I already gave a short answer to this, and I'm going to expand upon it here. First I going to assume that one is using the GR approach to this problem. The confusion appears to come from the idea that it is the force felt by each twin alone that determines the effect of the time dilation. This is not true. It is the potential energy difference that does. An example: The surface gravity of any planet is found by g=GM/r² (with M being the planet's mass and r it's radius. For the Earth this works out to about 9.8 m/sec² For Uranus, it works out to about 8.9 m/sec² , meaning that Uranus has a slightly smaller surface gravity thn the Earth's. But a clock on the surface of Uranus would run slower than one on Earth. This is because the Gravitational time dilation is due to the difference in gravitational potential, which for a planetary gravity field works out as Gp = -GMm/r. It turns that the gravity potential of Uranus' surface is about 3.6 times of that at the surface of the Earth. Gravitational potential can be thought of as "how deep" you are in a gravity field, or how much work it would take to climb up out of it. Now, how does this apply to the twin in the Ship? When he is at the "braking" and "return acceleration" phase of his trip ( the "turn- around") he can consider himself "at rest within a gravitaitional field" (his engines are just keeping him from falling in response to this field. ). As he looks around him, he sees, however, that everything else (including the Earth) is falling in response to this field. and everything is falling uniformly.(No matter how far something is "above" or "below" him they fall at the same acceleration; unlike a planetary field where the acceleration falls off with height.) This gravitational field must be uniform throughout. Thus the gravity potential difference between himself and the Earth is equal to the amount of energy it would take to accelerate an object over the distance between himself and the Earth at 1g. So the time dilation he will see depends both on the intensity of his acceleration and the distance between the Earth and himself. The manner of the time dilation will depend on the Earth's position in respect to this accelleration. While he is accelerating away from Earth or braking to a stop upon his return to Earth, he will see "Earth time" slow down. While he is braking at the far end and accelerating for the Return trip he will see it speed up. The point to remember is during the first two cases the twin is closer to the Earth than he is in the second two cases, so the "speed up" will be greater than the slow down. In a case where the twin accelerates, coasts, brakes, accelerates coasts, and brakes, you have to apply both the SR dilations and GR dilations to the accelerating and braking sections, and just the SR dilations to the coasting sections. When its all put together, more Time will have elapsed on Earth than for him. (Even though at times he will see Earth time as passing slower than his, the other times where he sees it moving faster, will more than compensate.)
The viewer on earth: If the viewer on earth could view the traveler in real time, the traveler would be moving very slow, EX. the traveler blinks his/her eye Please Register or Log in to view the hidden image! the blink of an eye by the traveler, viewed by the viewer, would take a long time to complete, possibly an 10 minuets or more. The viewer at .9c If the traveler could view the viewer on earth in real time, the viewer on earth would be moving very fast, EX. The traveler is viewing a birds-eye-view of N.Y. City at night. He or she would be seeing what looks like a sky-cam on fast forward. This is how I understand how this would work. If I’m wrong then Please send links (pfcgrogan@aol.com )to help me better understand this situation.
Tom2, First, I'd like to thank you for that pdf, it's awesome. After reading the explanation for the twin paradox in the pdf, two things still confuse me: 1) Why did the author draw the wordline the way he did? In my illustration (see attachment) I drew both the authors representation of the worldline (Example 1) and my representation of the wordline (Example 2). I think my representation is more valid because it shows all six parts of the travelling twin's journey. 2) The author assumes that L is the one that is changing frames. However, from L's frame of reference, it is M that is changing frames (even though L is feeling the acceleration/deceleration). Why didn't the author show how the wordline looks from L's frame of reference??
Janus58, Here is where you lost me: How exactly does distance influence the time dilation for the travelling twin? Remember, the force resulting from acceleration is a local phenomena, so no long distance interation is necessary. If distance effects time dilation, what interaction is used to transmit the distance so that the twin knows how fast he/she should age?? You seem to be implying something like "quantum entanglement" where the molecules in the travelling twin's body "somehow" always know the distance between them and the Earth. Please elaborate how this is possible.
Different Janus, I like the fact that I am seeing something different here. I don't know yet if it satisfies my thoughts but at least you are saying something that I have never heard before. That is that the gravity affect and acceleration affect on time is "energy well" dependant. Not specific force dependant. Why is it that I have never see this said before. Is this conventional? Do all physicsts accept this view or is this your independant addition.?
agree this seems new, uncertain what I think. Q: If someone was twice as far away as earth, when the twin arrives at the earth, would he measure twice the time dilation? Is it distance away or distance TRAVELLED you are looking at - assuming it is the second one. Glad to see something new!Please Register or Log in to view the hidden image!
