# Spacetime curvature replaces acceleration?

In my understanding, not only mass but any kind of energy/ force bends spacetime. So is it correct to say that every object in the world moves along geodesic? If the object is submit to a force, it will not feel an acceleration: the force will bend the underlying spacetime --> the geodesic of the object will change --> its worldline in spacetime will change, but only because the curvature of spacetime has changed. It will still move along a geodesic. So acceleration, in some sense, simply doesn't exist anymore.

Is this interpretation correct?

• I'm sure this has already been answered somewhere on Physics SE, but I don't know where, so I'll post a quick hint: Worldlines that are geodesics describe free-fall; an observer following such a worldline feels weightless. Worldlines that are not geodesics describe deviations from free-fall, and such an observer feels weight. – Chiral Anomaly Aug 29 '19 at 13:38
• @Chiral Yes but my question is how a worldline cannot be a geodesic? If there is any force, it will simply change the curvature and so the geodesic path of the particle will change as well. But it will still be a geodesic curve. Where am I going wrong? – Federico Toso Aug 29 '19 at 13:51
• Federico, when I read "its worldline in spacetime will change, but only because the curvature of spacetime has changed", the impression I get is that you haven't conceptualized spacetime correctly. Reading "It will still move along a geodesic" strengthens that impression. Are you picturing an object as moving in spacetime? – Alfred Centauri Aug 29 '19 at 15:55
• @alfred Yes I am. Is this wrong? Isn't a worldline traced by the motion of a particle through spacetime? – Federico Toso Aug 29 '19 at 15:59
• Federio, if you're thinking of a point particle as point moving on a worldline in spacetime, then yes, I believe this is wrong. For example, see this answer from Ben Crowell: "Objects don't move through spacetime. Objects move through space. If you depict an object in spacetime, you have a world-line. The world-line doesn't move through spacetime, it simply extends across spacetime." - In other words, the depiction of a point particle in spacetime is not as a point but, rather, a curve in spacetime – Alfred Centauri Aug 29 '19 at 16:31

Mathematically, for a particle of mass $$m$$ and charge $$q$$ in an EM field $$F_{\mu\nu}$$, the geodesic equation is modified to
$$\frac{d u^\mu}{d\tau} + \Gamma^\mu{}_{\nu\lambda} u^\nu u^\lambda = \frac{q}{m} F^\mu{}_\nu u^\nu.$$
If there was no electromagnetic field, the right hand side would be zero, and you would just have the geodesic equation with the gravitational field $$\Gamma^\mu{}_{\nu\lambda}$$. If there is an EM field, then two things happen. For one, its energy-momentum influences the gravitational field, so that $$\Gamma^\mu{}_{\nu\lambda}$$ changes. But it also directly applies a force on the particle, given by the right hand side of the equation, and this effect is usually much bigger. The particle doesn't follow a geodesic anymore.