Transcription
I love Project Hail Mary, but the movie glosses over what I love about Andy Weir's novels, and that's the science. I guess the general public is not into math and science, but I am, and I think fans of this channel are too, so let's do the math and nerd out even harder than the novel does on relativity, mass ratios, and the Astrophage infection rate.
The movie mentions briefly that it takes him four years to get to Tau Ceti, but Tau Ceti is 12 light years away. So if you can't travel faster than the speed of light, how can you get there in four? Well that's where time dilation comes in. We can't travel faster than the speed of light, but something weird happens the closer you get to it. Time slows down. And if you keep accelerating, time keeps slowing down. Constant acceleration is like the eighth wonder of the world. If you can keep it up, it's like compounding interest. Extremely far away, things become closer and closer because of this time dilation, which begs the question to me, how long would it take to get the Betelgeuse or the edge of our galaxy or the nearest neighbor galaxies? The answer is that we can travel anywhere in the universe within a human lifetime.
Let's see it in action. Tau Ceti is 12 11.9 light years actually below us in relation to the galactic plane. But first, before we get to Tau Ceti, let's start with our closest neighbor. Let's start with Alpha Centauri and see what a hypothetical trip from Earth to Alpha Centauri would be like on the Hail Mary. This orange tube is not a wormhole. It's a representation of how fast time is going for our ship relative to Earth. We start the trip accelerating at 1.5 Gs. And if we get up to relativistic speeds extremely quickly, after only five months of accelerating, we're already going more than half of the speed of light. And while five months have passed for us on the ship, time has already slowed down and six months have passed on Earth. By the midpoint, we're traveling 97% of the speed of light and a year and a half have passed on the ship, but more than two and a half have passed on Earth. If we want to come to a stop at Alpha Centauri, it's here where we have to flip around and burn to slow down. And our time dilation start going back to normal as we slow down to normal speeds. And if we don't slow down, if we keep accelerating, time keeps getting slower. And that's the key. As we keep accelerating, our velocity tapers off because we can never get past the speed of light. It's asymptotic. We just keep adding nines to our 99.9 whatever of the speed of light. But the time dilation doesn't taper off. It's not bounded. The more we keep accelerating, the more time slows down. From the ship's perspective, it's exponential. Every additional year of acceleration, the time dilation roughly doubles.
If we compare the trip to Alpha Centauri to the trip to Tau Ceti, we can see this in action. The trip to Tau Ceti only takes about one year longer than the trip to Alpha Centauri, even though it's three times farther away. Rocky's trip from 40 Eridani took about the same tube because they're about the same distance from Tau Ceti than we are. They're about 10 light years. So the time dilation is pretty much the same if they took the same flip and burn profile. Which they didn't. But to really see the power of this, we need to go much further. Betelgeuse is about 500 light years away. With the same level of acceleration, the trip would only take 8.5 years. At the midpoint, we will reach 99.999% of the speed of light. I've scaled up the travel lines, but as often the case with cosmological distances, things are getting hard to see. To really understand the exponential nature of this, we have to change the way we're looking at this. We've got to move the stars around here. Let's compress everything based on how long it takes to travel to them. In order to illustrate that, let's add in some more stars. Like the Pleiades Cluster, which is a famous star cluster slightly closer than Betelgeuse. And the Orion Nebula, which is 1000 light years further than Betelgeuse. And when we engage our time dilation, this is what happens. Remember the farther away an object is, the more time dilation affects it. So the Pleiades and Betelgeuse are only slightly further than Tau Ceti and Eridani. And the Orion Nebula, which was 1000 light years further, is now just slightly farther than Betelgeuse. Betelgeuse took 8 and a half years to get to, and the Orion Nebula only takes 9 and a half. You can kind of now see how it's getting squished up around the edge. And that's the same thing that's going to happen for the rest of the stars in our galaxy. Our entire galaxy becomes a squished shell right on the edge of our doorstep.
Other galaxies too get pushed right to the edge. The large Magellanic Cloud, which is 160,000 light years away, takes 16 years. Andromeda, our closest large galaxy, which is normally 2.5 million light years away, takes just under 20 years to get to. The Virgo Cluster, a cluster of a couple thousand galaxies, which is 54 million light years away, would take just 23 years and 7 months. And finally, the edge of the observable universe, 46.5 billion light years away, would take just 32 years and 3 months. At which point we will be traveling 99.99999999999999999999921. The Speed of Light, 46.5 billion years would have passed on Earth because we're basically going the speed of light.
