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Energy-momentum pseudotensors of the gravitational field in the general theory of relativity Home Energy-momentum pseudotensors of the gravitational field in the general theory of relativity. Therefore, from expression 2. Using identity equivalent form 3. With the help of the Einstein--Hilbert equations in the form 3. The choice of the energy--momentum pseudotensor of the gravitational field depends to a large extent on the inclinations of authors and, as a rule, is realized on the basis of secondary properties. We shall study the properties of the "energy--momentum" quantities introduced in Einstein's theory for the examples of the determination of the "inertial mass" of a spherically syrmnetric source and the computation of the radiation "intensity" of small perturbations of the metric.
To be specific we shall carry out all computations on the basis of the symmetric pseudotensor of Landau--Lifshitz 3. Absurdity of the Definition of Inertial Mass in the General Theory of Relativity The equality of inertial and gravitational mass of the mass body Einstein considered to be an exact law of nature which must be reflected in his theory. At present it is commonly assumed to be proved that in the general theory of relativity the gravitational mass or, as it is sometimes called, the heavy mass of a system consisting of matter and the gravitational field is equal to its inertial mass.
Such an assertion is contained in the General relativity and gravitational waves. Read more. General Theory of Relativity. Relativity, The general theory. Einstein's General Theory of Relativity. Electrodynamics in the General Relativity Theory. The universe of general relativity.
The theory of relativity. Relativity: The Special and the General Theory. The special and the general theory. Relativity the special and the general theory. Relativity The Special and the General Theory. Hence, in the DGPP prescription, the gravitational term can be expressed as a tensor with respect to the coordinate transformations on the background space. The theory can be extended to include the matter contribution. If L m is the Lagrangian associated with the matter fields the corresponding energy-momentum tensor will be defined, in analogy with 21 as:.
Admitting the hypothesis of minimal coupling between the matter and the gravitational field, the free field equation 17 can be generalized to the form:. The symmetry properties of imply in the identity. Hence, in the DGPP formalism the invariance of the conservations laws with respect to coordinate transformations is compatible with the tensor nature of the conserved quantities, unlike the pseudotensorial formalism.
It will be shown in this section how the pseudotensors, defined originally in term of the Einsten metric g m n and its derivatives may be fully expressed in term of the elementary tensor fields h m n and defined in the DGPP formalism. Consider first the original expressions of the pseudotensors in term of the metric g m n :.
Defined as fields on the DGPP background space the pseudotensors, at each point, can be written in a local inertial coordinate system in which the connections are zero at a point and where the covariant derivatives are ordinary derivatives. To express the pseudotensors in an arbitrary coordinate system we can use the background connection see eq.watch
Energy–momentum pseudo–tensor for gravity
Hence, all the pseudotensors admit a purely tensorial representation in the DGPP formalism. We would like to remark that the procedure of replacing the ordinary derivative by the covariant one yields nontrivial results non null tensors only in the DGPP framework, where the background connection , instead of is used, so that in this case the covariant derivative of g m n is not zero. We must remember that the DGPP framework is just an alternative view of the same theory, i. The answer comes if we note that the coordinate transformation invariance of GR is translated to invariance under gauge transformations on h m n and in the DGPP formalism.
In the case of a finite transformation, the change in mn is given by. On the form 43 the transformation acts only in the field mn , letting invariant the background metric g m n x. In this sense, one can interpret the transformation 45 as a pure gauge transformation. Analogously the pure gauge transformation for the field x is.
It is possible to show that the dynamical equations for the gravitational field 17 are invariant under the transformations 45 and 48 supposing that the background space is Ricci-flat. On the form 44 , 46 , 47 and 49 however, the transformation is not a pure gauge transformation , since it acts on the field as well as on the background metric.
In fact, these are the usual transformations on tensorial fields resulting from a general mapping of the manifold on which they are defined. Hence, all tensors in the manifold, in particular the energy-momentum tensors defined in the last section, will transform in the usual homogeneous way:.
However, the situation is completely different for the case of transformations 45 and The tensors do not transform in the usual way 50 but contains extra inhomogeneous terms which brings the possibility of annulling them. The DGPP energy-momentum tensor, for example, transform in this case according to:. Hence, it is always possible to find gauge transformations 45 and 48 which makes the energy-momentum tensors defined previously to be null.
