Yet another spacetime-bending layman question

[dodo]

TL;DR Summary

TL;DR: why the spacetime deformation that bends the trajectory of a ball tossed up and coming down is not bending my own body the same way?

Hi,
apologies if questions of the same form have been asked by laymen multiple times. (You must all be tired of this!) I tried to search on the forum, but I did not find my exact question answered. For context, I have some university math background, but when it comes to physics I am pretty much a layman.

If I toss a ball from my hand and catch it back (let's say the trajectory is not exactly vertical, so I have to move my hand a little to catch the ball back), it describes a very narrow arc of ellipse (the focus is at Earth's center of mass, so modelling it with the focus at infinity as an arc of parabola is not much of a relative error). Imagine now that I take a piece of paper, and draw vertical parallel lines representing inertial lines under the spacetime deformation; the ball would follow one straight line segment in this diagram, but my hand would have to be drawn simultaneously at both ends of that line segment. Ultimately my question is, then, why is my body not torn by this extreme deformation.

I understand (I think) how, in classical mechanics terms, a tidal force moves the water much more than it moves the land, as the latter is held by cohesion forces stronger than the water's. But I fail to see how this would apply to the ball's inertial trajectory bending to such narrow curve (which would be a straight line away from me, if I were in interstellar space away from big masses), and that bending being the result of spacetime deformation (and hence as much an imaginary artifact as a force as "centrifugal force" is), while not apparently affecting my own body (and my apartment) the same way.

A drawing of my body on that paper with straight lines would help. As we determined, my hand is at two places. I am ignoring the time dimension on this drawing (plus mapping 3 spatial dimension onto 2) but that "mapping" would need to be multivalued, in order to take the same 3D point to 2 different positions on the paper! So please feel free to imagine how the missing dimensions would illustrate (if at all) the issue.

Thanks again for your patience.

 

[dodo]

Post correction:
I did assume that the ball's trajectory was not exactly vertical; an exact vertical trajectory represents a degenerate ellipse, which may explain the apparent puzzle of my hand being at two places on the paper. Let's continue assuming that the trajectory is a proper ellipse, that I cannot make that diagram for an exact vertical fall any more than I can divide by zero, and retract the notion that the mapping is multivalued; we simply have 2 very close 3D points mapped to the ends of a line segment on the paper. I'm still trying to imagine how my body looks on that paper, which may help with the question "what does gravity do to my body".

 

[PeterDonis]

why the spacetime deformation that bends the trajectory of a ball tossed up and coming down is not bending my own body the same way?

There are several responses to this.

First, the ball is being treated as a point particle in your post. Your body is not. Deformation of an object that is not a point particle is simply a different thing from "deformation" (bad word--see below) of the trajectory of a point particle--or, if you want to be more precise, the trajectory of the center of mass of an object. The ball itself could be deformed by tidal gravity even though its center of mass was in free fall, following a straight line in spacetime. (More on that below.) And looking at the ball's trajectory in space, which is what you are doing, tells you nothing useful about spacetime curvature. So the comparison you are trying to make is not valid.

Second, the ball's trajectory in spacetime is not "deformed". It's a straight line. Your trajectory in spacetime is "deformed", i.e., curved. However, the reason your spacetime trajectory--or, more precisely, the trajectory of your center of mass--is curved is not that the spacetime geometry is doing it; it's that the Earth's surface is doing it, pushing you upwards and preventing you from freely falling as the rock does. That's not tidal gravity.

Third, "spacetime deformation", spacetime curvature, is tidal gravity. To see the effects of tidal gravity, you can't just look at one trajectory. You have to look at multiple trajectories of neighboring freely falling objects to see if they converge or diverge; or, you need to look at an object, like a ball, as an extended object and see if there are internal stresses in the object even though its center of mass is in free fall. But you can't do the latter type of test on an object, like you yourself, that is not in free fall; your body has internal stresses simply due to being pushed upwards by the Earth's surface. In order to see if there were any effects--"deformation"--on your body due to tidal gravity, you would need to be in free fall so the effect of being pushed was not there.

 

[PeterDonis]

why is my body not torn by this extreme deformation.

What makes you think the tidal gravity in the Earth's vicinity is "extreme"?

 

[Dale]

TL;DR Summary: TL;DR: why the spacetime deformation that bends the trajectory of a ball tossed up and coming down is not bending my own body the same way?

I am ignoring the time dimension on this drawing (plus mapping 3 spatial dimension onto 2)

That is the main problem. When you are talking about general relativity and how it equates tidal gravity with curvature, it is essential to recognize that it is curvature of spacetime and not merely curvature of space.

In this view the path of the ball is a straight line in spacetime called a geodesic. The path of your hand is a very gently curved line. The curve is so gentle that the radius of curvature is around 1 light year. Such gentle curvature poses no risk of damaging your hand.

 

[Ibix]

Ultimately my question is, then, why is my body not torn by this extreme deformation.

As others have mentioned, it's actually the ball that travels in a straight line (usually called a geodesic when referring to paths in curved spacetime) and you that follows the curved path. You have to remember that we're talking about spacetime here, and you've only thought about the spatial projection of the trajectory, which is why you think the ball's path is curved (you're in good company, by the way - literally everyone before Einstein would have agreed with you).

Let's say you throw the ball 5m up, so its flight time is 2s. You travel 2s in time while never being more than 5m from the ball's geodesic. It turns out that the correct scale factor to convert distances into times is $c$, the speed at which light travels. So we can say that you travel 2s into the future while never being more than $\frac 5{3\times 10^8}\mathrm{s}$ away from a straight line path. That is, your curved path is less than two parts in one hundred million away from the straight line when you take into account all four dimensions. Gravity near Earth is very very weak.

As Peter and Dale point out, it's actually tidal gravity that tears you apart and that isn't correlated with "plain old" gravity in general. That's a bit messier to characterise, though, so I'm sticking with a simpler concept that might point out what you are missing: the time part of spacetime.

 

[PeterDonis]

The path of your hand is a very gently curved line.

And this curved line has nothing to do with the spacetime curvature in your vicinity. To probe that you would need to compare the worldlines of nearby freely falling objects to see if they converge or diverge.

 

[DaveC426913]

...my hand would have to be drawn simultaneously at both ends of that line segment. Ultimately my question is, then, why is my body not torn by this extreme deformation.

Is the OP ultimately referring to (a very mild form of) spaghettification? (When you fall into a black hole you are compressed laterally).

On Earth, this is due to the tiny deflection of the pull from the Earth between two points some distance apart, yes?

As in: my left shoulder and my right shoulder - two feet apart - are pulled toward each other because "down" for each of them is not strictly parallel. But they're parallel to within one part in a million! (2 feet over 2 million feet.)

 

[Ibix]

Is the OP ultimately referring to (a very mild form of) spaghettification? (When you fall into a black hole you are compressed laterally).

I think the OP is doing so in that sentence, yes, and confusing it with the curvature of the spatial projection of a geodesic which is what the rest of the post is about.