Hamilton's principle/Explainers/Gravity from an action - Revision history

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			{{Hamilton's principle map}}

	This page explains how gravity comes out of the same method. It uses no		This page explains how gravity comes out of the same method. It uses no

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

	[[Hamilton's principle/Explainers/Electromagnetism from an action]] built		[[Hamilton's principle/Explainers/Electromagnetism from an action\|Electromagnetism from an action]] built
	electromagnetism from one requirement: a certain adjustment must have no physical		electromagnetism from one requirement: a certain adjustment must have no physical
	effect, at every point independently.		effect, at every point independently.
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	An object with nothing pushing it travels as straight as the geometry allows. That		An object with nothing pushing it travels as straight as the geometry allows. That
	is the shortest-path idea from		is the shortest-path idea from
	[[Hamilton's principle/Explainers/Where the minimum fails]], now applied to space		[[Hamilton's principle/Explainers/Where the minimum fails\|Where the minimum fails]], now applied to space
	and time together.		and time together.

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	The volume's final argument, explaining why any of this works at all, is at		The volume's final argument, explaining why any of this works at all, is at
	[[Hamilton's principle/Explainers/Sum over histories]].		[[Hamilton's principle/Explainers/Sum over histories\|Sum over histories]].

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{{Technical|Hamilton's principle/Applications/The Einstein-Hilbert action}}

This page explains how gravity comes out of the same method. It uses no
mathematics. The technical page it goes with is named above.

== The demand ==

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

[[Hamilton's principle/Explainers/Electromagnetism from an action]] built
electromagnetism from one requirement: a certain adjustment must have no physical
effect, at every point independently.

Gravity comes from a similar demand, applied to something more basic.

'''Nothing physical may depend on your choice of coordinates.'''

Coordinates are labels. Where you put the origin, which way the axes point, what
units you use — all of that is your bookkeeping, not the world's. And you should be
free to choose differently in different places.

Insist on that, write the simplest cost rule that respects it, and general
relativity comes out.

== What is being varied ==

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

There is one thing to decide first: what the theory is about.

For electromagnetism it was a field spread through space. For gravity, the thing
that varies is the '''shape of space and time itself''' — specifically, the rule
for measuring distances and durations from place to place.

That rule can differ from place to place. Where it does, the geometry is curved.

So gravity is not a force acting in space. It is a statement about the shape of
space, and the cost rule is a cost rule for shapes.

== Free fall is going straight ==

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

The consequence for motion follows without extra assumptions.

An object with nothing pushing it travels as straight as the geometry allows. That
is the shortest-path idea from
[[Hamilton's principle/Explainers/Where the minimum fails]], now applied to space
and time together.

Near a massive body, the geometry is curved. Straight in that geometry looks bent
from a distance. That bending is what we call gravity.

'''A falling apple is not being pulled. It is going straight through a curved
geometry.''' Nothing distinguishes free fall from doing nothing at all — which is
why astronauts float, and why you feel nothing during a fall until you land.

== The edge of the problem ==

Something happens here that did not happen anywhere else in this volume, and it is
worth flagging.

Every calculation so far quietly discarded a leftover term at the boundary of the
region considered. The usual justification is that nothing is being nudged at the
edge, so the term vanishes.

For gravity it does not vanish. The cost rule involves shapes in a way that leaves
something behind at the boundary even when the edge is held fixed.

The fix is to add a matching term by hand that cancels it.

This is not tidying. That boundary term turns out to carry the '''energy of the
gravitational field''', and treatments that discard it get the equations right and
lose everything that depends on the value of the total. The warning issued much
earlier in this volume — that a leftover term must be argued away rather than
assumed away — is paid for here.

== Where energy goes wrong ==

Now the deepest consequence, and it is the one Noether's second theorem was
written for.

For ordinary theories, energy conservation is a real dynamical statement with a
definite local meaning. You can point at a region and say how much energy is in
it.

For gravity you cannot.

Every attempt to define the energy stored in the gravitational field at a point
gives an answer that depends on your choice of coordinates. Change coordinates and
the energy density changes, though nothing physical has.

For a long time this looked like a failure of ingenuity. It is not. Noether proved
that '''any''' theory with this kind of local symmetry has exactly this property:
its energy conservation becomes an identity, true by construction, rather than a
dynamical law.

There is no local gravitational energy to find. The searching failed because the
thing does not exist.

That is a rare kind of result — a proof that a research programme cannot succeed,
with the reason given.

'''Energy has not stopped being conserved.''' For a system that settles down at
large distances, a perfectly good total energy can be defined, and it lives in
exactly the boundary term of the previous section. What has gone is the ability to
say where the energy '''is'''.

== A prediction you can photograph ==

The volume's central point turns up in the sky.

A massive galaxy bends light passing near it. Light from a distant object behind it
can therefore reach us by more than one route, curving round different sides.

We see several images of the same object. Astronomers photograph them regularly.

Each route is a legitimate straightest path. Only one is the quickest. The others
are perfectly good paths that are not shortest — which is exactly what this volume
has been saying since its first page, now in a photograph.

The routes have different lengths, so the light on them left at different times.
When the distant object flickers, the images flicker one after another, sometimes
weeks apart.

== What the method achieved ==

Stand back and look at what was assumed.

Not that gravity attracts. Not an inverse square law. Not that mass bends light.
Only that the laws must not depend on your choice of labels, and that the cost rule
should be the simplest one respecting that.

Out came a theory that predicts the bending of light, the precession of Mercury,
the slowing of clocks in a gravitational field, and gravitational waves — all
confirmed, some of them decades later.

That is the strongest case for the whole approach in this volume. Write down what a
theory must ignore, and the theory writes itself.

== Where to go next ==

The volume's final argument, explaining why any of this works at all, is at
[[Hamilton's principle/Explainers/Sum over histories]].