I do not understand how you can have acceleration without changing position (at 10:06). Acceleration is the derivative of speed, which is the derivative of position change. If the position change is zero, how can the acceleration be non-zero?
Position is not an absolute notion: you need to answer "position with respect to what?".
If the thing you're measuring position against is also accelerating, then you need to apply some acceleration of your own to stay still with respect to it.
The terms you want to look up are "proper acceleration" and "coordinate acceleration". The curvature of spacetime means the thing I'm measuring position against is moving relative to me (c.f. the example of two people walking in parallel across the Earth, nevertheless eventually meeting: the curvature means that even though neither of them is measuring an acceleration, nevertheless they are accelerating towards each other), so I need to have some internal ("proper") acceleration of my own to counteract the fact that our geodesics are moving away from each other.
Your position in spacetime is changing. You're going straight in spacetime, but spacetime is curved by the mass of the Earth so you're following that curve into the center.
The surface of the Earth keeps you from actually falling in, and is therefore pushing you away or upwards from the center. This is the acceleration acting on your straight line path through curved spacetime. This is the deviation from your geodesic.
So some flat earth arguments are actually correct if general relativity is correct, namely that gravity is an illusion and that the real reason we are stuck to the earth is that the earth is accelerating toward us at 9.8 m/s^2
I don't understand it. How can it accelerate towards anyone on its surface? Where does it get energy to accelerate? We generally need to burn fuel to accelerate something in the space.
This is about relativity. We're going straight in space-time, but space-time itself is curved because of the heavy mass nearby (the Earth). This is visualized by the rocket curving towards the planet in the video, and the bent sheet experiment where the balls spiral towards the center.
So we're curving in towards the center of the mass of the Earth, but the reason we don't end up in the core of the Earth is because the surface stops us. The Earth is "pushing" us away from the center, and that's the acceleration. It's accelerating you off your straight line path, and this is the deviation from the geodesic.
I thought that the Earth curved spacetime, and since an inertial observer approaching the earth would follow a straight line through curved spacetime, they'd appear to be following a curved line through space towards the earth.
What's confusing me here is the notion that when two objects collide, they accelerate into each other. Why and how is force constantly applied after the collision? My intuition is falling down here, and none of the resources I've looked at so far have explained why the acceleration happens.
1st sentence is correct and is the same as what I said, sorry if I wasn't clear.
Remember that spacetime = space + time dimensions. The object is always travelling through time, and the curvature of spacetime is converting some of that speed through time into speed through space. That's what you perceive as motion (caused by gravity).
Time and space are linked together. The faster you go through space, the "slower" you go through time (as in you experience it slower). This is very measurable and even used to alter timings for satellites GPS readings. You can take an atomic clock on a plane and age slower than someone who just stayed on the ground.
So the spacetime curvature is continuously converting some of your temporal motion into spatial motion, until that's stopped by the surface of the Earth which is constantly "accelerating" to stop you from going further.
As to why we always move through time, that's beyond my understanding at this point but it's a fundamental axiom of physics.
Yeah, sorry, I was restating what I understood you had said, in case I'd misunderstood any part of it, and then explaining where I was lost.
Your explanation makes sense, and it sounds like I'd need to understand the maths behind relativity to be able to really understand how objects behave in spacetime.
This amounts to a confusion over notation: "proper acceleration" (e.g. as measured by an accelerometer) vs "coordinate acceleration" (the acceleration an observer observes an object to be undergoing).
The acceleration an observer sees you undergoing is the same as the inherent "proper" acceleration you're undergoing, minus the acceleration of their coordinate frame with respect to yours. For me to stay still with respect to you, if you're in a frame that is accelerating away from me, I need some proper acceleration to catch up and counteract the fact that our frames are diverging. But if spacetime is curved, your frame probably is accelerating relative to mine - c.f. the example on the Earth's surface, where our frames inexorably accelerate towards each other as we move parallel to each other. So for me to stay still with respect to you, I need to have some proper acceleration to balance out the coordinate acceleration derived from the fact that our frames are moving in a curved space.
My understanding of General Relativity[1] is that mass distorts space-time, so an object traveling in a "straight line" through distorted space-time will curve with that distortion.
If the object's velocity isn't enough to traverse the curved space-time, it will move toward the center of the mass generating the distortion and fall out of the sky.
If the object is traveling quickly enough, it can continue traversing the distorted space-time and orbit that mass.
If the object is traveling even more quickly, it will traverse the distorted space-time and continue on without orbiting the mass.
In all three cases, from the perspective of the object traversing the distorted space-time, it continues to travel in a straight line, as it's the space-time that's distorted.
A (flawed) analogy would be riding a bicycle between the peaks of two identically sized hills. Starting at the top of the first hill, you coast down increasing your velocity.
Once you reach the bottom of the first hill and head up the second, your velocity decreases.
If your velocity at the bottom of the first hill is too small, you'll go up the second hill and as your velocity reaches zero, you roll back toward the bottom of the hill.
You will pick up velocity and then roll back up the first hill, then down again, then back up the second, etc. until you end up stopped at the bottom of the hill. This is akin to falling to the center of the distorting mass.
If your velocity is high enough to carry you back up to the top of the second hill and then stop, you'll roll back down and get to the bottom with the same velocity you had coming down the first hill. You'll then oscillate between the tops of both hills. This is akin to orbiting the mass.
If your velocity at the bottom of the first hill is enough to carry you past the top of the second hill, you'll just keep going after reaching the top of the second hill. That's akin to flying by the mass.
It's a flawed analogy, because in a curved space-time the directional portion of the motion vector doesn't change.
As John Wheeler[0] simplified it:
"Mass tells space-time how to curve, and space-time tells mass how to move."
