Dr Photon:
Tom Roberts wrote:

Conservation of energy is indeed a "murky subject" when one considers
non-local measurements. Locally there's no doubt that energy is
conserved. This is a non-local example.

By "locally" I mean over a region of spacetime small enough
so that variations in the metric and curvature tensors can be
neglected compared to the measurement resolution.


For example, a high-energy interaction at Fermilab in the D0
detector is clearly a local measurement; throwing a baseball
is not. The former covers a region of ~20 meters in spacetime,
the latter covers ~10^9 meters (~3 light-seconds) in spacetime.

I guess there will always be a problem with comparing what's "over
here" with what's "over there", but what about the following:

If there is a way that conservation of energy is not conserved
non-locally, is there any way to get at it? So if I threw a baseball at
an electric generator (via all sorts of weird spacetime curvatures, if
necessary), that then powered a light bulb, which directed its light

Here's the basic problem with defining a conserved energy, which
addresses your question regarding noether's theorem, too. Energy is
defined as the conserved quantity associated with invariance under
time translations. That comes straight from noether's theorem.

The metric has 16 components, 10 of which are independent, so
the metric has 10 degrees of freedom. You will be able to define
a conserved energy if you can make the metric time independent.
You are free to make a change of coordinates to do that, but a
coordinate transformation only has 4 degrees of freedom. In general,
that is not sufficient to remove the time dependence. That means you
cannot define a quantity called the energy such that it is conserved.

Bilge wrote:

Here's the basic problem with defining a conserved energy, which
addresses your question regarding noether's theorem, too. Energy is
defined as the conserved quantity associated with invariance under
time translations. That comes straight from noether's theorem.

according to such *definitions*, then energy can never be created or
destroyed. It intrigued me in the paper Bill Hobba referenced, that the
author seemed to be *looking* for a conserved quantity (ok, not that I
followed all the tensors...). But still it seems to make sense that in
some (currently unknown) circumstance that energy *could* be created?
Otherwise we are saying it is an impossibility by definition.

The metric has 16 components, 10 of which are independent, so
the metric has 10 degrees of freedom. You will be able to define
a conserved energy if you can make the metric time independent.
You are free to make a change of coordinates to do that, but a
coordinate transformation only has 4 degrees of freedom. In general,
that is not sufficient to remove the time dependence. That means you
cannot define a quantity called the energy such that it is conserved.

which taken by itself seems to imply that energy *could* increase. I
appreciate Tom's point that if something is undefined, then you can't
talk about it increasing or decreasing. Bill pointed out that if the
metric is static then energy will not increase, and went on to imply
that if the metric does change and the local energy increases, then
that energy must have come from somewhere else. (for example, moving
masses around may change the metric and increase the potential where I
am, but it must have taken energy to move the masses around).

However, I am still a little concerned about defining energy as "that
which does not increase in time". As far as Noether and Lagragians go,
I guess that is saying that if the Lagrangian needed to describe a
system changed with time, then energy *could* increase, but that would
require changing the laws of motion. Ok, this has never been observed,
but is it a-priori impossible?

br

Bilge wrote:
The metric has 16 components, 10 of which are independent, so
the metric has 10 degrees of freedom. You will be able to define
a conserved energy if you can make the metric time independent.
You are free to make a change of coordinates to do that, but a
coordinate transformation only has 4 degrees of freedom. In general,
that is not sufficient to remove the time dependence. That means you
cannot define a quantity called the energy such that it is conserved.

The usual way of stating this requirement is that the manifold must have
a timelike Killing vector (that guarantees the 4-fold freedom of a
coordinate transfom is indeed sufficient to remove the time dependence
from all metric components; it of course does this by limiting the
permitted form of those components). This GREATLY limits the set of
manifolds with this property. In such a manifold, using the timelike
Killing vector as the time coordinate, nothing moves, so it's not useful
as a model of the real world.

BTW the metric components have 10 ALGEBRAICALLY-independent
degrees of freedom, but they are not completely independent.
In fact, the Einstein field equation and the Bianchi identities
are sufficient to determine them all (given suitable boundary
conditions, as these are differential equations, not algebraic
ones).


Tom Roberts

Bilge wrote:

Here's the basic problem with defining a conserved energy, which
addresses your question regarding noether's theorem, too. Energy is
defined as the conserved quantity associated with invariance under
time translations. That comes straight from noether's theorem.