This would require a lot of energy. Actually, in this example, it would require a lot of matter because that's what we're doing. That's what Astrophage is doing. It's converting matter into energy. Astrophage is like a little E equals MC squared bioengine. So how much matter do we need for this trip to the edge of the universe? About 500 sextillion tons, which is the mass of 87 Earths. And this starts asking the question, how much mass is too much? Because we're already idealizing a lot of things here. In modern rocket engineering, a mass ratio of 20 to 1 is considered realistic. That means the fuel weighs 20 times more than the ship. And in fact, that's exactly what the mass ratio of the Project Hail Mary was in the novel. The ship needed 2 million kilograms of Astrophage to reach Tau Ceti in four years. And the ship weighed 100,000 kilograms, 20 to 1. Problem is, that was a mistake by Andy Weir. Two million kilograms is what was needed if it was a flyby. To stop at the destination, you need to square that number. So what it really needed was 400 million kilograms. I reached out to Andy Weir and he responded and confirmed this, which would mean a mass ratio of 400 to 1, which is a little unreasonable. But we do have an option to save some fuel. We don't need to accelerate for the whole period. And that's what Andy Weir says he changed. He said he added a coast phase. Okay, so let's do the math. Using the same specs from the book, 1.5 Gs of acceleration. If we coast for 85% of the time, that is we accelerate for about one light year, coast for 10, and then decelerate for one. We'll get there in six years and eight months. Confusing to me, in the movie they didn't fix this. There's a point at which at the end the computer says it's going to take four years, two months, and 11 days, which is almost exactly a 50% coast phase. So that seems intentional to me. Like maybe that was Andy Weir's fix to do a 50% coast phase. But with a 50% coast phase, the mass ratio becomes 124 to 1, which would mean the ship would have had to have weighed 16 tons dry or used more astrophage. And in the movie they say it still uses 2 million kilograms of astrophage. But honestly, there are so many assumptions we're making here that 124 to 1, maybe that's plausible in this universe holding astrophage. You can play around yourself with these numbers on my website. This is the calculator on my website.
Either way, I'm still inspired that it's still physically possible to reach other galaxies within the known laws of physics. Because there could be other methods of generating insane amounts of energy in the far future, like a Kugelblitz drive or something else crazy. And we could still get there without wormholes or breaking physics. While I'm in the mode of being an armchair physicist, there is one other flaw in Project Hail Mary that I thought was interesting. In the story, Tau Ceti is the only star nearby Earth which is not dimming due to astrophage infection. Canonically in the novel, as they're tracing back where it came from, they say that the sun was infected by WISE 08550714, which was infected by Sirius, which was infected by Epsilon Eridani. They also say that WISE infected Wolf 359, Lalande 21185, and Ross 128. From this, they deduced that the range at which astrophage can survive interstellar distances is eight light years, and thus the range that it can infect other stars is eight light years. Problem is, in real life, WISE is 11 light years from Sirius. So that's a no go. The shortest distance from Tau Ceti to Sol in less than eight light year hops would have been Tau Ceti to Lacaille 9352, Lacaille 8760, Ross 154, Barnard's star, and Sol. I feel like I'm playing Elite Dangerous. But that wasn't actually the problem that I thought was interesting. The problem is that if eight light years is indeed the infection range, 90% of stars in our galaxy have another star within eight light years. In my opinion, probable that astrophage would have spread throughout the entire galaxy like a plague infecting nearly every star. It sort of depends on the distribution of stars, and there are some filters that could have limited it, but I just thought it was interesting to visualize what an eight light year range spread would have looked like in our nearby stars. This is using the ATHYG Stellar catalog, by the way, which is not a full representation of all the stars in our neighborhood, but it's pretty close.
The point is, this is an awesome sci-fi. I saw an interview with the directors of the movie who said that the story is about saving the galaxy by making a friend. But the novel is deeper than that. It's about making a friend who's a complimentary thinker to you, who's not the same but gets you and you get him. It makes me feel less alone, I guess. Although I did have to go to another star system and find a literal alien to feel less alone. But having seen the movie and seen reactions from all different types of people, it's starting to illustrate to me the difference between me and the normies. It's starting to. I still need some time to process it, but let me know what you guys think.
If you want to dig even deeper into the math and science behind Project Hail Mary, I would recommend Brilliant. Brilliant is like the teacher I wish I had. I never did. I don't learn by reading textbooks. I learn by doing it myself, which is exactly what Brilliant teaches you via interactive problem solving and a host of fields you'd need to science your way out of interstellar problems. They have courses in math, science, coding, and AI, all crafted by world class teachers from MIT, Harvard, and Stanford. They have a course on exponential functions, which is one of the core concepts underlying this time dilation and astrophage infection rates. But there are a ton of other interesting topics for kids and adults. They're designed for ages from 10 to 110. You can learn for free on Brilliant for a full 30 days by going to my link, brilliant.org slash the overview effect, scan the QR code on screen, or click the link in the description. And Brilliant has also given my viewers 20% off an annual premium subscription, which gives you limited access to everything in Brilliant. I'm extremely selective with what I choose to throw on this channel. I say no to almost everything. Brilliant is one of the few that I cannot recommend enough. Thank you, Brilliant.