Humans Reaching Andromeda?

Question

In a general astronomy class, the professor said that Andromeda, being 2 million light years from the Milky Way, would never be reached by humans. But it seems to me that we may be able to reach a significant fraction of light speed using technology like what was envisioned in the Orion project -- we would have to improve on Orion technology but the point is, near light speed does not seem out of the question.

If this is so, then with relativistic time-dilation, some brave astronauts would experience an arbitrary short amount of subjective time and so could reach Andromeda or even galaxies farther away than that. Given near light speed, is there any reason a spacecraft could not travel that far?

EDIT: I want to clarify that given the technology to reach the closest star in a reasonable length of time and return, it seems like we get any distance as a "free" side effect -- is that the case? I realize that the big difference is that while a trip at .99c to Alpha Centauri can even be in theory a roundtrip, to Andromeda the astronauts would know millions of years had passed on Earth so the return would probably not be undertaken but as far as reaching the nearest galaxy, that seems almost as feasible as reaching the nearest star.

Answer 1

You could certainly get to Andromeda if you can travel fast enough, but how are you going to travel fast enough? You could run the calculations - at $0.2c$ (which is way faster than any artificial object has travelled before), relativistic corrections would be insignificant, and it takes $10$ million years to get to Andromeda. That's longer than recorded human history. Even if you go much faster at $0.99c$, the Lorentz factor is still only about 7, Andromeda is still $280,000$ light years away, and it still takes too long to get there.

If you want to get to Andromeda in a human lifetime, then you need to move fast enough that the distance to Andromeda after length contraction is, say, $10$ light years. That takes a Lorentz factor of $200,000$, corresponding to a speed of $0.99999999998$c. The amount of energy you need would be $\gamma mv^2 \approx 2 \times 10^{24} J$ for a $100kg$ human astronaut, or roughly the same order of magnitude as the amount of solar energy the Earth receives a year.

It may be possible in principle, but it's just not happening with current technology.

Answer 2

It seems that this is the core of the question:

given the technology to reach the closest star in a reasonable length of time and return, it seems like we get any distance as a "free" side effect

That's not how it works, unfortunately.

The rough analysis is that, in the frame of reference of the ship, the distance between the Earth and the star (or galaxy) gets length-contracted by a factor of the Lorentz factor $$\gamma=\frac{1}{\sqrt{1-v^2/c^2}},$$ and the star travels towards the ship at the speed of light, so it covers the reduced distance quickly.

For speeds that are "close" to the speed of light, the Lorentz factor can be very large. The first problem is that the details of how close and how large, quantitatively speaking, do matter. If you want to get to Andromeda, the same principle applies, but you need $\gamma$ to be much larger, which means that you need to be going much closer to the speed of light. (As calculated in Allure's entirely-correct answer, to go to Andromeda in a human lifetime you need your speed to match $c$ to one part in around $10^{11}$.)

And the second problem is that this costs energy:

• From a newtonian perspective, our intuition says that two objects traveling at speeds $v_1=0.99c$ and $v_2=0.99999999999c$ are going at about the same speed, so their newtonian energies $\frac12mv_1^2$ and $\frac12mv_2^2$ are very similar.
• This newtonian intuition is, unfortunately, dead wrong. Relativity limits objects' ability to travel faster than $c$, but it does not prevent them from accelerating, no matter how close they are to $c$; instead, it makes it progressively harder and harder to accelerate, the closer you get to $c$.
• As a result, every additional "$9$" in your velocity as a fraction of $c$ costs additional energy. And, by a nice twist, the measure of this energy is precisely the Lorentz factor, multiplied by the characteristic $mc^2$, i.e., $E=\gamma mc^2$.

So, if you want to go so fast that $\gamma$ is large, you need to take the mass of your ship, convert that into energy by multiplying it by $c^2$ to get a gynormous energy, and then multiply that by the large factor $\gamma$ that you originally set as a target.

As mentioned already: going to Andromeda in a human lifetime is indeed possible in principle, but it doesn't come "for free" once you figure out how to get to nearby stars, and it certainly isn't gonna happen using technology that we currently know about.

