Carlo Rovelli's analogy of gravity as running down the beach into the sea where the resistance of water makes us fall down into the waves

Question

In "The Order of Time" (page 12), Carlo Rovelli explains how things fall due to the slowing down of time and uses the following metaphor:

"Where time passes uniformly, in interplanetary space, things do not fall. They float. Here on the surface of our planet, on the other hand, the movement of things inclines naturally towards where time passes more slowly, as when we run down the beach into the sea and the resistance of the water on our legs makes us fall headfirst into the waves."

An extract can also be found here:
https://www.theguardian.com/books/2018/apr/14/elastic-concept-order-of-time-carlo-rovelli
I am having a hard time to understand the intuition behind this analogy.
In the water analogy, the speed of the person and different resistances against different heights of the body cause the person's tipping over. How does this relate to a point mass in space-time? What is the relationship between the elements of this analogy (resistance of water, height of the person, running speed, tipping over, etc.) and gravitational pull between two point particles?

levitopher · Accepted Answer

I think he is talking about the depth of the water. The key connection he's making is that objects naturally want to fall towards other objects - that's the force of gravity, although in other places in the book he generally talks about gravity from the GR perspective, where objects generally move towards regions of higher curvature. The point is that this is the same direction of increasing proper time - in the direction of higher curvature. So that's the connection.
The analogy to water I think relates the slope of the beach to the curvature of spacetime, and the depth of the water to the passage of proper time. Out of the water ("flat ground") there is no motion (due to gravity) and nothing impeding the passage of time. As you enter the water, time starts to slow down due to the resistance of the water. If you were pointlike, you would roll faster when the beach slopes more, and as you get further under the water, your proper time would slow down more.
Now since you are not pointlike (an extended object), different parts of your body can experience different passages of proper time. That's what happening when part of your body is in water - your feet experience less proper time then your head, and you "fall into the water".

Carlo Rovelli's analogy of gravity as running down the beach into the sea where the resistance of water makes us fall down into the waves

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