Fascinating thread. I can't add anything about the main topic, but about the gravity part.

In the EFE (field equations) of GR, the source term is a rank-2 tensor of 4 dimensions (3space + 1 time), the stress-energy tensor, which can visualized as a 4x4 matrix. That sucker includes the energy content, the currents of that energy (similiar to how EM depends on the charge plus the currents of that source charge), plus the momentum tensor, which includes the scalar pressure of a system.

The rows and columns are thus tabulating how much energy and momentum in this direction are flowing this way and that. The term in the upper left corner, T_00, is the energy density.

The solution gives you the space-time metric, basically the "shape" of space-time due to the given mass-energy-momentum distribution. And what that can be is a lot more complex than Newton.

The Newtonian limit comes when all other terms go to zero, and you're left with T_00. T_00 is basically the total energy, with the mass equivalent being that over c^2. So basically, the gravity comes from all the energy that's there, rest energy plus kinetic energy plus radiant energy plus any other energy that's there.

And note that means the energy that's actually there, not the energy before it all fell together from infinity. That will include the Newtonian notion of gravitational binding energy. So in general, bound systems will have less total energy than if the components were unbound (assuming it shed most of that binding energy, which most systems do, I'm certain -- you could imagine letting something collapse in a closed box and somehow holding all the energy inside)

In GR, the notion of Newtonain gravitational (binding) energy basically doesn't mean anything -- you've got what you've got at any point. That may be confusing, let's elaborate.

Consider a little test mass, 'm' sitting there stationary down in a Schwarzschild well. The (coordinate) rest energy the distant observer says is there gets modified by the metric, and will be less than mc^2,
mc^2*sqrt(1 - R/r), in fact. And so the "lost binding energy" is automatically accounted for. If 'm' is stationary at some point r, then there's less energy there than if were sitting out at infinity. It had to shed it.

Now, suppose we dropped that test mass 'm' in from infinity. It would start out with E = mc^2, and the total energy would remain the same as it fell. We'd just say that some of that E was "converted" to kinetic energy.Our T_00 term would include additional contributions from the kinetic part (basically something that can be made to look like the SR gamma factor, although it is modifed by the metric). If we stop the falling mass at some r, we've got to do something with the remaining kinetic energy, and that's the Newtonian binding energy. If it stays on the scene in some fashion (say gets converted to photons), then it's still there as more sources beyond the 'm' sitting there. When it leaves, as photons tend to do, well it's gone.

-Richard

That went on longer than I intended, but there's there's some more little tricky things here about the energy. The distant observer (at infinity, technically) says the little 'm' sitting at r has less rest energy than it did at infinity.

However, the local observer right there at 'r' says the rest energy remains mc^2. How can that be? Well, that's his slower clock rate relative to us. He would say it had *more* than mc^2 rest energy at infinity, as clock rates are higher there according to him.

But both observers, who can solve the EFE in the their own coordinates come up with the same invariant space-time manifold. They would say the "total energy" of the system is different, yet agree on the invariants.

And that gets us to an important, tricky point. In GR, the "total energy" of a global system, and hence the notion of the total mass cannot be defined, generally, in any invariant manner! *LOCALLY*, there's no problem --defer to a local observer for the local energy density (and that's what the stress energy tensor is actually doing, although it gets complicated, well beyond me). The problem is integrating (adding it all up) globally. In general, no two different observers will come up with same thing.

However, there are some attempts at defining that total energy or mass in some way to be useful under certain circumstances and restrictions. A few well known ones are the ADM mass, the Komar mass, the Bondi mass, for instance, which are various ways to do it (which I don't understand, so don't ask me to 'splain them! ).

So it's sort of flusterating, but that's GR for you. A given stress-energy distribution makes a given field, but how much total energy (and momentum, etc, etc) is there is a relative, coordinate dependent thing.

-Richard

Hmmpphh. I read a little about the Komar mass. I knew the details would get complicated.

In EM, note than div E ~ rho, the charge density, so integrate div E up over up some region to get the total charge enclosed. Or convert that to a surface integral of the flux of E.

Now, the Komar mass is basically, do something similiar in GR to come up with the total "gravitational charge" enclosed in a region. Lots of pretty tensor math, there, but it's the same thing, just bigger and more complicated.

That can give you a scalar number involving integrals of the stress-energy tensor. And the result depends on the *pressure* as well as the energy. So this notion of "gravitational charge" is actually more than some notion of E/c^2!

