What happens in a frame of reference of a car is that a gravity field emerges and pushes the passengers to their seats, and the coffee to the edge of the cup. It might be more convenient to speak of acceleration, but the underlying reality is the same, -- the warped time/space.
Wrong. Thirty-five years ago you could still find a physicist talking about a cetrifugal force. But that force doesn't exist; it only appears to exist; it is a fictitious force. A gravity field does not emerge to push you into your seat. The gravitational field is there all along. And since it doesn't accelerate you at all, the force of gravity is in equilibrium, and therefore the net gravitational force is zero.
What you feel as a mysterious appearance of a gravity force is only a contact force of the seat of your car pushing against your body. Since your body is not accelerating in either the horizontal or vertical directions, your body pushes back with a contact force. What you feel is that interplay between contact forces, and what eventually accelerates you forward is the friction forces between your body and the seat, not a mysteriously created gravity force.