Revised: July 24, 1998

Solution to: "Standard Twin Problem" as posed by Perez-Franco.

PROBLEM:

Robert and Eka are 20 years old when Robert decides to go on a round trip to a planet 20 light years away. He travels at a velocity of .8c How old are the twins when Robert returns? Also what will Eka observe, as compared to Robert, during his actual trip?

SOLUTION:

Since it takes time for the moving image of Robert to reach Eka, she will perceive Robert where he used to be and what he used to look like, at some time in the past. That is to say, Robert will have traveled an additional distance during the time it took for Robert's historic image to reach Eka, even though the image traveled towards Eka at the speed of light (c). Consequently Robert's observed parameters such as: the distance he has traveled, the velocity at which he is traveling and his aging rate etc., differ from the actual distance he has traveled, his actual velocity and his actual aging rate.

The actual distance, X_ACT, Robert has traveled at time t is:

X_ACT = t v [1] where v is his actual velocity.

Next let us derive the observed distance X_OBS Robert has traveled at time t.

For simplicity's sake, while developing the formula, we shall call the actual distance traveled X_ACT = Y and the observed distance traveled X_OBS = X

The length of time it took Robert's image to reach Eka is: X / c

During that time Robert traveled an extra distance of X / c times v.

It then follows that:

Y = v t and Y - X = X v / c

When we substitute we get:
v t - X = X v / c. or,
X v / c + X = v t
X (v / c + 1) = v t
X (v + c) / c = v t or,
X_OBS = t v c / (v + c) [2]

It next follows from [1] that the observed velocity V_OBS is:
X_OBS divided by time t or, V_OBS = v c / (v + c) [3]

The observed time t_OBS required for Robert to reach the planet is equal to the actual distance traveled divided by his observed velocity or,

t_OBS = X / V_OBS = v t / [v c / (v + c)] or
t_OBS = v t (v + c) / v c or,
t_OBS = t (v + c) / c [4]

However in general, t_OBS is the time at which Eka makes an observation or:

t_OBS = X_OBS/V_OBS or

t_OBS is equal to the observer's real time t.

Robert's observed age A_OBS, (disregarding his starting age), is equal to the actual time it took Robert to reach a particular location. For example, Robert's age when he gets to the planet is equal to the distance traveled, divided by the actual velocity at which he traveled towards the planet. By the time Eka sees him arrive that will be his "observed" age, or

A_OBS = X/V = t

Alternatively, at any given time the actual age of the moving historic image of Robert when it reaches Eka is equal to the observed length of time it took Robert to get there, minus the time it took for his image to reach Eka, or:

A_OBS = t_OBS - X/C = t((V + C)/C) - Vt/C or:
A_OBS = (tV + tC - Vt)/C = t [5]

Finally the observed aging rate A'_OBS is equal to:
A_OBS divided by the observed length of time t_OBS it took to get there or,
A'_OBS = t / [t (v + c) / c] or,
A'_OBS = c / (v + c) [6]

WE NOW HAVE ALL THE FORMULAS NEEDED TO SOLVE THE PROBLEM AS FOLLOWS:

ROBERT AS OBSERVED BY EKA ON THE WAY OUT:

The actual time required for Robert to reach the planet is [1] or,
t_ACT = X_ACT / v = 20 c / .8 c = 25 years.

The observed velocity of Robert while traveling towards the
planet is [3] or,
V_OBS = v c / (v + c) = .8 c² / (.8 c + c) = .8 c / 1.8
V_OBS = 0.444444....c

The observed time required for Robert to reach the planet is the actual distance traveled divided by his observed velocity [4] or,
t_OBS = t (v + c) / c = 25 (.8c + c) / c = 45 years.

The number of years Robert appears to have aged when he reaches the planet is:
A_OBS = t_ACT or,
A_OBS = 20 c / .8 c
A_OBS = 25 years.

Alternatively:
A_OBS = t_OBSC/(V + C) [5]
A_OBS = 45C/(.8C + C)
A_OBS = 25 years.

The observed aging rate of Robert on his way out is:
A'_OBS = c / (v + c) [6] or,
A'_OBS = c / (.8 c + c)
A'_OBS = .5555..... times normal.

ROBERT AS OBSERVED BY EKA ON THE WAY BACK FROM THE PLANET:

When Robert appears to take off for home he actually left 20 years earlier, because that is the length of time it took for his image to reach Eka.

The actual time required for Robert to get back home is [1] or,
t_ACT = X_ACT / v = -20 c / -.8c which is 25 years.

The observed velocity of Robert while traveling home is [3] or,
V_OBS = v c / (v + c) or,
V_OBS = -.8 c² / (-.8 c + c)
V_OBS = -4 c

The observed time required for Robert to return home is [4] or,
t_OBS = t (v + c) / c = 25 (-.8 c + c) / c = 25 (.2 c) / c or,
t_OBS = 5 years.

The observed aging rate of Robert during his observed 5 year trip home is [6] or,
A'_OBS = c / (v + c)
A'_OBS = c / (-.8c + c)
A'_OBS = 5 times normal.

This means that Robert appears to have aged: t_OBS times A'_OBS or,
5 x 5 = 25 years on his way home from the planet, which is the actual time it took him to turn around and get home. See [1] .

IN SUMMARY:

Both Robert and Eka aged 25 years during the time it took Robert to reach the planet and 25 years during the time it took him to get back home. Since the twins were 20 years old when Robert left, they will be 70 years old after he gets back.

Finally, it doesn't make any difference if Eka observes Robert during his trip or if Robert observes Eka. Because, even though he is moving, Robert will consider himself at rest in his spaceship at all times. Consequently it will appear to him as if Eka is moving away from him, instead of the other way around. In other words it doesn't matter which observer is being accelerated and slowed down because it doesn't change any of the above equations.

As far as special relativity is concerned, again "time dilation" is not needed as explained in my first paper: "Helical Particle Waves" of my "Selected Papers", since relativistic particles can continue to gain energy because they turn into helical wave particles, at linear speeds in access of about .6 c. This is further discussed in my final paper: "Time Dilation, Fact or Fiction"

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