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1 Introduction

In the framework of general relativity (GR), as a consequence of the Einstein Equivalence Principle (EEP), the gravitational field has non-tensorial strengths \(\varGamma ^i_{kl} = \{^i_{kl}\}\) and admits no energy-momentum tensor. One can only attribute to this field gravitational energy-momentum pseudotensors. The leading object of such a kind is the canonical gravitational energy-momentum pseudotensor \(_E t_i^{~k}\) proposed already in past by Einstein. This pseudotensor is a part of the canonical energy-momentum complex \(_E K_i^{~k}\) in GR.

The canonical complex \(_E K_i^{~k}\) can easily be obtained by rewriting Einstein equations to the superpotential form

$$\begin{aligned} _E K_i^{~k} := \sqrt{\vert g\vert }\bigl ( T_i^{~k} + _E t_i^{~k}\bigr ) = {}_F\! U_i^{~[kl]}{}_{,l}\,, \end{aligned}$$
(1)

where \(T^{ik} = T^{ki}\) is the symmetric energy-momentum tensor for matter, \(g = det[g_{ik}]\), and

$$\begin{aligned} _E t_i^{~k} =&\,{c^4\over 16\pi G} \bigl \{\delta _i^k g^{ms}\bigl (\varGamma ^l_{mr}\varGamma ^r_{sl} - \varGamma ^r_{ms}\varGamma ^l_{rl}\bigr )\\&+ g^{ms}_{~~,i}\bigl [\varGamma ^k_{ms} - {1\over 2} \bigl (\varGamma ^k_{tp}g^{tp} - \varGamma ^l_{tl}g^{kt}\bigr )g_{ms}\\&- {1\over 2}\bigl (\delta ^k_s \varGamma ^l_{ml} + \delta ^k_m \varGamma ^l_{sl}\bigr )\bigr ]\bigr \}; \end{aligned}$$
$$\begin{aligned} _F {U_i^{~[kl]}} = {c^4\over 16\pi G}{g_{ia}\over \sqrt{\vert g\vert }}\bigl [\bigl (-g\bigr )\bigl (g^{ka} g^{lb} - g^{la} g^{kb}\bigr )\bigr ]_{,b}. \end{aligned}$$
(2)

\(_E t_i^{~k}\) are components of the canonical energy-momentum pseudotensor for gravitational field, and \(_F {U_i^{~[kl]}}\) is von Freud superpotential.

$$\begin{aligned} _E K_i^{~k} = \sqrt{\vert g\vert }\bigl (T_i^{~k} + _E t_i^{~k}\bigr ) \end{aligned}$$
(3)

are components of the Einstein canonical energy-momentum complex for matter and gravity.

In consequence of (1) the complex \(_E K_i^{~k}\)satisfies local conservation laws

$$\begin{aligned} {_E K_i^{~k}}_{,k}\equiv 0. \end{aligned}$$
(4)

In very special cases one can obtain reasonable integral conservation laws from these local conservation laws. Additionally, one can also introduce the canonical superenergy tensors. This was done in past in a series of our articles (see, e.g., [1–11] and references therein).

It appears that the idea of the superenergy tensors is universal: to any physical field having an energy-momentum tensor or pseudotensor one can attribute a corresponding superenergy tensor.

2 The Canonical Superenergy Tensors

Here we give a short description of the general, constructive definition of the superenergy tensor \(S_a^{~b}\) applicable to gravitational field and to any matter field. The definition uses locally Minkowskian structure of the spacetime and, therefore, it fails in a spacetime with torsion, e.g., in a Riemann-Cartan spacetime.

In the normal Riemann coordinates NRC(P) we define (pointwise)

$$\begin{aligned} S_{(a)}^{(b)}(P) = S_a^{~b} :=(-) \displaystyle \lim _{\varOmega \rightarrow P}{\int \limits _{\varOmega }\biggl [T_{(a)}^{(b)}(y) - T_{(a)}^{ (b)} (P)\biggr ]d\varOmega \over 1/2\int \limits _{\varOmega }\sigma (P;y) d\varOmega }, \end{aligned}$$
(5)

where

$$\begin{aligned} T_{(a)}^{(b)}(y) :=&T_i^{~k}(y)e^i_{~(a)}(y) e_k^{(b)}(y),\\ T_{(a)}^{(b)}(P):=&T_i^{~k}(P) e^i_{~(a)}(P)e_k^{~(b)}(P) = T_a^{~b}(P) \end{aligned}$$

are physical or tetrad components of the pseudotensor or tensor field which describes an energy-momentum distribution, and \(\bigl \{y^i\bigr \}\) are normal coordinates. \(e^i_{~(a)}(y),\) \(e_k^{~(b)} (y)\) denote an orthonormal tetrad \(e^i_{~(a)}(P) = \delta _a^i\) and its dual \(e_k^{~(a)}(P) = \delta _k^a \), parallelly propagated along geodesics through \(P\) (\(P\) is the origin of the NRC(P)).

