The Relativistic Rocket Equation

I hope you got yourself an A on this term paper, quite creative and unique in my limited knowledge. Well this would have you findable in a google search real quick. Hope that is a wanted side effect

@TomKerbal I meant an actual war game conducted by military powers, sorry!
Anyways, the way I define complexity lies between possibility and the calculation of said possibilities. The many combinations of a pack of skittles is just as insane as that of chess, but chess is harder to compute, making it more complex. However, a single result from a difficult equation (say, one for calculating the way space time curves in the presence of a Kerr black hole) is still complex because of raw computation rather than possibilities.
I define complicated more similarly to convoluted than to complex; something complex is at its base state whilst something complicated can be simplified.

@LeMagicBaguette
Interesting... how do you define complexity ? Take the Newtonian gravity equation for example. Very simple, but only if you have 3 bodies they become extremly complicated to solve (we can use a computer with a numerical solver in our game here, that's really nice). So even if the rules are simple that does not mean a system can not be very complex in my opinion. But you are right, there is a difference between complexity and how complicate to solve (what the english term for it? DeepL says "complexity" as well, but that's strange...). In my understanding high complexity means a high number of possible thinks that can happen (like chess), and complicated means it is very difficult to predict the future behavior (like chess ;-). A war game like World of Warships (which I play very often) is not easy to compare with chess. For me playing chess is much more stressful than playing WoWS, but I do not know what that means concerning to the question of complexity. I do not play chess too often then ;-)

@TomKerbal I wouldn’t say chess is super complex; rather, it’s size of board is what gives it so many possibilities, but the rules are simple. And yes, I like chess.
With a war game, a single warship can occupy tons of spaces.

@LeMagicBaguette Great! But... what do you mean ? :-) Sorry, I am not a native english speaker, perhaps I missed something ?
Do you like chess ?
It's really complex, far more combinations possible than atoms in universe exist. My favorite opening: english.
1. c4

@TomKerbal it was mostly just in context. It’s like chess is complex compared to noughts and crosses and a war game is complex to chess.
Although the derivation is a bit cracked.

To support the discussion once again a little more positive: There are considerations to send frozen human DNA to Proxima Centauri and to build up a human colony there with the help of robots. For this purpose a spaceship is to be built, which is to be accelerated from the earth with laser light half of the distance, and the remaining half is to be decelerated (no idea, how). On board a super AI, which is to wind itself through all meteroid storms at 1/10 c gentle cruising speed. Have you heard about this, or what do you think about it ? I'm staying out of it this time ;-)

@LenatezSpaceAgencys Sorry you are right, I should not have written this; even this equations here do not look too much complicated it's derivations clearly are. So sorry for this.

In my opinion it's much more important to be able to go for moon and mars and place colonies there. There will be asteroids killing all live at earth in the future, perhaps 100 000 years or so, and Burce Willies will probably not br here to help! There is no way to stop gigants >10km or even more. What then ?

@LeMagicBaguette
That's simple math, at least for this field of science, it's school mathematics. One question I would have is, why v_e and a can be set as constant simultaneously. But there are some references written here, I will check.

I don’t understand why there’s such complex math being posed for a simple* game.
I love it.

@sacr3dbac0n0
Perhaps my first serious answer here:
As far as I remember I have to agree to my honorable colleague @mundindel that the curve you plot is not valid to the space pilot but only for his twin waiting on earth observing his courageous sibling. Due to this relativity time deluting and size reducing things space raider twin will spectacularly demolecurize while arriving at Alpha Centauri in 1 million years of our time or so. That's at least what I remember from my "Einstein has stolen so many ideas" - professor lectures at university... but a long time ago, so maybe I throw some things mixed up here ;-)

Oh one more thing! I nearly forgot... For all of your brave optimists, this will be very, very hard now! I am so sorry. Really. But that is so good! I have to tell you. Please forgive me!
Some years ago the astrophysicists discovered that they have a 95% -- error in their equations, if they want to describe the universe as a whole. As we physicists are (very practical, at least some of us), they invented a kind of matter, which can't be measured directly, and which is (not more, perhaps only a little) 95% of the mass of the whole universe. Extremely practically
(I mean that with the not being able to detect). The equations were then clearly better. Since this appears to them then nevertheless somewhat suspiciously (at least some of them), they called the whole thing then (very aptly(?)... is this the correct term? ): " D.a.r.k Matter" . Huuuuuuuuuu.... :)
Sounds like a mediocre (whtat's that DeepL?) science fiction movie :-) . And shall I tell you something: Yes ! That's exactly how it should be done, I think that's absolutely right.
[Switching DeepL out, using my own brill(i)ant english]
Buuut one should never claim to understand this, I think.
(Additionally you should never try to put a pile of even darker matter in your study room as decoration :-D )

So much optimism here :-)
That's great. I really enjoy that!
I apologize that I can't share it. Humans simply can not understand the laws of nature, most people who didn't study physics did not know. So we do not know why masses do this gravity thing, why the speed of light has its specific value, why the electron has it's charge and so on. We simply understand nothing which is fundamental. We can only describe nature with formulas, and if we are lucky we can predict some things which will happen if we put other things together in a specific way - so that we can build rockets for example, that do not explode (exeption is known, it has been a big one ;) ). But nevertheless: Yes! Many wonders will be detected and it may lead us to the stars, perhaps one day. Sorry for not understanding the clear explanation of the non-twin-paradox :-D
There is no hope for me anyway. I didn't even understand the behavior of the highly relativistic electrons I did calculate in my plasma simulation many years ago - and the results of the simulation have at least led to some puplications and an invitation to the Berkeley Labs... so I am very impressed by every people who understands this :-)

