Movie Monday Vol. 1: ‘Prometheus’

Posted by That Scientist on July 9, 2012 · 4 Comments 

Starting from this Monday forward, I’ll be waxing cinematic each week with a post analyzing a particular aspect of a motion picture. I’m not one of those moviegoers who looks at any science fiction film and instantly exclaims “that’s impossible!” at any plot point inconsistent with our reality. Nah, I prefer to accept a film on the story’s terms, as long as it remains consistent. (Superman can’t cast expelliarmus and Harry Potter can’t conceal his identity with eye-glasses.) After all, isn’t science fiction about ideas?, about “what if”?

This week, I’ll be starting with a recent release, Ridley Scott’s 2012 quasi-prequel Prometheus.

Prometheus has earned its share of online criticism for illogical plot turns, overall implausibility, and inconsistency. One object of such criticism is that the film’s spaceship travels 327 trillion kilometers from Earth to the moon LV-223, in what one character states was “2 years, 4 months, 18 days, 36 hours, 15 minutes” or about 872 days–Many commentators have incorrectly identified this as impossible faster-than-light travel.

If you reach Neptune, you’re about 1/74,000^th the way to LV-223. At the scale of this image (when expanded), LV-223 would be over 11 kilometers away.

The sun is not to scale here. If you scaled the sun to a ballpoint pen ball, then the our nearest star Proxima Centauri would be about 12 miles away.

Is that kind of journey possible? Well, technically yes it is.

327 trillion kilometers is about 34.56 light years, meaning light from LV-223 would take 34.56 years to reach the Earth (or likewise from Earth to LV-223). Light travels in empty space at about 299,792,458 meters per second. At that rate you could circle the Earth 7.5 times in one second. Physicists usually call the speed of light ‘C’.

According to Albert Einstein’s special theory of relativity, the speed of light is constant for all observers all the time. (Imagine a ball that flies at 50 mph no matter how to view it. You can speed up toward the ball, but the flight of the ball appears unchanged to you.) There are many strange consequences to that simple fact. For example:

1. No object with mass may reach or surpass the speed of light. The kinetic energy of an object approaching the speed of light approaches infinity (impossible).
2.  Traveling at high, near-C velocity will cause you to travel through time at a slower rate than stationary observers. Yes, time moves differently for all observers. This phenomenon is called time dilation.
3. Traveling at near-C velocity will cause space itself to shorten for the high-velocity observer! The distance you travel changes. This phenomenon is called length contraction.

These consequences are weird and counter-intuitive. It also means that if you traveled really fast on a trip away from and back to Earth you could “time travel” into the future. If you spent a single year on a ship blazing along at .99995*C then when you return to Earth 100 years will have passed!

If nothing can surpass the speed of light (not even nuisance neutrinos) then how on Earth can the ship travel 34.56 light years in under two and a half years?

The ship must have traveled at such a high velocity that the observers inside the ship only experience 872 days due to the time dilation. Let’s take a look at how this would work.

The formula for time dilation factor is:

Where Δt is the time dilation factor, v is average velocity and C is the speed of light.

To compress 34.56 Julian years into 872 days, the dilation factor must be 14.476. To undergo major time dilation such as this, the ship would need to travel an average velocity of .99761 C—almost the speed of light!

The ship would need some time to accelerate and get up to speed. When it approaches LV-223 the ship will also need time to slow down to rest. The maximum acceleration a well-trained, well-equipped human can withstand is about 10 g’s, or 98.067 ms^-2. (Let’s say in the future they have suits that allow them to do that for an extended time.) Under constant propulsion force, the ship would reach .99761 C in 35.38 days! (That would be like riding in the loop of an extremely fast roller coaster for a month straight, nonstop.) So with 70.8 days just for changing speeds, that leaves 801 days to cruise along at near-C velocity. Also, after using up time to accelerate, now we need to raise the cruising velocity to .998 C to make up that time.

NASA’s Space Shuttle weighed in at about 2 million kilograms, so let’s just say the good ship Prometheus has that same mass.

The ship at cruising, with a mass of 2 million kg., would have a kinetic energy around 2.19 trillion trillion joules! (One joule is about the energy released dropping an apple from a height of 3 feet. One million joules is the kinetic energy of a heavy pickup truck driving 70 mph. One trillion trillion joules is about the total energy from the sun that strikes the Earth over three months.)

The ship needs fuel though: Let us say that the ship has a thermodynamically perfect matter/antimatter engine, sort of like the Star Trek ‘warp engine’. (This is impossible in reality. A perfect engine would be cool to the touch, by the way.) Based on the famous E = mc² equivalency, the ship would need to carry some 59 million kg of matter/antimatter fuel to speed up and slow down–significantly heavier than the ship itself. Now accounting for depleting fuel but also carrying fuel to propel the deceleration, the ship and its fuel would then cruise with a kinetic energy of about 2.66 trillion trillion joules–that’s about 100 times the chemical energy of Earth’s entire fossil fuel reserve!

If you were to witness the Prometheus go by, the people inside (if somehow you could spot them) would appear to be moving very slowly. What’s more, the ship itself would appear to be squashed along its direction of travel.

This whole analysis has completely ignored the effect of gravity, which in reality does of course have an effect. According to general relativity, the Prometheus could have slowed time by passing near a massive, dense object, such as a black hole, but this would also require tons more fuel to escape.

So, yes, technically the special theory of relativity suggests that 2-year, 35 light year voyage of the Prometheus is physically possible. But it is very, very far from plausible, especially in only the year 2091 CE. (In the film, once at LV-223, they claim the year is 2093, but on Earth the year would in fact be 2126.)

More complications:

1. In reality, the Prometheus, handling years of life support for dozens of crew would probably be rather heavier than 2 million kg
2. Matter/antimatter engines, fusion engines, and ultra-efficient fission engines do not exist, nor does the fuel needed to run such things, especially to run them at such energies.
3. Thermodynamics requires the engine must waste some energy. In reality much, much more fuel would be necessary, because of that waste.
4. A real ship would certainly not be able to accelerate at 10 g’s for 35 days. The passengers would not survive such forces for long.
5. To avoid carrying fuel, a ship could use a solar sail or some kind, but this would accelerate extremely slowly, prolonging the journey significantly, and would grow less effective further away from the sun. Similar problem holds for slowing to a stop–how could such a ship avoid whizzing right past its destination?
6. Voyager 1 is right now the farthest man-made object from Earth. In nearly 35 years it has reached about 18 billion km from Earth (four times further than Neptune from us). At that rate, Voyager 1 may just reach LV-223 in another 636,000 years. Due to time dilation effects, Earth has aged about 2 seconds more than Voyager 1 since the time it was launched in 1977.

Filed under Mathematics, Movie Monday, Physics · Tagged with Einstein, film, movies, physics, Prometheus, special relativity