Experimental tests of the stress energy tensor

[relativityfan]

hi,

I wonder if the stress energy tensor formulation of general relativity has been well experimentally verified, since the schwarzchild metric is used most of the time.
for example, is there any experimental evidence that an increase of the pressure with the same energy density, increases gravity?
has anything like that been experimentally verified?

 

[Mentz114]

You should be aware that 'pressure' referred to in the SET is momentum flux per unit time through a unit area ( if I remember correctly) so not quite the same as force/area.

It's not the sort of thing that can be experimentally verified with current technology. The gravitational field of anything smaller than a mountain is too slight to be measured accurately enough.

 

[bcrowell]

You should be aware that 'pressure' referred to in the SET is momentum flux per unit time through a unit area ( if I remember correctly) so not quite the same as force/area.

But the rate of transfer of momentum per unit time is force, so I think it actually is pressure.

It's not the sort of thing that can be experimentally verified with current technology. The gravitational field of anything smaller than a mountain is too slight to be measured accurately enough.

I doubt that it can be verified using controlled laboratory experiments, but I think it has been well tested by astronomical observations.

Cosmological models depend explicitly on the pressure part of the stress-energy tensor. The universe has gone through epochs when it was dominated by radiation, then matter, and now by the cosmological constant. During the radiation-dominated epoch, the size of the pressure components of the stress-energy tensor was of the same order of magnitude as the size of the mass-energy component. If you omit them, I don't think you get good agreement with cosmological observations.

Neutron stars are relativistic objects, so I think the pressure terms may also have an effect on neutron star models.

On a more model-dependent basis, the stress-energy tensor has to have pressure components that act as sources, because otherwise Lorentz invariance would be broken. We have extremely high precision laboratory tests of Lorentz invariance. But of course this is less direct than the cosmological tests.

 

[Mentz114]

But the rate of transfer of momentum per unit time is force, so I think it actually is pressure.

It has the same dimensions as pressure, but that is an accident.

[P]= [MLT^-2]/[L²] = [ML²T^-2]/[L³] = [energy density]

 

[Ich]

It has the same dimensions as pressure, but that is an accident.

Force/Area an accident? It's the SI unit. And every non-SI unit I'm aware of, too. And it's how you measure it. And it's what engineers need to know.
Force/Area is simply pressure, not by accident, but by definition.

 

[Mentz114]

Force/Area an accident? It's the SI unit. And every non-SI unit I'm aware of, too. And it's how you measure it. And it's what engineers need to know.
Force/Area is simply pressure, not by accident, but by definition.

You've completely missed the point.

The quantity under discussion has the units of pressure, but it is not a force acting on a surface.

My problem is that if you have a large cloud of gas contracting under gravitational pressure, there is no containing vessel, or stationary surface for any force to act on, so I can't accept that pressure, in the engineering sense ( steam engines ?) gravitates.

I'm trying to work out the equivalent in terms of the average force exerted on each other by colliding particles. The only pressure I can come up with with is the average change in momentum divided by the collision cross section, per unit volume. Now that makes sense, because it is proportional to an energy density, and we know energy gravitates.

The weakest point in my argument is that energy is not a vector, but if the collision cross sections are oriented, I think this difficulty can be overcome.

 

[PAllen]

You've completely missed the point.

The quantity under discussion has the units of pressure, but it is not a force acting on a surface.

My problem is that if you have a large cloud of gas contracting under gravitational pressure, there is no containing vessel, or stationary surface for any force to act on, so I can't accept that pressure, in the engineering sense ( steam engines ?) gravitates.

I'm trying to work out the equivalent in terms of the average force exerted on each other by colliding particles. The only pressure I can come up with with is the average change in momentum divided by the collision cross section, per unit volume. Now that makes sense, because it is proportional to an energy density, and we know energy gravitates.

The weakest point in my argument is that energy is not a vector, but if the collision cross sections are oriented, I think this difficulty can be overcome.

So you think a a gas planet floating in space has no pressure? That's a new one on me. I have no knowledge of whether the stress-energy tensor term under consideration corresponds to what would normally be considered pressure, but I certainly know you don't need a container or surface to have fluid pressure.

 

[Ich]

The only pressure I can come up with with is the average change in momentum divided by the collision cross section

...which is pressure in the engineering sense, force per area. It is not an "accident", that's all I want to say.

 

[Mentz114]

...which is pressure in the engineering sense, force per area. It is not an "accident", that's all I want to say.

Sure. Anyway I short circuited my own argument by finding a 'pressure' in the gas cloud that makes physical sense. So I admit, pressure ( defined as above ) gravitates. I just wanted to understand it in terms of energy.