Now, that only works for a *stationary* space-time, which means basically that there is some observer who can write the metric so that it does not depend on time. (The meaning of "stationary" vs "static" are sort of backwards from what I'd think. Stationary means not dependent on time, while static means no time-space cross terms, ie no frame dragging. Static further implies there is a well defined "rest frame" for the source, and a static space-time is always itself stationary, but a stationary space-time does not have to be static).

Now, "stationary" is an invariant notion, actually, even though some observers might see things changing with time! It has a rigorous definition, but basically it means you can find some coordinate system where nothing is changing with time. If no such system exists, then the space-time is "truly dynamic".

When that is the case, the Komar mass is out the window because the result would be, well, changing with time itself. A notion of some invariant "gravitational mass" of the system should not change with time. When the space-time is stationary, the result is invariant and constant.

Komar is one way to do it. There are others.

-Richard

Here's a little more info, if anyone is interested. The Komar, ADM, and Bondi masses all converge to the same thing in the weak field, Newtonian limit, and so do capture, at least in that limiting sense, the notion of the gravitational mass ("charge").

In that limit, this mass will be just M - E/c^2, where E is the Newtonian brinding energy, and M is the mass "at infinity", the regular Newtonian mass before it all fell together in whatever gravity well it did.

In the non-weak field regimes, it gets complex and all "depends on what you mean by mass and what 'is' is" and all that good stuff.

The Komar mass works for stationary space-times. When the space-time is stationary, the notion of energy/mass is well defined and an invariant can be pulled out for it.

In non-stationary cases, it doesn't work, unfortunately. However, if the space-time is asymptotically flat, one can define the energy/mass "at infinity", which I presume means in the coordinates of some observer (or class of them) at infinity. That's what the ADM and Bondi masses do.

The ADM mass is done at spatial infinity, while the Bondi mass is done at "null" (light-like) infinity. What that latter really means don't ask me. I think it means roughly this. We've got some coordinates. Spatial infinity means let the space coordinates go to infinity. Temporal infinity would be let the time coordinate go to infinity. Null infinity would be, I think, follow a null geodesic, a light world line, out to its infinity.

At any rate, there is a difference. The Bondi mass of a system will not include any energy that a system looses to infinity via gravitational radiation, while the ADM mass will include it. The Komar mass couldn't even be defined in such dynamic situations.

Similiar things can be done with angular momentum in these special cases, but I gather that gets even more complex.

However, the rub is that in non-stationary, non-asymptotically flat space-times, none of these, total mass, L, etc, can be defined in any meaningful, invariant manner.

A question I have now is what does Lambda, the cosmological constant contribute to these masses, and if so, what would it be for deSitter space-time, which is static?

-Richard

I was thinking more about this. If you start from this end and you can calculate the radiant energy in the Sun very accurately, and as we already know very accurately the rate at which it is coming out, can't we get the average time taken to high precision, bypassing the uncertainties of path lengths in random walks?

Hi, Rtomes!

Like everything dealing with a trillion-trillion-trillion photons interacting with a trillion-trillion-trillion atoms, molecules, subatomic particles, atomic and thermonuclear processes, I don't think there's a single answer.

I believe that some photons take about one second to escape. I also believe that there are some that have bouncing around in there for millions of years.

As to an average?

I haven't a clue!

Has Grant tackled this? He and Astromark are usually pretty good on this kind of stuff. I only know the generalities, not the details.

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If I set the budget, we'd have Ares and more. Unfortunately, I don't set the budget, and Ares is just too expensive and too far out for us to accomplish our goals within the budget we were given.

If we halt the ISS, all versions of Ares, and transport Orion and Altair aboard DIRECTv3's Jupiter family of Shuttle-Derived Launch Vehicles, we just might make it back to the Moon by 2020.

I'm not a solar scientist, Rtomes. I do know enough to know that this is a vastly over-simplified approach to the problem, which is incredibly more complicated than this.

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If I set the budget, we'd have Ares and more. Unfortunately, I don't set the budget, and Ares is just too expensive and too far out for us to accomplish our goals within the budget we were given.

If we halt the ISS, all versions of Ares, and transport Orion and Altair aboard DIRECTv3's Jupiter family of Shuttle-Derived Launch Vehicles, we just might make it back to the Moon by 2020.

170k years rings a bell with an article I read about 15 years ago.

As does your other comment. I seem to recall some article about the sun (or any star) stopping and restarting it's fusion cycle, and how we wouldn't notice it.

So, that raises another question - does the 11,000 sunspot have anything to do with a similar 11,000 on/off fusion cycle for the sun?

