Given time dilation from traveling near light speed, would a spacecraft be able to reach any arbitrary distance including to another galaxy?

Question

Not much more than in the title except that when I took an astronomy class the professor said, Andromeda is 2 million lightyears away so it is safe to say humans will never reach there. But with time dilation, why would a human not only not be able to reach this galaxy but it could take from the passenger POV only a year or something? Or is this some practical issue like a propulsion system that received power from Earth lasers would eventually fail because the lasers would be too attenuated?

Answer 1

In your own reference frame (in the ship) you age at the normal rate.

Time will appear dilated from the frame of reference of another observer. For example, an observer on earth will measure the time you take according to

$$t’ = \frac{t}{\sqrt{1 - \frac{v^2}{c^2}}}$$

where $t’$ is the time measured on earth, $c$ is the speed of light but $t$ is the time elapsed in your own frame of reference. This means that as stated, you will age as normal but people on earth will see this process take longer depending on how fast you are travelling.

Answer 2

You are correct that it is, in principle, possible to reach any place in the galaxy in a short amount of time if you do not care about earth time.

From earth, it will seem that due to time dilation you experience the 200 million years it takes to get you there as only one year.

For you, it seems that due to length contraction the large distance to Andromeda shrinks to a light year.

Of course, the propulsion system is absolutely nuts, nothing we have can create enough thrust to accelerate a spacecraft (weighing at least a metric ton) to these speeds.

Answer 3

Even if you had a perfect lossless energy transmission system, the energy requirements are enormous. It's hard to comprehend how much energy it takes to accelerate something to relativistic speeds, but here's a rough calculation that may provide some insight.

To achieve a time dilation / length contraction Lorentz factor of $\gamma=2$, you need to be travelling at $c\sqrt3/2\approx 0.866c$. Of course, to get to the Andromeda galaxy you need a much higher $\gamma$, but let's investigate the energy cost of merely getting up to $\gamma=2$.

Now you can't just add KE to the ship, you have to conserve momentum & throw something in the opposite direction (like the reflected light in your laser propulsion system), but let's get a rough estimate ignoring that energy cost for now.

At $\gamma=2$, the ship's total energy is twice its rest mass, i.e., its KE equals its rest mass. According to the IEA, the global energy production for 2018 was 14421 Mtoe (millions of tons of oil equivalent). 1 Mtoe $\approx 4.1868\times 10^{16}$ joules. Converting that to its mass equivalent using $E=mc^2$, we get just under 6718 kg for 14421 Mtoe.

In other words, it would take the current annual energy production of the whole planet to create the kinetic energy in a body of mass 6.7 metric tons travelling at 0.86c. And as I said, that's ignoring the energy required for the reaction mass.

In this answer I do that more accurate calculation, assuming we have an ideal lossless antimatter engine that uses light for the reaction "mass" (and doesn't radiate any heat in any other directions). I derive the equation $$\gamma = \frac{1+k^2}{2k}$$ where $k$ is the final mass of the ship relative to its initial mass. $k=2-\sqrt 3\approx 0.268$ gives us $\gamma=2$. That is, the ship loses 73.2% of its mass getting up to 0.866c.