Voyager 1: to the edge of the Solar System

Quote: weaselman

Well ... kinda ... yes and no. There is a subtle point here that makes this statement wrong.
While both twins are inertial, their views are equivalent and symmetrical, you can't tell which one is moving. If you ask twin A, he'll tell you that B's clock is slower than his, but B will tell you that A's clock is slower.
The symmetry breaks when one of the twins turns around (accelerates) to come back and compare the clocks. Now there is an objective difference between the two twins, because one of them stayed inertial all the time, and the other one did not.
The proper time of the inertial frame is always longer than a non-inertial one. This is just a geometric property of the Minkowsky space, similar to the familiar notion of a straight line being the shortest distance between two points in the familiar Eucledian space.

There's a lot wrong with the whole response, and I'm pretty sure you're not going to change your mind. But for others who find the discussion interesting ...

Just because one twin turns around doesn't mean there's any change in speed. And, even when there is, it's not like hsi speed goes away or relativistic effects end during acceleration and then return when acceleration ends. And, the earthbound twin is undergoing constant acceleration - centripetal and gravitational - and yet his frame of reference is somehow considered "proper"? He only appears to be not moving because he's not moving relative to the earth, but the earth is moving - spinning on its axis, going around the sun, following the sun around the galaxy, following the movement of the galaxy across space - only possibly one of which (the last) involves no acceleration. His path through space is herky-jerky, constantly changing direction and velocity (i.e., accelerating) relative to anything you select (outside of the surface of the earth).

Relativity explains this. Every motion and speed is relative to something. There's only one constant - the speed of light. Everything - time, space, dimension - is relative and changes to accomodate it. Only the speed of light is constant.

Quote: ItsCalledSoccer

Just because one twin turns around doesn't mean there's any change in speed.

Sure, it does. Not in speed, but in velocity. The direction changes.

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And, even when there is, it's not like hsi speed goes away or relativistic effects end during acceleration and then return when acceleration ends.

I am not sure what exactly you call "relativistic effects", and why they are supposed to end with acceleration.
The role of acceleration is not to "end relativistic effects", but to break the symmetry between the two twins - as long as they are both inertial, there is no difference between them, each can objectively insist that his clock is faster than the other's.

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And, the earthbound twin is undergoing constant acceleration - centripetal and gravitational - and yet his frame of reference is somehow considered "proper"?

I don't know what you mean by a "proper" frame of reference.

I was talking about proper *time* - it exists in any frame, inertial or not, you can think of it as a sort of a "distance" between two events in space-time. This is analagous to the familiar Eucledian geometry. Consider two points in "ordinary" space. There are (infinitely) many paths connecting them, but one of them is "special", because it has the shortest length of them all - it's a straight line.
In space-time, points can be thought of as events that occur at the given time at a particular place in space. You can connect any two points by various paths (called world-lines), that represent different observers. The "length" of a given path is the proper time of the given observer (how much time has elapsed on this observer's clock between the two events).
Now, the interesting property of the space-time is that *longest* path between any two events is always the one, representing the inertial observer.

Back to the twins. Once both of them meet in the same reference frame, and compare their clock, they will discover that the inertial clock has registered a longer interval, than the one that experienced acceleration, because the proper time for an inertial observer is always the fastest.

As for the effects of gravity, and rotation of the earth, obviously, we chose to disregard them in this case as insignificant, compared to the effects, experienced by the other twin. If the first guy was waiting for his brother near a massive black hole, he could very well end up being younger than his brother when they met again (because in that case he could no longer be considered inertial).

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Relativity explains this. Every motion and speed is relative to something. There's only one constant - the speed of light. Everything - time, space, dimension - is relative and changes to accomodate it. Only the speed of light is constant.

As long as you are talking about constant velocity motion, this is correct (except for "changing time, space and dimension to accomodate it", which doesn't make any sense). Acceleration is a whole different beast - unlike inertial motion, you actually can tell whether or not you are being accelerated without reference to other objects. The notion of acceleration is not therefore relative.

