I have a beginners question concerning the Einstein field equations.
Although I've read some basic texts about G.R. and understand some of the formula's and derivations I still have the feeling there's something I don't properly understand. Hope my question is not too vague.
The Einstein field equations state: $G_{\mu \nu } =\kappa T_{\mu \nu }$.
I am told that these equations can be justified on the basis of two fundamental properties:
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They are coordinate independent, meaning that these are tensors therefore they can be defined without reference to a specific coordinate system. For example 'Gravitation' by Misner, Thorne and Wheeler devotes a paragraph on the importance of 'coordinate independent thinking'.
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The beauty of above equations is that they generate a curvature of spacetime on the left side in which the stress energy tensor on the right is automatically conserved: ${T^{\alpha\nu}}_{;\nu}=0$.
Now how does this compare to classical mechanics?
Newtons laws imply that we are describing our system with respect to an inertial frame of reference.
Hamiltonial mechanics gives us a way to use more general coordinates. But still the assumption is that spacetime is flat and we always know how to transform back to our 'normal' inertial coordinates.
A typical approach in classical mechanics would be to specify some distribution of mass at $t=0$ as a boundary condition, then use Newton or Euler-Lagrange to calculate the evolution of this distribution in time thereby arriving at a (conserved) value for the stress-energy tensor at every time and at every point in space.
In general relativity the equation $G_{\mu \nu } =\kappa T_{\mu \nu }$ somehow suggests to me that our 'boundary condition' now requires us to specify some $T_{\mu \nu }$ for all of spacetime at once (meaning at every point and at every time). With that we can now calculate the curvature of our spacetime, which in turn guarantees us that stress-energy is automatically conserved.
I know I'm probably wrong, but it seems to me that in this way we can specify just any stress energy tensor on the right (since it's automatically conserved anyway). After solving the field equations we are done, only to conclude that (since we've calculated that spacetime must now be curved in some way) we don't exactly know in what coordinates we are working anymore.
Also having to specify $T_{\mu \nu }$ over all of spacetime suggests to me that we have to know something about the time evolution of the system before we can even begin to calculate the metric (which in turn should then be able to tell us about the time evolution of the system??).
Is it perhaps that we have to start with postulating some form of $T_{\mu \nu }$ with certain properties, and then solve for $g_{\alpha\beta}$ and work our way back to interpret how we would transform our spacetime coordinates to arrive at acceptable coordinates (local Lorentzian coordinates perhaps)? So that we have now rediscovered our preferred local Lorentzian coordinates after which we can look at our $T_{\mu \nu }$ again to draw some conclusions about it's behaviour?
So my question is:
Is there a straightforward way to explain in what way the Einstein field equations can have the same predictive value as in the classical mechanical case where we impose a boundary condition in a coordinate system we understand and then solve to find the time evolution of that system in that same coordinate system?
I know the question is very vague, I wouldn't ask if I wasn't confused. But I hope that maybe someone recognizes their own initial struggle with the subject and point me in the right direction.
Best Answer
There are no Lorenzian coordinates in General Relativity. The properties of space-time, such as geodesic lines, conservation laws and singularities, can be described in any suitable coordinate system.
Usually Einstein's equations are solved for a specific type of matter, for example electromagnetic field, for which: $$ T_{\mu\nu}=\frac{1}{4\pi}\left(F_{\mu\alpha}F_{\nu}{}^{\alpha}-\frac{1}{4}g_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}\right) $$ Then the equation $T^{\mu\nu}{}_{;\nu}=0$ leads to Maxwell's equations: $F^{\mu\nu}{}_{;\nu}=0$. Einstein's equations are solved for both metric $g_{\mu\nu}$ and electromagnetic field tensor $F_{\mu\nu}$. Both $g_{\mu\nu}$ and $F_{\mu\nu}$ must be specified on a spacelike hypersurface (such as $t=0$ in Minkowski space; these are initial conditions) and at the spatial infinity (boundary conditions); Einstein equations allow to find the evolution of the system in the future.
Also, as a side remark, equation $T^{\mu\nu}{}_{;\nu}=0$ isn't a conservation law in the common meaning. If a space-time has a symmetry with respect to transformation $x'^{\mu}=x^\mu+\xi^\mu$, then vector $\xi^\mu$ satisfies the Killing equation: $$ \xi_{\mu;\nu}+\xi_{\nu;\mu}=0 $$ Then $(T^{\mu\nu}\xi_{\nu})_{;\mu}=0$, and $$ J=\int T^{0\nu}\xi_{\nu}\sqrt{-g}d^3x $$ will be a conserved quantity. If there are no Killing vectors, then there are no conservation laws.