Since all (nonaccelerating?) frames of reference are equivalent..... 1) Two objects approach each other directly each at just over half the speed of light. From the point of view of one, why isn''t the other approaching faster than light? 2) A fast spaceship leaves earth and the astronaut ages slower than the family he leaves behind. From his point of view he is static and his family is moving away quickly, aging slowly. When he gets home who has aged the most? ******** I am not the Iraqi information minister. GSACP : GameDev Society Against Crap Posting To join: Put these lines in your signature and don''t post crap!

This wouldn''t happen to be homework would it?

1) It''s all a matter of the way time is percieved. Time is different for each of the objects, meaning that the combined percieved speed wouldn''t exceed the sped of light.

2) ...

Basically your first assumption was incorrect. Frame of reference depends entirely on the (relative) velocity.

For the first question you want to start off in a reference frame where both objects have equal velocity in opposite directions. So object A has velocity α*c and object B has velocity -α*c. Then use the Lorentz transformation to find the relative velocity of A in the reference frame of B. I suggest you do the maths yourself.

The second question is just the twin paradox. Google it.

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Original post by Mastaba
This wouldn''t happen to be homework would it?

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Who the hell gets this for homework?

Ive googled for the twin paradox but I still see no answer. There is mention that if, say, they send an EMR synchronisation signal from home, the home clocks will seem too slow on the outward journey and too fast on the home journey, but since c is finite, isn''t this just a case of catching up with projectiles? And what''s this about different inertial frames in the turnaround?

********


I am not the Iraqi information minister.

GSACP : GameDev Society Against Crap Posting
To join: Put these lines in your signature and don''t post crap!

The closer an object moves to the speed of light, the more its spacetime is warped. Time slows down, distances change. I don''t remember the equations for it, but apparent velocities at any point of reference are always less then or equal to c.

The twin paradox can be explained by the fact that it cannot be a non-accelerating point of reference. To get the astronaut to a speed close to c, you need to accelerate to it first, then decelerate to get back. I don''t remember the specifics on this either.

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Original post by walkingcarcass
Who the hell gets this for homework?

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Who? Anyone taking an intermediate physics course.

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Original post by walkingcarcass
Since all (nonaccelerating?) frames of reference are equivalent.....

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All inertial (nonaccelerating) frames of reference are equivalent with respect to the laws of physics, not necessarily the observations made in them.

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Original post by walkingcarcass
1) Two objects approach each other directly each at just over half the speed of light. From the point of view of one, why isn't the other approaching faster than light?

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I assume you mean that the each object is observed to be travelling at a speed greater than half the speed of light on a line passing through the other object in a frame of reference in which they appear to be equidistant from the point where they will meet. In this case, they are approaching eachother faster than the speed of light, ie. in the current frame of reference the distance between them is shrinking at a rate faster than c. This is fine, as neither of the objects is moving faster than c in this frame of reference.

In the frame of reference of either of the objects, however, the other object is not moving at > c. Due to the effects of the Lorentz transformations (or the spacetime warping that they quantify), each object sees the other object moving toward it at a speed less than c. The velocities are transformed when you change frames of reference.

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Original post by walkingcarcass
2) A fast spaceship leaves earth and the astronaut ages slower than the family he leaves behind. From his point of view he is static and his family is moving away quickly, aging slowly. When he gets home who has aged the most?

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Adapted from http://www.physics.queensu.ca/%7Ephys242/pdf/Lecture9.pdf

Say there are two planets, 20 light years apart, at rest with respect to eachother (in the same inertial frame). The spaceship leaves earth (one of the planets) at 4/5 c travelling towards the other planet. In the frame of reference of earth, this journey takes 25 years. In the frame of reference of the ship, the journey takes 15 years, because the distance between the planets is no longer 20 light years. The spaceship is in a different frame of reference from the earth, one moving at 4/5 c with respect to earth, so distances measured in the earth's frame of reference are shorter when measured in the spaceship's frame of reference. When the spaceship arrives at the other planet, it turns around and comes back. It's still travelling at 4/5 (but in a different frame of reference from the earth, or the one it was in when it was heading away from earth), so it still takes 15 years to get back to earth, in the spaceship's frame of reference. In earth's frame of reference, it takes the ship 25 years to get back. When the ship arrives, it's taken 50 years total in earth's frame of reference, and 30 years ship time to make the journey.

