Exoplanets

Will humans ever be able to travel to the stars and colonize habitable exoplanets?
Closest habitable exoplanet Proxima Centauri b is 4.25 light-years away. The fastest spacecraft is NASA robotic spacecraft Parker Solar Probe that was launched in 2018 it has a max speed of 192 000 meters per second which is 0,0006 percent of light speed. With that speed, it will take 7000 years to reach Proxima Centauri b. We need to get to a speed at least 10 percent of light speed to reach Proxima Centauri in a reasonable time. A reasonable time is 50 years if we consider the wait dilemma. In 50 years we will be able to build space ships that can reach the target destination much faster, so it will be more beneficial to wait.
We will never reach the stars using rocket engines because we would need more fuel than it is mass in the observable universe to reach 10 percent of light speed.
The rocket equation is 
 
$$\Delta v =v_{e}\ln \frac{m+f}{n} $$
Where m is the mass of the vehicle, ve is the exhaust velocity of the rocket and f  is the mass of the fuel. If we solve f  from the rocket equation, we get:
$$f =me^{\frac{\Delta v }{v_{e}}}-m $$
If  we assume that the mass is equal to 1000 kg and exhaust velocity of 210,000 m/s which is the speed of our very best reaction engines, we get:
$$1000e^{\frac{3000}{21}}-1000 \approx 1*10^{65} $$
 Compare that with the total amount of mass in the observable universe that is  \(1.5*10^{53}\)kg. 

It would be theoretically possible to make the trip using nuclear fusion by detonating many nuclear bombs behind the spacecraft. Physicist Freeman Dyson and Ted Taylor at General Atomic worked with this idea in a project called Orion in 1958, but the Partial Test Ban Treaty of 1963 that forbid nuclear bombs in space ended the project. Another idea is to use a fusion rocket. Project Daedalus was a study conducted between 1973 and 1978 to design an unmanned interstellar spacecraft to reach Barnard's Star 5.9 light-years away. The trip was estimated to take 50 years, but the technology behind fusion rockets is complicated and we are not there yet. 

A more efficient method is to use antimatter rockets. When a subatomic particle collides with its respective antiparticle a large fraction of the rest mass is converted into energy. Depending on how much antimatter we could make it would be possible to reach a speed of 80 percent of light speed. The travelers on a ship with that speed only need to wait 3.2 years to arrive at Proxima Centauri b.
Because of time dilation from Einstein's special relativity :
$$t =\frac{t_{o}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$$

$$\Rightarrow \frac{4.25}{0.8}*\sqrt{1-0.8^{2}}=3.2.$$
Antimatter is produced in many experiments at CERN, but far from the amount that is needed it takes a lot of energy and it is hard to contain. 

Credit Gerhard Janson pixabay

Another option is to travel faster than light. The Mexican theoretical physicist Miguel Alcubierre created an idea in 1994 which a spacecraft could travel at warp speed by using Albert Einstein's field equations in general relativity.
A signal in space cannot move faster than light, but the space itself can expand faster than light. The idea is that space is moving the ship just like a surfers board is moved by the water waves. This is mathematically possible, but one needs negative gravity. Negative gravity is produced by negative mass, not anti-matter which has positive gravity. Scientists have never observed negative gravity or mass, but there is nothing in the Einstein field equations that forbid it. Negative mass is equal to negative energy.  Dark energy is postulated to act in opposition to gravity and will perhaps work as negative energy. Dark energy explains why the universe is expanding at an accelerating rate and 72 percent of total energy in the universe consists of dark energy. If scientists figure out how to get this dark energy, perhaps warp drive like in Star-trek will be possible in the future.

The most promising option to travel to Proxima b is to use a solar sail read about the project here Breakthrough Starshot