Energy conservation of some system may be _equivalent_ to the time
translation symmetry of the systems Lagrangian but it is not
necessarily DEFINED that way. The difference is important because lack
of said symmetry does not imply energy conservation violation. Energy
is DEFINED as an abstract quantity remaining constant throughout a
closed systems time-evolution. If such quantity changes, then it is not
energy. Lack of time-translation symmetry reveals more on the character
of GR/Noether's thereom than it does on energy conservation.

Schoenfeld wrote:
Energy conservation of some system may be _equivalent_ to the time
translation symmetry of the systems Lagrangian but it is not
necessarily DEFINED that way.

Sure it is. This is 2005, not ~1900.


The difference is important because lack
of said symmetry does not imply energy conservation violation.

Actually it does (insofar as "energy" can be defined at all in such
systems). That's why the definition changed once Noether's theorem
explained the inconsistencies. Of course in classical mechanics the
Lagrangian is time invariant, so the question never came up. shrug

Noether's theorem says that for every continuous symmetry of
the Lagrangian there is a corresponding conserved quantity,
and for every conserved quantity there is a corresponding
symmetry. It also specifies how either can be determined from
the other.


Energy
is DEFINED as an abstract quantity remaining constant throughout a
closed systems time-evolution.

Not true. See above -- what you claim simply is not possible. If your
"abstract quantity" is conserved, there is a symmetry of the Lagrangian
corresponding to it, and if that symmetry is not time translation
invariance, then your "abstract quantity" is not energy. shrug

In modern physics, energy is specifically defined as the conserved
Noether current related to time translation invariance of the system's
Lagrangian. This is not "abstract" at all. shrug


If such quantity changes, then it is not
energy.

If there is no time translation invariance of the Lagrangian, then there
is no useful definition of "energy".


Lack of time-translation symmetry reveals more on the character
of GR/Noether's thereom than it does on energy conservation.

Your statements reveal more about your lack of knowledge about modern
physics, than about energy conservation.


I know of no simple examples of systems with a time dependent
Lagrangian. But an advanced example is any system in GR that emits or
absorbs gravitational radiation. In some sense, gravitational radiation
can carry "the ability of the system to do work" into or out of an
ostensibly closed system. It can, in general, zoom out to or come in
from spatial infinity, so there is no possibility of "closing the system".


The Lagrangian for GR includes the term \integral R dV, where R is the
Ricci scalar and dV is the invariant volume element; the integral
extends over the entire manifold. A sufficient condition for this term
to be time-translation invariant is the existence of a timelike Killing
vector (and its use as the time coordinate); necessary conditions are
more complicated....


Tom Roberts

Tom Roberts wrote:
Schoenfeld wrote:
Energy conservation of some system may be _equivalent_ to the time
translation symmetry of the systems Lagrangian but it is not
necessarily DEFINED that way.

Sure it is. This is 2005, not ~1900.

There is no reason to define energy in that way.

hint: The work exchanged in an adiabatic process depends only on the
initial and the final state and not on the details of the process.


The difference is important because lack
of said symmetry does not imply energy conservation violation.

Actually it does (insofar as "energy" can be defined at all in such
systems). That's why the definition changed once Noether's theorem
explained the inconsistencies. Of course in classical mechanics the
Lagrangian is time invariant, so the question never came up. shrug

That is a school-boy reasoning error. "If today is tuesday implies Tom
has to go to school" does NOT mean that "if today is not tuesday then
Tom does not have to go to school". There is a fundamental difference
between an equivalency and a definition, a difference which you should
learn.


Noether's theorem says that for every continuous symmetry of
the Lagrangian there is a corresponding conserved quantity,
and for every conserved quantity there is a corresponding
symmetry. It also specifies how either can be determined from
the other.

More accurately, it says that certain symmetries imply certain
conserved quantities. The difference is important.


If your
"abstract quantity" is conserved, there is a symmetry of the Lagrangian
corresponding to it,

That's not necessarily true. Noether's thereom applies only to theories
describable by a Lagrangian (or a Hamiltonian). In other cases, there
is no bijection between the set of symmetries and the set of conserved
quantities. For this reason there is NO reason to define energy in
general terms, as the time-translation symmetry of the Lagrangian.

and if that symmetry is not time translation
invariance, then your "abstract quantity" is not energy. shrug

Again NOT generally true.


In modern physics, energy is specifically defined as the conserved
Noether current related to time translation invariance of the system's
Lagrangian. This is not "abstract" at all. shrug

That's false again. Some theories don't carry Lagrangian of Hamiltonian
formalisms, so you can't express energy with the definition you gave.



