Calculating communication timing with a space ship traveling 50 light years at near light speed

Question

So forgive me is this is a better question for somewhere else since this is ultimately a math and physics question, but:

I am writing a story where a group of new graduates are sent to work at a company branch office on a planet 50 light years away on a ship at near light speed, only to find when they arrive that while they were under the effect of time dilation, back on Earth the company had an Enron-esque implosion and has been closed for years.

Obviously if I need to stretch the laws of physics a bit to make the narrative work, I can do that, but I am trying to make this actually work out with real astrophysics and relativity as much as possible, so I have been doing some of my own research, but there are a few things I am not able to figure out.

What I know based on my own research (feel free to correct me if I messed any of this stuff up):

• The ship could follow a flightpath where it accelerates at 1 G until it gets to around 99.9% of light speed, coasts, and then decelerates at 1 G to slow down again at the destination. Radio communication occurs at light speed, so it takes only 50 years for messages from Earth to reach the destination.

• According to this calculator: https://www.omnicalculator.com/physics/space-travel To travel 50 light years, the ship would take about 52 years relative to Earth and the destination, while the travelers would experience about 7 years from their perspective due to time dilation.

• At some point in their journey, the ship would be outrunning any communications sent by Earth and wouldn't be able to learn from Earth that the company closed.

• it would take 50 years for the people already at the branch office to get the news from the home office that the company went under.

• Even if/when knowledge of the company's bankruptcy reaches them, turning the ship around and heading home isn't possible for both physics reasons and because they aren't the ships only customer.

• Obviously there are a bunch of human factors that would affect if and when the knowledge would reach them that I am ignoring for this question like would anyone at an imploding company even remember to send a message out, could someoneone at the colony be able to read between the lines and figure out the company was doomed before the news actually arrived and send a message to the incoming ship, etc.

EDIT/ My apologies about breaking the multiple question rule: to rephrase, "What are the relativistic timing effects of messages sent from Earth to the destination planet in order for news of bankruptcy to arrive before the ship does?" (I will leave my previous section however just because one of the replies reference it directly) /EDIT

What I am not so sure about is:

1. Would a ship under time dilation still be able to receive understandable messages from Earth and the destination, or would the time dilation its under render the messages unusable?

2. Am I correct that there is not really a physical way the branch office they are traveling to could learn about the bankruptcy before the ship does, because the ship is between the two so obviously any message sent from Earth would have to reach the ship before it does the planet?

3. At what point in the 52 objective years after they left would the bankruptcy have to have occurred for them to receive the news at exactly the point where they have, or practically almost have, arrived?

4. To the travelers on the ship, at what point in what to them seems like a 7 year journey would they be able to get news of the company's collapse?

Thanks in advance

Answer 1

You have a few things wrong

I think this is because the site you are using does not assume you decelerate to the destination but rather keeps accelerating until you whip past at nearly the speed of light.

If you accelerate to the midpoint at 1g (0.105 light years/year²), you then need to flip over and start decelerating. To get to the midpoint takes 33.18 years in the source and target's frames of reference (assuming they are at rest with respect to one another), and you hit a top speed of 0.961c. So, the whole journey takes 66.35 years. This is known as a hyperbolic path. It’s called a hyperbolic path because it’s hyperbolic - it approaches but never reaches a light-like asymptote. If it were to keep accelerating, that light-like path would be an event horizon - any signal emitted from Earth after that would never reach the spacecraft.

Given the parameters, we can plot the world lines of the Earth, the Branch office, and the ship on a spacetime diagram. The Earth and the Branch office are both vertical because they don't move in space, and the ship is the hyperbolic lines joined together in the middle at the flip-and-burn manoeuvre. I have also plotted the world lines of 4 signals sent from Earth 5 years apart; by convention, these run at 45 degrees so that a unit in space (x-axis) is equal to a unit in time (y-axis) - in this case, the units are light-years and years respectively.

Where the signal lines cross the world lines of the ship and the Branch office is when each receives the signal. As you can see, a signal sent 15 years after departure will be received about a year before the ship arrives and virtually simultaneously because, at this point, the ship is travelling at only around 0.1c and is very close to the destination.

However, a signal sent 5 years earlier will reach the ship about 10 years before it arrives, and one 5 years before that, about 53 years before it arrives.

Answer 2

Fairly straightforward

Note - technically the question is phrased as 4 different questions, which violates the site's one question per post rule. However, it could be worded as a single question that would give you what you need and it is all closely linked - so here goes. (Rephrase as something like: "What are the relativistic timing effects of messages sent from Earth to the destination planet in order for news of bankruptcy to arrive before the ship does?")

1. Any messages the ship received while travelling at close to c away from (and relative to) its point of origin/transmission will be considerably red-shifted - this actually makes them easier to read in a way because they are coming in over a greater period of time, but also means that the ship needs a more sensitive antenna since they are lower energy - see the next point. If we can build a starship capable of reaching relativistic speeds, we can definitely build software and hardware to handle the redshift or blueshift as applicable.

2. If the ship is travelling in a practically straight line between the point of origin and destination then any signal will reach the ship before it reaches the destination. However, this does not mean that the ship can read it - the message could be encrypted or, as per the previous point, the ship may not have a sufficiently large and sensitive antenna to receive the signal. Further, if the ship is in its acceleration or deceleration phases then there will be vibration etc that may reduce the sensitivity of the communications systems. So, if you want the ship to be unaware of the message until after it arrives, that is achieveable.

3. The ship is taking 52 Earth-frame years to travel the distance and light will take 50 Earth-frame years to travel the same distance, assuming that the two solar systems have no significant relative motion. If it takes more than 2 years for the Earthside company to go broke, the news will not arrive until after the ship has.

4. What point in the time-dilated travellers' 7 subjective-year journey they will receive the message depends on exactly when the message is sent relative to their departure. If you want to simplify matters then have it sent 1 year and 11 months after they departed - that way they are so close to arriving and have decelerated so much that their will be minimal time dilation differences between the ship and the destination planet. Or you can make it even simpler and use one of the options from point 2 to prevent the ship from receiving the message in-flight and they only are advised of the news as they reach near-real-time communications range of the destination planet.