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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?

Qmechanic
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Remi.b
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2 Answers2

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The short answer is that there isn't conservation of energy in general relativity.

Newtonian gravity and general relativity both have fully self-consistent descriptions of a rock falling to the ground, and both are, for all practical purposes, in perfect agreement with experiment for this situation. But that doesn't mean that they use the same description of the process.

GR doesn't actually have conservation of energy in all cases. The simplest description of the falling rock within GR is that the rock has a certain velocity vector, and that velocity vector remains constant. (Meanwhile the dirt is accelerating up toward the rock.)

If you want to describe the falling rock using energy and GR, you can, but it gets complicated. One approach is to exploit a timelike Killing vector in order to get a conserved quantity of the motion for a test particle. This quantity can be interpreted as an energy, but it doesn't split nicely into kinetic and potential energy.

In GR, we also have certain conserved measures of energy that are conserved in asyptotically flat spacetimes, which basically means spacetimes that can be approximated as some matter isolated within a certain region of space. An example of such a quantity is the ADM energy. The earth-rock system can be approximated this way, and ADM energy is constant.

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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.

J.G.
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