Strangeness Nuggets to Strangeness Condensation

[Garth]

Einstein's field equations imply energy conservation.

Einstein's field equations imply energy-momentum conservation, not, in general, energy conservation.

Energy is a frame dependent quantity, you have to specify a specific frame to make any measurement of energy meaningful it, and then, if there is not a time-wise Killing vector, it is not conserved.

Energy-momentum is a frame independent quantity and in GR is conserved:

T^{\mu}^{\nu};_{\mu} = 0

Garth

 

[marcus]

Einstein's field equations imply...

Energy ... is not conserved.

IIRC there is a Usenet FAQ about this, that I used to get via John Baez site. the fact that Gen Rel doesn't conserve energy has bothered people and they have used various kludges, like "pseudotensor", to get some form of energy-conservation.

but two naive observations are:

if our universe is infinite then it does not have a total energy! So then the question is hard to ask. On the other hand, if it is finite then according to the standard cosmology it's total energy is even now growing rapidly as it expands------since Lambda is a constant energy density (around 0.6 joules per cubic kilometer)

a constant energy density like our current Lambda is usually assumed as the mechanism of inflation----the same as today but souped-up

===============
also here is another thought: when we are talking about "big bounce" where QG replaces classical Gen Rel, and gets rid of the singularity, then classical Gen Rel is no longer in control------it breaks around the former singularity

so if classical Gen Rel is no longer in control at that point, then maybe we should be asking not does Gen Rel have an energy conservation law but DOES QUANTUM GRAVITY have an energy conservation law? what sort of quantities does one expect to be conserved in the socalled Planck regime right around the moment of the bounce?

 

[marcus]

would anybody like to discuss how, even in classical GR, the energy-momentum tensor could be conserved GLOBALLY, that is as an integral over some spatial slice?

I mean to distinguish this from having a quantity defined at a point that is conserved locally.

if we are talking about a global quantity belonging to the whole universe (as I think hossi was talking) then how do you define it? what time evolution conserves it? what sort of machinery do you need to make the idea of conservation meaningful? how do you integrate?

it would be great if someone wants to explain, simply what the mathematical setting would be to have either energy or energy-momentum globally conserved on a universe-wide basis----even in just CLASSICAL gen rel.

 

[Garth]

Even in an infinite universe you can always work with a representative volume of average cosmological density.

Garth

 

[marcus]

Even in an infinite universe you can always work with a representative volume of average cosmological density.

Garht

I know, Garth
but it is just one more complication (the volume has to expand with the universe) so suppose we keep it simple and consider a finite universe!

I still have my question. You can extend it to cover infinite case if you want:

would anybody like to discuss how, even in classical GR, the energy-momentum tensor could be conserved GLOBALLY, that is as an integral over some spatial slice?

I mean to distinguish this from having a quantity defined at a point that is conserved locally.

if we are talking about a global quantity belonging to the whole universe (as I think hossi was talking) then how do you define it? what time evolution conserves it? what sort of machinery do you need to make the idea of conservation meaningful? how do you integrate?

it would be great if someone wants to explain, simply what the mathematical setting would be to have either energy or energy-momentum globally conserved on a universe-wide basis----even in just CLASSICAL gen rel.

 

[Garth]

I know, Garth
but it is just one more complication (the volume has to expand with the universe) so suppose we keep it simple and consider a finite universe!

I still have my question. You can extend it to cover infinite case if you want:

would anybody like to discuss how, even in classical GR, the energy-momentum tensor could be conserved GLOBALLY, that is as an integral over some spatial slice?

I mean to distinguish this from having a quantity defined at a point that is conserved locally.

if we are talking about a global quantity belonging to the whole universe (as I think hossi was talking) then how do you define it? what time evolution conserves it? what sort of machinery do you need to make the idea of conservation meaningful? how do you integrate?

it would be great if someone wants to explain, simply what the mathematical setting would be to have either energy or energy-momentum globally conserved on a universe-wide basis----even in just CLASSICAL gen rel.

The general idea is that as energy-momentum is a frame independent quantity it is conserved on any space-like surface. The key idea, as taught in MTW, is the 'boundary of a boundary is zero' (Gravitation Box 15.1 Pg 365).

Garth

 

[marcus]

I still think someone should spell out, conserved in what version of time, on what foliation by spacelike surfaces etc. And explain using online sources to which we all have access. But probably that is not going to happen. So I will give a page by John Baez and Michael Weiss called
Is energy conserved in General Relativity?
http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html

According to them, there is no simple answer, it depends on special conditions and choices, but at least it is something available online about energy conservation in GR.

this page is part of the Usenet Physics FAQ.
(probably everyone here has consulted it at one time or another)

they refer to several chapters in the famous textbook MTW, to which Garth also referred. I hunted for MTW online one time. never found it.

anybody have a good online reference besides this Baez and Weiss one that they want to recommend?
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Garth, what I suspect is that when you integrate energy-momentum you are going to get a whole lot of cancellation that DEPENDS on the folliation by spacelike surfaces that you choose. And so the end result is going to be that you get NOTHING YOU CAN IDENTIFY AS ENERGY that is being conserved. this probably
agrees more or less with what you were saying------energy is not conserved in GR.
(in other words energy-momentum is conserved but it doesn't do you any good----but maybe you have some other way to say this)

 

[Garth]

I still think someone should spell out, conserved in what version of time, on what foliation by spacelike surfaces etc. And explain using online sources to which we all have access. But probably that is not going to happen. So I will give a page by John Baez and Michael Weiss called
Is energy conserved in General Relativity?
http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html

