- #1
Grasshopper
Gold Member
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- TL;DR Summary
- Since you don't appear to have "coordinate" acceleration, yet still feel the ground pressing up against you, are you only accelerating through time, but not space?
Disclaimer: I'm not actually sure which acceleration is proper and which is coordinate, and I can't recall the source for the half-remembered equation. I spent some time going through my bookmarks, but it was to no avail. Sorry about that.
So, general relativity removes the idea of gravity as a force, and instead, when you're standing on the ground (and therefore not in free fall), you have a force upon you (normal force), but no counter force balancing it. That must mean you are accelerating. However, you don't appear move through space. Of course you clearly are "moving" through time.
Could it be that the acceleration you experience is entirely through time?
Why do I feel this way? I saw an equation that looked something like this:
$$ \frac{dr^2}{dτ^2}= a^r - Γ^r u^2 = 0$$
I've neglected the subscripts and wrote pseudomath here, but the point is, it looks like total acceleration equals a particular type of acceleration minus Christoffel symbols times the square of a four velocity. I don't have the source at the moment, but if I recall correctly, the only component that mattered in this particular velocity was the time component. I believe the others were zero (which makes sense since you're not moving through space in your own reference frame).So, interpreting this pseudomath, it seems to say that you ARE feeling an acceleration, but you're not moving through space because your felt acceleration is canceled out by that term which includes the time component of the four-velocity squared (a spacetime curvature term, perhaps?). It would appear, then, that your motion is all tied up in motion through time, and it cancels out the acceleration, meaning you don't change spatial coordinates.
How far off is that? Please feel free to correct and instruct as much as you want. I don't mind seeing sophisticated math so long as you're not writing paragraphs in it without explanation as you go. Thanks as always!
So, general relativity removes the idea of gravity as a force, and instead, when you're standing on the ground (and therefore not in free fall), you have a force upon you (normal force), but no counter force balancing it. That must mean you are accelerating. However, you don't appear move through space. Of course you clearly are "moving" through time.
Could it be that the acceleration you experience is entirely through time?
Why do I feel this way? I saw an equation that looked something like this:
$$ \frac{dr^2}{dτ^2}= a^r - Γ^r u^2 = 0$$
I've neglected the subscripts and wrote pseudomath here, but the point is, it looks like total acceleration equals a particular type of acceleration minus Christoffel symbols times the square of a four velocity. I don't have the source at the moment, but if I recall correctly, the only component that mattered in this particular velocity was the time component. I believe the others were zero (which makes sense since you're not moving through space in your own reference frame).So, interpreting this pseudomath, it seems to say that you ARE feeling an acceleration, but you're not moving through space because your felt acceleration is canceled out by that term which includes the time component of the four-velocity squared (a spacetime curvature term, perhaps?). It would appear, then, that your motion is all tied up in motion through time, and it cancels out the acceleration, meaning you don't change spatial coordinates.
How far off is that? Please feel free to correct and instruct as much as you want. I don't mind seeing sophisticated math so long as you're not writing paragraphs in it without explanation as you go. Thanks as always!