I refer you to the old twin paradox, one twin remains on Earth whilst the other travels to, say, the Andromeda galaxy at close to the speed of light, (note that at this speed it is possible, due to the contraction of space, to reach Andromeda in less than a human lifetime) and then returns to Earth to meet up with their twin, the paradox ofcourse being, who is older? if either. I know that the answere is found in the fact that the twin who travelled to Andromeda had to accelerate to whatever velocity he travelled at and also decelerate and turn around, but I don't understand why this resolves the issue... apparently the maths is slightly more complicated than that of constant velocity but i'm fairly sure I can handle it. simply, how does the acceleration destroy the paradox? And please do show the maths behind it. Thanks
Acceleration is not needed to explain the paradox. The traveling twin had to stop and then reverse course. To illustrate: traveler goes to a nearby star (10 light years away) at almost the speed of light. Applying special relativity to the distance along his path, in his own coordinate system he gets there almost instantly. He then stops and returns to earth at the same speed with a very short elapsed time in his coordinate system. Meanwhile twenty years have passed on earth. The only important acceleration effect is the requirement to stop and then go in the opposite direction.
yes i know, but why does that destroy the paradox, i dont know the maths of relativistic acceration yet so i cant solve it myself, for example the maths of constant velocity revolve arounf the ratio 1:1/SQRT(1-(v/c)^2) but this doesnt work for acceration. according to the observer on the Ship the Earth's clocks run slowly and when they return they will have aged more than the people on the Earth... who is right? i have been told the answere lies in the deceleration and subsequent acceleration in the opposite direction but how does this solve the problem?
The reason is that the time dilation formula used to work out how fast each twin is ageing in the other's frame is only valid from the perspective of an inertial (i.e. non-accelerating) observer. The twin who stays on Earth is always inertial so the time dilation formula always tells him how fast the travelling twin is ageing compared to him, so integrating that correctly gives the travelling twin's age when he returns to Earth. On the other hand, the travelling twin accelerates, so he can't naively integrate the time dilation formula for the earth-bound twin over the entire trip and claim the result should be his age, because the formula wasn't valid over the entire trip. It's violently invalid during the acceleration phase. Well there isn't really anything to show. Everything that happens according to relativity is already fully specified in inertial coordinate systems. If you carefully specify an accelerating coordinate system you can work out how the earthbound twin ages from the perspective of the accelerating twin throughout the entire trip. Working this out isn't really interesting for a couple of reasons though. The first is that since the calculation is just based and adapted from what happens in an inertial coordinate system, you're guaranteed from the beginning that you're never going to get a mismatch or paradox. Assuming the space-travelling twin travels at constant velocity away from Earth, accelerates, and returns again at constant velocity, then you'll associate an accelerating coordinate system with his acceleration phase that transitions smoothly between his inertial rest frames before and after the acceleration. So any result you predict from such an accelerating coordinate system will have to be compatible with any predictions made in inertial frames by construction. The second is that non-inertial coordinate systems don't have the same physical significance that inertial frames do anyway. It's clear without even doing any calculation that, to avert a paradox, the earthbound twin has to age very rapidly in the space-travelling twin's frame while he's in the acceleration phase. This is accounted for by the relativity of simultaneity effect: as the space-travelling twin accelerates, he passes through many different inertial frames and his notion of simultaneity - which is what he uses to compare his age with the age of his earthbound twin - constantly changes. In a sense, his space, or "simultaneity" axis rapidly sweeps over the earth-bound twin's worldline. For an analogy, this is similar to the way a distant object, such as the moon, appears to move a great distance from your perspective if you turn on the spot, when it's really just your orientation that has changed. It's worth pointing out that, by general relativity's equivalence principle, if we expect to see clocks run faster from our perspective when we accelerate toward them, we also expect them to run faster if they're higher up in a gravitational field than we are. This is just GR's gravitational time dilation effect.
Here's my version. There is no paradox. Everything adds up nice and pretty. The Rocket ship Twin brothers are standing next to each other on Earth. One brother is showing the other brother his new rocket ship. He says "this baby will go from 0 to 93,000 miles per second (.5c) in one second." He explains to the brother that an acceleration rate of 93,000 miles/sec^2 is the rate of change of velocity. If an object's initial velocity is zero, and the object accelerates (thrusts) at the rate of 93,000 mi/sec^2 for one second, one second later the object will have traveled a distance of 46,500 miles, and will be traveling at the velocity of 93,000 mi/sec. The brother says, "cool, can I take a spin?" The other brother says, "no problem." The brother immediately jumps into the rocket, ready to go. The rocket ship has on board data acquisition systems that will record the exact distance traveled per time interval, and a state of the art time device. The ship also has a accelerometer that records the acceleration of the ship at all times in every direction. The brother flips the switch that activates the data acquisition systems and the max thrust engines at the same time. He turns off the thrust when one second has elapsed. He is now traveling at the velocity of 93,000 mi/sec. The brother continues to travel at that velocity for 10 seconds at which time he reverses thrust and "decelerates" at the rate of 93,000 miles/sec^2 for a duration of 1 second. His velocity is now zero miles per second and he is 1,023,000 miles away from his brother. He just traveled a total of 1,023,000 miles in the duration of 12 seconds. The brother decides to get some sleep. Exactly 8 hours after arriving, the brother activates all systems, including max thrust that starts the return journey. The brother accelerates at the same rate (93,000 miles/sec^2) for one second. After one second has elapsed he turns off thrust. He again travels at the velocity of 93,000 miles per second for 10 seconds, at which time he "decelerates" at the rate of 93,000 miles per second (reverse thrust). Total elapsed time of return travel is 12 seconds, and again, the distance traveled is 1,023,000 miles. The ship traveled 1,023,000 miles in 12 seconds in one direction, and 8 hours later traveled 1,023,000 miles in 12 seconds in the opposite direction. Actual travel distance- 2,046,000 miles Actual travel time- 24 seconds Layover time- 8 hours Total time- 8 hours 24 seconds Max acceleration- 93,000 mi/sec^2 Min acceleration-0 Maximum velocity- 93,000 mi./sec Minimum velocity-0 Average speed- 85,250 mi./sec
Best advice, bestofthebest: Ignore everything Motor Daddy says. This set of web pages, http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_paradox.html describes the twin paradox from many different perspectives. They all end up with the same result, so you can pick the explanation that best suits you personally. It is a good idea, however, to understand all of them because each looks at relativity from a slightly different aspect.
Thanks D H, i was going to ignore him/her since they seem to have no crasp on what special relativity is and that time slows down at high speeds and space contracts.
