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Kostik
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- TL;DR Summary
- In making the variation ##\delta I=0##, where the ##g_{\mu\nu}## are considered the 'coordinates' of the Lagrangian, Dirac assumes the ##g_{\mu\nu}## and their first derivatives are constant on the boundary. How does this correspond to the usual action principle method?
Dirac derives Einstein's field equations from the action principle ##\delta I=0## where $$I=\int R\sqrt{-g} \, d^4x$$ (##R## is the Ricci scalar). Using partial integration, he shows that $$I=\int L\sqrt{-g} \, d^4x$$ where ##L## involves only ##g_{\mu\nu}## and its first derivatives, unlike ##R##. Clearly ##L\sqrt{-g}## is both the action density in four dimensions as well as the Lagrangian density in three dimensions, since $$I=\int L\sqrt{-g} \, d^4x = \int dt \int L\sqrt{-g} \, d^3x.$$ Dirac considers the ##g_{\mu\nu}## to be the 'coordinates', and their time derivatives ##g_{\mu\nu,0}## the 'velocities', so by the action principle he determines the actual 'path' ##g_{\mu\nu}## which turns out to be Einstein's field equations (in the same way that the Euler-Lagrange equations determine the equation of motion).
When performing the variation ##\delta g_{\mu\nu}## he assumes ##g_{\mu\nu}## is constant on the boundary of the four-dimensional volume ##D## in the action integral
$$I=\int_D R\sqrt{-g}d^4x.$$ This parallels the classical variation method where ##I=\int_{t_0}^{t_1} L ( q^i, {\dot{q}}^i ) \, dt##, and one assumes ##q^i(t_0)=q^i(t_1)## for all paths, i.e., all paths are constant at the endpoints (the 'boundary').
However, Dirac also assumes that we keep "the ##g_{\mu\nu}## and their first derivatives constant on the boundary." This is essential in two separate places where the divergence theorem is used, because there are Christoffel symbol terms that contain derivatives of ##g_{\mu\nu}##.
How does one explain the need to assume that both ##g_{\mu\nu}## and their first derivatives are constant on the boundary (other than the fact that it's required for the action principle method to work)? Transitioning from the usual 'principle of stationary action' in 1D to the same principle in 4D, is it clear how the spatial derivatives of the 'coordinates' ##g_{\mu\nu}## come into play?
When performing the variation ##\delta g_{\mu\nu}## he assumes ##g_{\mu\nu}## is constant on the boundary of the four-dimensional volume ##D## in the action integral
$$I=\int_D R\sqrt{-g}d^4x.$$ This parallels the classical variation method where ##I=\int_{t_0}^{t_1} L ( q^i, {\dot{q}}^i ) \, dt##, and one assumes ##q^i(t_0)=q^i(t_1)## for all paths, i.e., all paths are constant at the endpoints (the 'boundary').
However, Dirac also assumes that we keep "the ##g_{\mu\nu}## and their first derivatives constant on the boundary." This is essential in two separate places where the divergence theorem is used, because there are Christoffel symbol terms that contain derivatives of ##g_{\mu\nu}##.
How does one explain the need to assume that both ##g_{\mu\nu}## and their first derivatives are constant on the boundary (other than the fact that it's required for the action principle method to work)? Transitioning from the usual 'principle of stationary action' in 1D to the same principle in 4D, is it clear how the spatial derivatives of the 'coordinates' ##g_{\mu\nu}## come into play?
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