Of course I have said why, and the reason is highlighted in red. It is true by definition. Frames of reference can be divided into two types: Those that are accelerated, and those that are not, right? Let's take those frames that are accelerated, attach the label "non-ineratial". Since accelerated and non-accelerated and inertial and non-inertial are both mutually exclusive, that means that non-accelerated frames must be (*drumroll*) inertial. By definition. Tom edit: fixed color brackets
There are not six parts, though. No, he knows it because it is built into the problem statement. What makes a 'frame' a 'frame'? It is not a physical set of rods and clocks. A velocity is what determines a frame. One does not change frames without changing velocities. One does not change velocities without an acceleration. One does not undergo an acceleration unless one is under the action of a force. The ship is under the action of a force, the Earth is not. No. Both observers agree that there is no change in the velocity of the Earthbound twin. You'd have to ask him. Tom
Re: Different As to why you've never heard it before, I don't know, but it is conventional. Look at the formula for Gravitational Time Dilation : t = t0/sqrt(1-2GM/Rc²) GM/R is a representation of gravitational potential. If the formula were "force" dependant you would use GM/R². That way, if you wanted to measure the Time dilation at the surface of the Earth you could just replace GM/R² with "g" (9.8 m/sec²) and get: t =t0/sqrt(1-2g/c²) and the result would be just "force" dependant As it is, in order to insert g into the formula you get: t =t0/sqrt(1-2gR/c²) Which means the time dialtion is not only dependant on the "local" acceleration, but the clock's distance from the center of the field as well. Of course the formula given assumes that you are talking about a mass generated field which falls off by the square of the distance, if you are talking about a different type of field (such a a uniform one) you have to make the appropiate adjustments.
Okay, the first thing to remember is: "Time dilation always happens to the other guy." The thing is, from the "traveling" twins perspective he doesn't experience time dilation, his clock always runs normal. It is the Earth that undergoes the dilation, and the degree of dilation the Twin sees the Earth undergo depends upon how far the Earth is away. The whole point of the Twin Paradox is to reconcile what the Stay at home and Traveling Twins see so that they at the end, they agree which aged more. From the Stay at home twins perspective, he merely sees his twin move away and come back under his own power. He sees the other's clock as moving more slowly due to his relative velocity, and expects less time to to have expired for his twin then for himself. The traveling twin see a more complicated picture because he has to take the effects of the accelerations he feels into account (Which if you are using the GR approach are due to gravitational fields) While he is accelerating from the Earth he sees two effects, one SR, which has the Earth's time undergo slowing, and the other GR, which also has the Earth's time slowing. (Again remember, from this twins perspective, he is merely standing still in a gravity field while everything else around him is falling). Then he coasts. Now he sees just the SR effect. but he also sees a length contraction effect where the distance between Earth and Alpha Cen has shrunk, So the time it takes for him to cross this distance will be less than that measured by his twin. (not because his clock has slowed down from his perpective, but because the distance traveled has shortened. ) Now he starts to brake. The SR effect lessens as he losses speed, but the GR effect has reversed so that he sees Earth time speed up. The further away the Earth is, the Greater the Speed-up the Twin will see. Remember, from his perspective, the Earth is at the "top" of a very long and uniform-in-strength gravity field, and he is closer to the "bottom". Since clocks higher in a gravity field run faster than clocks lower, he see the Earth clocks as running fast. And since The Earth is a lot "higher" in respect to him now than it was at the begining of the trip, the speed-up will outpace the Slow-down he saw when he started off. Once he has stopped, he now starts to accelerate back to the Earth. The SR effect begins to increase but the GR effect still over-rules. Coasting again, same effects as before, the trip takes less time because the distance is less. Braking to a stop at Earth You get the same GR effect you got when leaving and the SR effect decreases until you come to a stop. Stay at home twin says Traveling twin aged x amount of time because, according to him, the other's clock slowed down. Traveling twin agrees that he aged x amount of time, but because, for him, the trip was shorter in distance. Stay at home twin says that some time greater than x time passed for him because that's how much time he experienced during the duration of the trip. Traveling twin agrees that that same amount of time greater than x passed for his brother, because his brother underwent a series of combined slowdowns and speed ups in time rate with the speed-ups taking the edge so that over the duration it adds up to a total expired time of just that much greater than x. Both Twins agree which aged more and by how much; end of Paradox.
Hoodlum: <i>If you are in a spaceship (in zero gravity) and you place a ball in front of your head so that it floats perfectly still (relative to the ship), then you accelerate, the ball will hit you in the face thanks to the law of inertia. What you saw was the ball accelerating toward your face, however, if you have two acceleration measuring devices (accelerometers?), one on the ball and one on the ship, then you will measure a change in motion on behalf of the ship, and not the ball.</i> Yes. That tells you that the ball is in an inertial frame of reference and the spaceship is not. <i>So the obvious conclusion is that the ship changed motion, not the ball, even though relative to the inside of the ship the ball "appears" to the naked eye to move, the ball will "appear" to the "accelerometer" NOT to move.</i> Yes. That's what relativity is all about. It lets us transform between different viewpoints. If you view the ball's motion from an inertial frame, it does not move. If you view it from a non-inertial frame it appears to accelerate. <i>There is no "imaginary" force, because the velocity of the ship can be measured to change, and the inertia of the ball can be measured not to change, despite all appearences to the contrary.</i> A correct version of this statement is: there is no "imaginary" force on the ball <b>when it is viewed from an inertial frame of reference</b> (e.g. one which is not accelerating). When viewed from a non-inertial frame, the ball appears to accelerate, and Newton's second law requires an inertial force in order for that to happen. <i>If an object in your frame of inertial reference breaks the law of inertia, your frame of inertial reference has been accelerated.</i> Yes, and an accelerated frame is, by definition, non-inertial. <i>Whenever you see an "imaginary force" the only explaination is a change in the movement of you inertial reference frame.</i> Yes, but it can often be difficult to separate inertial forces from real forces. Inertial forces can kill you, so they are hardly "imaginary".
