Summer Reflections: Stars, Eclipses, and Relearning Physics
This essay is a bit of a brain dump - a collection of threads that wove together over the summer break. It started with a book, wound through some physics I’d half-forgotten, peaked at a waterfall during totality, and ended with me in Blender trying to recreate what I’d seen.
Project Hail Mary
I read Project Hail Mary in three days over Christmas so I could be a book snob when the movie drops. Worth it.
Weir’s formula is problem, science, solution, repeat - and he executes it well. The pacing keeps you turning pages, and the science feels plausible even when it’s fiction. Coming from Sanderson’s Stormlight Archive - which I enjoyed far more as a feat of writing, with its intricate multi-threaded plots and mystery woven across characters - I appreciated that Weir was attempting something different. Physics itself becomes the character, and at times the antagonist. The universe isn’t backdrop - it’s what he’s writing about.
What won me over was the consistency. Given one impossible premise, the rest feels like it could be true in our universe. Weir clearly did his homework. And without spoiling anything, there are moments of genuine wonder and connection that caught me off guard - emotional beats I wasn’t expecting from a “science problem” book.
What stuck with me most wasn’t the plot - it was the scale. The vast distances involved, the time dilation, the sheer audacity of interstellar travel. It made me want to brush up on my physics.
The Universe is Impossibly Large (Or Is It?)
For a long time I assumed the universe was simply too big to explore. The nearest star is 4.24 light-years away - light itself takes over four years to get there. The Milky Way spans 100,000 light-years. Andromeda is 2.5 million light-years out. A human lifetime is maybe 80 years. The math doesn’t work.
But it turns out the reality is weirder than intuition suggests.
The Speed of Light is Fixed
You’re on a train going 100 kph. You shine a flashlight forward. How fast is the light going?
Intuition says: 100 kph + the speed of light. But every experiment we’ve ever done says no - the light still goes at exactly c. Not c + 100 kph. Just c.
This seems wrong, but it’s what the universe does. If I’m standing still and you fly past me at half the speed of light, we both measure the same light beam going at c. Not c minus your speed. The same c.
The Geometry This Creates
Light is the same speed for everyone. That’s the experimental fact.
But think about what that means. I’m standing still. You fly past me at half the speed of light, chasing a light beam. You should measure the light going slower, right? You’re chasing it. But you don’t. You measure exactly the same speed I do.
The only way this works is if something else changes to compensate. If the speed stays the same but you’re moving, then either the distance the light travels must be different for you, or the time it takes must be different. Or both.
It’s both. Distance and time are not fixed. They stretch and squeeze depending on how fast you’re moving.
This creates a problem. If you and I can’t agree on distances, and we can’t agree on times, what can we agree on? We need some combination of distance and time that stays the same for everyone.
Here’s where light helps again. Light travels at c - let’s use units where c = 1, so light goes 1 light-second of distance in 1 second of time. For light, distance equals time. Which means for light, distance minus time equals zero.
That zero is the same for everyone. I measure the light going 1 light-second in 1 second: 1 - 1 = 0. You, flying past me, measure different values for the distance and the time - but you also get zero when you subtract them. That’s the invariant.
For anything slower than light, you cover less distance than time elapsed. So t² - x² isn’t zero - it’s some positive number. Call it τ²:
That τ (tau) is “proper time” - the time on your own clock, the time you actually experience. You and I disagree about t. We disagree about x. But we agree on τ.
This is hyperbolic geometry. In ordinary space, distance is x² + y² = r² - that plus sign gives you circles. In spacetime, the minus sign in t² - x² = τ² gives you hyperbolas instead.
Hyperbolas have a special property: they curve toward a diagonal line but never touch it. That diagonal - where t² - x² = 0 - is light. You can get 99.9999% of the way there. You can never arrive.
You’ve Seen This
Before you think this is all abstract physics: you’ve seen hyperbolas today. Every lamp with a circular shade casts light that hits the wall in a hyperbolic curve. The cone of light intersects the flat wall, and the edge of the light traces out the shape.
Light escapes from both ends of the shade in cones. Where those cones hit the flat wall, the edge of the light forms a hyperbola. Drag the slider - a wider shade opening makes wider, more curved light edges.
The same geometry that governs your desk lamp governs the universe. I’d just… forgotten it. Or maybe never properly learned it in the first place. Reading Project Hail Mary forced me to actually relearn this.
What This Means for Time
Here’s where it gets strange. The spacetime formula t² - x² = τ² tells us something profound: if you travel distance x in time t (as measured by someone watching you), but your experienced time is τ, then these quantities are related by that hyperbolic geometry.
If you travel distance x in time t, your speed is v = x/t. Substituting x = vt into the formula:
That ratio - t/τ, how much slower your clock runs compared to a stationary observer - is called γ (gamma):
At everyday speeds, v²/c² is tiny, so γ is close to 1. Your time and my time are basically the same. But as you approach light speed, v²/c² approaches 1, and γ blows up toward infinity.
At 90% of c, γ ≈ 2. Your clock runs half as fast as mine. At 99% of c, γ ≈ 7. At 99.99%, γ ≈ 70.
The Galaxy Becomes Reachable
This changes everything about interstellar travel.
If you could accelerate continuously at 1g (a comfortable Earth-like gravity), you’d keep speeding up, getting ever closer to c, with time dilation increasing the whole way. A trip to Proxima Centauri (4.24 light-years) would take about 3.5 years for you - though 6 years pass on Earth. To Andromeda? About 28 years on the ship - but 2.5 million years back home. At the midpoint of that trip, you’d be moving at 99.99999…% of c, with γ near 90,000.