So, despite the theories and relativistic calculations being mostly correct and proving that reaching and traveling the speed of light is impossible, we have to be optimistic and aggravate in other scenarios ahead that await us to be discovered and unraveled. Traveling to nearby stars is undoubtedly one of humanity's dreams and perhaps with a few centuries or even decades of development of new technologies it will be possible to come close to it. I'm very supportive of the idea that we might one day understand much of our known universe and that this will help us make it real. On the other hand, rockets with great capabilities to make our race multiplanetary (not just the starship) are under development or will emerge with increasing demand. And I believe it! I believe that with focus and unity we will achieve these goals. What do we have where it's still not possible to achieve this, but we will achieve something close, I believe that. (yes, I'm quite optimistic about human capacity but I'm quite realistic that we're only going to actually achieve this if we're really willing to change it)

@TomKerbal 😆 The twin paradox is not a paradox, and has clear explanation. Key word - acceleration (or perceived gravity which is equivalent).
Unruh effect is only significant for extremely high accelerations which no physical body can sustain anyway. Space gas friction and blue shift would be incomparably worse problems.
As for the deep space travel, there's a paradox (forgot the name), that at long travel distances, newer rockets launched later will pass older ones launched earlier, hence there's no point of flying to the stars unless the technology is already advanced enough.

@mundindel
What happens to the traveller (except fromt beeing ripped to molecules) when he deaccelerates to the observer to say Hello ? Or vice versa of course. So Start, go to lightspeed (near), brake to zero, return, go to lightspeed again, brake again, say hello.
Twin paradox ? Very paradox, because from the POV of the traveler the observer is the traveler, if you know what I mean.
I didn't never understood all this nice little pictures in brain explaining relativistic effects. Never seem to be plausible to me. Only the physical equations compared to real experiments count for me, false or true. In school they told me an electron is a small ball orbiting an other ball. Later this became clouds and wave and probability fields and so on... I gave up at the wave particle dualism. Now I am grown and I do not believe in anything they told me ;-) Just equations, comparing with real nature, wrong, false. No pictures in my head, please .-) We have the Schrödinger equation solving all problems in Atomic Physics, that's great. So traveler sees a lot of Doppler effects. Looks great ! Measure and compare the frequencies with theory, fit ? Good. Equation is usefull. Unruh Effect, is not there ? Uuuuu... bad equation, but great for mathematicians, we can learn a lot from it, but it does not fit to the real universe. Same for String theory...
But perhaps they will discover the Unruh effect... let's see.
I will never ever reach these dimensions in my life. So great.
Additionally if you are outside the visible part of the universe, you travel faster than light (the observer from earth would say, if he could see you). AND, if you are in water, go to warp, just asking Mr. Tscherenkow (rip).
now i am at operating temperature

@sacr3dbac0n0 Is this Matlab you are working with ? Really interesting, don't misunderstand me !
To this antimatter thing: It is really really hard to produce it. And if you have it, it's even harder to store it in electromagnetic traps. Same pessimism here for Nuclear Fusion (please don't ask my why). So, perhaps a wormhole somewhere not more than a few Lightdays away ? Yes, why not (exept for the 1000 reasons this is extremly unprobably). This could be at least a theoretical possible method to reach out for the stars :-)
So there is always hope.

And don't forget the Unruh - Effekt :-)
Lightsstrips in Warp, that's it. They are building up an experiment to prove that this effect really exists. I think I read this in the last American Scientific.
Back to topic:
My opinion to interstellar travel of human mankind: Never. Will never happen. Sorry to say, but look how far the Voyager probes have come, that is a few light hours or so. Yes that's really far for us, but the next star is about 5 Ly away.
Before they develop the warp drive, human mankind will destroy themself because they cut all the trees down. O.k., I am the pessimist, please convince me of the opposite :-)
For solar system the next star is Proxima Centauri (4.25Ly). Distance from Earth to Neptune is ~1/10000 from that. If you define the diameter of the solarsystem by the means of the Oort cloud that factor shrings to ~1/3 .... (If I haven't even miscalculated there....) If you define the solar system by the means of interstellar space (bordered by the Heliopause, what Voyager 1 just crossed) it is ~1/2000 .
So you know the story of the two Voyager probes ? It was a perfect planet constellation for their travel out of the system. I don't know how often such a constellation will exist in the near future.

It's a slippery topic 😁
Long story short, it all depends on the observer's POV:
- for the traveler, there's no speed limit and no relativity effects, plain Newtonian physics, and you can be flying faster than speed of light, measured over your local clock;
- for the external observer however, it will look like traveler's speed is asymptotically coming close to speed of light, and all relativity effects are in place (inertial vs gravitational mass shift, dimensional distortion, etc.).
For simplicity, it can be said that traveler's time observed externally starts slowing down during acceleration, hence all relativity effects. But locally, it won't be sensible.
So in terms of simulated high-speed flight, all local processes should stay linear, only global time will accelerate in reverse proportion to the gravitational time dilation formula.