The "Twin Paradox" with a marshmellow

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by Dennis Kean

Several centuries down the road, when our technology has evolved and we become capable of near light speed, one of the twins is sent on a 10 year journey, (earth's frame) to inspect a black hole at a distance of 50 trillion kilometers away and bring back some data.  The sum total distance is 100 trillion Kilometers.  The ship is capable of a speed of 298,400 kilometers per second and no more and has sufficient fuel as manufactured by EXXON to make the trip back

The scientists do their calculations and determine from the distances that indeed it will take 10.61 years for the round trip. The ship is designed to enter an orbit around the BlackHole and spin around for the return home effortlessly. As is well known, time dilation is cumulative and whatever transpires during the spin around the black hole, is just another time dilation that accumulates in the total sum, though the math may require GR math. But since the GR calculations do not contribute significantly, compared to the dilation of the trips to and fro, all that is required will be the straight line trips and the SR calculations certainly do not require much effort.
 

Earth's frame:
Speed of light  = 299,790 Km/s
Ship Velocity   = 298,400 Km/s
Distance(total) = 100,000,000,000,000 Km
Time            = 335,120,643 seconds
Time            = 10.61 years

However, it is also clear from Einstein's SR that the twin will experience a time dilation as well as a distance dilation.  They do their math and calculate that the contraction at the maximum speed will be 1/sqrt(1-v^2/c^2) or

Contraction = 10.3966
(Where v=Ship Velocity and c=Speed of Light)

As the twin whirls around the sun and zooms by the earth snapshots are taken from a series of signposts lined up to test the veracity of time dilation.  They all record and confirm that when the ship reaches the max speed the clock on the wall in the ship is running exactly 10.3966 times slower!

Wow!  "Einstein was so right, that insightful little gremlin" is heard in the halls of NASA!

Conversely, the pilot notices the same with his autocamera and computer which calculates all time delays of light beam arrivals that in the earth's frame the clocks are running 10.3966 times slower!  "Hot diggidy...!" he exclaims.  "That hairy little midget Einstein was so perceptive"!  It all works so perfect!

NASA runs the computations again and evaluates that the twin will age exactly 1.02 years going to the black hole and back, even though in NASA's time frame it will take 10.61 years.  After ten years they plan to put the twins side by side and compare to see what 9.59 years of age difference looks like.  The scientists are rubbing their hands and a glee is pervading silently throughout the camp.  Click click clack clack and the results are in!
 

The twin, however, having seen the earth clocks slowing down, as he passed by the earth near the speed of light, is struck with a morbid epiphany! He was looking forward to seeing the future more than his twin brother.  That was why he volunteered for the mission, to begin with.  Now, however, he realizes that if the earth's clocks are running slower, as he remains in his frame of reference, he is not going to be the twin who ages less!  "PASTA LA PIZZA!!, how can this be????"  Of all the luck......!

He realizes for the first time that Einstein's SR equations do not present a confluent future reality!  They calculate what will happen to the individual in the other frame, but the person in the other frame will never experience the predicted calculations.  Why not? It becomes clear to him that according to SR he will age faster and his brother, instead, will age 10 times slower, just like the Big Ben clock on the NASA's main building indicates!  Great sorrow overtakes him as his plans seem to have worked in a completely opposite way!

Can you help this twin to appease his mind and show him where he is wrong in his math, so he can relax for the trip and be assured that Einstein was not a mental case?  Evaluate for him how long his brother will age during this 1.02 years that he will be away, now that his brother on planet earth is receding away at the speed of 298,400 Km/s .  After all relativity is relative and we can start from his [the flying twin's] frame of reference just the same way we started from the earth's reference.  Right???  Here are the necessary values to get you going
 

Try calculating how far his brother on the earth is from the black hole, to be sure you are on the right track.  L' = L (sqrt(1-v^2/c^2))  Just imagine that you are starting from scratch and doing the analysis for the first time!   It should be a breeze for anyone who has taken algebra in school!  Any takers?  Don't be shy.  Apply what you have learned in school and help the worried twin find his peace!  Also, remember that you cannot use any values from the earth's frame.  You must do all your math entirely with the figures supplied in the ship's frame of reference!  Fair is fair!  Any argument that the earth's frame values are needed to get a proper answer implies that you cannot use the earth's frame either without the need of some other frame that is more fundamental to get the ship's frame of reference!  Are we clear on that?  After all, we are all moving in space and if we cannot do this calculation based alone on the values of the frame we are in, then SR is useless.  But SR is relativity and we all know what that means, so let's use it as if it were discussing relativity and not some Ether reference based system!

Good luck and please write back if you get stuck.  (Also, don't accept ad hominems from the professor!)

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