# Gravity in a spacetime with 2 indistinguishable dimensions, with all spacetime directions equivalent

A spacetime with 2 indistinguishable dimensions and all spacetime directions equivalent would have the signature (++) meaning that there would be no difference between spacelike and timelike directions and that if the spacetime was flat it would be Euclidean instead of Lorentzian.

I understand that, in 3+1 dimensional spacetime, moving with constant velocity in a uniform gravitational field is equivalent to accelerating at a constant rate and in a constant direction in $$0$$ gravity.

If we assume, in a spacetime with 2 indistinguishable dimensions and all directions equivalent, that moving with constant velocity in a uniform gravitational field is equivalent to accelerating at a constant rate and in a constant direction in $$0$$ gravity, and that the spacetime is flat in $$0$$ gravity, then the spacetime in a uniform gravitational field must also be euclidean as the spacetime in a uniform gravitational field must be equivalent to the spacetime in $$0$$ gravity in order for there to be the equivalence principle. An object accelerating at a constant rate and constant direction will trace out a spacetime circle. This means that if there is a massive body that has a uniform gravitational field around it, and this object moves at a constant rate and constant direction, it will trace out a spacetime circle and so must not be a geodesic. Also for a massive body given an initial spacetime position and initial spacetime velocity there would be two different ways for the massive body to trace out a spacetime circle.

Does this mean that the equivalence principle, in which moving with constant velocity in a uniform gravitational field is equivalent to accelerating at a constant rate and constant direction in $$0$$ Gravity, doesn't work in a spacetime with 2 indistinguishable dimensions and every spacetime direction equivalent?

I know that the hyperbolic plane has constant negative curvature, and in the hyperbolic plane all directions are indistinguishable, and all points are indistinguishable so I'm wondering if the spacetime around a massive body in a spacetime with two indistinguishable dimensions and all spacetime directions equivalent would have the geometry of the hyperbolic plane?

Would the gravitational field around a massive body be uniform in 2 indistinguishable dimensions with all directions equivalent depending on how we define a uniform gravitational field in this case?

• In your third paragraph, you bring up the notion of a uniform gravitational field. It's not clear to me why you think this notion is relevant, nor does general relativity offer anything with all the properties we would expect from a uniform gravitational field. On this topic, you may be interested in section 7.5 of my GR book, lightandmatter.com/genrel – Ben Crowell Sep 18 '19 at 23:32