Using this coordinate system, the pseudotensors are calculed in Ref. Performing the coordinate transformation on the time variable. We can write the two asymptotically flat line element used above in spherical coordinates r , q , f. For the line element 52 we obtain. One observes that, as before, the metrics 57 and 58 are related by the coordinate transformation Let us now calculate how the energy-momentum tensors changes in the DGPP formalism. The metric of the background, which is flat spacetime, is given by.
On the localization of the gravitational energy
From equations 10 and 57 we obtain the following non-null components of the gravitational field:. It must be stressed that the same background flat metric 59 was used to determine the new potential, in agreement with the form 43 where the background metric is not changed. Using first the potential 60 and the background metric 59 one obtain for this prescription.
For the gravitational field 61 we get. We would like to emphasize that the change in L 00 from equation 65 to equation 66 is not due to the fact that we are dealing with a component of a tensor. This change can be calculated by using the transformations 46 and 47 yielding. In this last section we summarize the conclusions about the pseudotensorial and DGPP formalisms based on the results we have obtained in the previous sections.
First, from the relations obtained in section II, and from the definition 32 of the DGPP energy-momentum tensor one conclude that, although having exhibited a gravitational energy-momentum tensor, which is a necessary condition to express a local quantity of energy in a form independent of the coordinate system, the DGPP formalism does not solve, at last, the problem of the localization of the gravitational energy because the DGPP energy-momentum tensor is not invariant under the pure gauge transformation 45 , and it can be put to zero under a suitable choice of this kind of transformation.
In fact, in the DGPP formalism both the pseudotensors and the DGPP energy-momentum tensor can be defined from tensorial superpotentials U m n b , which are gauge dependent and differ between them by certain anti-symmetric tensorial term Y m n b so that the conservation laws are not violated. The expressions below show the tensorial superpotentials in the various prescriptions as well as the relations that can be established between them:.
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Hence, the fact that the DGPP energy-momentum is a real tensor is not important, since the pseudotensors can also be written as tensors in this formalism. What is crucial is the dependence of all these quantities under the pure gauge transformations 45 , Furthermore, according to the results of section IV, these gauge transformations are equivalent to the general coordinate transformations defined on the GR spacetime. Hence, one conclude that in spite of their intrinsic conceptuals differences, the pseudotensorial and DGPP formalisms have analogous properties and therefore constitute essentially equivalent forms to represent the conservation laws of gravitational systems.
This does not mean that the DGPP formalism is useless. Note that the manifold mapping group MMG is, at the same time, the covariant group and the symmetry group of GR . In this sense, the role of the DGPP formalism is to split these two distinct aspects of the MMG by defining a background metric over which the fields propagate.
The covariance role of the MMG manifests itself in the tensorial nature of fields and equations of motion in the DGPP formalism, which is trivial since any theory can be set covariant. The symmetry role of the MMG, which is one of the most important features of GR, appears in the DGPP formalism as the symmetry of the equations of motion under the pure gauge transformation One should also note that if the field h m n does not always appear together with g m n and also with in their self-coupling and coupling with matter, the theory looses its gauge symmetry, and there is no sense in nullifying the energy-momentum tensor by means of a gauge transformation: the gravitational energy turns out to be localizable.
The theory can not be formulated in geometrical terms, a preferred reference frame is present the one adapted to g m n , and the Equivalence Principle is not satisfied there is at least one type of particle which follows the geodesics of g m n , not of g m n as the others. This is the only way to have a notion of localizable gravitational energy in some alternative theory of gravity which necessarily will not satisfy the Covariance Principle in the sense that the MMG is no more the symmetry group of the theory  and the Equivalence Principle.
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For an example of a theory constructed on these lines see Ref. Hence, the DGPP formalism, although equivalent to GR, not only clarify some important aspects of it but also helps us to envisage alternative routes to describe the gravitational field. Anderson, Principles of Relativity physics , chap.
Komar, Phys. Grishchuk, A. Petrov and A. Popova, Commun. Brown and J. York Jr.