The book "The Dancing Wu Li Masters" page 189 talking about General Relativity says "A geodesic is not always a straight line".
Is that true? What is a definition of "straight line" that makes sense in this context?
The naive definition of "straight" is a three dimensional Euclidean sense. However, that is problematic. You'd need to project our four dimensional universe onto a "flat" three dimensional space. Presumable this would be done by choosing a pretty big inertial frame of reference and setting the Time axis to a constant. But choose a different frame, or different constant, and suddenly "straight" means something different.
Perhaps it makes more sense to define "straight" by projecting our four dimensional universe onto a four dimensional Euclidean space. But I think this will not work. Consider how difficult it is to project a curved 2D space onto a flat 2D space, such as making a map of the Earth. Except in special cases, the "straight lines" on the map do not correspond to any useful concept of "straight" on the globe. So I think in 4D the result will be not any better.
I am starting to think that the statement "A geodesic is not always a straight line" is simply nonsense, because there is no concept of "straight" except for the geodesic itself.
Perhaps five dimensions? Is there a projection of our four dimensional universe onto a five dimensional Euclidean space that would give us a sensible definition of "straight"?
Sorry if this is off-topic or too "philosophical".