This is not the same as if you'd considered the frame of reference of the ship the whole time. By changing its velocity, the ship changes the times that clocks at other locations read in its frame of reference. For example, a second ship leaves earth at the same time as the first ship arrives at the planet, as measured in the frame of reference of the ships, and at the same speed (4/5 c). In the ships' frame of reference, this event happens at time 15 years (since that's how long it took the ship to reach the planet in its frame of reference). When the first ship changes it's velocity to match that of the planet, it changes its frame of reference. In the frame of reference of the planet and earth, the second ship does not leave the earth at the same time that the first ship arrives at the planet. In the earth's frame of reference, the second ship leaves 9 years after the first ship. Thus, when the first ship slows down when it arrives at the planet in its frame of reference, the second ship's observed position will change from 15 light years away (at earth) to 4/5*9 = 36/5 = 7.2 light years away, while earth's observed position will change to 20 light years away. (I think I have that right... I might have made a mistake however). Similarly, if the first ship could read earth's clocks before and after changing its speed, and frame of reference, the reading would change, even after factoring in the time it woulk take light to reach the location of the ship. Before changing speeds, the ship could read earth's clocks with a telescope, add 15 years (the time it takes light to travel 15 light years, which is the distance the ship is from the earth in its frame of reference) and get a time: 15 years. If the ship then changed speeds and reference frames to match the planet, and took another reading of earth's clocks, added 20 years (the time taken for light to reach the planet in this reference frame), it would get a new time: 25 years. (Again, I think I have that right, but I might not.)

So, the ship could then change its speed, and return to earth, and have a bunch more of the same effects, and end up taking 30 years total ship time, and 50 years total earth time.

OK?

[edited by - Geoff the Medio on July 17, 2003 3:43:28 PM]

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Original post by Mastaba
I''ve googled for the twin paradox but I still see no answer. There is mention that if, say, they send an EMR synchronisation signal from home, the home clocks will seem too slow on the outward journey and too fast on the home journey, but since c is finite, isn''t this just a case of catching up with projectiles?

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Whoa. I have no idea what they are on about there.

Basically the twin paradox (which isn''t really a paradox) tries to make some argument involving a symmetry of the two situations for the astronaut and the stay-at-home. From the stay-at-home''s point of view the astronaut is moving so her clocks will run slower and from the astronaut''s point of view the stay-at-home is moving so his clocks will run slower.

Now using the Lorentz transform the time between ticks of the second hand for each person, we''ll call them ΔT_A and ΔT_S for Astronaut and Stay-at-home, are related by

ΔT_A = ΔT_S/sqrt(1-v²/c<sup>2)

This only contains a v2 = (-v)2 so it''s irrelevant whether we are talking about the outbound or homeward journey, contrary to what you read.

As lots of people have mentioned the "symmetry" is incorrect. The astronaut is subject to an acceration when she departs, turns around, and stops on coming home. General Relativity (as opposed to Special Relativity which uses the Lorentz Transforms) applies to accelerating reference frames like the Astronaut''s and describes how the astronauts clock will slow down.

Why can''t you just say that the Stay-at-home has accelerated and the Astronaut stayed put? Simple, imagine they both have a marble sitting on their kitchen tables (it''s a big spaceship). The astronaut''s marble will roll off when she accelerates, the Stay-at-home''s won''t. So there is definately no symmetry in their two situations.

Of course there is the question of what makes the marble fall off. General Relativity says it is due to the gravitational field of everything in the universe acting on it but my mind boggles whenever I think about that</sup>

Y''know there''s a great story by Heinlein about the twin effect. It also postulates telepathy between twins, and how the time differential would still make communication difficult.

Read it, and you''ll never lack for understanding about this. In the end, the twin that leaves Earth ends up marrying someone young enough to be his brother''s granddaughter, which answers your question.

Then there''s Ender''s Game, in which Mazer Rackham goes in a circle at relativistic velocities in order to stay alive long enough to train the commander of the counterattack. And of course, Ender himself, who stays alive 3000 years by not staying on any one planet for very long.