# Definition of Negative Pressure in the Stress-Energy Tensor

1. Jan 6, 2012

### putongren

I've been trying to find out what the definition of negative pressure in the context of General Relativity. There are some information on negative pressure and I found this:

https://www.physicsforums.com/archive/index.php/t-299495.html

The starter of that thread also doesn't know what the physical meaning of negative pressure.

According to General Relativity, positive mass, positive energy, and negative pressure leads to repulsive gravity.

I read somewhere that repulsive gravity is related to the negative pressure that exists on the cosmological scale or inside spherical objects such as planets.

I guess maybe I should explain why I want to know the definition of negative pressure. There's a researcher who's trying to create a device that can spin electrons in a crystal lattice. Since spinning an object that possesses gravitational mass creates a gravito-magnetism (a second gravitational field so to speak), spinning electrons would also create a minuscule gravitational field as well. If you spin these astronomically-large number of electrons in a crystal lattice in the same direction at the same frequency, a coherent gravitational field will emerge. It would seem that the gravitational field is being amplified around the crystal lattice. I think some of you might be already familiar with this technology that is being developed.

I was wondering if it's possible to amplify the negative pressure at a given point so to create an amplified coherent gravitational repulsive field. Technology that can produce this effect would have an amazing effect on space exploration development.

Last edited: Jan 6, 2012
2. Jan 6, 2012

### Mentz114

I don't get it - pressure has the units of energy density and in GR it represents a flux through a unit surface in unit time. To be negative, it must be a negative flux surely, and so implies negative energy. Or, can the direction make it negative ( that seems to be coordinate dependent).

On repulsive gravity - this can happen in other (ultra-relativistic) scenarios like the Schwarzschild radial geodesic where d2r/dt2 has two terms of opposite sign and for large enough dr/dt the negative term dominates. This is why in holonomic coordinates the geodesic doesn't reach the horizon. Won't help with the electrons though.

3. Jan 6, 2012

### putongren

Hey thanks for the answer. I have a very surface-level understanding of GE. I still don't understand your last paragraph Mentz. What is holonomic?

4. Jan 6, 2012

### Mentz114

Holonomic coordinates are those used to write the metric, as opposed to local frame coordinates. Holonomic coords describe what is seen by 'the observer at infinity'. In local frame coords the faller just sails through the horizon in a finite time as recorded on their clock.

So the repulsive gravity I refered to is coordinate dependent to that extent.

5. Jan 6, 2012

### Staff: Mentor

Negative pressure means you have a perfect fluid stress-energy tensor where the pressure is negative. The general form of a perfect fluid stress-energy tensor is given, for example, here:

http://en.wikipedia.org/wiki/Fluid_solution

The only example I'm aware of where such a negative pressure is actually postulated to exist, physically, is vacuum energy density. Ned Wright explains why if the vacuum has a non-zero (positive) energy density, it has to have a negative pressure, here:

http://www.astro.ucla.edu/~wright/cosmo_constant.html

"Dark energy" is a form of vacuum energy density, so it has negative pressure; that's why it causes the expansion of the universe to accelerate. That's presumably the "cosmological scale" negative pressure you're referring to.

I'm not aware of any case of negative pressure being postulated inside massive objects like stars or planets. Do you have any references?

I'm not. Are there any publicly visible references?

Since, as I said above, I'm not aware of any case of negative pressure inside an object, I don't think there's anything to be amplified.

6. Jan 6, 2012

### putongren

http://www.scansite.org/scan.php?pid=157

I lost the article about negative pressure existing inside spherical objects. Sorry.

7. Jan 6, 2012

### Staff: Mentor

I was hoping for some actual papers; this article says "Li’s theory has passed through the scientific quality-control peer review process" so there should be some somewhere. On arxiv.org would be nice.

I can't comment much on the article itself since it doesn't give any real technical information. It makes some statements that, taken as written, are incorrect, but that may be either translation problems (I get the impression that English is not the author's first language) or a non-technical person trying to interpret a technical theory that they don't really understand (the article's author is not the person actually doing this research). That's one reason why it's usually better to look at an actual paper that is written for a technical audience. However, from what I can see, it does not look like the effect being described is due to negative pressure.

