# Humans Reaching Andromeda?

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.

• Hi all -- I have removed comments attempting to answer the question or discuss various topics. Please use comments to seek clarification or suggest improvements to the question. Thanks! Jun 2, 2021 at 11:16

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.

• And the mass of the spaceship would be several orders of magnitude more than the mass of the human, so the energy requirements would be even worse. Jun 2, 2021 at 0:49
• You left out two big problems: how to slow down at the other end, and how to not get destroyed by relativistic dust particles en route. I think even the blueshifted CMBR may be lethal at that speed. Also, your energy requirement is only correct for something like ground-based laser propulsion. A rocket requires a minimum of $2γ$ fuel/payload for $v\approx c$, and that's with the highest efficiency permitted by SR; a real rocket, even Orion, is much less efficient. If you want to slow down at the other end you need at least $4γ^2$ fuel/payload (realistically far more). Jun 2, 2021 at 1:12
• @Allure: once a ship reaches near c, why is Andromeda harder to reach than Alpha Centauri? Of course, the odds of encountering a problem would go up but assuming there was no accident between here and Andromeda, distance is not the reason why the galaxy could not be reached., it seems, because of time dilation. Jun 2, 2021 at 2:07
• @releseabe check the calculations above. 0.99c is "near c", but it's not enough to get to Andromeda in reasonable time. 0.999999999999c is also "near c", but it takes a humongous amount more energy to actually attain. Jun 2, 2021 at 2:11
• @releseabe At relativistic speeds, it's better to think in terms of $\gamma$ rather than $v$ or $v/c$. Sure, $v=0.99c$ is fast, but it's snail pace compared to, for example, the speed of neutrinos we detect from the Sun, which have $\gamma$ on the order of a million. Jun 2, 2021 at 2:39

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.

• I think this is it: getting sufficient time dilation is the problem and increasing the speed to get this dilation is very difficult, much harder near c. Jun 2, 2021 at 23:37
• +1 I suddenly realized the answer I wrote might be too technical, and was about to ask if I needed to make it more elementary; then I discovered you've already written it. Jun 4, 2021 at 6:50

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

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

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

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

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.

• I think you are off by quite a few orders of magnitude. What value do you think $\gamma$ has? Jun 1, 2021 at 22:58
• Agree with G. Smith, the numbers in this answer seem off by several orders of magnitude. To make Andromeda 20 light years away you need a Lorentz factor of 100,000, which corresponds to a very large speed indeed. Jun 2, 2021 at 0:42
• You should get about $10^{29}\text{J}$ with your assumptions. It looks like you made two mistakes: your speed is too slow, perhaps due to roundoff error (it should be something like 299792457.9... m/s, not 299792000 m/s), and you entered it in Wolfram Alpha as 299792 m/s, which is only 0.001c. You don't need to calculate the speed; you can just take $γ = Δt/Δτ \approx 10^5$. Jun 2, 2021 at 1:21
• I have edited my answer using the correct value of $v$ for the energy calculation, though I was not able to find whether and where I went wrong in the calculation for $v$. The new energy value seems a bit more reasonable. Jun 2, 2021 at 8:44
• @benrg You were right, though I don't know what went wrong when solving the equation for $v$ in Wolfram Alpha – doing it more or less manually finally gave the right result. I have corrected my answer. Jun 3, 2021 at 9:03

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.

• According to the classic relativistic rocket article, the 2 million LY journey to Andromeda takes 28 years, and an ideal engine consumes 4100 tonnes of fuel per kg of payload. That is, the engine is 100% efficient, and works by annihilation of matter with antimatter. And of course, if the rocket has to carry all that fuel, the total mass will be enormous, due to the Tsilovsky rocket equation. Jun 2, 2021 at 2:22
• Oops! That figure of 4100 T per kg is for a journey where the acceleration is +1 g so you arrive at just under c. If you do the midpoint flip, the fuel requirement is 4.2 billion T per kg. Jun 2, 2021 at 2:29
• Actually, those fuel calculations already incorporate the Tsilovsky equation. Sorry. Jun 2, 2021 at 2:57

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.

• I think u are almost certainly mistaken. Traveling near c, a human would experience an arbitrarily small passage of time while traveling to any destination. This is a staple of science fiction and has been proven in real life (with muons for example) over and over. Jun 1, 2021 at 22:09
• That would be true if they started at the speed of light relative to the Andromeda galaxy, and kept going past it. But not if they started (relatively) at rest, sped up, and then slowed down. Jun 1, 2021 at 22:15