The Simplist Explanation of the Twin Paradox

Most work done recently of this type is commonly rejected by the scientific community. The only real changes that have been done is the Holographic Principle made by Leonard Susskind. That only says differently than previous conceptions of the idea is that an observer falling in a black hole would see time speeding up around them. Then if they returned to a stationary observer, that person would appear to be older than them. They would have actually aged less. Their personal clock would slow down.

But, the light clock example hasn’t been recognized to ever be solved accurately. There is a correct derivation for gravitational time dilation in Sean Carrols book, The Particle at the End of the Universe. He goes on a lot about online chat forums in physical as well. It was an interesting side note in the book. I wouldn’t trust a lot he says about electronics.

It can be solved correctly by setting up the equation of a light triangle to be;

(vt)^2 + (ct’)^2 = (ct)^2

In this situation, the dilated time is on the shorter end of the light triangle. This mathematically forces the amount of time measured to be a lower quantity, due to side b of the triangle being a shorter distance. It is denoted to be prime, since it cannot be exactly the same value or the triangle would be zero in size, dimensionless.

The equation could be manipulated to where you could completely determine the amount of distance an object traveled by only knowing the difference of squares of their dilated time.

d = c sqrt( t^2 - t’^2 )

 

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The first time I read your above post, I think I misinterpreted your above comments. With my second reading, I THINK you are talking about each twin merely observing the other twin with a telescope (or perhaps by looking at a TV image that is constantly being transmitted from each of them to the other twin). You are right that there won't be any instantaneous age changes in that case.

But I'm talking about something completely different ... I'm talking about each twin figuring out what the current age of the other twin is "RIGHT NOW". I.e., each twin wants to know how old the other twin is at each instant of their own life. They KNOW that isn't what they see on a TV image sent from their twin, because that image has taken a long time to get from their twin to them ... it's way out of date ... it doesn't show their twin's current age.

The answer I wanted to get is very easy for the home twin to get. Since she's always inertial (unaccelerated) she can use the simple time dilation equation for an inertial observer. That equation says, for example, that when their relative speed is 0.866 ly/y, he will be ageing half as fast as she is. He concludes the same thing (that she is ageing half as fast as he is) during his two constant speed portions of his trip. But at the reunion, they obviously can't each be younger than the other. The home twin's answer is the correct one, because she never accelerates. But during his instantaneous turnaround, he accelerates with an infinite acceleration that has an infinitesimal duration. The only way they can agree about their respective ages at the reunion is for him to conclude that she ages by a large amount instantaneously during his turnaround.

 

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You seem confused about how this paradox actually works. They don’t return to be the same age. The twin that accelerated returns younger than the other twin.

If somehow a ship could instantaneously turnaround with the same velocity, they would begin actually being the younger twin, and they would observe the other twin to actually be older after that point in time.

 

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I didn't say the twins are the same age at their reunion ... I said that "The only way they can agree about their respective ages at the reunion is ...". There is a difference.