So you think a a gas planet floating in space has no pressure?

Didn't I just derive ( in principle) an expression for the pressure in the cloud ?

 

[bcrowell]

A good example is the radiation-dominated early universe. For a photon gas in thermal equilibrium, $P=\rho/3$. This is the pressure that all that cosmic background radiation would make on any exposed surface in the early universe. In the FRW equation $\ddot{a}/a=-4\pi/3(\rho+3P)$, the mass-energy-density and pressure terms contribute equally. If you didn't include the pressure term, you'd be off by a factor of 2 in predictions of the universe's rate of deceleration. Since cosmology is getting to be a high-precision science these days, I suspect that observations are good enough to say that pressure really does gravitate in this sense.

 

[pervect]

I'd have to say that we don't understand enough about the matter and energy distribution of the universe yet to say that we have good experimental confirmation that pressure causes gravity from cosmological observations. I'm not as certain as to what we can tell from neutron star observations - for instance,we know GR calculates the correct rate of decay for neutron binary inspiral, but it's not clear to me how much this directly confirms that specific point in question. Back to the main point.

Right now we believe that most of the universe is made out of dark matter that we can't see, and we observe that the universal expansion is accelerating, and we attribute this to dark energy that we're still looking for. With out any intent to be alarmist (I don't think there is any serious challenge to the Einstein field equations at the current time), due to this uncertanity, I don't think we can say we know enough about the distribution of matter and energy in the universe to say that cosmological observations directly confirm Einstein's field equations.

 

[bcrowell]

I'd have to say that we don't understand enough about the matter and energy distribution of the universe yet to say that we have good experimental confirmation that pressure causes gravity from cosmological observations.

Really? Maybe 15 years ago, but as of today, I'd be surprised if $\ddot{a}/a$ could be off by a factor of 2 for the radiation-dominated era without running into severe contradictions with high-precision CMB observations. After all, the universe's photon density isn't hard to measure like its dust density is.

Actually, I think there's probably going to be a different issue, which is that the form of the source terms in the field equations is strongly constrained by Lorentz invariance and the equivalence principle. All viable test theories have Lorentz invariance baked in, and most of them also have some form of the e.p., so it's going to be quite difficult -- I'm tempted to say impossible -- to devise an empirical test that can be interpreted solely as a test of the pressure term in the field equations. This is the same difficulty we have with verifying the speed of propagation of gravitational effects; nobody can test it, because propagation at c is included in all viable test theories, not just GR.

As an example of what I mean by the tight constraints imposed by Lorentz invariance and the e.p., see http://www.lightandmatter.com/html_books/genrel/ch08/ch08.html#Section8.1 , "Example 3: A failed attempt at tinkering".

Does anyone know of a good, freely accessible, recent review paper on experimental tests of GR? The best source I have is "Was Einstein Right?," which is a popular-level book, and which is way out of date.

 

[Mentz114]

A good example is the radiation-dominated early universe. For a photon gas in thermal equilibrium, $P=\rho/3$. This is the pressure that all that cosmic background radiation would make on any exposed surface in the early universe. In the FRW equation $\ddot{a}/a=-4\pi/3(\rho+3P)$, the mass-energy-density and pressure terms contribute equally. If you didn't include the pressure term, you'd be off by a factor of 2 in predictions of the universe's rate of deceleration. Since cosmology is getting to be a high-precision science these days, I suspect that observations are good enough to say that pressure really does gravitate in this sense.

That example adds weight to my supposition that we are looking for/at energy. For the photon gas the condition $P=\rho/3$ comes from the necessity that the trace of the SET should be zero ( because it's a source free radiation solution), which is not true for general fluids.

But I'm not challenging 'pressure gravitates'. I suppose eventually the data will decide.

Pervect's post reminds me that the cosmological constant is back in fashion to explain the observations and that adds $\Lambda g_{mn}$ to the perfect dust SET, which looks some pressure terms, but we refer to it as energy.

 

[bcrowell]

In the PPN formalism http://en.wikipedia.org/wiki/Parameterized_post-Newtonian_formalism , I guess what we're discussing is parametrized by $\zeta_4$, which equals 0 in GR. WP lists an upper limit on this parameter of 0.006 from an experiment by Kreuzer. A reasonably recent review of the experimental status of GR is this one by Will: http://relativity.livingreviews.org/Articles/lrr-2006-3/ . Will's table 2 in section 3.2 classifies $\zeta_4$ as measuring nonconservation of momentum. Section 3.7.3 discusses $\zeta_4$.