According to them, there is no simple answer, it depends on special conditions and choices, but at least it is something available online about energy conservation in GR.

this page is part of the Usenet Physics FAQ.
(probably everyone here has consulted it at one time or another)

they refer to several chapters in the famous textbook MTW, to which Garth also referred. I hunted for MTW online one time. never found it.

anybody have a good online reference besides this Baez and Weiss one that they want to recommend?
===================

Garth, what I suspect is that when you integrate energy-momentum you are going to get a whole lot of cancellation that DEPENDS on the folliation by spacelike surfaces that you choose. And so the end result is going to be that you get NOTHING YOU CAN IDENTIFY AS ENERGY that is being conserved. this probably
agrees more or less with what you were saying------energy is not conserved in GR.
(in other words energy-momentum is conserved but it doesn't do you any good----but maybe you have some other way to say this)

That is what I am saying (in GR).

In GR it is the curvature of space-time, particularly the dilation of time, but also the fact that the space-time continuum itself can carry energy away through gravitational waves, that affects the conservation of energy.

There are global definitions of a conserved quantity called energy of a system in GR but these all only hold in the absence of curvature or gravitational waves, i.e. at a 'null inifintiy' away from a gravitating mass.

The energy of a particle is also conserved if there is a time-like Killing vector i.e when the metric components in the observer's frame of reference do not depend on time.

Thus, treating the Earth as a static field, we find that the total energy of an object falling towards the Earth is conserved. The effect of the increase in its kinetic energy (it is in free fall and no forces are acting upon it - no work is being done on it) is compensated by the change of time dilation acting on the object (as measured by an Earth clock), as it enters into the Earth's stronger gravitational field.

However if we look at it from the falling object's POV it is the Earth that is falling towards the observer and accelerating.

In this frame of reference the Earth's total energy increases as it appears to be freely falling towards the observer.

Now the metric components as measured by that falling observer do change with time, there is no time-like Killing vector. The kinetic energy of the Earth, measured by that observer, increases with time, however this time it is not compensated by time dilation, for time dilation now acts in the opposite sense and makes matters worse.

In this case energy is not conserved.

But note that in the first case, when energy is conserved, the frame of reference (the Earth) in which it is conserved is the one co-moving with the Centre of Mass of the system.

It is this thought: that local energy conservation requires a particular frame of reference and a time-like Killing vector, which are both provided by the the CoM of the system and also it is this same frame that is selected by Mach's Principle, which lies at the heart of my own work in http://en.wikipedia.org/wiki/Self_creation_cosmology . (Note: A Plea: Please be free to sensibily edit that Wikipedia article as its neutrality is questioned because I am the main author.)

Garth

 

[pervect]

The following historical article about Noether's theorem demonstrates some of the difficulties energy conservation has in GR.

There are some semantic differences with the Baez paper, hopefully they won't be too confusing.

http://arxiv.org/PS_cache/physics/pdf/9807/9807044.pdf

I would recommond Wald's "General Relativity" for a discussion of energy in GR - howver, it's a textbook, not an online reference. It's probably easier to get a hold of than MTW's book. There are at least 3 commonly used defintions of the energy and/or mass of a system in GR. These apply in different circumstances.

1) static space-times. These have a timelike translation symmetry which gives rise to a conserved energy directly by Noether's theorem. This gives rise to the Komar mass. This is the simplest sort of mass to deal with.

2) asymptotically flat space-times, considered at null infinity. This gives the Bondi mass.

3) asymptotically flat space-times, considered at spatial infinity. This gives the ADM mass.

The Bondi and ADM mass are very similar in that they both apply in asymptotically flat space-times. A brief summary of the difference besides "null infinity" vs "spacelike infinity" would be that the later includes the energy in gravitational radiation, while the former does not. Thus the Bondi mass allows one to study how a system loses mass via gravitational radiation, while the ADM mass gives a more strictly conserved quantity.

There are apparently some other useful notions of mass in GR, such as Dixon mass, but I don't know a lot about them (check the archive logs).

 

[marcus]

Hi there,

found this old thread while googling for cns... Have a very simple question. Let's assume that creation of universes through black holes somehow works, and that while doing so the parameters of the (cosm)SM change a little bit each time. What happens to the total energy of the universes?

...
...
Hoping for enlightenment,

B.

Just checking. Did we answer hossi's (Bee's) question?

Bee is the one who found this thread and reactivated it. With her post #27 of 5 July.

correct me if I am mistaken but I think we more or less agree that the answer to her question is NOTHING happens to the total energies of the various universes-----they keep on being determined by their inflationary epochs and whatever else determines their total energies. Is that right?

And is Bee still around, I wonder.

 

[pervect]

I gather you don't have MTW, but I'll quote it anyway.

There is no such thing as the energy (or angular momentum, or charge) of a closed universe according to General Relativity"

This can be found in the subject index, under "mass energy, no meaning of, for closed universe".

Open universes won't fare much better, being infinite.

It boils down to the technical conditions for being able to define a mass not being met. One needs either asymptotic flatness, or symmetry in time, to define a mass in GR. This is what the sci.physics.faq is trying to explain.

[add]
It might be possible to construct a cosmology where a universe did have a well defined mass, but it wouldn't be a standard cosmology. You'd need either a static universe, or one that had an 'edge' where the universe was surrounded by a vacuum. A universe in an infinite (preferably) or very large vacuum region, of such a density that it is not "closed", should be asymptotically flat, allowing its mass to be defined. The principle that cosmology is homogeneous and isotropic everywhere prohibits edges.

Garth's theory isn't standard GR, so these remarks don't apply directly to SCC.