Quasi-local definitions of energy in general relativity

Defining energy is a surprisingly difficult problem in general relativity. For instance, the energy density of the gravitational field of a planet at a particular point could be determined by a comoving observer measuring the kinetic energy of a freely falling object. Due to the equivalence principle both the object and the observer fall at equal rates. Therefore, the observer would not assign any energy to the object. Other observers like an observer who is at rest with respect to the planet would measure different values. This raises the question of how energy depends on the choice of an observer which violates the philosophy of general relativity whose tensorial equations are independent of the used reference system. In classical electrodynamics the stress-energy tensor is a measure of the energy and momentum transported by the electromagnetic field due to a source distribution j. A similar construction in general relativity leads to the so-called Bel-Robinson tensor Tμνρσ [1, 2, 3] which can be thought of as being induced by a stress-energy tensor Tμν . Its physical meaning however remains unknown since it does not even have units of energy density. This is a consequence of the equivalence principle which equates the gravitational mass (the ”charge” of gravity) with the inertial mass. The source term, i.e. jμ in electrodynamics and Tμν in general relativity, does not contain the energy of the gravitational field. However, since the equations of general relativity are non-linear there may be a non-linear contribution to the stress-energy. For instance, gravitational waves do not pass through each other without distortion. Due to the absence of Stokes theorem for second ranked tensors conserved quantities do not exist. Landau and Lifshitz were able to prove that the stress-energy-momentum pseudotensor