If such quantity changes, then it is not
energy.

If there is no time translation invariance of the Lagrangian, then there
is no useful definition of "energy".

Profoundly wrong.


Lack of time-translation symmetry reveals more on the character
of GR/Noether's thereom than it does on energy conservation.

Your statements reveal more about your lack of knowledge about modern
physics, than about energy conservation.

Schoenfeld wrote:

[snip]
Tom Roberts wrote:
In modern physics, energy is specifically defined as the conserved
Noether current related to time translation invariance of the system's
Lagrangian. This is not "abstract" at all. shrug


That's false again. Some theories don't carry Lagrangian of Hamiltonian
formalisms, so you can't express energy with the definition you gave.

QM uses Hamiltonians, and GR uses Lagrangians (and I'm sure the
equations could be rearranged if needs be). They describe everything we
know so far, what theories do you refer to that *can't* be expressed
these ways?

br

Tom Roberts wrote:

Schoenfeld wrote:
Energy conservation of some system may be _equivalent_ to the time
translation symmetry of the systems Lagrangian but it is not
necessarily DEFINED that way.


Sure it is. This is 2005, not ~1900.

My complaint is that we know QM and GR have to be modified, so there is
a small window for the possibility for the creation of energy. Say
under an extremely high E-field (a single electron orbiting a Z=200
nucleus, for example), what if the vacuum really becomes unstable and
has a small runaway effect which spits out a million particles for no
currently known reason? (ok I'm making this up, but for the sake of
argument...). Can we say that energy wasn't created, *by definition*?
Seems a bit over the top.

What if such a runaway condition accidentally happened in the centre of
the Sun, which turned into a gamma-ray burster and fried Earth?

Of course I'm not arguing that we can actually create energy, or the
above situations will happen, but I must admit discomfort in going from
"this is the way the universe seems to work" to saying "this is the way
the universe *does* work, *by definition*".

It reminds of the case of *defining* the speed of light as constant.
Sure it's consistent with everything we currently know, but I don't get
the point of *defining* it that way rather than saying that's just the
way it happens to be (as far as we know).

br

Tom Roberts:
Bilge wrote:
The metric has 16 components, 10 of which are independent, so
the metric has 10 degrees of freedom. You will be able to define
a conserved energy if you can make the metric time independent.
You are free to make a change of coordinates to do that, but a
coordinate transformation only has 4 degrees of freedom. In general,
that is not sufficient to remove the time dependence. That means you
cannot define a quantity called the energy such that it is conserved.

The usual way of stating this requirement is that the manifold must have
a timelike Killing vector (that guarantees the 4-fold freedom of a
coordinate transfom is indeed sufficient to remove the time dependence

The ``usual'' way of stating anything depends upon the person
or persons to whom you are stating it.

from all metric components;

If that were the case, every metric would be invariant under a
time translation.

it of course does this by limiting the
permitted form of those components). This GREATLY limits the set of
manifolds with this property. In such a manifold, using the timelike
Killing vector as the time coordinate, nothing moves, so it's not useful
as a model of the real world.

BTW the metric components have 10 ALGEBRAICALLY-independent
degrees of freedom, but they are not completely independent.

I have no idea what you mean by that. My best guess is that you are
trying to say that because several degrees of freedom are coupled,
the number of degrees of freedom is reduced, which simply isn't
true. The number of independent degrees of freedom is not a function
a function of whether or not those degrees of freedom are separable.

In fact, the Einstein field equation and the Bianchi identities
are sufficient to determine them all (given suitable boundary
conditions, as these are differential equations, not algebraic
ones).

Huh? Who said anything about not being able to determine any
metric components? Quie honestly, I havent the slightest idea how
any of this is connected to anything I wrote.

Schoenfelch:

Bilge wrote:

Here's the basic problem with defining a conserved energy, which
addresses your question regarding noether's theorem, too. Energy is
defined as the conserved quantity associated with invariance under
time translations. That comes straight from noether's theorem.

Energy conservation of some system may be _equivalent_ to the time
translation symmetry of the systems Lagrangian but it is not
necessarily DEFINED that way.

Find another hobby. This one isn't working for you.

The difference is important because lack
of said symmetry does not imply energy conservation violation. Energy
is DEFINED as an abstract quantity remaining constant throughout a
closed systems time-evolution. If such quantity changes, then it is not
energy. Lack of time-translation symmetry reveals more on the character
of GR/Noether's thereom than it does on energy conservation.