# Could I, within my lifetime, reach any star I wanted if I went fast enough?

Disclamer: I'm not talking about FTL travel here. I'm also not talking about any weird space warping mechanics like wormholes and such.

I've always thought that if a star was 4 light years away, then it would be impossible to reach it with less than 4 years of travel time. Therefore, any star more than 100 light years away would require multiple generations on the ship in order to get there even if they travelled at the speed of light(or close to it). The people who set off on the mission wouldn't be alive when the spaceship arrived at the star.

But the other day I had a realisation that, for anyone travelling close to the speed of light(0.999c), the length between them and the star would contract and they could get there almost instantly(in their reference frame). This also makes sense for someone observing from earth; they would see me take about 100 years to get to this star, but, because I'm going so fast, they would see me barely age. By the time I had got to the star, they would observe me still being about the same age as I was when I set off, even though they are 100 years older. In my reference frame, I would have also barely aged and I would have reached a star that's 100 light-years away in a lot less than a 100 years of travel time.

So is this assessment correct? Could I reach a star that's 100 light years away in my lifetime by going close to the speed of light?

It would be good to get an answer in two parts: In a universe that's not expanding, and in a universe that is expanding.

• See math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html for some journey times (and the relevant equations) of a relativistic rocket accelerating at 1 g. It also mentions the insane amount of fuel required. ;) Apr 10 at 15:14
• @PM2Ring: Fuel calculation is wrong. physics.stackexchange.com/q/321835/55856 Apr 11 at 2:59
• @Joshua "All models are wrong, some are useful". IMHO, the calculations on that page are a useful starting point. But yes, they assume a simple flat spacetime, and ignore things like space gas & dust, photons from the CMB (and elsewhere), and the technical difficulties of building & fueling an antimatter powered rocket. And the possible benefits of using massive black holes for the Oberth effect. Apr 11 at 3:43
• Yes, to the person travelling at that speed, c is effectively "infinite" speed. Travelling at c would allow you to go anywhere in the universe without any time on your own clock passing. Travelling "faster" than 'c' would only be possible if your own clock was allowed to run backwards - hence the causality paradox of FTL travel.
– J...
Apr 11 at 13:59
• funny, I was just reading the penultimate chapter of "how to" by Randall Munroe which answers this question, while omitting the issues mentioned by @PM2Ring (although it does point out that due to expansion you would never be able to reach certain galaxies ahead of you) Apr 11 at 14:36

No, you cannot. The do-not-attain speed is 99.99999999999999999998% of the speed of light, at which point your interactions with the cosmic background radiation are blue-shifted to the proton-degneration resonance energy. This will erode your spacecraft and you to something rather unrecognizable over a distance of 160 million light years. This limit is known as the Greisen-Zatsepin-Kuzmin limit and also known as the zevatron limit since a proton at that speed has about 1 ZeV energy and energy rather than velocity is what's observed in catching a cosmic ray.

See the Greisen-Zatsepin-Kuzmin limit.

Thus, your maximum Lorentz factor is $$5 \cdot 10^{10}$$ and therefore your maximum attainable distance is $$5 \cdot 10^{12}$$ light years.

While this exceeds the radius of the current observable universe ($$4 \cdot 10^{10}$$ light years), after taking into account the expansion of the universe, there are stars within our future light cone that you cannot reach.

• @Joshua Even intergalactically, there's still one particle per cubic meter. So near the speed of light, that's still about 300 million ZeV of cosmic rays every second. Apr 11 at 6:45
• Won’t you be able to accelerate again after travelling the first few hundreds of billions of light years, because the microwave background will have red-shifted much more by that time? Apr 11 at 12:16
• @Joshua What!? Then there is a special universal reference frame, and every introductory book or article on relativity is just lying!? Mind blown. Isn't the impossibility of FTL entirely based on there being no universally special reference frames? Doesn't this change absolutely everything? Why is nobody talking about this!? Apr 11 at 18:55
• @JounceCracklePop No, this is well understood, and even asked and answered on this very site: physics.stackexchange.com/questions/25928/… Apr 11 at 19:19
• @JounceCracklePop Everything we can see is in roughly the same reference frame. No suns or galaxies moving around at relativistic velocities relative to their neighbors. The fastest large objects in the universe are rogue planets moving only a millionth of light speed, and those are the extreme outliers. That doesn't mean that, in relativistic terms, the universe's reference frame is in any way "special", any more than my cat's is, or Pluto's. But it does mean the universe's intergalactic particles are way deadlier if we're a long way from the universe's not-at-all-special reference frame. Apr 12 at 0:06

I think your argument is correct. For example, in the Earth reference frame version, you would have aged $$L/c\gamma$$, where $$L$$ is the distance from Earth to the star in Earth's frame, and $$\gamma=(1-v^2/c^2)^{-1/2}$$. Since $$v$$ can be arbitrarily close to $$c$$, then $$\gamma$$ can be arbitrarily large, so your age can be arbitrarily small for fixed $$L$$. (Of course, it takes increasing huge amounts of energy to increase $$v$$ as you get close to $$c$$, which will kill any attempt to do this in practice, but I take it you are ignoring this constraint.)

