### 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 [432, 185]. 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 [432] and the actual pseudotensor (Einstein, Møller, Bergmann, or
Landau–Lifshitz) [185].
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. (10.8) and (10.9). 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.