Abstract Generated abstract
The paper examines how conservation laws in Einstein’s theory of gravitation are connected with the transformation properties of the gravitational-field Lagrangian. Starting from the Lagrangian form of the Einstein equations, it computes the variation of the Lagrangian density under an infinitesimal change of coordinates in two ways: first using the field equations and Einstein’s energy momentum pseudotensor, and then using the known transformation law of the Lagrangian. Comparison of the two expressions yields a general divergence identity involving arbitrary functions of the coordinates. Particular choices of these functions reproduce Einstein’s conservation laws and related conservation laws discussed in earlier formulations.
Full Text
UDC 530.12:531.51
PHYSICS
I. G. FICHTENGOLTS
ON CONSERVATION LAWS IN EINSTEIN’S THEORY OF GRAVITATION
(Presented by Academician V. A. Fock on 11 XII 1965)
The paper considers the connection between conservation laws in Einstein’s theory of gravitation and the properties of the Lagrangian function of the gravitational field.
We shall start from the Einstein gravitational equations in Lagrangian form:
\[ \frac{\partial \mathcal L}{\partial g_{\mu\nu}} - \frac{\partial}{\partial x_\alpha} \frac{\partial \mathcal L}{\partial(\partial g_{\mu\nu}/\partial x_\alpha)} = -\varkappa \sqrt{-g}\,T^{\mu\nu}, \tag{1} \]
where
\[ \mathcal L = \sqrt{-g}\,g^{\mu\nu} \bigl(\Gamma_{\mu\alpha}^{\beta}\Gamma_{\nu\beta}^{\alpha} - \Gamma_{\mu\nu}^{\alpha}\Gamma_{\alpha\beta}^{\beta}\bigr) \tag{2} \]
is the Lagrangian function of the gravitational field; \(g_{\mu\nu}\) is the fundamental tensor; \(g\) is the determinant composed of the components of the fundamental tensor \(g_{\mu\nu}\); \(\Gamma_{\mu\nu}^{\alpha}\) are Christoffel symbols of the second kind; \(\varkappa\) is Einstein’s gravitational constant, and \(T^{\mu\nu}\) is the mass tensor. Greek indices take the values \(0,1,2,3\). Summation over identical Greek indices from 0 to 3 is assumed.
For the purpose of deriving conservation laws, following works \((^{1-4})\), let us pass from the variables \(x_0, x_1, x_2, x_3\) to new variables \(x'_0, x'_1, x'_2, x'_3\), putting
\[ x'_\beta = x_\beta(x_0,x_1,x_2,x_3,\varepsilon), \tag{3} \]
where \(\varepsilon\) is a certain parameter, and
\[ (x'_\beta)_{\varepsilon=0}=x_\beta. \tag{4} \]
Taking into account that the Lagrangian function (2) depends only on \(g_{\mu\nu}\) and their first derivatives with respect to the variables \(x_\alpha\), and noting that
\[ \left( \frac{\partial}{\partial \varepsilon} \frac{\partial g'_{\mu\nu}}{\partial x'_\alpha} \right)_{\varepsilon=0} = \frac{\partial}{\partial x_\alpha} \left( \frac{\partial g'_{\mu\nu}}{\partial \varepsilon} \right)_{\varepsilon=0} - \frac{\partial g_{\mu\nu}}{\partial x_\beta} \frac{\partial}{\partial x_\alpha} \left( \frac{\partial x'_\beta}{\partial \varepsilon} \right)_{\varepsilon=0}, \tag{5} \]
we obtain