The galaxy is traversable in a human lifetime. You just can’t come back to the same Earth you left. A round trip to Andromeda means 56 years for you, 5 million years for everyone else.
This works in reverse too. If someone left Earth 65 million years ago - when dinosaurs roamed - and flew in a straight line at near-light speed, they could arrive back today having aged only a few decades. The key is straight line. No turning around required. They’d just need somewhere very far away to aim for, and enough fuel to get up to speed. (This is part of what makes Halo’s slipspace lore work, incidentally.)
The problem is stopping, or changing direction. A one-way flyby to Andromeda takes only about 15 years if you accelerate the whole way. But then you’re hurtling past at over a billion kph with no way to slow down. Gravitational slingshots don’t work at relativistic speeds - even a black hole barely bends your path. To actually stop and explore, you’d need to carry fuel to decelerate. And to come home, you’d need fuel to accelerate again. The tyranny of the rocket equation makes this basically impossible - each maneuver multiplies your fuel requirements exponentially.
Why It Takes So Much Energy
The same geometry that bends time also bends energy. Total energy of a moving object is:
At rest (γ = 1), this is just E = mc². Mass itself is stored energy.
Kinetic energy is total energy minus rest energy:
Where does E = γmc² come from?
Just as spacetime has an invariant “distance” (t² - x² = τ²), energy and momentum have an invariant too. This isn’t a coincidence - energy and momentum are the time and space components of motion through spacetime.
Position in spacetime is (t, x). As you move, you trace a path through spacetime. The “velocity” through spacetime has two parts - how fast you move through time, and how fast you move through space. Energy is related to movement through time; momentum is related to movement through space.
The invariant for position is: t² - x² = τ²
The invariant for energy-momentum follows the same pattern:
That same minus sign appears. Same geometry, different quantities.
Rearranging: E² = (pc)² + (mc²)²
At rest (p = 0), this gives E = mc². For a moving object, momentum is p = γmv. (Why γmv instead of just mv? Because momentum must be conserved in all reference frames, and the γ factor is needed to make that work.) Working through the algebra gives E = γmc².
At low speeds, γ ≈ 1 + ½v²/c², so KE ≈ ½mv² - the familiar formula. But at high speeds, γ dominates. Each additional 9 in your velocity (99% → 99.9% → 99.99%) roughly triples your energy cost. The curve never levels off - it steepens forever. You’re not climbing a hill with a top - you’re following a hyperbola that gets endlessly closer to an unreachable asymptote.
This is the challenge of math you don’t apply. It atrophies. I could do calculus in university, could manipulate equations and solve problems. But without regular use, it fades to vague recognition. “Oh yeah, hyperbolas, those curve things.”
I’d learned about hyperbolas. I’d learned about spacetime. I’d never connected them. And when I tried to remember what a hyperbola even looked like, I pictured 1/x - the curve that swoops through two quadrants. That’s not a hyperbola. That’s the reciprocal function. A hyperbola is x² - y² = 1. Same asymptotes, different curve. I’d been carrying the wrong picture for years.
The deep understanding evaporates. There’s something humbling about relearning it.
Totality at Niagara Falls
All this thinking about the universe - light travelling, distances too vast to comprehend, time stretching - reminded me of the first time I’d felt cosmic scale rather than just calculated it.
In April 2024, we drove to Niagara Falls to witness the total solar eclipse.
There’s nothing quite like it. Photos don’t capture it. Videos don’t capture it. The experience of standing in the shadow of the moon, watching the sun’s corona blaze around a perfect black disk, feeling the temperature drop and the world go quiet - it’s primal. You understand, viscerally, why ancient peoples thought the world was ending.
The falls added something. The constant roar of water, the mist, and then this cosmic event overhead. Scale meeting scale.
Making Eclipses in Blender
Later, I saw Avatar: Fire and Ash. One scene features a solar eclipse - the corona flaring around the edges - and it made me want to capture what I’d seen. Not the photos, which don’t do it justice, but the image in my mind’s eye.
Volumetric corona attempts - getting the falloff right took iteration
The technical challenge was interesting: the light falloff, the atmospheric scattering, the way the corona doesn’t just stop but fades gradually into space. Blender’s volumetrics helped, but it took iteration. I’m still not happy with it. The real thing was better.
The Thread
What connects a book, Blender renders, a road trip, and forgotten math?
Looking up, I think. Looking out.
Project Hail Mary made me think about the distances between stars. The eclipse made me think about our place in the solar system. The physics made me think about the shape of spacetime itself. And relearning hyperbolas reminded me that understanding isn’t permanent - it requires maintenance.
We’re on a small rock orbiting an average star in one arm of an ordinary galaxy. The nearest other star is 4.24 light-years away - close enough that we could theoretically reach it, far enough that it would take a generation ship or relativistic speeds. The rest of the galaxy stretches 100,000 light-years across. And beyond that, a universe that’s actively moving away from us.
The Psalmist looked at the same sky:
When I consider your heavens, the work of your fingers, the moon and the stars, which you have set in place, what is mankind that you are mindful of them, human beings that you care for them?
— Psalm 8:3-4
Further Reading
- I finally understood why time even exists! - This video helped me understand the concepts covered here