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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 in the space ship reference frame the exoplanet would would be approaching from an initial distance of 4.0 ly/7= 0.57 light year away, and the time elapsed when it passes by the exoplanet passes by the ship would be 0.57 ly/0.99 c= 0.5757..years or about 6.91 months. In the earth reference frame the space craft passes by the exoplanet at its closest approach (4 ly/0.99c) or 4.0404..years would have elapsed..and spacecraft traveled exactly 4.0 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 clock at rest in earth reference frame , 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: Event A-clock on spaceship reads that it leaves earth, and event B-clock on spaceship records the instant the rocket passes by exoplanet at 0.99c relative to earth-exoplanet frame. 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 with 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 successful pass by).... Would the interval C-D be measured as 56.28 years by the ships clock?

Further let us consider an event E also measured in Earth frame 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. What would the time interval 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(C-B-E is the sequence of event in ships reference frame), while according to earth E happened BEFORE the ship passed exoplanet (C-E-B in earth's reference frame,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-B be reversed in Earth relative spaceship's reference frame? I am a high school student, who has just little knowledge about special relativity (I haven't seen a derivation yet, just know that lengths contract, and time dilates when comparing between reference frames) so apologies if this understanding or naming conventions are different.

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The question was too long to read all the way, but it went off the rails in the second paragraph: "from the perspective of the space ship it would be traveling"

Rather: in the spaceship frame, which is at rest and not traveling, the exoplanet is approaching at $\beta=0.99$, from an initial distance of $d'=4/\gamma$ ly.

Then: when doing relativity, it is best to be clear. Really clear.

Step 1: Define frames. $S$ is at earth, $S'$ is the space ship.

Step 2: Origins are coincident at the beginning.

Note that events are points in space-time, so everyone agrees on them. Different frames assign different coordinates, but the events are the same.

Step 3: Label Events...usually chronologically (if possible) in the unprimed (so-called at-rest) frame.

Naming the clocks A and B is just confusing...$S$ and $S'$ have clocks. The C is the start? No.

Here are the events:

  1. $E_0$ Ship leaves Earth, at Earth
  2. $E_1$. At Earth when ship arrives at planet in ship frame
  3. $E_2$ Ships arrives at planet at Earth in earth frame
  4. $E_3$ Ship arrives at planet at ship

Now assign coordinates. Let's say planet is $L$ distance away.

Labeling the events in the $S$ frame as $(t, x)$, and in the $S'$ frame as $(t', x')'$:

$$ E_0 = (0, 0) = (0, 0)'$$

$$ E_2 = (L/\beta, 0) $$

$$ E_3 = (L/\beta, L) = (L/\beta/\gamma, 0)'$$

The hard one is:

$$ E_1 = (L/\beta/\gamma^2, 0) $$

Note that $E_2$ and $E_3$ are simultaneous in $S$, while $E_1$ and $E_3$ are simultaneous in $S'$.

From here: use the Lorentz transformation.

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  • $\begingroup$ E0 and E3 is pretty clear. E2 and E1 is not clear. Could you please rewrite it to be clearer. Also let us consider event E5 (not in chronological order)...2 years after launch in earth frame hurricane destroys launch pad. i.e E5= (2 years, 0)= (2*γ=14 years,?)'. What would be t prime for this event in ship frame (14 years as I calculated), and would it be After the ship encounter exoplanet in the ship frame? Is the order of Events E0-E3-E5 in ship frame but E0-E5-E3 in earth frame? $\endgroup$ Jun 19, 2023 at 20:43
  • $\begingroup$ The ones I posted are straightforward, at some point, you just have to do the Lorentz transforms from a frame in which you know the coordinates to the one you're not sure of. It helps to put them on a Minkowski diagram, though it $\beta=0.99$, that's not practical. $\endgroup$
    – JEB
    Jun 20, 2023 at 1:54

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