Massachusetts Institute of TechnologyDepartment of PhysicsPhysics 8.962 Spring 2000Stress-Energy Pseudotensors andGravitational Radiation Powerc2000 Edmund Bertschinger. All rights reserved.1 IntroductionIn a curved spacetime there is, in general, no globally conserved energy-momentum.Aside from the case of scalars like electric charge, t ensors defin ed in different tangentspaces cannot be added in nonflat spacetime.However, if we are willing to drop the requirement that all our equations be tensorequations, then it is possible to define a globally conserved energy-momentum. Unlike atensor equation, the form of the conservation laws we derive will change depending onthe coordinate system. However, that makes t hem no less valid. Consider, for example,the three-dimensional equation ∇ · B = 0 and its integral form,HdS · B = 0. Whenwritten in Cartesian coordinates these equations have an entirely different form thanwhen they are written in spherical coordinates. However, they are equally correct ineither case.The approach we will follow is to derive a conserved pseudotensor, a two-ind exobject that transforms differently than the components of a tensor. Unlike a tensor, apseudotensor can vanish at a point in one coordinate system but not in others. Theconnection coefficients are a good example of this, and the stress-energy pseudotensorswe construct will depend explicitly on the connection coefficients in a coordinate basis.Despite this apparent defect, pseudotensors can be quite useful. In fact, they arethe only way to defin e an integral energy-momentum obeying an exact conservation law.Moreover, in an asymptotically flat spacetime, when tensors can be added from differenttangent spaces, the integral energy-momentum behaves like a four-vector. Thus, we canuse a pseudotensor to derive the power radiated by a localized source of gravitationalradiation.12 Canonical Stress-Energy PseudotensorThe stress-energy tensor is not unique. Given any Tµνsuch that ∇µTµν= 0, one mayalways define other conserved stress-energy tensors by adding the divergence of anotherobject:Tµν→ Tµν+ ∇λSµνλwhere Sµνλ= −Sµλν. (1)Clearly the stress-energy tensor need not even be symmetric.Recall the equation of local stress-energy conservation:∇µTµν= (−g)−1/2∂µ√−gTµν− ΓλνµTµλ= 0 . (2)Because of the Γ terms, Gauss’ theorem does not apply and the integral over a volumedoes not give a conserved 4-vector.However, as we will see, it is possible to define a pseudotensor τµνwhose conservationlaw is ∂µ(√−gτµν) = 0 instead of ∇µτµν= 0. The two equations are identical in flatspacetime but the first one can be integrated by Gauss’ law while the second one cannot.Moreover, there are many different conserved stress-energy pseudotensors, just as thereare many different conserved stress-energy tensors.This section will show how to construct conserved stress-energy pseudotensors andtensors, illustrating the procedure for scalar fields and for the metric. The key resultsare given in problem 2 of Problem Set 7.2.1 Stress-energy pseudotensor for a scalar fieldWe begin with a simple example: a classical scalar field φ(x) with actionS[φ(x)] =ZL(φ, ∂µφ) d4x . (3)The Lagrangian density depends on φ and its derivatives but is othewise ind ependentof the position. In this example we suppose that L includes no derivatives higher thanfirst-order, but this can be easily generalized. Note that if S is a scalar, then L mustequal a scalar times the factor√−g which is needed to convert coordinate volume toproper volume. We are not assuming flat spacetime — the tr eatment here is valid incurved spacetime.Variation of the action using δ(∂µφ) = ∂µ(δφ) yieldsδS =Z"∂L∂φδφ +∂L∂(∂µφ)∂µ(δφ)#d4x=Z(∂L∂φ− ∂µ"∂L∂(∂µφ)#)δφ(x) d4x +Isurf∂L∂(∂µφ)δφ(x) d3Σµ. (4)2The surface term comes from integration by parts and is the counterpart of the pδqendpoint contributions in the variation of the action of a particle. Considering arbitraryfield variations δφ(x) that vanish on the boundary, the action principle δS = 0 gives theEuler-Lagrange equation∂µ"∂L∂(∂µφ)#−∂L∂φ= 0 . (5)Now, by assumption our Lagrangian density does not depend explicitly on the co-ordinates: ∂L/∂xµ= 0. This implies the existence of a conserved Hamiltonian den-sity. To see how, recall the case of particle moving in one dimension with trajectoryq(t). In this case, time-indepence of the Lagrangian L(q, ˙q) implies dH/dt = 0 whereH = ˙q(∂L/∂ ˙q) − L.In a field theory q(t) b ecomes φ(x) and there are d = 4 (for four spacetime dimensions)parameters for the field trajectories instead of just one. Therefore, instead of dH/dt = 0,the conservation law will read ∂µHµ= 0. However, given d parameters, there are dconservation laws not one, so there must be a two-index Hamiltonian density Hµνsuchthat ∂µHµν= 0. Here, ν labels the various conserved quantities.To construct the Hamiltonian function one must first evaluate the canonical momen-tum. For a single particle, p = ∂L/∂ ˙q. For a scalar field theory, the field momentum isdefined similarly:πµ=∂L∂(∂µφ). (6)In a simple mechanical system, the Hamiltonian is H = p ˙q − L. For a field theory theLagrangian is replaced by the Lagrangian density, the coordinate q is replaced by thefield, and the momentum is the canonical momentum as in equation (6).The canonical stress-energy pseudotensor is defined as the Hamiltonian densitydivided by√−g:τµν≡ (−g)−1/2[(∂νφ)πµ− δµνL] . (7)The reader may easily check that, as a consequence of equation (5) and the chain rule∂µf(φ, ∂νφ) = (∂f/∂φ)∂µφ+ [∂f/∂(∂νφ)]∂µ(∂νφ), the canonical stress-energy p seudoten-sor obeys(−g)−1/2∂µ√−gτµν= 0 . (8)2.2 Stress-energy pseudotensor for the metricThe results given above are easily generalized to an action that depends on a rank (0, 2)tensor field gαβinstead of a scalar field. Let us suppose that the Lagrangian densitydepends only on the field and its first derivatives: L = L(gαβ, ∂µgαβ). (This excludes theEinstein-Hilbert action, which depends also on the second derivatives of the metric. Wewill return to this point in the next section.) The Euler-Lagrange equations are simplyequation (5) with φ replaced by gαβ.3Because our field has two indices, the canonical momenta have two