Everyone talks about positive curvature when they talk about the EFE, and for good reason. I'd just like to know what would change in the equation if it accounted for negative curvature of spacetime instead. Would anything change, or is it just a matter of plugging in negative values instead?
I understand it seems strange that anyone would want to know this, but I'm just generally interested. What is different if you consider negative instead of positive curvature? How does one set the metrics to negative (if that makes sense)?