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{{Plebeian|Hamilton's principle/Explainers/Gravity from an action}}

The '''Einstein-Hilbert action''' produces the field equations of general
relativity from a variational principle. It is the last application in this
volume and it is where three threads left open earlier converge: the surface term
that
[[Hamilton's principle/Mathematics/Several independent variables]] could not
discard, Noether's second theorem, and her classification of energy conservation
as proper or improper.

== The action ==

[[File:OPL-HP-F-14.png|thumb|right|400px|Two stationary light paths around a mass.]]

The dynamical variable is the metric <math>g_{\alpha\beta}</math>. Requiring the
action to be a scalar under arbitrary coordinate transformations, built from the
metric and its derivatives, leaves very little freedom. The simplest choice is

:<math>S_{\text{EH}} = \frac{1}{2\kappa}\int R\,\sqrt{-g}\; d^4x
\qquad\qquad\mathrm{(1)}</math>

with <math>R</math> the Ricci scalar, <math>g</math> the determinant of the
metric, and <math>\kappa = 8\pi G/c^4</math>. The factor
<math>\sqrt{-g}\,d^4x</math> is the invariant volume element; without it the
integral would not be a scalar.

'''(1) contains second derivatives of the metric.''' <math>R</math> is built from
the Christoffel symbols of
[[Hamilton's principle/Applications/Geodesics]] and their derivatives, so
<math>\mathcal{L}</math> depends on <math>g</math>, <math>\partial g</math> and
<math>\partial^2 g</math>. This is the exception flagged on
[[Hamilton's principle/Applications/Classical field theory]], where first-order
Lagrangians were said to be the rule.

It escapes the usual instability because the second derivatives enter
'''linearly''' and can be moved into a total divergence. The field equations come
out second order, not fourth. That is a special property of the Ricci scalar and
not a general feature of higher-derivative actions.

== Varying ==

[[File:OPL-HP-F-45.png|thumb|right|400px|Free fall as a straight path in a curved geometry.]]

Write the variation of (1) using
<math>\delta\sqrt{-g} = -\tfrac{1}{2}\sqrt{-g}\;g_{\alpha\beta}\,
\delta g^{\alpha\beta}</math> and
<math>R = g^{\alpha\beta}R_{\alpha\beta}</math>:

:<math>\delta S_{\text{EH}} = \frac{1}{2\kappa}\int
\left[ R_{\alpha\beta} - \tfrac{1}{2}R\,g_{\alpha\beta} \right]
\delta g^{\alpha\beta}\,\sqrt{-g}\; d^4x
\;+\; \frac{1}{2\kappa}\int
g^{\alpha\beta}\,\delta R_{\alpha\beta}\,\sqrt{-g}\; d^4x
\qquad\qquad\mathrm{(2)}</math>

The second integral is the one that matters. The variation of the Ricci tensor is
a total divergence,

:<math>g^{\alpha\beta}\,\delta R_{\alpha\beta} = \nabla_\gamma v^\gamma
\qquad\qquad\mathrm{(3)}</math>

for a vector <math>v^\gamma</math> built from
<math>\delta g</math> and <math>\nabla\delta g</math>, so by the divergence
theorem it becomes a surface integral over <math>\partial\Omega</math>.

'''That surface integral does not vanish''' when only <math>\delta g</math> is
required to vanish on the boundary, because <math>v^\gamma</math> also contains
the '''normal derivative''' of <math>\delta g</math>, which is unconstrained.
This is precisely the second case named on
[[Hamilton's principle/Mathematics/Several independent variables]]: with second
derivatives in the Lagrangian, two integrations by parts are needed and the
surface term involves the derivative of the variation.

Setting the first integral to zero gives the '''Einstein field equations in
vacuum''':

:<math>R_{\alpha\beta} - \tfrac{1}{2}R\,g_{\alpha\beta} = 0
\qquad\qquad\mathrm{(4)}</math>

and adding a matter action with stress-energy tensor <math>T_{\alpha\beta}</math>
gives the full equations,
<math>G_{\alpha\beta} = \kappa T_{\alpha\beta}</math>.

== The surface term ==

[[File:OPL-HP-F-52.png|thumb|right|400px|The boundary term in one dimension and in a field.]]

The standard repair is to add a compensating boundary term to (1),

:<math>S = S_{\text{EH}} + \frac{1}{\kappa}\oint_{\partial\Omega}
K\,\sqrt{\vert h \vert}\; d^3x
\qquad\qquad\mathrm{(5)}</math>

with <math>K</math> the trace of the extrinsic curvature of the boundary and
<math>h</math> the induced metric. Its variation cancels the unwanted term, and
the variational problem is then well posed with <math>\delta g = 0</math> on the
boundary alone.

This is not cosmetic. The boundary term carries the gravitational energy in
several standard constructions, and it is what makes the action finite for
asymptotically flat spacetimes after a subtraction. A treatment that discards the
surface term silently gets (4) right and loses everything that depends on the
value of the action.

'''The general lesson,''' stated on
[[Hamilton's principle/Mathematics/Boundary terms and natural boundary conditions]]
and now realised: a boundary term must be argued away, not assumed away. Here it
cannot be argued away and must be cancelled by hand.