Wikipedia has a pretty decent discussion on this. It defines "proper time" as "time measured by a single clock between events that occur at the same place as the clock," which is pretty much right. It then goes on to apply an example to the twin paradox:

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For a twin "paradox" scenario, let there be an observer A who moves between the coordinates (0,0,0,0) and (10 years, 0, 0, 0) inertially. This means that A stays at x = y = z = 0 for 10 years of coordinate time. We find that being "at rest" in a special relativity coordinate system means that proper time and coordinate time are the same.

Let there now be another observer B who travels in the x direction from (0,0,0,0) for 5 years of coordinate time at 0.866c to (5 years, 4.33 light-years, 0, 0). Once there, B accelerates, and travels in the other spatial direction for 5 years to (10 years, 0, 0, 0). The total proper time for observer B to go from (0,0,0,0) to (5 years, 4.33 light-years, 0, 0) to (10 years, 0, 0, 0) is 5 years. Thus it is shown that the proper time equation incorporates the time dilation effect.

Now, none of us should depend on Wiki for our understanding of relativity, but this example is pretty good in that it shows that "proper time" changes depending on who's carrying the clock, and that "proper time" incorporates time dialation.

I would also point out ...

* it never mentions acceleration, only velocity. Acceleration doesn't appear in Lorentz equations. Acceleration could be implied if you imagine his path as up-and-back, but if it's helpful, imagine it as a big circle rather than an up-and-back. His speed never changes, only his direction, so he's accelerating centripetally the whole time, kind of like the guy on earth.
* it puts in quotes "at rest", I think implying that "at rest" means "at rest relative to [whatever]". The space traveler is "at rest" relative to his spaceship. If you think of the earth as a kind of "spaceship" like Sagan did, then the homebody is "at rest" relative to a spaceship as well.

May the science gods forgive me for referring to Wikipedia ...

Quote: ItsCalledSoccer

it shows that "proper time" changes depending on who's carrying the clock, and that "proper time" incorporates time dialation.

Exactly.

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* it never mentions acceleration, only velocity. Acceleration doesn't appear in Lorentz equations.

It doesn't appear there, because it is assumed to be zero. Lorentz equations are only applicable to inertial (unaccelerated) frames. They are also symmetrical between all observers.

You cannot use Lorentz equations to tell which of the twins is younger when they meet for the second time.
Doing the calculations in the twin A's coordinates, you will find that twin B should be younger, but changing the point of view to B's, you'll come to the opposite conclusion. Thus the "paradox" (they did not call it "twin *paradox* for nothing).

To explain away the contradiction ("paradox") one needs to notice that as long as both twins remain inertial, it will never be possible for them to ever meet and compare their clocks. If one of the twins turns around to return, his path can no longer be described by the Lorentz equations, that are only applicable to inertial frames, and thus the symmetry is broken.

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Acceleration could be implied if you imagine his path as up-and-back, but if it's helpful, imagine it as a big circle rather than an up-and-back. His speed never changes, only his direction, so he's accelerating centripetally the whole time, kind of like the guy on earth.

Note that centripetal acceleration is proportional to the square of the tangential speed. When the latter is close to the speed of light, the acceleration necessary to maintain circular motion would become enormous. If you wanted that acceleration to be comparable to what is experienced by the guy staying on Earth, the radius of the circle would become really huge (way bigger than the radius of the universe). In theory, yeah, that could happen, but it still would not cancel the difference in the clock rates though - accelerations are a lot trickier than inertial movement, to account for all effects properly you'd have to integrate each twin's movement over the entire world line, and I can't think of a simple way to explain this qualitatively.
This is why the way twin paradox is usually formulated is that one of the brothers goes straight away, and then abruptly turns back, so that Lorentz equations can be used everywhere except for the single point of his path.

Quote: weaselman

Exactly.

We may be saying the same thing but there's still something that confuses me ...

If proper time depends on who's carrying the clock ... what if both are carrying a clock? Wouldn't both be right? Which was kind of my whole point ... they're both right. But you said, no, they're not both right. And yet they are.

