# Lapse and shift in ADM decomposition

Poisson in Relativist's Toolkit and also other authors in various papers state explicitly that after one does the 3+1 decomposition, the lapse and shift $N$ and $N^a$ are non-dynamical variables, and that they can be chosen arbitrarily, since they only specify the foliation.

I understand that they are lagrange multipliers and hence they get multiplied with something that vanishes after we obtain the equations of motion (by varying the Lagrangian). But they and their derivatives exist in the constraint equations, arent these considered as "dynamics"?

For example, in the 3+1 decomposition, we define the angular velocity at the horizon as $N^k |_{r_+}$ and the temperature as $\frac{(N^2)'}{\sqrt {g_{rr}N^2}} |_{r_+}$ . Does that mean that the only dynamical quantity in these expressions is the $g_{rr}$ component? What happens if I change lapse and shift? I dont think one could say that they are completely arbitrary, since I cannot just set them to be zero on will.

Another example, if I have a Kerr black hole spacetime (or any specific spacetime for that matter), can I really choose the lapse and shift arbitrarily? What is the real meaning of them being "arbitrary" and "non-dynamical"? And do they have any connection to gauge?

Comment to the question (v7):

1. In the context of an action formulation, if the Euler-Lagrange (EL) equations contain time-derivatives of a variable $\phi^{\alpha}$, then the variable $\phi^{\alpha}$ is called a dynamical variable; else $\phi^{\alpha}$ is an auxiliary variable. (Spatial derivatives are irrelevant for this classification.) Lagrange multipliers are typically auxiliary variables.

2. In the Hamiltonian formulation, a Lagrange multiplier is indetermined (determined) for a first (second) class constraint, respectively. Gauge-fixing choices are subjected to pertinent rank conditions. For further information, see e.g. Refs. 1-2.

3. The ADM formalism is a Hamiltonian formulation of GR. The lapse $N$ and shifts $N^i$ are Lagrange multipliers for a set of first class constraints, cf. Refs. 3-5.

References:

1. P.A.M. Dirac, Lectures on QM, (1964).

2. M. Henneaux and C. Teitelboim, Quantization of Gauge Systems, 1994.