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If you could measure Earth's gravity with infinite precision, and did so both with Earth and its full population, and Earth with no people, would there be a direct difference in gravity? Is this even a valid way to think of gravity?

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I think the question has more to do with when a person walks around the surface, drives their car, or takes an elevator, or when an a cat jumps off a tree, or a bird flies through the sky, does that change the Earth's gravity, and the answer, as noted in the other answer, is yes, but very slightly.

Earth's mass and angular momentum is constant (if we ignore atmospheric loss, radiation out and radiation in and meteors/space dust in). What's not consistent is Earth's shape and variation in density, and Earth's shape and variation in density, does, very subtly, change it's gravity in the sense that Earth's gravity is lumpy, not smooth.

How big an effect Humans have on the planet's gravity by walking or driving around or building stuff is close to negligible. The largest effects on Earth's gravitational lumpiness are tectonic, not man made. Parts of Japan, for example, sank nearly 3 feet during the big Earthquake in 2011. The largest man made effect is the melting of glaciers driven by climate change.

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Yes, it would change the attractive force by 10^-14 if there was a world without people.

Depending on the distance, you can measure gravity of people and buildings, in fact they used gravity to measure gaps in the pyramids of Giza once.

Today gravity surveys can detect caves and reservoirs of 2m to 50m at 2m to 150m depth, so in the future, we will be able to discover many caves and fossils with nanotechnology gravimeters. http://physicsworld.com/cws/article/news/2016/mar/30/new-gravimeter-on-a-chip-is-tiny-yet-extremely-sensitive

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