\[ \begin{aligned} \left(\frac{\partial \mathcal L^*}{\partial \varepsilon}\right)_{\varepsilon=0} ={}& \mathcal L\,\frac{\partial}{\partial x_\beta} \left(\frac{\partial x'_\beta}{\partial \varepsilon}\right)_{\varepsilon=0} + \frac{\partial \mathcal L}{\partial g_{\mu\nu}} \left(\frac{\partial g'_{\mu\nu}}{\partial \varepsilon}\right)_{\varepsilon=0} + \frac{\partial \mathcal L}{\partial(\partial g_{\mu\nu}/\partial x_\alpha)} \frac{\partial}{\partial x_\alpha} \left(\frac{\partial g'_{\mu\nu}}{\partial \varepsilon}\right)_{\varepsilon=0} \\ &- \frac{\partial g_{\mu\nu}}{\partial x_\beta} \frac{\partial \mathcal L}{\partial(\partial g_{\mu\nu}/\partial x_\alpha)} \frac{\partial}{\partial x_\alpha} \left(\frac{\partial x'_\beta}{\partial \varepsilon}\right)_{\varepsilon=0}. \end{aligned} \tag{6} \]
Here
\[ \mathcal L^* = \mathcal L' \left| \frac{D(x'_0,x'_1,x'_2,x'_3)} {D(x_0,x_1,x_2,x_3)} \right|, \tag{7} \]
\[ \mathcal L' = \sqrt{-g'}\,g'^{\mu\nu} \bigl(\Gamma_{\mu\alpha}^{\prime\beta}\Gamma_{\nu\beta}^{\prime\alpha} - \Gamma_{\mu\nu}^{\prime\alpha}\Gamma_{\alpha\beta}^{\prime\beta}\bigr). \tag{2′} \]
If we now make use of the gravitational equations (1) and the equalities
\[ \left( \frac{\partial g'_{\mu\nu}}{\partial \varepsilon} \right)_{\varepsilon=0} = - g_{\mu\cdot}\, \frac{\partial}{\partial x_\nu} \left( \frac{\partial x'_\beta}{\partial \varepsilon} \right)_{\varepsilon=0} - g_{\nu\cdot}\, \frac{\partial}{\partial x_\mu} \left( \frac{\partial x'_\beta}{\partial \varepsilon} \right)_{\varepsilon=0}, \tag{8} \]
then we shall have
\[
\left(\frac{\partial \mathcal L^{*}}{\partial \varepsilon}\right)_{\varepsilon=0}
=
\left(\mathcal L\delta^\alpha_\beta
-
\frac{\partial g_{\mu\nu}}{\partial x_\beta}
\frac{\partial \mathcal L}{\partial(\partial g_{\mu\nu}/\partial x_\alpha)}
\right)
+2\varkappa\sqrt{-g}\,gT^\alpha_\beta
\frac{\partial}{\partial x_\alpha}
\left(\frac{\partial x'_\beta}{\partial\varepsilon}\right)_{\varepsilon=0}
\]
\[
{}-2\frac{\partial}{\partial x_\alpha}
\left[
g_{\mu\beta}
\frac{\partial\mathcal L}{\partial(\partial g_{\mu\nu}/\partial x_\alpha)}
\frac{\partial}{\partial x_\nu}
\left(\frac{\partial x'_\beta}{\partial\varepsilon}\right)_{\varepsilon=0}
\right],
\tag{9}
\]
where \(\delta^\alpha_\beta=1\) for \(\alpha=\beta\) and \(\delta^\alpha_\beta=0\) for \(\alpha\ne\beta\). Consequently, taking into account Einstein’s conservation laws,
\[ \frac{\partial}{\partial x_\alpha}\sqrt{-g}\,(T^\alpha_\beta+\theta^\alpha_\beta)=0, \tag{10} \]
where
\[ \theta^\alpha_\beta = \frac{1}{2\varkappa\sqrt{-g}} \left( \mathcal L\delta^\alpha_\beta - \frac{\partial g_{\mu\nu}}{\partial x_\beta} \frac{\partial\mathcal L}{\partial(\partial g_{\mu\nu}/\partial x_\alpha)} \right) \tag{11} \]
is Einstein’s energy–momentum pseudotensor, we arrive at the conclusion that
\[ \left(\frac{\partial \mathcal L^{*}}{\partial\varepsilon}\right)_{\varepsilon=0} = \frac{\partial}{\partial x_\alpha} \left\{ 2\varkappa\sqrt{-g}\,(T^\alpha_\beta+\theta^\alpha_\beta) \left(\frac{\partial x'_\beta}{\partial\varepsilon}\right)_{\varepsilon=0} - 2g_{\mu\beta} \frac{\partial\mathcal L}{\partial(\partial g_{\mu\nu}/\partial x_\alpha)} \frac{\partial}{\partial x_\nu} \left(\frac{\partial x'_\beta}{\partial\varepsilon}\right)_{\varepsilon=0} \right\}. \tag{12} \]