== Noether's second theorem, realised ==

The action (1) is invariant under '''all''' coordinate transformations
<math>y^\alpha = p^\alpha(x)</math> — a group depending on four arbitrary
functions, with their first derivatives entering. That is a
<math>G_{\infty 4}</math>, so
[[Hamilton's principle/Mathematics/Noether's second theorem]] applies and there
must be four identities among the field equations.

They are the '''contracted Bianchi identities''':

:<math>\nabla_\alpha\!\left( R^{\alpha\beta}
- \tfrac{1}{2}R\,g^{\alpha\beta} \right) = 0
\qquad\qquad\mathrm{(6)}</math>

holding identically, for every metric, before any field equation is imposed.
Noether names general relativity as the second of her two examples of Theorem II,
and identifies exactly these <math>n</math> dependencies among the Lagrange
expressions and their first derivatives.

Three consequences, each an instance of a general statement from that page.

'''Four of the ten field equations are redundant.''' Ten components,
four identities, four gauge functions — leaving two propagating polarisations,
the same count as electromagnetism on
[[Hamilton's principle/Applications/The electromagnetic Lagrangian]] reached by
the same arithmetic.

'''The evolution is not determined.''' Four functions remain free for all time.
That is the coordinate freedom, and the initial-value problem for (4) is
constrained rather than free.

'''Conservation of matter energy is forced.''' Applying <math>\nabla_\alpha</math>
to <math>G_{\alpha\beta} = \kappa T_{\alpha\beta}</math> and using (6) gives
<math>\nabla_\alpha T^{\alpha\beta} = 0</math>. As with charge conservation in
electromagnetism, the symmetry of the action requires the conservation law rather
than permitting it.

== Improper energy conservation ==

The sharpest consequence is the one Noether wrote the paper for.

Her §6 result, stated on
[[Hamilton's principle/Mathematics/Noether's second theorem]], is that the energy
relations of a theory are '''improper''' exactly when the theory is invariant
under an infinite group containing the displacement group. General relativity is
such a theory: translations are a special case of arbitrary coordinate changes.

Therefore gravitational energy conservation is improper. The conserved quantity
is a combination of the field equations and their derivatives — an identity
wearing the costume of a conservation law — rather than a dynamical statement of
the kind that holds in classical mechanics or special relativity.

This is why there is no local gravitational stress-energy tensor. The various
pseudotensors in the literature are coordinate dependent, and their coordinate
dependence is not a defect of the constructions: it is what Noether's theorem
predicts. Global energy can still be defined for spacetimes with suitable
asymptotic structure, and the boundary term (5) is where that definition lives.

Noether adds that for the Hilbert assertion to hold literally, general relativity
must be read as any theory invariant under such a group. The property is
group-theoretic, and gravity is one instance rather than the reason.

== Geodesics recovered ==

The matter action for a free test particle is its proper time,

:<math>S = -mc\int ds
\qquad\qquad\mathrm{(7)}</math>

which is the arc-length functional of
[[Hamilton's principle/Applications/Geodesics]] in the spacetime metric. Its
extremals are geodesics, so '''free fall is a geodesic''' and (6) of that page is
the equation of motion under gravity.

Conjugate points then acquire a physical reading. Neighbouring geodesics
refocused by curvature are light rays refocused by a gravitational lens, and the
multiple images a lens produces are the several stationary paths between source
and observer. Only the first-arriving image corresponds to a minimal path; the
others are stationary and are not minima, in the sense established on
[[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points]].

That is the volume's central distinction appearing as an observation rather than
an argument.

<blockquote>
Provenance: every equation on this page is standard and is derived or stated
here; none is transcribed from a primary source. The characterisations of Noether
1918 §6 and of her general-relativity example are paraphrased from the
ESTABLISHED source under the translation rule on
[[Hamilton's principle/Mathematics/Noether's first theorem]]. Hilbert's 1915
paper and Einstein's field-equation papers are not sources of this volume and are
not cited. See [[Hamilton's principle/Sources]].
</blockquote>

== Next ==

[[Hamilton's principle/Applications/The Feynman path integral]] closes the
applications track.

== References ==

<references />

← Older revision		Revision as of 04:58, 16 September 2026
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	volume and it is where three threads left open earlier converge: the surface term		volume and it is where three threads left open earlier converge: the surface term
	that		that
	[[Hamilton's principle/Mathematics/Several independent variables]] could not		[[Hamilton's principle/Mathematics/Several independent variables\|Several independent variables]] could not
	discard, Noether's second theorem, and her classification of energy conservation		discard, Noether's second theorem, and her classification of energy conservation
	as proper or improper.		as proper or improper.
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	'''(1) contains second derivatives of the metric.''' <math>R</math> is built from		'''(1) contains second derivatives of the metric.''' <math>R</math> is built from
	the Christoffel symbols of		the Christoffel symbols of
	[[Hamilton's principle/Applications/Geodesics]] and their derivatives, so		[[Hamilton's principle/Applications/Geodesics\|Geodesics]] and their derivatives, so
	<math>\mathcal{L}</math> depends on <math>g</math>, <math>\partial g</math> and		<math>\mathcal{L}</math> depends on <math>g</math>, <math>\partial g</math> and
	<math>\partial^2 g</math>. This is the exception flagged on		<math>\partial^2 g</math>. This is the exception flagged on
	[[Hamilton's principle/Applications/Classical field theory]], where first-order		[[Hamilton's principle/Applications/Classical field theory\|Classical field theory]], where first-order
	Lagrangians were said to be the rule.		Lagrangians were said to be the rule.

