Friday, September 21, 2018

Gravity Wells are Sombreros

You've probably seen an illustration of a gravity well, something like this:

Or maybe you've seen one with Earth like this image:

Whenever I saw these gravity well depictions in school I mistakenly assumed they depicted gravity.  They don't.  These gravity wells depict gravitational potential, not gravity.  Gravitational potential tells you how much time is warped and how slow local time will progress due to general relativity.  It is not, however, correlated to what local gravity is.  These "gravity wells" would probably be better named "gravitational potential wells".  

Another problem is that these illustrations work fine for distances far away from massive bodies, but close this is an inaccurate picture of how gravity works. Xkcd's well known depiction relates planet radius to gravity, but this doesn't show how gravity works in the region of a massive body.

What do gravity wells look like near massive bodies? To illustrate, let's envision an empty area of space with a one light year radius and a single celestial body, something like Earth, in the center.

In this miniverse where is the least gravity and where is the most?

Based on the gravity well depiction, one might assume edge of this miniverse would have the least gravity and the center the most.

If that was your answer, get ready to have your mind blown.  

Now let's consider Earth. Where is the most gravity in Earth? Could it be the Dead Sea, the North Pole, the Mariana Trench, or Mt. Everest?

No, the most gravity experienced is almost 3,000 km below Earth's surface about halfway to the core. The least gravity would be the center of the planet. Consider this graph of Earth's gravity according to the Preliminary Reference Earth Model (PREM).

The points of both the most and least gravity are below the planet's surface!

The answer to our previous hypothetical miniverse was wrong. A miniverse with an Earth like mass would have the point of least gravity at the center and the area of most gravity below the body's surface.

To match traditional gravity well depictions Earth's gravity graph can be flipped so that low gravity is up and high gravity is down. Mirroring the graph then shows a path down to the core and back up again. The result is a sombrero gravity well cross section of Earth's gravity. Note the highest point in our gravity well, the point of least gravity, is inside of the planet and not on the edges. The edges will approach zero gravity, but will not reach it.

This also means the least gravitational time dilation in close proximity to Earth would be at the center of the planet. As long as no other body interferes with the gravity well this holds true as far out into space as the body is old.

To match the traditional 3D depiction, Earth's gravity well looks something like this.

Gravity wells are not just dips, they are shaped like sombreros.

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