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## Main Question or Discussion Point

Consider the following scenario:

There is a triplet of persons A, B and C. Person A stays on earth, while person B and C both go onto two different space missions, the directions parallel to each other. Person B travels at 0.45*c, person C at 0.9*c. The space missions are both set to take exactly one earth-year, that is, person B and C will be back after 1 year having elapsed for person A.

How much time elapsed for person B and C when they come back?

For person B: sqrt(1 - 0.45^2) ~ 0.8930 years

For person C: sqrt(1 - 0.9^2) ~ 0.4359 years

Now person B calculates how much time elapsed for person C. As they are inertial reference frames, wouldn't time for C speed up by a factor of:

1/sqrt(1 - (v2 - v1)^2/c^2)

Where v2 and v1 are the two velocities, 0.45c and 0.9c? That is, it would slow down by a factor of:

sqrt(1 - (0.9*c - 0.45*c)^2/c^2) = sqrt(1 - 0.45^2)

Since the time elapsed for person B is sqrt(1 - 0.45^2), would he not calculate the time elapsed for person C as:

sqrt(1 - 0.45^2)*sqrt(1 - 0.45^2) = 1 - 0.45^2

So wouldn't person B consider person C (1 - 0.45^2) ~ 0.7975 years older, even though person A would consider person C 0.8930 - 0.4359 ~ 0.4571 years older.

What's the error in my logic here?

Thanks in advance

There is a triplet of persons A, B and C. Person A stays on earth, while person B and C both go onto two different space missions, the directions parallel to each other. Person B travels at 0.45*c, person C at 0.9*c. The space missions are both set to take exactly one earth-year, that is, person B and C will be back after 1 year having elapsed for person A.

How much time elapsed for person B and C when they come back?

For person B: sqrt(1 - 0.45^2) ~ 0.8930 years

For person C: sqrt(1 - 0.9^2) ~ 0.4359 years

Now person B calculates how much time elapsed for person C. As they are inertial reference frames, wouldn't time for C speed up by a factor of:

1/sqrt(1 - (v2 - v1)^2/c^2)

Where v2 and v1 are the two velocities, 0.45c and 0.9c? That is, it would slow down by a factor of:

sqrt(1 - (0.9*c - 0.45*c)^2/c^2) = sqrt(1 - 0.45^2)

Since the time elapsed for person B is sqrt(1 - 0.45^2), would he not calculate the time elapsed for person C as:

sqrt(1 - 0.45^2)*sqrt(1 - 0.45^2) = 1 - 0.45^2

So wouldn't person B consider person C (1 - 0.45^2) ~ 0.7975 years older, even though person A would consider person C 0.8930 - 0.4359 ~ 0.4571 years older.

What's the error in my logic here?

Thanks in advance