Give me a couple of trillion euros - and in 100 years our people will be on Proxima Centauri. Already with modern technologies, you can begin to build such a ship, if there were a customer. I think of it like this: A thin (1mm thick) spinning disc, 10 kilometers in diameter, made of pure carbon (graphene). On one side (front), the disc is a trap for hydrogen atoms (and other elements in much smaller quantities) as it sweeps through space. On the reverse side, the disk is a reflecting wall that receives an impulse from thermonuclear explosions that are produced behind it at some distance (say, 15 kilometers). We are not yet able to maintain a continuous controlled fusion reaction, so ordinary uncontrolled thermonuclear explosions, which we have been producing for 60 years, but with the lowest possible power (up to 1 ton in TNT equivalent), will do. In this case, the fuse will not be a uranium explosion. The fuse will be a laser beam, broken by mirrors into thousands of beams and concentrated on a portion of hydrogen, heating it to 100 million degrees and causing a fusion reaction. How to build such a disk? Something like how cotton candy is produced. In a high orbit with a tremendous angular velocity (limited by the strength of the future disk), a cylinder with superheated vaporous pure carbon rotates, carbon vapors are ejected from nano-sized holes, its atoms and in a vacuum line up into ideal fibers. We get a disc with a diameter of 10 km from nano-fibers, conducting current. Using the method of electrostatic attraction, we deposit carbon atoms onto this framework, we obtain a super-strong graphene disk. Where can we get 100 thousand cubic meters of carbon in space (this is the volume of a disk - equal, by the way, to the annual production of an average coal mine)? - At the equator (almost) in Africa there is Mount Kenya (6 km). The land is cheap there. We build a rail overpass with a vacuum tube at the foot with an ascent uphill, 100 kilometers long, accelerate cargo projectiles with pure carbon on it with an electromagnetic force to the 2nd cosmic speed and catch in high orbit (geostationary), where we build a disk ... him to rotate? The main task of rotation is to concentrate atoms trapped in flight in the interstellar gas from the center to the outer ring in order to collect them there for use as "fuel" and other purposes. Each hydrogen atom in space, burying itself in the wall of the disk, without entering into a chemical reaction with the carbon of the wall, is physically captured by the atomic forces of carbon and spins (almost like air by the impeller of a centrifugal pump), receiving centrifugal acceleration, pushing it to the edges, where they can be easily caught in an annular trough, where there will be a pressure of accumulated hydrogen of several hundred atmospheres and prepare a new portion of them for a thermonuclear explosion. (One cubic millimeter of space contains one hydrogen atom. With a disk area of 100 sq. km. and a speed of 10,000 km per second ( 3% of light) we get 1.5 mol of hydrogen per second, or 3 grams, that is, 250 kilogramsammonia per day, of which 2.5 kg is heavy hydrogen, deuterium, which is more than enough for the m-i reaction.) The remaining 99% of ordinary hydrogen surrounds the charge, like a bubble, and are the working medium of the explosion, transmitting its momentum onto the disk together with helium and other explosion products. Part of the energy of the explosions is used to charge the capacitors for the operation of the igniter lasers, as well as to spin the disk and other purposes in flight, for which thermocouples and photocells are built into the disk, generating an electric current from the thermal and light (also - neutron energy) energy of the explosions the axes opposite the explosion are two coaxial turbines of opposite rotation, generating current ... In addition, the rotation of the disk itself is necessary to ensure the rigidity of the thin disk (for example, a cardboard rotating disk can cut wood) and the inviolability of its trajectory due to the gyroscope effect. The rest of the rooms can be attached to the disc using frictionless magnetic couplings and long cables. Acceleration, flight and deceleration will take 100 years, during the flight, the crew will improve the apparatus according to instructions from the Earth from substances captured in space, making parts on 3D printers, which will increase the speed and reduce the travel time. The disk has a big plus: its rear part needs to be profiled with prismatic rings so that a Fresnel prism is obtained (like the RATAN-600 flat radio telescope), that is, the disk has acquired the properties of a parabolic "mirror" of a radio telescope, with which it is necessary to receive and transmit messages to the Earth. Again, all the technologies listed here have already been mastered.
If the question is about the Hawking and Milner project, then it is planned to launch tiny satellites, accelerate them to a speed of about 160 million km / h (about 0.15 light), which, if successful, can reach Proxima Centauri in about 20 years.
If the question is about a hypothetical spacecraft with astronauts on board, then the reasoning is as follows:
First, everything depends on the speed at which the flight will take place. Currently, the distance to Proxima Centauri is no less than 4.241 light years. Because a physical object cannot reach the speed of light, the minimum time it takes to reach Proxima Centauri is not less than 4.241 years, and most likely more. It helps a little that Proxima Centauri is approaching us at a speed of about 20 km / s.
Now about more complex matters.
I. The theory of relativity.
Due to the fact that at such distances (and maybe speeds) it is already difficult not to take into account the theory of relativity, there are many refinements that distinguish the theory of relativity from the intuitive Newtonian mechanics.
If, for example, it is possible to accelerate the spacecraft with the astronauts to a speed of 0.99 light instantly and upon arrival also instantly stop it (and even if they get to their destination unharmed), then it will take several times:
The time that has passed for astronauts. It will be much less than 4.241 years. You can count how many times you can take the formulas (I don't remember by heart - I repent).
Time for an observer from Earth.
2.a. Estimated time of arrival. Take distance, divide by speed.
2.b. Time to receive confirmation of arrival. For example, upon arrival at Proxima Centauri, the automation sends a signal to Earth that the ship has arrived. The signal will go back 4.241 years and on Earth we will learn about the event through this time.
3.a. Let's say we send a signal from the Earth that the ship takes off with its start. Thus, from the moment of receiving the signal about the departure of the ship until the arrival of the ship, approximately 0.04241 years will pass, i.e. only about two weeks.
II. Acceleration and deceleration.
If the ship does not accelerate instantly, then this moment must be taken into account. Time will certainly rise.