In deriving equality (12), the equations of gravitation (1) were essentially used. We shall now calculate the same quantity \((\partial\mathcal L^*/\partial\varepsilon)_{\varepsilon=0}\), but without taking account of the equations of gravitation. For this purpose we shall use the transformation law of the Lagrangian function (2). As is known (see (5)),
\[ \mathcal L'\left| \frac{D(x'_0,x'_1,x'_2,x'_3)} {D(x_0,x_1,x_2,x_3)} \right| = \mathcal L+ \frac{\partial}{\partial x_\alpha} \left(P^\nu_{\mu\nu}\mathfrak G^{\mu\alpha} - P^\alpha_{\mu\nu}\mathfrak G^{\mu\nu}\right), \tag{13} \]
\[ P^\alpha_{\mu\nu} = \frac{\partial x'_\sigma}{\partial x_\mu} \frac{\partial x'_\tau}{\partial x_\nu} \frac{\partial^2 x_\alpha}{\partial x'_\sigma\partial x'_\tau}, \qquad \mathfrak G^{\mu\nu}=\sqrt{-g}\,g^{\mu\nu}. \tag{14} \]
Since
\[ \left(\frac{\partial P^\alpha_{\mu\nu}}{\partial\varepsilon}\right)_{\varepsilon=0} = - \frac{\partial^2}{\partial x_\mu\partial x_\nu} \left(\frac{\partial x'_\alpha}{\partial\varepsilon}\right)_{\varepsilon=0}, \tag{15} \]
we have, according to (13),
\[ \left(\frac{\partial\mathcal L^{*}}{\partial\varepsilon}\right)_{\varepsilon=0} = - \frac{\partial}{\partial x_\alpha} \left\{ \frac{\partial\mathfrak G^{\mu\nu}}{\partial x_\mu} \frac{\partial}{\partial x_\nu} \left(\frac{\partial x'_\alpha}{\partial\varepsilon}\right)_{\varepsilon=0} - \frac{\partial\mathfrak G^{\mu\alpha}}{\partial x_\nu} \frac{\partial}{\partial x_\mu} \left(\frac{\partial x'_\nu}{\partial\varepsilon}\right)_{\varepsilon=0} \right\}. \tag{16} \]
Comparing (12) and (16), we arrive at the following conservation laws
\[ \frac{\partial}{\partial x_\alpha} \left\{ 2\varkappa\sqrt{-g}\,(T^\alpha_\beta+\theta^\alpha_\beta)\varphi^\beta - 2g_{\mu\beta} \frac{\partial\mathcal L}{\partial(\partial g_{\mu\nu}/\partial x_\alpha)} \frac{\partial\varphi^\beta}{\partial x_\nu} + \frac{\partial\mathfrak G^{\mu\nu}}{\partial x_\mu} \frac{\partial\varphi^\alpha}{\partial x_\nu} - \frac{\partial\mathfrak G^{\nu\alpha}}{\partial x_\mu} \frac{\partial\varphi^\beta}{\partial x_\nu} \right\}=0, \tag{17} \]
\[ \varphi^\alpha=(\partial x'_\alpha/\partial\varepsilon)_{\varepsilon=0} \]
are arbitrary functions of the variables \(x_0, x_1, x_2, x_3\). If arbitrary constants are chosen as \(\varphi^\alpha\), then from (17) Einstein’s conservation laws (10) follow at once. Setting in (17) \(\varphi^\alpha=c_\beta\mathfrak G^{\alpha\beta}\), where \(c_\beta\) are arbitrary constants, we obtain the conservation laws considered in \((^6)\). If, however, using the equations of gravitation, one eliminates the mass tensor from (17), then one arrives at the conservation laws considered in \((^{4,7,8})\).
Leningrad Institute
of Precision Mechanics and Optics
Received
1 XII 1965
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