8. Jan 6, 2012

### putongren

The negative pressure effect part is just something I made up. Ever since reading this article, I just thought maybe it was possible to build a machine that could propel spacecraft using small-scale gravitational repulsion. Just wide-eyed day dreaming.

Sorry about not having any technical details. My main purpose in this post is just to know whether that idea is conceptually feasible.

9. Jan 6, 2012

### Antiphon

You can do thought experiments using unusual mass distributions to simulate this type of field.

For example, hollow out a small sphere in the core of the earth. Because you have surrounded the empty space in the sphere with a large mass distribution, the hollow core will behave as if it's got exactly the pressure you describe.

How? Well, place a cluster of marbles in the exact center and release them. What happens? They will all be "repelled" from the center as if there were a finite mass there but with a repulsive gravity field.

10. Jan 6, 2012

### PAllen

So far as I know, they will do no such thing. The metric inside a spherical mass shell is flat Minkowski.

11. Jan 6, 2012

### Antiphon

You're right. We can introduce a little asymmetry and fake it for illustration.

The earth has a hollowed out sphere as before. The earth has also been sliced down the center and a 1 meter thick annulus of styrofoam is sandwiched between the halves.

Now the marbles will look as if they are being repelled from the center (but along the axis connecting the two halves of the earth).

12. Jan 6, 2012

### PAllen

Ok, that seems to work, but I don't see the relevance to negative pressure terms in the stress energy tensor. G and T tensors would be identically zero throughout the empty region (though no longer flat Minkowski).

13. Jan 6, 2012

### Staff: Mentor

Gravitomagnetism is not "gravitational repulsion", so no, it doesn't sound feasible to me.

14. Jan 6, 2012

### pervect

Staff Emeritus
Negative pressure is just tension. Consider a bar surrounding the x-axis. Then if the bar is under tension T_xx is negative. Similarly, if the bar is in compression, T_xx is positive.

Or consider a spherical pressure vessel. If you fill the vessel with compressed gas, then T_{theta,theta} and T_{phi,phi} are negative because the pressure vessel is under tension.

15. Jan 6, 2012

### Staff: Mentor

Negative pressure is tension. It is very common for solutions involving solid shells or other similar structures to have tension.

EDIT: I see that pervect already mentioned that.

16. Jan 6, 2012

### Staff: Mentor

Oops, good point! I should thought more carefully about what I was trying to say.

In the cases described above the pressure is not isotropic; T_rr is still positive. So the stress-energy tensor is not of the "perfect fluid" form. AFAIK there is no case other than vacuum energy where a *perfect fluid* SET has negative pressure.

And, of course, the "negative pressure" (i.e,. tension) in the cases described above does not cause "gravitational repulsion". Gravity around the bar or the spherical pressure vessel is still attractive. Again, AFAIK there is no case other than vacuum energy where negative pressure actually causes "gravitational repulsion" (as in the accelerating expansion of the universe).

17. Jan 7, 2012

### timmdeeg

Negative pressure and thermodynamics.

The first law (assuming adiabatic expansion) states dU = -pdV.
In the case of expanding an ideal gas, the work done is taken from it's internal energy, therefore the sign of dU is negative. As dV is positive, also the pressure p is positiv.

However, if the expansion is driven by a cosmological constant thus providing the internal energy needed, then dU is positiv. And the vacuum pressure negativ accordingly.

18. Jan 7, 2012

### pervect

Staff Emeritus
Let me clarify this just a bit - I was rereading it and I wasn't sure if it was clear that I meant that the stress energy tensor in the shell, i.e. the pressure vessel, was negative.

The pressure in the gas filled interior would be positive, and most likely be isotropic.

19. Jan 7, 2012

### Staff: Mentor

I assume you mean the theta-theta and phi-phi *components* of the SET. The other nonzero components (T_rr and T_00, the energy density) will be positive (and T_00 will be much larger in magnitude than any of the others).

20. Jan 7, 2012

### Staff: Mentor

This is basically the argument Ned Wright gives on the page I linked to in post #5.