13.1 Calculation of tidal heating

According to astronomical observations, there is intense volcanic activity on the moon Io of Jupiter. One possible explanation of this phenomenon is that Jupiter is heating Io via gravitational tidal forces (like the Moon, whose gravitational tidal forces raise the ocean’s tides on the Earth). To check if this is really the case, one must be able to calculate how much energy is pumped into Io. However, gravitational energy (both in Newtonian theory and in general relativity) is only ambiguously defined (and hence, cannot be localized), while the phenomena mentioned above cannot depend on the mathematics that we use to describe them. The first investigations intended to calculate the tidal work (or heating) of a compact massive body were based on the use of various gravitational pseudotensors [432Jump To The Next Citation Point, 185Jump To The Next Citation Point]. It has been shown that, although in the given (slow motion and isolated body) approximation the interaction energy between the body and its companion is ambiguous, the tidal work that the companion does on the body via the tidal forces is not. This is independent of both the gauge conditions [432Jump To The Next Citation Point] and the actual pseudotensor (Einstein, Møller, Bergmann, or Landau–Lifshitz) [185Jump To The Next Citation Point].

Recently, these calculations were repeated using quasi-local concepts by Booth and Creighton [94]. They calculated the time derivative of the Brown–York energy, given by Eqs. (10.8View Equation) and (10.9View Equation). Assuming the form of the metric used in the pseudotensorial calculations, for the tidal work they recovered the gauge invariant expressions obtained in [432, 185]. In these approximate calculations the precise form of the boundary conditions (or reference configurations) is not essential, because the results obtained by using different boundary conditions deviate from each other only in higher order.

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