Let us imagine a spaceship traveling to a planet around a star 4.0 light years away exactly at 99 percent the speed of light (Lorentz factor of around ~7) relative to earth. For simplicity let us assume the distance between earth and this exoplanet is constant at exactly 4.0 ly (earth exoplanet forms a inertial reference frame) Further we assume that there is no change in gravitational potential such as moving away from the gravity well of our solar system into the exoplanet star system,importantly no acceleration and deceleration is involved
According to my calculations from the perspective of the space ship it would be traveling 4 ly/7= 0.57 light year, and the time elapsed when it passes by the exoplanet from its perspective would be 0.57 ly/0.99 c= 0.5757..years or about 6.91 months. From the perspective of earth when the space craft lands on the exoplanets (4 ly/0.99c) or 4.0404..years would have elapsed..and spacecraft traveled exactly 4 ly. Are the calculations based on the stated assumptions correct so far?
What I think a Lorentz factor of 7 means, for every second on spacecraft clock, 7 seconds would elapse on the spacecraft, and vice versa (that is one second by earth's clock 7 second would transpire on the spaceship) Is this interpretation of Gamma of 7 correct?
Let us consider two events A clock on spaceship reads that it leaves earth and B-clock on spaceship records the instant the rocket passes by exoplanet at 0.99c relative to it and earth. The ship records A-B interval as 0.5757 years (this is the proper time). According to set of clock rest with earth this interval would be Delta T 7*0.5757=4.04 years. Let us reverse scenario: Event C (clock on earth records spacecraft leaving, this is coordinated wtih A), and Event D (clock on earth records time it receives confirmation spacecraft has passed exoplanet..this is 8.04 years later, 4.04 years for the spacecraft planet+4 years to get the signal back of succesful passby).... Would hte interval C-D be measured as 56.28 years by the ships clock? Its the symmetry/reciprocity of time dilation in special relativity I am having trouble understanding.
Further let us consider an event E also measured in Earth proper time where 2 years after the departure of the spaceship a hurricane destroys the launch pad. The proper time by Earth clock for this is 2 years, but would the time elapsed between C-E according to a set of clocks moving at the same velocity (i.e at rest relative to) the spaceship (which is still traveling at 0.99c into the galaxy relative to earth after passing exoplanet) be measured at 7*2=14 years? That is according to the ship Event E transpired AFTER the ship passed the exoplanet, while according to earth it was BEFORE the ship passed exoplanet (i.e the order of event is reversed according to the frame of reference).
TLDR, Clarify what would the event C-D and C-E be measured by a set of clocks moving with the same velocity as spaceship, and would the order of events C-E be reversed in Earth, and spaceship's reference frame? Are the questions clear?