We have

$$\begin{aligned} e^i_{~(a)}(y) e_i^{~(b)}(y) = \delta _a^b. \end{aligned}$$
(6)

For a sufficiently small 4-dimensional domain \(\varOmega \) which surrounds P we require

$$\begin{aligned} \int \limits _{\varOmega }{y^i d\varOmega } = 0, ~~\int \limits _{\varOmega }{y^i y^k d\varOmega } = \delta ^{ik} M, \end{aligned}$$
(7)

where

$$\begin{aligned} M = \int \limits _{\varOmega }{(y^0)^2 d\varOmega } = \int \limits _{\varOmega }{(y^1)^2 d\varOmega } = \int \limits _{\varOmega }{(y^2)^2 d\varOmega }=\int \limits _{\varOmega }{(y^3)^2 d\varOmega } \end{aligned}$$
(8)

is a common value of the moments of inertia of the domain \(\varOmega \) with respect to the subspaces \(y^i = 0,~~(i= 0,1,2,3)\).

As \(\varOmega \) we can take, e.g., a sufficiently small ball centered at \(P\):

$$\begin{aligned} (y^0)^2 + (y^1)^2 + (y^2)^2 + (y^3)^2 \le R^2, \end{aligned}$$
(9)

which for an auxiliary positive-definite metric

$$\begin{aligned} h^{ik} := 2 v^i v^k - g^{ik}, \end{aligned}$$
(10)

can be written in the form

$$\begin{aligned} h_{ik}y^i y^k \le R^2. \end{aligned}$$
(11)

A fiducial observer O is at rest at the origin P of the Riemann normal coordinates NRC(P) and its four- velocity is \(v^i =*~ \delta ^i_o\), where \(*\) means that equation is valid only in special coordinates. In [3] \(\sigma (P;y)\) denotes the two-point world function introduced by J.L. Synge [12]:

$$\begin{aligned} \sigma (P;y) =*{1\over 2}\bigl (y^{o^2} - y^{1^2} - y^{2^2} -y^{3^2}\bigr ). \end{aligned}$$
(12)

The world function \(\sigma (P;y)\) can be defined covariantly by the eikonal-like expression [12]

$$\begin{aligned} g^{ik} \sigma _{,i} \sigma _{,k} = 2\sigma ,\quad \quad \sigma _{,i} := \partial _i\sigma , \end{aligned}$$
(13)

together with requirements

$$\begin{aligned} \sigma (P;P) = 0,\quad \quad \partial _i\sigma (P;P) = 0. \end{aligned}$$
(14)

The ball \(\varOmega \) can also be given by the inequality

$$\begin{aligned} h^{ik}\sigma _{,i} \sigma _{,k} \le R^2. \end{aligned}$$
(15)

Tetrad components and normal components are equal at P, so, we will write the components of any quantity attached to P without tetrad brackets, e.g., we will write \(S_a^{~b}(P)\) instead of \(S_{(a)}^{(b)}(P)\) and so on.

If \(T_i^{~k}(y)\) are the components of an energy-momentum tensor of matter, then we get from (5)

$$\begin{aligned} _m S_a^{~b}(P;v^l) = \bigl (2{\hat{v}}^l {\hat{v}}^m - {\hat{g}}^{lm}\bigr ) \nabla _l \nabla _m {} {\hat{T}}_a^{~b} = {\hat{h}}^{lm}\nabla _l \nabla _m {}{\hat{T}}_a^{~b}. \end{aligned}$$
(16)

Hat over a quantity denotes its value at P, and \(\nabla \) means covariant derivative.

Tensor \(_m S_a^{~b}(P;v^l)\) is called the canonical superenergy tensor for matter.