Again, not an original idea. I just read something along those lines, long ago...

Think of the engine on a B-25 Bomber... They sort of came and went for a while until the ignition actually took hold...

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If I set the budget, we'd have Ares and more. Unfortunately, I don't set the budget, and Ares is just too expensive and too far out for us to accomplish our goals within the budget we were given.

If we halt the ISS, all versions of Ares, and transport Orion and Altair aboard DIRECTv3's Jupiter family of Shuttle-Derived Launch Vehicles, we just might make it back to the Moon by 2020.

Well the radius of the Sun is 2.32 light seconds, so that seems rather unlikely. Even allowing for that little miscalculation and saying you mean a few seconds, ...

With an average path length of about 0.9 mm before getting scattered, I think that you will find the chance of any given photon taking only one second to escape is around about 1 in 2^(300000 km / 0.9 mm) or 1 in 10^100000000000. Even allowing for the large number of photons escaping the Sun and its long life, I think we can safely say that no photon ever flew straight out.

Yes, but the number you get is the average time a randomly chosen "erg" of radiant energy will take to escape. It's not terribly clear what else that means-- it's not a "photon", because the individual photons are being created and destroyed all the time. The "random walk" picture tries to respect the quantum identity of each photon, but an "erg" emitted in the core breaks down into N gamma-ray photons at first, but cascades into the vastly more numerous M visible photons by the time they escape, and the majority of the time will be spent as X number of X-ray photons, but even that increases as we go from hard X-rays to soft X-rays, so which are we counting more? The energy weighting is different if you think in terms of photons versus "ergs", and that's what makes the former harder to quantify. The time you refer to is certainly the cleanest and most precise, but people like to imagine the individual photons "bouncing around".

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Logic is the grammar of truth.

Meaning and absolute certainty are incompatible, and profound meaning and absolute certainty are profoundly incompatible.

The only thing intelligence is capable of is recognizing itself.

I'm not sure about this 11,000 year sunspot cycle, the only one I know is an 11-year half-cycle. Certainly nothing that is happening on 11,000 year timescale in the core should ever affect anything going on outside the core, no evidence of it will survive the diffusion process on timescales much shorter than a million years.

__________________
Logic is the grammar of truth.

Meaning and absolute certainty are incompatible, and profound meaning and absolute certainty are profoundly incompatible.

The only thing intelligence is capable of is recognizing itself.

Not all photons are produced at the center! Some are produced on the surface due to simple heating!

__________________
If I set the budget, we'd have Ares and more. Unfortunately, I don't set the budget, and Ares is just too expensive and too far out for us to accomplish our goals within the budget we were given.

If we halt the ISS, all versions of Ares, and transport Orion and Altair aboard DIRECTv3's Jupiter family of Shuttle-Derived Launch Vehicles, we just might make it back to the Moon by 2020.

Ooops... Sorry. I'd been talking with someone on the phone about ice ages.

You're correct - for sunspots, it's 11 years, not 11,000.

__________________
If I set the budget, we'd have Ares and more. Unfortunately, I don't set the budget, and Ares is just too expensive and too far out for us to accomplish our goals within the budget we were given.

If we halt the ISS, all versions of Ares, and transport Orion and Altair aboard DIRECTv3's Jupiter family of Shuttle-Derived Launch Vehicles, we just might make it back to the Moon by 2020.

I assume the 170,000 year walk time is from the center of core to the top of the radiative zone.

Since the radiative zone is about 65% of this distance, then would the walk's time by about 110,500 years from the core's top to the tachocline (boundary between radiative zone and convective zone)? I would guess it is less assuming the mfp is less in the core region.

What is intriguing about this is the idea that we are seeing 100,000 year-old light (daughter photons, if you like) from the core.

Of course, if the core were to suddenly hiccup, then we would not have to wait that long since a pressure wave would not take that long to propogate. This wave, I suspect, would greatly change the mfp, and our photons could surf it, right?

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Lighten up! This is a stellar board!

No, it would be virtually all of the 170,000 years. The time is not proportional to the distance, because the mean free path is much larger the higher up you go. Also, in the convection zone, the photons are carried by gas motions like a person going up an escalator, and cover distances on speeds like the sound speed (which is tens of km/s). So the photons are dumped through the convection zone on timescales no longer than a single day.
Right, and then there's the convection.
One can think of it that way. It would be more strictly correct to say that the photons were "just born" in the photosphere, made out of energy that took 170,000 years to get there if you only count the free-propagation time (more like millions of years total), but these are all just pictures anyway so I don't think any of them are "wrong" in any serious sense.
The core photons couldn't surf the wave, they'd be left behind by it, but if the wave made it to the surface without dissipating first, you would see how the wave altered the surface right away. It would alter the escape of the photons near the surface, whose energy was released 170,000 years ago (or the few-million year number we got via other means is probably more relevant).

