- #1
Kostik
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- TL;DR Summary
- In Dirac's "General Theory of Relativity", Dirac derives Einstein's field equations and the geodesic equation from the variation ##\delta(I_g+I_m)=0## of the actions for gravity and matter. The two dynamical variables in the variation are ##g_{\mu\nu}## and ##p^\mu## which satisfies ##\delta p^\mu = (p^\nu b^\mu - p^\mu b^\nu)_{,\nu}## where ##b^\mu## is an arbitrary displacements of an element of matter. He discards a boundary term during partial integration that is not easily (?) justified!
In Dirac's "General Theory of Relativity", p. 53, eq. (27.11), Dirac is deriving Einstein's field equations and the geodesic equation from the variation ##\delta(I_g+I_m)=0## of the actions for gravity and matter. Here ##p^\mu=\rho v^\mu \sqrt{-g}## is the momentum of an element of matter. He makes arbitrary displacements of an element of matter ##b^\mu##. The two dynamical variables in the variation are ##g_{\mu\nu}## and ##p^\mu## which satisfies ##\delta p^\mu = (p^\nu b^\mu - p^\mu b^\nu)_{,\nu}=0##.
The action is over a 4-dimensional volume ##M##. Dirac integrates by parts: $$\int_M v_\mu (p^\nu b^\mu - p^\mu b^\nu)_{,\nu} \, d^4 x
= \int_M [v_\mu (p^\nu b^\mu - p^\mu b^\nu)]_{,\nu} \, d^4 x
- \int_M v_{\mu,\nu} (p^\nu b^\mu - p^\mu b^\nu) \, d^4 x .$$ Remember: on the boundary ##\partial M##: $$(p^\nu b^\mu - p^\mu b^\nu)_{,\nu}=0.$$ (It's unusual for a divergence to vanish on a boundary, but remember this is actually a variation ##\delta p^\mu = (p^\nu b^\mu - p^\mu b^\nu)_{,\nu}##. As ##p^\mu## is one of the dynamical variables, it is kept constant on the boundary. Hence, the variation ##\delta p^\mu=0## on the boundary.)
Dirac assumes the boundary term vanishes. How do we show this? $$\int_M [v_\mu (p^\nu b^\mu - p^\mu b^\nu)]_{,\nu} \, d^4 x = \int_{\partial M} v_\mu (p^\nu b^\mu - p^\mu b^\nu) \, dS_\nu = 0 \quad (\text{Show!})$$ where ##dS_\nu## is the oriented hypersurface element in 3D space.
If ##v_\mu (p^\nu b^\mu - p^\mu b^\nu)## were constant on the boundary, the result would be trivial. But this is not the case!
One can also write this integral: $$\int_M [v_\mu (p^\nu b^\mu - p^\mu b^\nu)]_{,\nu} \, d^4 x
= \int_{\partial M} v_\mu n_\nu (p^\nu b^\mu - p^\mu b^\nu) \, \sqrt{|h|} \, d^3 y$$ where ##n_\nu## is the "oriented unit normal vector" on ##\partial M##. (Here I am supposing that the boundary ##\partial M## can be parameterized by ##x^\nu = x^\nu (y^m)##, ##m=1,2,3## and ##h## is the determinant of the matrix ##h_{mn}=g_{\mu\nu}\frac{\partial x^\mu}{\partial y^m}\frac{\partial x^\nu}{\partial y^n}## ... but this is just linear algebra / change of variable stuff, and isn't important to the problem at hand.)
To repeat, the key fact that I have to work with is that, on the boundary ##\partial M##:
$$(p^\nu b^\mu - p^\mu b^\nu)_{,\nu}=0.$$
The action is over a 4-dimensional volume ##M##. Dirac integrates by parts: $$\int_M v_\mu (p^\nu b^\mu - p^\mu b^\nu)_{,\nu} \, d^4 x
= \int_M [v_\mu (p^\nu b^\mu - p^\mu b^\nu)]_{,\nu} \, d^4 x
- \int_M v_{\mu,\nu} (p^\nu b^\mu - p^\mu b^\nu) \, d^4 x .$$ Remember: on the boundary ##\partial M##: $$(p^\nu b^\mu - p^\mu b^\nu)_{,\nu}=0.$$ (It's unusual for a divergence to vanish on a boundary, but remember this is actually a variation ##\delta p^\mu = (p^\nu b^\mu - p^\mu b^\nu)_{,\nu}##. As ##p^\mu## is one of the dynamical variables, it is kept constant on the boundary. Hence, the variation ##\delta p^\mu=0## on the boundary.)
Dirac assumes the boundary term vanishes. How do we show this? $$\int_M [v_\mu (p^\nu b^\mu - p^\mu b^\nu)]_{,\nu} \, d^4 x = \int_{\partial M} v_\mu (p^\nu b^\mu - p^\mu b^\nu) \, dS_\nu = 0 \quad (\text{Show!})$$ where ##dS_\nu## is the oriented hypersurface element in 3D space.
If ##v_\mu (p^\nu b^\mu - p^\mu b^\nu)## were constant on the boundary, the result would be trivial. But this is not the case!
One can also write this integral: $$\int_M [v_\mu (p^\nu b^\mu - p^\mu b^\nu)]_{,\nu} \, d^4 x
= \int_{\partial M} v_\mu n_\nu (p^\nu b^\mu - p^\mu b^\nu) \, \sqrt{|h|} \, d^3 y$$ where ##n_\nu## is the "oriented unit normal vector" on ##\partial M##. (Here I am supposing that the boundary ##\partial M## can be parameterized by ##x^\nu = x^\nu (y^m)##, ##m=1,2,3## and ##h## is the determinant of the matrix ##h_{mn}=g_{\mu\nu}\frac{\partial x^\mu}{\partial y^m}\frac{\partial x^\nu}{\partial y^n}## ... but this is just linear algebra / change of variable stuff, and isn't important to the problem at hand.)
To repeat, the key fact that I have to work with is that, on the boundary ##\partial M##:
$$(p^\nu b^\mu - p^\mu b^\nu)_{,\nu}=0.$$