Answer 3

(Edited answer to address the edited question, corrected original answer and calculation below)

Generally speaking, if you somehow managed to get a spaceship moving arbitrarily close to the speed of light, the the astronauts in it could reach any destination in a short amount of time – from their perspective. This is because their time $t'$ is related to the time of a "stationary" observer $t$ by

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

and

$$\lim_{v\to c}\gamma=\infty\implies\lim_{v\to c}t'=0$$

So the problem is getting a spacecraft to move this fast. As I calculated below, one would need about $1.1\cdot10^{29}\mathrm{J}$ to move a $10000\mathrm{t}$ spacecraft to a high enough velocity.

---

Let the velocity at which the spacecraft travels be $v$. Then $$t=\frac{2.537\cdot10^6\mathrm{ly}}{v}\approx\frac{2.4\cdot10^{22}\mathrm{m}}{v}$$ is the time that elapses until the spacecraft reaches M31 as measured by the stationary observers on earth¹ and $$t'=\frac{t}{\gamma}=t\sqrt{1-\frac{v^2}{c^2}}$$ is the time measured by the astronauts.

The average age of an astronaut is 34 (Source). So we would want the astronauts on such a mission to only age around 20 years. So

$$t'=20\mathrm{y}=6.307\cdot10^8\mathrm{s}=t\sqrt{1-\frac{v^2}{c^2}}\\6.307\cdot10^8\mathrm{s}=\frac{2.4\cdot10^{22}\mathrm{m}}{v}\sqrt{1-\frac{v^2}{c^2}}\\\implies v\approx299792457.99\frac{\mathrm{m}}{\mathrm{s}}\simeq c=299792458\frac{\mathrm{m}}{\mathrm{s}}$$

^{Link to the computation}

So the required velocity for the astronauts to reach M31 alive is extremely close to the speed of light. If we assume the mass of the spaceship to be $10000000\mathrm{kg}$ which is listed on the Wikipedia page on Project Orion for an "Advanced interplanetary" vehicle, the energy required to get to this velocity is

$$E=(\gamma-1)m_oc^2\approx1.1\cdot10^{29}\mathrm{J}$$

^{Again, link to computation}

For comparison, this is about half of the rotational energy of the entire earth (Source).

Other problems could be the spacecraft not withstanding the extreme forces when accelerated to almost light speed or the spacecraft hitting debris at high velocities, which would probably damage it significantly.

---

¹ Assuming no acceleration is needed – which of course would be necessary, but I will ignore it for the sake of simplicity.

Note: It is not impossible that I made mistakes during this calculation. If someone notices them, please comment so I can try to correct them.

Answer 4

We can use the formula for an uniform $g$ acceleration:

$$x = \frac{c^2}{g}\cosh \left(\frac{g}{c}\tau\right) – \frac{c^2}{g}$$

where $\tau$ is the time for the crew. It would take about 15 years to travel half the distance keeping a comfortable $g$ inside the ship. Then, invert the direction of the engines, and travel more 15 years decelerating at a $g$ rate, if the intention is to disembark somewhere in the galaxy.

30 years is a long trip, but not impossible. What seems impossible is how to carry the fuel, while keeping the ship with a reasonable mass.

The next generation could be raised in the ship.

Answer 5

with relativistic time-dilation, some brave astronauts would experience an arbitrary short amount of subjective time

It’s understandable that one might think this way, but relativity doesn’t work like that. A hint is in the name, “relativity”: time dilation is about how fast time progresses for one observer relative to another observer.

In your example, you have only one observer: a group of brave astronauts, who are travelling (arbitrarily close to) the speed of light for a distance of 2,500,000 light years. So in the astronauts own rest frame, they are at rest (by definition) and the Andromeda galaxy is moving toward them at the speed of light. But since it’s over 2.5M light years away, it will take 2.5 million years for the galaxy to reach them. That’s a really, really long time – much longer than humanity has been in existence.

Time dilation comes into play when you compare two observers. It’s often used when you want to compare two observers who start at the same time and place, and then later compare notes about how much time has passed if they meet again at the same time and place. (Space and time are related to each other, so that second “and place” matters). That isn’t the case in your example, where the Brave Astronauts meet the Gallant Andromedans only once.