It is well known in General Relativity (GR) that a tensor describing the energy-momentum of the gravitational field does not exist; GR is not complete. Einstein realized (with Grossmann) in 1913 that gravity gravitates and there must be a tensor that describes that phenomenon. However, he could not produce that entity and introduced a pseudo-tensor instead in 1915. The futility and infinity of pseudo-tensors and other approaches to describe local gravitational energy led to the general statement first mentioned above. However, gravity still gravitates and we must be able to produce a tensor to describe that important fact. That was accomplished by extending a classic theorem of differential geometry in a Riemannian spacetime, the Berger-Ebin theorem, to a Lorentzian spacetime called the Orthogonal Decomposition Theorem (ODT) in the paper Modified General Relativity, and updated in the recent paper Modified General Relativity and quantum theory in curved spacetime published in Int J Mod Phys A. The ODT states that an arbitrary symmetric tensor in a Lorentzian spacetime w_ab can be decomposed into a linear sum of divergenceless tensors v_ab plus another tensor Phi_ab, which belongs to a subspace orthogonal to that of v_ab: w_ab=v_ab+Phi_ab. By returning to Einstein's original postulate of a total e-m tensor T_ab, which must be divergenceless and locally conserved, the matter e-m tensor \tildeT_ab is no longer divergenceless. A constant multiple of it can be set equal to an arbitrary symmetric tensor and decomposed by the ODT to give: k\tildeT_ab=v_ab+Phi_ab. Lovelock's theorem demands that in 4-D spacetime, the only divergenceless symmetric tensors consisting of a concomitant of the metric and its first two derivatives are the metric and the Einstein tensor. Thus, we arrive at Einstein's equation k\tildeT_ab=Lambda g_ab+G_ab+Phi_ab with a new tensor that describes the energy-momentum of the gravitational field. Why can I say that? Phi_ab is constructed from the Lie derivative of both the metric and a product of unit line element covectors. Lie derivatives have the unique property that a tensor constructed from them has the same value when the Lie derivative is expressed with covariant or partial derivatives. Thus, when the connection coefficients (Gamma's) vanish under free-fall, Phi_ab is invariant. Gravitational energy can be localized. It is easy to prove that Phi_ab vanishes if and only if X, the line element vector along which the Lie derivative is calculated, is a Killing vector. Of course, in general there are no Killing vectors unless a symmetry is involved. Line element vectors are virtually never used in the literature except for a few theorems on the evolution of time. It is imperative to understand that a Lorentzian spacetime does not exist without a line element field (X,-X): a non-compact paracompact manifold admits a Lorentzian metric g_ab if and only if it admits a line element field (Hawking and Ellis 1973).

It is well known in General Relativity (GR) that a tensor describing the energy-momentum of the gravitational field does not exist; GR is not complete. Einstein realized (with Grossmann) in 1913 that gravity gravitates and there must be a tensor that describes that phenomenon. However, he could not produce that entity and introduced a pseudo-tensor instead in 1915. The futility and infinity of pseudo-tensors and other approaches to describe local gravitational energy led to the general statement first mentioned above. However, gravity still gravitates and we must be able to produce a tensor to describe that important fact. That was accomplished by extending a classic theorem of differential geometry in a Riemannian spacetime, the Berger-Ebin theorem, to a Lorentzian spacetime called the Orthogonal Decomposition Theorem (ODT) in the paper Modified General Relativity, and updated in the recent paper Modified General Relativity and quantum theory in curved spacetime published in Int J Mod Phys A. The ODT states that an arbitrary symmetric tensor in a Lorentzian spacetime $w_{ab}$ can be decomposed into a linear sum of divergenceless tensors $v_{ab}$ plus another tensor $\Phi_{ab}$, which belongs to a subspace orthogonal to that of $v_{ab}$: $w_{ab}=v_{ab}+\Phi_{ab}$. By returning to Einstein's original postulate of a total e-m tensor $T_{ab}$, which must be divergenceless and locally conserved, the matter e-m tensor $\tilde{T}_{ab}$ is no longer divergenceless. A constant multiple of it can be set equal to an arbitrary symmetric tensor and decomposed by the ODT to give: $k\tilde{T}_{ab}=v_{ab}+\Phi_{ab}$. Lovelock's theorem demands that in 4-D spacetime, the only divergenceless symmetric tensors consisting of a concomitant of the metric and its first two derivatives are the metric and the Einstein tensor. Thus, we arrive at Einstein's equation $k\tilde{T}_{ab}=\Lambda g_{ab}+G_{ab}+\Phi_{ab}$ with a new tensor that describes the energy-momentum of the gravitational field. Why can I say that? $\Phi_{ab}$ is constructed from the Lie derivative of both the metric and a product of unit line element covectors. Lie derivatives have the unique property that a tensor constructed from them has the same value when the Lie derivative is expressed with covariant or partial derivatives. Thus, when the connection coefficients (Gamma's) vanish under free-fall, $\Phi_{ab}$ is invariant. Gravitational energy can be localized. It is easy to prove that $\Phi_{ab}$ vanishes if and only if X, the line element vector along which the Lie derivative is calculated, is a Killing vector. Of course, in general there are no Killing vectors unless a symmetry is involved. Line element vectors are virtually never used in the literature except for a few theorems on the evolution of time. It is imperative to understand that a Lorentzian spacetime does not exist without a line element field (X,-X): a non-compact paracompact manifold admits a Lorentzian metric $g_{ab}$ if and only if it admits a line element field (Hawking and Ellis 1973).