Tetrad formalism vs coordinate formalism example

One thing you can do with tetrads is express quantities everywhere in terms of what "natural" observers would measure at each point in spacetime.

To be more concrete, consider a spacetime foliated by slices of constant timelike coordinate. At each point, one can imagine the "normal observer" whose 4-velocity is the unit timelike normal to the constant-time slice (that is, the 4-velocity components in the coordinate basis are $u_mu = n_mu equiv -alpha delta^0_mu equiv -(-g^{00})^{-1/2} delta^0_mu$, in the language of the ADM formalism). This observer has the nice property that its constant-time surface coincides with the global constant-time surface; things occurring at the same time coordinate appear simultaneous for this observer.

If the stress-energy tensor has components $T^{munu}$ in the coordinate basis, we might think of $T^{00}$ as the energy density, by analogy with special relativity. However, the normal observer would see an energy density of $n_mu n_nu T^{munu} = alpha^2 T^{00}$. If instead we expressed stress-energy components in the normal observer's locally flat coordinates (denoted with primes), we would simply have energy density $T^{0'0'}$, with the lapse already accounted for.

This can give a nice physical interpretation to individual components of tensors: $T^{0'0'}$ is the energy density seen by a concrete, sensible observer, while $T^{00}$ is hard to interpret without all the other components of the tensor. With this particular tetrad, one also can carry over any complicated models of local phenomena that have been worked out in special relativity right over to GR.¹

The cost comes in relating quantities at different points, as when moving along trajectories or integrating over regions of spacetime. In many such cases, the fact that the tetrad is evolving with respect to the natural coordinate basis as you move around the spacetime makes these sorts of operations worse with tetrads.²

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¹In fact, this happens in my field, where we have more and better approximations to nonlinear behaviors in fluid dynamics in Minkowski spacetimes than in general spacetimes.

²Also, I wouldn't really describe tetrads as "coordinate free" (though a lot of people do). A tetrad is just a choice of bases for the tangent bundle that isn't the one basis you naturally get from the coordinate system at hand. But replacing the coordinate-induced bases isn't the same as getting rid of the coordinates themselves. Nor do you avoid working with components in a specific basis when dealing with tetrads (which is what some people mean when they say "coordinate free").