For the gravitational field, substitution of the canonical Einstein energy-momentum pseudotensor as \(T_i^{~k}\) in (5) gives

$$\begin{aligned} _g S_a^{~b}(P;v^l) = {\hat{h}}^{lm} {\hat{W}}_a^{~b}{}_{lm}, \end{aligned}$$
(17)

where

$$\begin{aligned} {W_a^{~b}}{}_{lm}=&\, {2\alpha \over 9}\bigl [B^b_{~alm} + P^b_{~alm}\\&- {1\over 2}\delta ^b_a R^{ijk}_{~~~m}\bigl (R_{ijkl} + R_{ikjl}\bigr ) + 2\delta _a^b{\beta }^2 E_{(l\vert g}{}E^g_{~\vert m)}\\&- 3 {\beta }^2 E_{a(l\vert }{}E^b_{~\vert m)} + 2\beta R^b_{~(a\vert g\vert l)}{}E^g_{~m}\bigr ]. \end{aligned}$$

Here \(\alpha = {c^4\over 16\pi G} = {1\over 2\beta }\), and

$$\begin{aligned} E_i^{~k} := T_i^{~k} - {1\over 2}\delta _i^k T \end{aligned}$$
(18)

is the modified energy-momentum tensor of matter.Footnote 1

On the other hand

$$\begin{aligned} B^b_{~alm} := 2R^{bik}_{~~~(l\vert }{}R_{aik\vert m)}-{1\over 2}\delta _a^b{} R^{ijk}_{~~~l}{}R_{ijkm} \end{aligned}$$
(19)

are the components of the Bel-Robinson tensor (BRT), while

$$\begin{aligned} P^b_{~alm}:= 2R^{bik}_{~~~(l\vert }{}R_{aki\vert m)}-{1\over 2} \delta _a^b{}R^{jik}_{~~~l}{}R_{jkim} \end{aligned}$$
(20)

is the Bel-Robinson tensor with “transposed” indices \((ik)\).

In vacuum \(_g S_a^{~b}(P;v^l)\) takes the simpler form

$$\begin{aligned} _g S_a^{~b}(P;v^l) = {8\alpha \over 9} {\hat{h}}^{lm}\bigl ({\hat{C}}^{bik}_{~~~(l\vert }{}{\hat{C}}_{aik\vert m)} -{1\over 2}\delta _a^b {\hat{C}}^{i(kp)}_{~~~~~(l\vert }{}{\hat{C}}_{ikp\vert m)}\bigr ). \end{aligned}$$
(21)

Here \(C^a_{~blm}\) denotes components of the Weyl tensor.

Some remarks:

  1. 1.

    In vacuum the quadratic form \(_g S_a^{~b} v^a v_b\), where \(v^a v_a = 1\), is positive-definite. This form gives the gravitational superenergy density \(\varepsilon _g\) for a fiducial observer O.

  2. 2.

    In general, the canonical superenergy tensors are uniquely determined only along the world line of an observer O. But in special cases, e.g., in Schwarzschild spacetime or in Friedmann universes, when there exists a physically and geometrically distinguished four-velocity field \(v^i(x)\), one can introduce, in a unique way, unambiguous fields \(_g S_i^{~k}(x;v^l)\) and \(_m S_i^{~k}(x;v^l)\).

  3. 3.

    It can be shown that the superenergy densities \(\varepsilon _g,\, \varepsilon _m\), which have dimension \({Joule \over (meter)^{5}}\), exactly correspond to the Appel’s energy of acceleration \({1\over 2}{\varvec{a}}{\varvec{a}}\). The Appel’s energy of acceleration plays the fundamental role in Appel’s approach to classical mechanics [13–15].

  4. 4.

    We have proposed, in our previous papers, to use the tensor \(_g S_i^{~k}(P;v^l)\) as gravitational energy-momentum tensor.

  5. 5.

    We have used the canonical superenergy tensors \(_g S_i^{~k}\) and \(_m S_i^{~k}\) to local (and also to global) analysis of some well-known solutions to the Einstein equations like Schwarzschild, Kerr, Friedmann, Gödel, Kasner, Bianchi I, de Sitter and anti-de Sitter solutions. The obtained results were interesting (see [1–5, 7, 8, 11]), e.g., in Gödel universes the sign of the superenergy density \(\varepsilon _s := \varepsilon _g + \varepsilon _m\) depends on causality \((\varepsilon _s <0)\) and non-causality \((\varepsilon _s >0)\), and, in Schwarzschild spacetime the integral exterior superenergy \(S\) is connected with Hawking temperature \(T\) of the Schwarzschild black hole: \(S = {8\pi k c^3\over 9\hbar G} T\). We have also studied the transformation rules for the canonical superenergy tensors under conformal rescaling of the metric \(g_{ik}(x)\) [8, 16].

  6. 6.

    The idea of the superenergy tensors can be extended to angular momentum (see, [3, 10]). The angular supermomentum tensors do not depend on a radius vector and, in gravitational case, they depend only on “spinorial part” of the suitable gravitational angular momentum pseudotensor.

  7. 7.

