Questions: Gravitational Energy and Pseudo-Tensors

Question 1

Why can't gravitational energy be described by a true tensor in general relativity?

• A. Because the Einstein field equations are nonlinear
• B. Because the equivalence principle allows gravity to be locally eliminated — a freely falling observer sees no gravitational field, hence no gravitational energy, at any single point
• C. Because gravitational energy is always exactly zero
• D. Because tensors can only describe electromagnetic energy, not gravitational energy

Answer: (B)

A tensor that is nonzero in one coordinate system is nonzero in all coordinate systems. But the equivalence principle says that at any point, coordinates can be chosen (freely falling frame) in which the gravitational field — and hence any candidate for gravitational energy density — vanishes. A quantity that can be made to vanish by coordinate choice at any point cannot be a tensor (unless it is identically zero everywhere, which it clearly isn't globally). This is why pseudo-tensors, which are coordinate-dependent, are the best one can do locally.

Question 2

The Landau-Lifshitz pseudo-tensor gives different values for the gravitational energy at a point depending on the coordinate system used. This means gravitational energy in GR is physically meaningless.

• T. True
• F. False

Answer: False

While the local density of gravitational energy is indeed coordinate-dependent and physically ambiguous, the total gravitational energy integrated over appropriate regions is well-defined and physically meaningful in certain contexts. The ADM mass (total energy of an asymptotically flat spacetime) is coordinate-independent and conserved. The Bondi mass measures total energy at null infinity, accounting for energy carried away by gravitational radiation. These global and quasi-local quantities have clear physical interpretations — it is only the local density that is ill-defined.

Question 3

Explain the distinction between ADM mass and Bondi mass, and what physical process accounts for their difference.

Model answer: The ADM mass M_ADM is the total energy of an asymptotically flat spacetime measured at spatial infinity — it includes all forms of energy (matter, radiation, gravitational binding). The Bondi mass M_Bondi is measured at future null infinity (J⁺) and equals the ADM mass minus the total energy radiated away as gravitational waves up to that retarded time. As a binary system inspirals and emits gravitational waves, M_Bondi decreases while M_ADM remains constant (energy is conserved, just redistributed between the source and the radiation). The Bondi mass loss formula dM_Bondi/du ≤ 0 (u is retarded time) proves that gravitational waves carry positive energy.

The Bondi mass loss formula was historically important because it settled the debate about whether gravitational waves are physically real and carry energy. The positivity of the energy flux (Bondi, van der Burg, Metzner, 1962; Sachs, 1962) established that gravitational radiation is a genuine physical phenomenon, not a coordinate artifact.

Question 4

In what sense is the non-localizability of gravitational energy consistent with the conservation law ∇^μ T_μν = 0 for matter?

Model answer: ∇^μ T_μν = 0 is a local conservation law for matter energy-momentum only. In flat spacetime, this integrates to global conservation of total matter energy. In curved spacetime, the covariant divergence cannot be integrated to give global conservation because there is no way to compare vectors at different points. Energy can transfer between matter and the gravitational field: for example, a ball falling in a gravitational field gains kinetic energy (matter T_μν) at the expense of gravitational potential energy. The pseudo-tensor formalism captures this by writing ∂_μ((-g)(T^μν + t^μν)) = 0, where the ordinary (not covariant) divergence does integrate to global conservation — but only at the cost of using the coordinate-dependent pseudo-tensor t^μν for gravitational energy.

This framework is self-consistent: matter energy-momentum is locally conserved (covariant divergence), and total energy-momentum (matter + gravity) is globally conserved in an appropriate sense (ordinary divergence with pseudo-tensor). The price of global conservation is loss of coordinate-independence for the gravitational contribution.