By such a small amount that it would never, ever matter, but it’s still cool to know!
This small difference does actually become relevant when trying to explain gravity, though.
As I’m sure you’ve heard, objects in free-fall do not feel an acceleration. Instead, objects in an inertial frame travel along straight world lines through curved spacetime “down” toward the gravitational source. Light is “pulled” into large objects because that’s what a straight line looks like to a light beam.
I have a question, though: why is that I can’t feel gravity during a free-fall, but I can feel my own weight when standing on the Earth? Isn’t that kinda weird?
(I’m wrapping the rest in a spoiler tag out of mercy to mobile users.)
Tap for more paragraphs.
Wouldn’t everything I just said suggest that if I can feel the direction of gravity, it must be because I’m accelerating somehow? If I were blind and in free-fall, I wouldn’t be able to tell what direction “down” even is, and yet, the muscles in my legs tell me very explicitly that they are “lifting” me.
The answer, or a very useful answer, anyway, is that something is accelerating: the ground. The stationary ground accelerates up through curved spacetime to push me. That’s how I can tell.
But how does the ground accelerate without moving? Shouldn’t the Earth expand if it’s accelerating?
We’re definitely approaching the limit of the things I can explain at the moment, but the key is time dilation. The fact that your head ages faster than your feet is the reason why the Earth doesn’t expand. It accelerates without moving because time is also curved.
You can think of it this way: If you were to graph the position of objects through time, the Earth’s surface might curve toward the y-axis, the time axis, because it is accelerating. But, the y-axis also curves “away” from the Earth’s surface in the same way, such that they maintain their distance. And so, the Earth doesn’t appear to move.
Your head ages faster than your feet do! :D
By such a small amount that it would never, ever matter, but it’s still cool to know!
This small difference does actually become relevant when trying to explain gravity, though.
As I’m sure you’ve heard, objects in free-fall do not feel an acceleration. Instead, objects in an inertial frame travel along straight world lines through curved spacetime “down” toward the gravitational source. Light is “pulled” into large objects because that’s what a straight line looks like to a light beam.
I have a question, though: why is that I can’t feel gravity during a free-fall, but I can feel my own weight when standing on the Earth? Isn’t that kinda weird?
(I’m wrapping the rest in a spoiler tag out of mercy to mobile users.)
Tap for more paragraphs.
Wouldn’t everything I just said suggest that if I can feel the direction of gravity, it must be because I’m accelerating somehow? If I were blind and in free-fall, I wouldn’t be able to tell what direction “down” even is, and yet, the muscles in my legs tell me very explicitly that they are “lifting” me.
The answer, or a very useful answer, anyway, is that something is accelerating: the ground. The stationary ground accelerates up through curved spacetime to push me. That’s how I can tell.
But how does the ground accelerate without moving? Shouldn’t the Earth expand if it’s accelerating?
We’re definitely approaching the limit of the things I can explain at the moment, but the key is time dilation. The fact that your head ages faster than your feet is the reason why the Earth doesn’t expand. It accelerates without moving because time is also curved.
You can think of it this way: If you were to graph the position of objects through time, the Earth’s surface might curve toward the y-axis, the time axis, because it is accelerating. But, the y-axis also curves “away” from the Earth’s surface in the same way, such that they maintain their distance. And so, the Earth doesn’t appear to move.
Isn’t that cool?
It is why physics was my favorite science class. Thanks for taking the time to write all that out