In terms of an expanding universe, what you would want to do is solve the geodesic equation for a massive particle in an FLRW spacetime, and then compute your proper time to travel on a path that led from Earth to the star. There are definitely cases where you could reach the star: if the time it took you to reach the star in the Earth frame (which is also the cosmic rest frame) is less that the Hubble time (the timescale over which $$L$$ changes significantly). There are also definitely cases where you could not reach the star: in an accelerating Universe like the one we find ourselves in now, if the star was outside of our cosmological horizon, you can never reach the star no matter how fast you travel. In between those extremes, the answer will in general depend on $$L$$ and the expansion history of the Universe.

Essentially, the expansion of the Universe will increase the amount by which you will age to reach the star compared to what you would get in a non-expanding Universe, for fixed $$v$$. If it's possible to reach the star in a finite amount of time, then you can make that age as small as you want by cranking up your velocity. However, depending on the expansion history, there can be some stars where you cannot reach them with any amount of time. These would be stars where light could not travel from Earth to the star.

• Separate from the energy issue, going arbitrarily fast would require an arbitrarily high proper acceleration, and the human body is not made to take an arbitrarily large number of $g$s. However, if I've done the math correctly (using Baez's relativistic rocket page) you could travel something like 250 petaparsecs in 80 years of proper time while accelerating (and decelerating) at 1 $g$, so the cosmological factors would come into play well before you had to worry about the duration of the voyage at limited acceleration. Apr 10 at 23:54
• @MichaelSeifert Agreed and good point. This is also discussed in J.G.'s answer (who also brought up the need to decelerate mid-flight). I have to admit, before this question I actually didn't realize how far you theoretically could travel with enough energy. Apr 10 at 23:56
• Busted by the Zevatron limit. Apr 11 at 3:02
• Earth's gravity should be just enough acceleration to reach anywhere in a human lifetime; the distance traveled becomes exponential in the proper time. Although you almost double the time if you need to stop, and keeping the front of the ship from being incinerated by the blue-shifted CMB is basically impossible. Apr 11 at 3:20
• The question is - if the star is millions of light years away, it might not be there anymore when one arrives, as it has aged several millions of years... Apr 12 at 5:50

What you've hit upon is that, while your speed $$c\beta=c\tanh w$$ with $$w$$ your rapidity is bounded, your proper velocity $$c\beta\gamma=c\sinh w$$ is not.

Unfortunately, even if you can get all the kinetic energy you want (let's not ask how), you can only safely accelerate so fast, which constrains how far you can travel. Say you accelerate at $$g$$ for artificial gravity, decelerating in the second half of the journey to arrive at rest: this result implies an $$80$$-year life can afford about $$e^{40}\sim10^{17}$$ light years, a few orders of magnitude more than the current radius of our Hubble zone, but then its expansion is accelerating.

How much does the Universe's long-term expansion spoil your plans? That depends. For example, a "big rip" hits an infinite scale factor in a finite time. Less dramatically, suppose dark energy eventually causes an exponential expansion, as in a de Sitter universe. Well, the above scenario would cover exponential ground in a static universe, so it depends on which exponential growth is faster.

Yes, assuming you can get close to the speed of light, you can get to the star in your lifetime.

The expansion of the Universe is not important for the length scale of 100 light years.

• Not sure why this is getting down-voted. Would the expansion of the universe matter on bigger scales? Like if I wanted to go to another galaxy?
– Augs
Apr 10 at 14:45
• You can't ever actually reach the speed of light. Perhaps you meant to say "assuming you can get arbitrarily close to the speed of light?" (@Augs I didn't downvote but I assume it's because of the phrase "assuming you can reach the speed of light"). Apr 10 at 14:46
• I believe so. Never say that here, and don't bother to complain that it is "splitting hairs", because that is the whole point of this site ;) I have not downvoted either (this time)! Apr 10 at 15:30
• I got sloppy there, edited my answer. Apr 10 at 15:38
• Not sure why you specifically pointed out the length scale of 100 light years; the whole point of this question was to figure out whether it is possible to travel longer distances because of length contraction. I recommend J.G.'s very good answer regarding the importance of the expansion of the Universe. Apr 13 at 17:16