[Physics] Does vacuum spacetime have an inherent curvature


I am a complete novice in physics beyond the high school level, so please excuse anything wrong in my question.

So I have recently read that according to General Relativity, the presence of mass in spacetime causes spacetime to become curved and that is how gravity arises.

Does vacuum spacetime have an inherent curvature? What I mean to say is that if we remove all kinds of matter and energy from the universe (somehow), and then try to determine the curvature of spacetime, will it be flat or will it be curved?

And if vacuum spacetime does have an inherent curvature, why or how does that curvature arise, given that nothing possessing energy or mass is present in the universe I have described above.

Best Answer

The Einstein equation allows spacetime to have an inherent curvature, but this is an adjustable parameter. That is, general relativity does not predict what the inherent curvature would be, only that it could exist and could take any value. The only way we can tell whether spacetime has an inherent curvature or not is by observation.

One of the terms allowed in the Einstein equation is a cosmological constant, normally written as $\Lambda$. If $\Lambda$ is non-zero then spacetime has a scalar curvature (the Ricci scalar) given by:

$$ R = 4\Lambda $$

And this is exactly the sort of curvature that you are asking about because it is "built in" to spacetime and exists even in a universe completely empty of matter or energy.

In fact does appear that the universe could have a cosmological constant. When we observe the motion of supernovae in the universe it looks as if the universe is expanding faster than it should, and this could be due to a cosmological constant. The trouble is that it could also be due to a form of energy called dark energy, and at the moment we cannot tell which (if either) of these is the case. One way to tell would be to see if the effect changes with the age of the universe. A cosmological constant would be ... well ... constant, while dark energy could change with time. However at the moment our measurements are not precise enough to tell if the effect has changed over the age of the universe.

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