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	the '''normal derivative''' of <math>\delta g</math>, which is unconstrained.		the '''normal derivative''' of <math>\delta g</math>, which is unconstrained.
	This is precisely the second case named on		This is precisely the second case named on
	[[Hamilton's principle/Mathematics/Several independent variables]]: with second		[[Hamilton's principle/Mathematics/Several independent variables\|Several independent variables]]: with second
	derivatives in the Lagrangian, two integrations by parts are needed and the		derivatives in the Lagrangian, two integrations by parts are needed and the
	surface term involves the derivative of the variation.		surface term involves the derivative of the variation.
Line 105:		Line 105:

	'''The general lesson,''' stated on		'''The general lesson,''' stated on
	[[Hamilton's principle/Mathematics/Boundary terms and natural boundary conditions]]		[[Hamilton's principle/Mathematics/Boundary terms and natural boundary conditions\|Boundary terms and natural boundary conditions]]
	and now realised: a boundary term must be argued away, not assumed away. Here it		and now realised: a boundary term must be argued away, not assumed away. Here it
	cannot be argued away and must be cancelled by hand.		cannot be argued away and must be cancelled by hand.
Line 115:		Line 115:
	functions, with their first derivatives entering. That is a		functions, with their first derivatives entering. That is a
	<math>G_{\infty 4}</math>, so		<math>G_{\infty 4}</math>, so
	[[Hamilton's principle/Mathematics/Noether's second theorem]] applies and there		[[Hamilton's principle/Mathematics/Noether's second theorem\|Noether's second theorem]] applies and there
	must be four identities among the field equations.		must be four identities among the field equations.

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	four identities, four gauge functions — leaving two propagating polarisations,		four identities, four gauge functions — leaving two propagating polarisations,
	the same count as electromagnetism on		the same count as electromagnetism on
	[[Hamilton's principle/Applications/The electromagnetic Lagrangian]] reached by		[[Hamilton's principle/Applications/The electromagnetic Lagrangian\|the electromagnetic Lagrangian]] reached by
	the same arithmetic.		the same arithmetic.

Line 152:		Line 152:

	Her §6 result, stated on		Her §6 result, stated on
	[[Hamilton's principle/Mathematics/Noether's second theorem]], is that the energy		[[Hamilton's principle/Mathematics/Noether's second theorem\|Noether's second theorem]], is that the energy
	relations of a theory are '''improper''' exactly when the theory is invariant		relations of a theory are '''improper''' exactly when the theory is invariant
	under an infinite group containing the displacement group. General relativity is		under an infinite group containing the displacement group. General relativity is
Line 180:		Line 180:

	which is the arc-length functional of		which is the arc-length functional of
	[[Hamilton's principle/Applications/Geodesics]] in the spacetime metric. Its		[[Hamilton's principle/Applications/Geodesics\|Geodesics]] in the spacetime metric. Its
	extremals are geodesics, so '''free fall is a geodesic''' and (6) of that page is		extremals are geodesics, so '''free fall is a geodesic''' and (6) of that page is
	the equation of motion under gravity.		the equation of motion under gravity.
Line 189:		Line 189:
	and observer. Only the first-arriving image corresponds to a minimal path; the		and observer. Only the first-arriving image corresponds to a minimal path; the
	others are stationary and are not minima, in the sense established on		others are stationary and are not minima, in the sense established on
	[[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points]].		[[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points\|the Jacobi condition and conjugate points]].

	That is the volume's central distinction appearing as an observation rather than		That is the volume's central distinction appearing as an observation rather than
Line 199:		Line 199:
	1918 §6 and of her general-relativity example are paraphrased from the		1918 §6 and of her general-relativity example are paraphrased from the
	ESTABLISHED source under the translation rule on		ESTABLISHED source under the translation rule on
	[[Hamilton's principle/Mathematics/Noether's first theorem]]. Hilbert's 1915		[[Hamilton's principle/Mathematics/Noether's first theorem\|Noether's first theorem]]. Hilbert's 1915
	paper and Einstein's field-equation papers are not sources of this volume and are		paper and Einstein's field-equation papers are not sources of this volume and are
	not cited. See [[Hamilton's principle/Sources]].		not cited. See [[Hamilton's principle/Sources\|Sources]].
	</blockquote>		</blockquote>

	== Next ==		== Next ==

	[[Hamilton's principle/Applications/The Feynman path integral]] closes the		[[Hamilton's principle/Applications/The Feynman path integral\|the Feynman path integral]] closes the
	applications track.		applications track.