Stress-energy tensor and active mass


by stevebd1
Tags: active, mass, stressenergy, tensor
stevebd1
stevebd1 is offline
#1
Nov17-08, 10:39 AM
P: 604
Imagine a 2.2 sol mass neutron star on the brink of collapse with a radius of 12 km, an average density of 0.605e17 kg/m^3 and an average EOS of ~1/7. Based on active mass (i.e. including for pressure), the stress-energy tensor (g) would be based on [itex]g=\rho c^2+3P[/itex] resulting in g≡3.143 sol.

The neutron star goes quark-nova, throwing off ~0.6 sol mass of matter and reducing to 1.6 sol with a radius of 9 km, an average density of 1.042e18 kg/m^3 and an average equation of state of ~1/8 (though density increases, pressure appears to drop once neutrons breaking down into smaller components, i.e. quarks, hence a marginally lower EOS). Based on active mass, the stress-energy tensor for the quark star is now g≡2.2 sol.

Allowing for the mass thrown off, this makes a difference between the ns stress-energy tensor and the qs stress-energy tensor of ≡0.343 sol. Allowing for the fact that the expelled matter may carry some of this away as kinetic energy (roughly ≡0.168 sol mass) still leaves 0.175 sol mass of stress-energy 'unaccounted' for. Is it possible this could result in a gravity wave? Where exactly does the stress-energy induced by pressure come from?


source for neutron and quark star specifics-
'Neutron star interiors and the equation of state of super dense matter' by F. Weber, R. Negreiros, P. Rosenfield
http://arxiv.org/PS_cache/arxiv/pdf/...705.2708v2.pdf page 3
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Naty1
Naty1 is offline
#2
Nov17-08, 03:36 PM
P: 5,634
Where exactly does the stress-energy induced by pressure come from?....here's a generalized description ...I happened to be reading about that very subject....pressure as a gravitational entity.

......Pressure is the flux of momentum:

from http://math.ucr.edu/home/baez/einstein/node5.html

here's a simple picture of a non accelerating ball of test particles:


Let be the volume of a small ball of test particles in free fall that are initially at rest relative to each other. In the vacuum there is no energy density or pressure, so , but the curvature of spacetime can still distort the ball. For example, suppose you drop a small ball of instant coffee when making coffee in the morning. The grains of coffee closer to the earth accelerate towards it a bit more, causing the ball to start stretching in the vertical direction. However, as the grains all accelerate towards the center of the earth, the ball also starts being squashed in the two horizontal directions. Einstein's equation says that if we treat the coffee grains as test particles, these two effects cancel each other when we calculate the second derivative of the ball's volume, leaving us with

And here's a view of an accelerating ball of test particles:

...consider a small ball of test particles, initially at rest relative to each other, that is moving with respect to the matter in the universe. In the local rest frame of such a ball, the right-hand side of equation (2) is nonzero. For one thing, the pressure due to the matter no longer vanishes. Remember that pressure is the flux of momentum. In the frame of our moving sphere, matter is flowing by. Also, the energy density goes up, both because the matter has kinetic energy in this frame and because of Lorentz contraction. The end result, as the reader can verify, is that the right-hand side of equation (2) is negative for such a moving sphere. In short, although a stationary ball of test particles remains unchanged in the Einstein static universe, our moving ball shrinks!

This has a nice geometric interpretation: the geometry in this model has spatial curvature. As we noted in section 2, on a positively curved surface such as a sphere, initially parallel lines converge towards one another. The same thing happens in the three-dimensional space of the Einstein static universe. In fact, the geometry of space in this model is that of a 3-sphere. This picture illustrates what happens:
go to http://math.ucr.edu/home/baez/einstein/node9.html




One dimension is suppressed in this picture, so the two-dimensional spherical surface shown represents the three-dimensional universe. The small shaded circle on the surface represents our tiny sphere of test particles, which starts at the equator and moves north. The sides of the sphere approach each other along the dashed geodesics, so the sphere shrinks in the transverse direction, although its diameter in the direction of motion does not change.
stevebd1
stevebd1 is offline
#3
Nov18-08, 06:39 AM
P: 604
Thanks for your reply Naty1. Pressure as a flux of momentum goes someway to explaining the other equation I've seen in relation to active mass (which is also shown to some extent on the Baez website under 'gravitational collapse'-

[tex]g=\rho+\frac{1}{c^2}(P_x+P_y+P_z)[/tex]

which explains the 3P and demonstrates the pressure as confined momentum/kinetic energy.

It still doesn't say exactly where this 'extra' energy comes from in the context of the 1st law of thermodynamics. I suppose this leads to the question where does the kinetic energy that is induced by velocity or rotation come from. Is it a residual energy brought on by what ever process was applied to get the object moving in the first place? i.e., the increase in pressure was brought on by the collapse of a star, hence some of the original stars mass was converted to confined momentum (i.e. pressure), a rapidly rotating object's kinetic energy (neutron star) was again, brought on by the collapse of a rotating star, the increase in kinetic energy brought on by conversation of angular momentum meaning some of the mass of the dying star is converted to kinetic energy (as a projectile fired from a gun absorbs the energy released from the chemical reaction from the gunpowder being struck).

If this is the case, then any change (i.e. drop) in a relativistic EOS that results in a significant drop in pressure might very well produce significant gravity waves.

Naty1
Naty1 is offline
#4
Nov18-08, 11:42 AM
P: 5,634

Stress-energy tensor and active mass


You are likely at the level where "we just don't know" comes to the fore.

I still find it incredible that without knowing the finest level of details, what is "really" going on, physicsts are able to concoct mathematics that predicts so many phenomena and confirm many with experimental evidence. The related part that's so crazy is the nature appears to follow mathematical constructs that we put together....

"Pressure is the flux of momentum",,,but who thought of that?? How many other thoughts were proposed, discarded and how did the "flux" interpretation win out???

In another thread here someone posted the components of GR tensors don't have physical interpretations but I fid that hard to accept...


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