| Summary: | An unambiguous definition of gravitational energy remains one of the unresolved issues of physics today. This problem is related to the non-localization of gravitational energy density. In General Relativity, there have been many proposals for defining the gravitational energy density, notably those proposed by Einstein, Tolman, Landau and Lifshitz, Papapetrou, Møller, and Weinberg. In this review, we firstly explored the energy–momentum complex in an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>n</mi><mrow><mi>t</mi><mi>h</mi></mrow></msup></semantics></math></inline-formula> order gravitational Lagrangian <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mo>=</mo><mi>L</mi><mfenced separators="" open="(" close=")"><msub><mi>g</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msub><mo>,</mo><msub><mi>g</mi><mrow><mi>μ</mi><mi>ν</mi><mo>,</mo><msub><mi>i</mi><mn>1</mn></msub></mrow></msub><mo>,</mo><msub><mi>g</mi><mrow><mi>μ</mi><mi>ν</mi><mo>,</mo><msub><mi>i</mi><mn>1</mn></msub><msub><mi>i</mi><mn>2</mn></msub></mrow></msub><mo>,</mo><msub><mi>g</mi><mrow><mi>μ</mi><mi>ν</mi><mo>,</mo><msub><mi>i</mi><mn>1</mn></msub><msub><mi>i</mi><mn>2</mn></msub><msub><mi>i</mi><mn>3</mn></msub></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mi>g</mi><mrow><mi>μ</mi><mi>ν</mi><mo>,</mo><msub><mi>i</mi><mn>1</mn></msub><msub><mi>i</mi><mn>2</mn></msub><msub><mi>i</mi><mn>3</mn></msub><mo>⋯</mo><msub><mi>i</mi><mi>n</mi></msub></mrow></msub></mfenced></mrow></semantics></math></inline-formula> and then in a gravitational Lagrangian as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi><mi>g</mi></msub><mo>=</mo><mrow><mo>(</mo><mover><mi>R</mi><mo>¯</mo></mover><mo>+</mo><msub><mi>a</mi><mn>0</mn></msub><msup><mi>R</mi><mn>2</mn></msup><mo>+</mo><msubsup><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>p</mi></msubsup><msub><mi>a</mi><mi>k</mi></msub><mi>R</mi><msup><mo>□</mo><mi>k</mi></msup><mi>R</mi><mo>)</mo></mrow><msqrt><mrow><mo>−</mo><mi>g</mi></mrow></msqrt></mrow></semantics></math></inline-formula>. Its gravitational part was obtained by invariance of gravitational action under infinitesimal rigid translations using Noether’s theorem. We also showed that this tensor, in general, is not a covariant object but only an affine object, that is, a pseudo-tensor. Therefore, the pseudo-tensor <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>τ</mi><mi>α</mi><mi>η</mi></msubsup></semantics></math></inline-formula> becomes the one introduced by Einstein if we limit ourselves to General Relativity and its extended corrections have been explicitly indicated. The same method was used to derive the energy–momentum complex in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mfenced open="(" close=")"><mi>R</mi></mfenced></mrow></semantics></math></inline-formula> gravity both in Palatini and metric approaches. Moreover, in the weak field approximation the pseudo-tensor <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>τ</mi><mi>α</mi><mi>η</mi></msubsup></semantics></math></inline-formula> to lowest order in the metric perturbation <i>h</i> was calculated. As a practical application, the power per unit solid angle <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ω</mo></semantics></math></inline-formula> emitted by a localized source carried by a gravitational wave in a direction <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>x</mi><mo stretchy="false">^</mo></mover></semantics></math></inline-formula> for a fixed wave number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">k</mi></semantics></math></inline-formula> under a suitable gauge was obtained, through the average value of the pseudo-tensor over a suitable spacetime domain and the local conservation of the pseudo-tensor. As a cosmological application, in a flat Friedmann–Lemaître–Robertson–Walker spacetime, the gravitational and matter energy density in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula> gravity both in Palatini and metric formalism was proposed. The gravitational energy–momentum pseudo-tensor could be a useful tool to investigate further modes of gravitational radiation beyond two standard modes required by General Relativity and to deal with non-local theories of gravity involving <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mo>□</mo><mrow><mo>−</mo><mi>k</mi></mrow></msup></semantics></math></inline-formula> terms.
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