If gravity is not a force, how come it can produce potential energy?

From my poor understanding of Einstein's theory of relativity, gravity is not a force but the consequence of mass curving the space-time continuum. As such, the moon revolving around the earth is actually going straight in the space-time continuum.

The concept of force is very related to the concept of energy. A force is something which changes the energy of a system by doing work on it. In kinematics, work as:

$$W=\int\vec{F}\cdot d\vec{x}$$

Consider power station using tidal power for example. These stations are creating electricity from potential energy in tides. This energy must comes from somewhere and it is confusing to me how all this water gained potential energy if there was no force caused work on it. I could somehow conceive that the concept of potential energy is an "illusion", just like the concept that gravity is a force but then, it seems surprising that we can convert this "illusion" into electricity!

If gravity is not a force, how come it can produce energy? How to make sense of potential energy under the theory of relativity?

• Essentially, because it is a force. My answer to another question explains my point: physics.stackexchange.com/a/357234/20427 – Dvij Mankad Apr 29 '18 at 17:23
• Several comments removed (some preserved). Remember that the purpose of comments is to improve the post they are attached to, not to carry on an extended discussion. – rob Apr 29 '18 at 19:10

Newton's first law is a special case of the second, with $\ddot{x}^i=0$ when no net force acts on the body. In general relativity, this becomes $\ddot{x}^\mu =-\Gamma^\mu_{\nu\rho}\dot{x}^\nu\dot{x}^\rho$, where this time $\dot{f}$ is the derivative of $f$ with respect to proper time, and the coefficients $\Gamma^\mu_{\nu\rho}$ are called Christoffel symbols and characterise the spacetime geometry. Here Roman indices such as $i$ run over components of $3$-vectors, while Greek indices such as $\mu$ run over components of $4$-vectors. In the weak field limit, and if the body is moving at a speed $\ll c$, we get $\ddot{x}^i\approx-\Gamma^i_{00}$ (the $\dot{x}^0$ approximates $1$, since proper and relativistic time are similar). This shows how a mass-independent acceleration of bodies - in other words, the gravity Newton knew - is recovered. Multiplying by mass gives a "force", from which we can compute potential energy as usual.