In other words, using the Wiki example and taking away all knowledge of time dialation, the traveler might say, "I've only been gone 5 years, how the hell did you get ten years older?" And the homeboy might say, "You haven't been gone 5 years, you've been gone 10." Both are right. Both have clocks. Both are on "proper time" according to themselves. Whose time is more "proper"? The one not accelerated? They're both accelerating, so that can't be.

Quote: weaselman

Note that centripetal acceleration is proportional to the square of the tangential speed. When the latter is close to the speed of light, the acceleration necessary to maintain circular motion would become enormous. If you wanted that acceleration to be comparable to what is experienced by the guy staying on Earth, the radius of the circle would become really huge (way bigger than the radius of the universe). But in theory, if that happened, there would not be any difference in age when the brothers met in this case.

If acceleration changes everything, which you are saying, then it doesn't matter the magnitude. Gravity is relativistic, not centripetal acceleration, even though they both have the effects of a force.

****

This is having the feel of getting circular, and is probably not a conversation worth having any more. Whatever reason you give for making the earth clock more "proper" than any other, it just can't make sense. Einstein makes sense.

Quote: ItsCalledSoccer

We may be saying the same thing but there's still something that confuses me ...

If proper time depends on who's carrying the clock ... what if both are carrying a clock? Wouldn't both be right?
Which was kind of my whole point ... they're both right. But you said, no, they're not both right. And yet they are.

Each clock is right in its own reference frame. If one twin stays inertial, and the other one "jumps into another reference frame", and then returns, then his clock is no longer displaying the proper time in the Earth reference frame.
ON EARTH, the clock that stays on Earth shows Earth's proper time, which is the closest thing to "the age of the universe" in any sane definition (ignoring the issue that the Earth is way younger of course), and is in this sense correct. All other (non-inertial) clocks are behind.

Think of it this way. Suppose, I am heading North, and you go North-East on a plane. Suppose, we each move 1 mile in our respective direction, and stop to look at each other. If I look at you perpendicularly to the line I travel, you will appear to be about .3 miles behind me. If you look at me the same way, you will conclude that I am behind.

This is exactly the same mechanism as the one that causes relativistic time dilation (it has to do with the relativity of simultaneity, basically, when trying to compare my clock to yours, I am looking at it perpendicularly to my time axis instead of yours).

But if now you you turn North-West, and walk another mile, and I continue walking North, then when we meet, we will notice that while you walked two miles, I have only walked about 1.4. This is again, the same mechanism as in the twin paradox (only, in case of space-time the "straight" - inertial - path is longer, not shorter, than the curved one).

Back to the question of who is right. The meaningful question to ask would be how far are we from the origin. You started at the origin and have walked two miles (this is your "proper distance"), while I have only walked 1.4, would you agree that in this case my "proper distance" is correct - i.e., we really are 1.4 miles away from the origin, not two miles? We are not "both right" - I was walking along a straight line, while you were not.

Same thing happens with our twins - one of them stays inertial (an analog of walking a straight line in "normal"
space), while the other one did not. The proper time of the inertial guy is therefore reflecting the "true" interval between the two events (just like my 1.4 miles was the "true" distance from the origin), and the other one is just his subjective measurement of how long it took him to travel.

Here's an interactive graphic set up by a couple of guys ... I don't know anything about them so I can't speak to their scientific standing, but it illustrates the paradox between us perceiving ourselves at the center of the universe and the possible existence of space outside of what we see that we can't possibly know about (yet).

In other words, it illustrates the hollow golf ball in the loaf analogy, or the loaf in the oven analogy, or whichever one works better for you.

But besides that,it's really interesting. Consider it my Christmas gift to y'all!

http://primaxstudio.com/stuff/scale_of_universe/

Now that's cool!

Quote: Calder

Now that's cool!

agree.

BTW there was an illustration in the History Channel's "the Universe" that showed if our solar system was represented by a dot the size of a period on a sentence, our Milky Way Galaxy would be the size of the United States.