doogly » Sat Jul 17, 2010 2:37 pm UTC

diagram.jpg

Are you familiar with this style of diagram? It allows everything to make sense!

The balls are the two red lines.

If you are in the rest frame of the rooms, the walls are just the vertical black lines. Since they are at rest they move entirely in the direction of your timelike coordinate (vertical). Spatial distances are entirely horizontal, your x coordinate. You measure the separation to be the horizontal black lines.

In coordinates where the left ball is at rest, you are in a boosted frame, so you use the orange t' and x' coordinates. You measure the spatial distances to be slices at a constant coordinate t', so these are the parallel orange lines.

Caution: diagram not to scale! The angle between x' and x and between t' and t should be the same, but here for some reason I drew t' to t much larger. It's just a sketch though.
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby makc » Sat Jul 17, 2010 6:12 pm UTC

doogly wrote: The angle between x' and x and between t' and t should be the same, but here for some reason I drew t' to t much larger. It's just a sketch though.
not at all, the angle is the same only if you choose c=1... which is the only reasonable choice, otherwise this diagram is useless :)


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