    As a result of averaging, the tensors \(_g S_a^{~b}(P;v^l)\) and \(_m S_a^{~b}(P;v^l)\), in general, do not satisfy any local conservation laws. Only in a symmetric spacetime or in a spacetime which has constant curvature one can get

    $$\begin{aligned} \bigl [_g S_a^{~b}(P;v^l)\bigr ]_{,b} = 0. \end{aligned}$$
    (22)
  8. 8.

    There exists an exchange of the canonical superenergy between gravity and matter in the following sense. Let us consider the consequence of (4)

    $$\begin{aligned} \bigl (\varDelta ^{(4)}_E K_i^{~k}\bigr )_{,k} =\bigl [ \bigl ( \varDelta ^{(4)} (\sqrt{\vert g\vert } _E t_i^{~k}\bigr )+ \varDelta ^{(4)}\bigl (\sqrt{\vert g\vert } T_i^{~k}\bigr )\bigr ]_{,k} = 0, \end{aligned}$$
    (23)

    where \(\varDelta ^{(4)} := (\partial _0)^2 + (\partial _1)^2 + (\partial _2)^2 +(\partial _3)^2\). The quantities (with total balance equal to zero)

    $$\begin{aligned} \varDelta ^{(4)}\bigl (\sqrt{\vert g\vert }_e t_i^{~k}\bigr ), ~~\varDelta ^{(4)}\bigl (\sqrt{\vert g\vert } T_i^{~k}\bigr ) \end{aligned}$$
    (24)

    have dimensions of the canonical superenergy and, when taken at the origin P of the NRC(P) and written covariantly, then they coincide with the canonical superenergy tensors \( _g S_i^{~k}(P;v^l), ~~_m S_i^{~k}(P;v^l)\) respectively.

  9. 9.

    Recently we have noticed that the total superenergy density is positive-definite or null for known stable solutions to the Einstein equations and negative-definite for unstable solutions. The physical meaning of this fact is under study.

Changing the constructive definition (5) to the form

$$\begin{aligned} <T_a^{~b}(P)>:= \displaystyle \lim _{\varepsilon \rightarrow 0}{\int \limits _{\varOmega }{\bigl [T_{(a)}^{~~~(b)}(y) - T_{(a)}^{~~~(b)}(P)\bigr ]d\varOmega }\over \varepsilon ^2/2\int \limits _{\varOmega }d\varOmega }, \end{aligned}$$
(25)

where \(\varepsilon := {R\over L}>0\) (equivalently \(R = \varepsilon L\)) is a real parameter and \(L\) is a dimensional constant:\([L] = m\), one obtains the averaged relative energy-momentum tensors. Namely, from (25) one obtains:

$$\begin{aligned} <_m\! T_a^{~b}(P;v^l)> = _m S_a^{~b}(P;v^l) {L^2\over 6} \end{aligned}$$
(26)

for matter and

$$\begin{aligned} <_g\! t_a^{~b}(P;v^l)> = _g S_a^{~b}(P;v^l){L^2\over 6} \end{aligned}$$
(27)

for gravity.

The components of the averaged relative energy-momentum tensors have correct dimensions but they depend on a dimensional parameter \(L\). How to choose the dimensional parameter \(L\)?

In [7] we have proposed a universal choice of the parameter \(L\). Namely, we have proposed \(L = 100 L_P \approx 10^{-33}\) m. Here \(L_P := \sqrt{{\hbar G\over c^3}}\approx 10^{-35}\) m is the Planck length.

Such choice of \(L\) gives the averaged gravitational relative energy-momentum tensors components which are negligible in comparison with the components of the energy-momentum tensor for matter. Consequently, with such choice of the parameter \(L\), these tensors play no role in evolution of the material objects and in evolution of the Universe. On the other hand, the choices can be made such that \(L ={2GM\over c^2}\) for a closed system with mass \(M\), \(L = \lambda \) for a gravitational wave of wave length \(\lambda \), and \(L = {2GM_U\over c^2} = {c\over H_0} = c t_0\) in cosmology, which lead us to the averaged relative energy densities of the same order as ordinary energy density of matter. Here \(M_U,\, H_0, \, t_0\) mean mass of the observed part of the Universe, actual value of the Hubble constant and an approximate age of the Universe respectively.

Of course, there exist other possibilities of choosing the length parameter \(L\).

3 Conclusion

On the superenergy level, or on the averaged relative energy-momentum level, there seems to be no problem with a suitable expression for energy. However, canonical superenergy tensors seem more fundamental than the corresponding averaged relative energy-momentum tensors because they do not depend on the choice of \(L\).