__________________
Logic is the grammar of truth.

Meaning and absolute certainty are incompatible, and profound meaning and absolute certainty are profoundly incompatible.

The only thing intelligence is capable of is recognizing itself.

I would think the millions of years is the more appropriate time if we are refering to the random walk time. [I doubt Nascar deducts the racer's pit time from their total elapsed time of the race? ] Both times are nice to know.

I was toying with the idea of the pressure wave altering the mfp so much so that the new, faster propogation rate would match that of the wave. [I used to surf a lot. ]

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Lighten up! This is a stellar board!

I am interested in the proportion of the total mass or energy content of the Sun that is radiation (or possibly the total relativistic content which you say is much higher). It doesn't matter that photons have a short life.

You can see the purpose of these questions in the ATM thread Explaining Planetary Alignments Relationship to the Sunspot Cycle

The numbers in the thread say that about 1/10 of the heat energy in the Sun is radiative. A far less amount (1/1,000,000 or so) of the mass-energy is in radiation.

__________________
Logic is the grammar of truth.

Meaning and absolute certainty are incompatible, and profound meaning and absolute certainty are profoundly incompatible.

The only thing intelligence is capable of is recognizing itself.

Hi all,
I've revisited this thread to ask a question about something said by a solar physicist on another forum in relation to Ray Tomes ideas about the sun's 'relativistic content'. In his ATM thread Ray states:
The solar physicist I've been trying to discuss this with says:
It could be that there is a problem with terminology here. Ray himself is at pains to point out that because no-one has considered this before, there isn't a lexicon of agreed terminology to describe the process he believes is happening.

I'd be very grateful if anyone in the know could comment on what this solar physicist has said and help shed some light on the question of whether the exchanges taking place between photons and matter in the interior of the sun would lead to there being a 'gradient' of 'relativistic content' from the core to the surface.

Thanks in advance for any replies.

An early theorem by Eddington is that to a rough but useful degree, the ratio of kinetic energy in gas to the energy in radiation is more or less constant throughout a star. It varies a great deal from star to star (see "Eddington parameter"), but not so much within a star. The starting point is to assume this ratio is constant, and then look at possible ways it can actually vary in specific cases, but one should not expect a steep change in it. I have no idea what that "energy production rate of a candle" business is meant to convey, I guess he's saying a candle has a very small volume but a lot of heat comes from that tiny volume. Still, the radiation temperature in the core of the sun is over 10 million Kelvin, so it's really quite hot and bright in there (X-ray bright), but the gas is also similarly hot-- and not at all relativistic, even the electrons.

1 million years is the time given in "Universe", a coffee-table book edited by Marin Rees that I was given, for a photon produced in the core to pass through the plasma of the Radiative zone.

Seems a suspiciously round number...

Earlier in the thread, a paper stating 170,000 years seemed to be the accepted consensus.

Hi Ken, many thanks for your reply. I've read through the thread a couple of times. Most of it still goes over my head, but I hope I've gleaned a little understanding.

One comment I would like some clarification on is where you said the following in reply to Ray:
Are you saying here that the 'negative gravitational correction' cancels out the small relativistic factor Ray is talking about?

I've probably framed that question badly, and highlighted the depth of my ignorance, but hey ho.

In a sense, yes. The "relativistic mass" correction is simply the kinetic energy, using E = mc^2. So the Sun has more relativistic mass than it would if it were the same except for being at absolute zero temperature. However, when an object in pressure equilibrium (like the Sun) loses heat to its surroundings, it contracts and gets hotter. The graviational potential provides the energy needed. So a contracted hot object like the Sun does indeed have a high kinetic energy content, but it has an even higher negative gravitational energy, so has less total energy than it started with prior to losing heat. Less energy means less mass-- in a sense, I'm saying the gravitational potential counts in the "relativistic mass" also, not just the kinetic energy. So rtomes has even the sign wrong in the relativistic mass correction! But more to the point, whatever its sign, it is tiny fcompared to the rest-mass-energy of the Sun.

Here is the paper you cite in full. You can see how they make the calculation: http://articles.adsabs.harvard.edu/f...pJ...401..759M

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