Question

8. When the previous astronaut returns from his trip, the observer on Earth a. is the same age as the astronaut because the same perspective applies to both. b. is older than the astronaut because of special relativity. c. is younger than the astronaut because of general relativity. d. is younger than the astronaut because of special relativity.

Answer 1

Let,

The astronaut was traveling with the speed of   ( towards or away does not matter ) with resect to earth.

distance measured in the frame of the earth of the point to which the journey has to be made

so total journey distance measured in earth's frame

( note that before leaving for journey while the astronaut was on the earth he will also agree on this distance )

distance measured by astronaut between earth and the point to which the journey has to be made, when the astronaut is moving with speed .

so total journey distance measured in earth's frame

By lorentz contraction the relation between these distances will be given as,

__________________________________________ relation 1

where,

speed of light

Since whatever the speed is it will be always less than , the lorentz factor    will always be greater than 1

that is, ( off-course for non-zero )

therefore,

Let,

time period measured by observer on earth ( that is in earth's frame ) for astronaut's round trip,

time period measured by astronaut for his round trip ( / journey )

will be given as,

____________________________________________ relation 2

and will be given as,

____________________________________________ relation 3

we have from relation 1,

Using this in relation 2 we get,

using relation 3 we get,

Since as mentioned earlier , we get,

  

So, we see that the time period measured by observer on earth for astronaut's complete journey is greater than the time period measured by astronaut for his journey,

that is time passed on earth is greater than the time passed on astronauts spaceship for his complete journey,

therefore the answer is,

b) the observer on earth is older than astronaut because of special relativity.

option a) is wrong because wee see that same perspective does not apply to both, in both frames the journey distances measured are different.

option c) is wrong because nowhere in this problem we are dealing with general relativistic calculations.

option d) is wrong because it is the opposite of the answer we found by doing calculations above,