Shyan said:
I sometimes feel physics likes to play tricks on us!
Physics tries to uncover the rules of the game in nature. GR is a theory of
constraint system. Because of the contracted Bianchi identity \nabla_{ \nu } G^{ \mu \nu } = 0, only six out of 10 (field) equations are dynamical ones. The remaining four are constraints. In fact, the Einstein equations involve \ddot{ g }_{ i j } but not \ddot{ g }_{ 0 \rho }. Indeed, the invariance of the action under a general coordinate transformations shows that an action principle and with it the field equations
cannot determine the metric tensor unless a coordinate system is specified in some non-covariant way.
So, in order to determine g_{ \mu \nu }, it is necessary to impose a
coordinate condition, explicitly violating general coordinate invariance. This procedure is often called
“gauge fixing”. However, this is an abuse of language: one must keep in mind that gauge fixing has no observable effects, while the choice of a coordinate system has physical meaning (detectable inertial forces depend on it). In fact, the properties of the solution of Einstein equation depend on the choice of a coordinate.
So, in order to make physically acceptable predictions, one must supplement the Einstein equation with a coordinate condition, itself regarded as
a field equation. On theoretical grounds, the most natural coordinate condition seems to be the
de Donder condition g^{ \mu \nu } \ \Gamma^{ \rho }_{ \mu \nu } = 0 , \ \ \ \ \ \ (1) which is also called harmonic coordinate condition: For any rank-2 tensor T^{ \mu \nu }, it is easy to show the following \nabla_{ \mu } T^{ \mu \nu } = \frac{ 1 }{ \sqrt{ - g } } \partial_{ \mu } ( \sqrt{ - g } \ T^{ \mu \nu } ) + \Gamma^{ \nu }_{ \rho \sigma } \ T^{ \rho \sigma } . So, taking T^{ \mu \nu } = g^{ \mu \nu }, and using \nabla_{ \mu } g^{ \mu \nu } = 0, we find g^{ \mu \nu } \ \Gamma^{ \rho }_{ \mu \nu } = - \frac{ 1 }{ \sqrt{ - g } } \ \partial_{ \sigma } ( \sqrt{ - g } \ g^{ \sigma \rho } ) , and the de Donder condition, Eq(1), now reads \partial_{ \sigma } ( \sqrt{ - g } \ g^{ \sigma \rho } ) = 0 . \ \ \ \ \ \ (2) This can be rewritten as \frac{ 1 }{ \sqrt{ - g } } \partial_{ \sigma } ( \sqrt{ - g } \ g^{ \sigma \mu } \ \delta^{ \rho }_{ \mu } ) = \frac{ 1 }{ \sqrt{ - g } } \partial_{ \sigma } \left( ( \sqrt{ - g } g^{ \sigma \mu } \ \partial_{ \mu } ) x^{ \rho } \right) = 0 . Thus, the term “
harmonic coordinate condition” is understood, because the operator ( - g )^{ - 1 / 2 } \ \partial_{ \sigma } ( \sqrt{ - g } \ g^{ \sigma \mu } \ \partial_{ \mu } ) is nothing but the Riemannian form of the flat space
d’Alembertian operator \eta^{ \rho \sigma } \partial_{ \rho } \partial_{ \sigma }.
The privileged nature of harmonic (de Donder) coordinate condition is explained nicely in Fock’s textbook:
(i) Note that the condition (1) is covariant under the transformation group GL(4). Indeed, in an arbitrary coordinate system \{ \bar{ x }^{ \mu } \}, one can show that \bar{ g }^{ \mu \nu } \ \bar{ \Gamma }^{ \sigma }_{ \mu \nu } = \frac{ \partial \bar{ x }^{ \sigma } }{ \partial x^{ \rho } } ( g^{ \mu \nu } \ \Gamma^{ \rho }_{ \mu \nu }) - g^{ \mu \nu } \frac{ \partial^{ 2 } \bar{ x }^{ \sigma } }{ \partial x^{ \mu } \partial x^{ \nu } }. \ \ \ (3) Under GL(4), the second term vanishes implying the covariant nature of the condition (1), i.e. the de Donder condition holds in all coordinate frames reached by the action of the group GL(4).
(ii) It is always possible to impose the de Donder condition. Indeed, suppose that g^{ \mu \nu } \Gamma^{ \rho }_{ \mu \nu } \neq 0. According to (3), we can always find coordinate system \bar{ x } such that \bar{ g }^{ \mu \nu } \bar{ \Gamma }^{ \sigma }_{ \mu \nu } = 0. This is done by solving the following 4 uncoupled second order PDE’s g^{ \mu \nu } \frac{ \partial^{ 2 } \bar{ x }^{ \sigma } }{ \partial x^{ \mu } \partial x^{ \nu } } = \frac{ \partial \bar{ x }^{ \sigma } }{ \partial x^{ \rho } } g^{ \mu \nu } \ \Gamma^{ \rho }_{ \mu \nu } . \ \ \ \ (4)
(iii) If the de Donder condition holds in a frame x, it also holds in all frames \bar{ x } satisfying g^{ \mu \nu } \frac{ \partial^{ 2 } \bar{ x }^{ \sigma } }{ \partial x^{ \mu } \partial x^{ \nu } } = 0 . \ \ \ \ \ (5) Indeed, from (3) and (4) we can show that (5) is a necessary and sufficient condition for the de Donder.
Finally, repeat the above argument replacing g^{ \mu \nu } by the energy-momentum tensor T^{ \mu \nu } or (equivalently from Einsten equation) by the Einstein tensor G^{ \mu \nu }. Such class of coordinate frames is called “non-rotating” frames. This class can be viewed as the INERTIAL FRAMES of curved spacetime. Recall that unlike the case in special relativity, in curved spacetime an inertial frame relative to an observer, is not necessarily inertial relative to another observer (however, it is non-rotating). You can easily see that in such frames, the matter energy-momentum tensor density satisfies GLOBALLY an ORDINARY conservation law \partial_{ \mu } ( \sqrt{ - g } T^{ \mu }{}_{ \nu } ) = 0. It follows from this equation that, for physical system confined to a spatial volume V, i.e. T_{ \mu \nu } = 0 outside V, the integrals P_{ \nu } = \int_{ V } d^{ 3 } x \ \sqrt{ - g } \ T^{ 0 }{}_{ \nu } , define 4 time-independent physical quantities and transform as a vector under the group of linear coordinate transformations.
Sam
