will be . This imagined spaceship could offer round trips to Proxima Centauri lasting about 7.1 traveler years (~12 years on Earth clocks), round trips to the Milky Way's central black hole of about 40 years (~54,000 years elapsed on earth clocks), and round trips to Andromeda Galaxy lasting around 57 years (over 5 million years on Earth clocks). Unfortunately, sustaining 1-gee acceleration for years is easier said than done, as illustrated by the maximum payload to launch mass ratios shown in the figure at right. In curved spacetime In the language of general relativity, the components of an object's acceleration four-vector A (whose magnitude is proper acceleration) are related to elements of the four-velocity via a covariant derivative D with respect to proper time : Here U is the object's four-velocity, and Γ represents the coordinate system's 64 connection coefficients or Christoffel symbols. Note that the Greek subscripts take on four possible values, namely 0 for the time-axis and 1-3 for spatial coordinate axes, and that repeated indices are used to indicate summation over all values of that index. Trajectories with zero proper acceleration are referred to as geodesics. The left hand side of this set of four equations (one each for the time-like and three spacelike values of index λ) is the object's proper-acceleration 3-vector combined with a null time component as seen from the vantage point of a reference or book-keeper coordinate system in which the object is at rest. The first term on the right hand side lists the rate at which the time-like (energy/mc) and space-like (momentum/m) components of the object's four-velocity U change, per unit time τ on traveler clocks. Let's solve for that first term on the right since at low speeds its spacelike components represent the coordinate acceleration. More generally, when that first term goes to zero the object's coordinate acceleration goes to zero. This yields Thus, as exemplified with the first two animations above, coordinate acceleration goes to zero whenever proper-acceleration is exactly canceled by the connection (or geometric acceleration) term on the far right. Caution: This term may be a sum of as many as sixteen separate velocity and position dependent terms, since the repeated indices μ and ν are by convention summed over all pairs of their four allowed values. Force and equivalence The above equation also offers some perspective on forces and the equivalence principle. Consider local book-keeper