Relativity, Time Dilation and multiple reference frames

I am doing some hard sci-fi worldbuilding and really want to get as much correct as I can, but the effects of high sub-light speed travel on the travellers is doing my head in. I've pored literally over 100 hours over reading, video and podcasts on time dilation, special and general relativity, the twin 'paradox' etc etc. Most of these sources are particle physicists of some kind but I'm getting conflicting or incomplete information (also entirely likely I've just not understood). So, here I am praying for your help! I don't need the math, just a clear understanding of the physics.

The Context:

3 people on a space station travelling at a constant velocity. Emily, Max and Beth. Dave is on a distant planet. Assume no acceleration for anyone at the moment.

Emily remains on the space station.

Max travels away from the space station in a straight line at 90% light speed. His destination (Dave) is 10 light years away.

Beth travels away from the space station in a straight line at 70% light speed, but at a 45' angle to Max.

Question 1: (One-way with constant velocity) When Max arrives at Dave, what time has passed for Emily, Max, Beth and Dave?

Question 2: (Return with constant velocity) Max reaches Dave and immediately returns. When Max arrives back at Emily, what time has passed for Emily, Max, Beth and Dave?

Question 3: (One-way with acceleration) Max and Beth accelerate at 1G. When Max arrives at Dave, what time has passed for Emily, Max, Beth and Dave?

Question 4: (Return with acceleration) Max and Beth accelerate at 1G. Max reaches Dave and immediately returns. When Max arrives back at Emily, what time has passed for Emily, Max, Beth and Dave?

Here's hoping you clever-cats can set me straight! Many thanks!

• Better place to ask this Q would be Physics. It's precisely defined enough to hope for a precise answer from someone who does science, not World building. – Adrian Colomitchi Mar 4 '20 at 8:50
• I agree with @AdrianColomitchi. I wouldn't mention " I don't need the math, just a clear understanding of the physics". You can't understand physics without math. – L.Dutch - Reinstate Monica Mar 4 '20 at 9:32
• @L.Dutch-ReinstateMonica you can use nice light-cone diagrams in this case, perhaps, but I guess the important thing here is that the "clear understanding" is the maths. – Starfish Prime Mar 4 '20 at 10:26
• This is a maths question, unfortunately it’s my day off and I’m a little stoned. The question has caught me off guard and I’m laughing at myself a bit. Can you give us more information about the vectors? Even if this was taking place on a 2-Dimension plane I still think I need a little more to start drawing triangles. – Darius Arcturus Mar 4 '20 at 11:56
• Your mention of the 1G needs a little context too, sorry. – Darius Arcturus Mar 4 '20 at 11:57

Q1: Outbound trip 10ly at 0.9c with one a potential observer at 0.7c

• Max arrives at Dave's location in 4,055 days, according to Dave's clock. Dave is 4,055 days older. If Max sent a message at light-speed informing Dave of the journey, Dave received this message only 405 days ago.
• Emily has also counted 4,055 days pass by since Max and Beth left. If Max send an "Arrived Safely" message at this time, it will be another 3,650 days until Emily receives it.
• Max is 770 days older than when he began the trip.
• At this point in time, Beth is 2,068 days older

Q2: Return trip at 0.9c with potential observer at 0.7c

• When Max returns to at Emily, she will have counted off 8,110 days since Max initially left, according to Dave's clock. Emily is 8,110 days older since Max embarked on the trip. If Max sent a "Coming Home" message at light-speed informing Dave of the journey, Emily received this message only 405 days ago.
• Dave has also counted 4,055 days pass by since Max left for home. If Max send an "Arrived Home Safely" message at this time, it will be another 3,650 days until Dave receives it. If Max has been diligently communicating both ways Dave would have counted a total of 4,460 days between Dave receiving Max's "Coming to See You" and "Arrived Home Safely" messages.
• Max is 1,540 days older than when he began the trip.
• At this point in time, Beth is 4,136 days older

Q3: Q1 with 1g acceleration (rounding 1g to 10 m/s/s, and rounding c up to 3$$\times 10^8$$ for my own benefit) (also assuming relativistic effects on mass don't cause the engines to provide less acceleration)

• Beth arrives at her cruise velocity of 0.7c in 21 million seconds (243 days). Roughly 213 days have passed for Beth.
• Max arrives at his cruise velocity of 0.9c in 27 million seconds (312 days). Roughly 248 days have passed for Max. During acceleration, 0.385 light-years of the total trip distance were traveled.
• Max will cruise the remaining 9.24 light-years in 3,747 days relative to Dave and Emily's clocks / 712 days on Max's clock.
• Deceleration on arrival takes an equal 312 days as Dave or Emily would measure it, but only 248 as Max would measure it.
• The total trip time, then is : 4,371 days as Dave and Emily would measure it; 1,336 days as Max would.

Q4:

• Beth continues at her cruise velocity unchanged.
• Max accelerates and decelerates for the first and last 312 days of the trip, counting only 248 days on his own clock for the speed-up and slow-down.

You said you aren't interested in the math, so i will not bore you with numbers:

Q1: For Max, less time will have passed than for a stationary observer. For the others, the question is ill defined, since simultaneity is relative as well. So you can't say "When Max arrrives" for anyone but Max and Dave, and Dave didn't experience Max' start. If you mean "how much time will have passed until they calculate that Max has arrived", obviously somewhat more than 10 years for the stationary observers, somewhat less for Beth (her angle is irrelevant) and least of all for Max.

Q2: in this case, Emily has experienced both start and end of Max' journey. She will have experienced more time than he has.

Q3: Same as Q1, but calculation is more complicated (that's general relativity due to acceleration, and we'd need to calculate his end velocity). Anyway, when he arrives at Dave with highly relativistic speed, Dave will have a Very Bad Day.

Q4: same as Q2&Q3.

note that Beth is irrelevant since she's never again interacting with any of the others.

Since you have multiple frames of reference, unless the precise times are important to the story (i.e. a murder mystery which depends on the difference in times to establish an alibi or something), then the only sensible way to deal with this is to write each character in their own POV.

Max will experience no real change in subjective time, but when he arrives he will see notably different time stamps from the other characters emails, with the largest divergence being the people who have the greatest gamma between themselves and himself (gamma being the factor used to calculate changes in frames between two different objects). This is also known as the Lorentz factor

The basic formula

You know intuitively that the gamma between a moving object and one at rest will be far higher than between two moving objects, so the highest gamma is between the character moving at .9 c and the one at rest, the second highest gamma is between the character moving at .7 c and the one at rest, and the lowest is between the character moving at .9 c and the one moving at .7 c

Since the math is a bit complex, if you actually need to calculate the different frames of reference, then try an online calculator:

https://www.vcalc.com/wiki/vCollections/Lorentz+factor

http://www.calctool.org/CALC/phys/relativity/gamma

• "The gamma between a moving object and one at rest will be far higher than between two moving objects" isn't always true. Two spaceships that take off from a stationary planet and fly in opposite directions will be moving faster wrt to each other than either one is wrt to the planet, so the gamma is highest between the two moving objects. The direction of motion plays a big role here. – Nuclear Hoagie Mar 4 '20 at 18:59