There is strong theoretical evidence that ζ4, which is related to the gravity generated by fluid pressure, is not really an independent parameter – in any reasonable theory of gravity there should be a connection between the gravity produced by kinetic energy (ρv2), internal energy (ρΠ), and pressure (p). From such considerations, there follows [275] the additional theoretical constraint
6ζ4 = 3α3 + 2 ζ1 − 3ζ3. (58 )

A classic test of Newton’s third law for gravitating systems was carried out in 1968 by Kreuzer, in which the gravitational attraction of fluorine and bromine were compared to a precision of 5 parts in 105.

Will doesn't give an actual reference for Kreuzer, so maybe it was unpublished?

[EDIT] Will has a paper analyzing the Kreuzer experiment: http://adsabs.harvard.edu/full/1976ApJ...204..224W

[EDIT] The Kreuzer paper is Phys Rev 169 (1968) 1007.

 

[bcrowell]

Here's my understandnig of the Will paper reanalyzing the Kreuzer experiment. Kreuzer basically verified that bromine and fluorine obeyed Newton's third law when interacting gravitationally in a Cavendish balance. This means that they tested whether active gravitational mass (the ability to make gravitational fields) is proportional to passive gravitational mass (the response to gravitational fields) for these two substances. If you let ζ4 be nonzero, then active gravitational mass depends on the pressure in a nonstandard way, which differs from GR's predictions. The electromagnetic interactions in a bromine nucleus and a fluorine nucleus produce different amounts of pressure. Therefore if ζ4 was nonzero (or, more generally, if some linear combination of PPN parameters was nonzero), we would get violations of Newton's third law in experiments like the Kreuzer experiment.

 

[pervect]

Really? Maybe 15 years ago, but as of today, I'd be surprised if $\ddot{a}/a$ could be off by a factor of 2 for the radiation-dominated era without running into severe contradictions with high-precision CMB observations.

There is (or maybe was) at least one dark-horse theory, Self Creation Cosmology (SCC) http://arxiv.org/abs/gr-qc/0212111, a Branse-Dicke variant, that could get away without any inflation at all.

The author, Garth Barber, used to post on PF, I'm not sure if he's still around. However, if the gravity probe B results are as accurate as they claim, SCC is dead, or at least in serious trouble - it predicts a significant difference in the geodetic precession from GR.

 

[PAllen]

There is (or maybe was) at least one dark-horse theory, Self Creation Cosmology (SCC) http://arxiv.org/abs/gr-qc/0212111, a Branse-Dicke variant, that could get away without any inflation at all.

The author, Garth Barber, used to post on PF, I'm not sure if he's still around. However, if the gravity probe B results are as accurate as they claim, SCC is dead, or at least in serious trouble - it predicts a significant difference in the geodetic precession from GR.

I thought the Gravity Probe B results were an unfortunate failure due to unforeseen anomalies, at least for frame dragging. Are you saying the geodetic effect (which was supposed to be much larger) is accepted as verified with confidence by Gravity Probe B?

That would be good news to me. I was depressed about the cancellation of funding due to 'no confidence' in the process of factoring out noise from the frame dragging signal.

 

[bcrowell]

Gravity Probe B gave a very robust measurement of the geodetic effect.

IIRC frame dragging was also confirmed by GPB, although that was more at the limits of its sensitivity.

 

[Ich]

Does anyone know of a good, freely accessible, recent review paper on experimental tests of GR?

http://arxiv.org/abs/gr-qc/0510072

The author, Garth Barber, used to post on PF, I'm not sure if he's still around. However, if the gravity probe B results are as accurate as they claim, SCC is dead, or at least in serious trouble - it predicts a significant difference in the geodetic precession from GR.

Yes, but now it postdicts the https://www.physicsforums.com/showthread.php?p=2908466#post2908466".

 

[Garth]

http://arxiv.org/abs/gr-qc/0510072Yes, but now it postdicts the https://www.physicsforums.com/showthread.php?p=2908466#post2908466".

Hi - I'm still around though not posting as often as I used to, lots more able people to do that!

Actually the OP question is a very good question in that all the standard tests of GR (bending of light, precession of perihelion, radar echo delay etc.) only test the behaviour of test particles (Mercury!) and photons in a vacuum, i.e. they are testing
$$R_{\mu\nu} = 0$$.

The stress energy tensor is used in the Schwarzschild solution to model the gravitating body but then you fit that model to first order to the Newtonian solution - so it might be wrong (I personally don't think so) and the fitting procedure then compensates for the error. If so then different second order effects should show up, which is what Gravity Probe B is testing, so far without any deviation from GR.

Of course the early universe tests the non-vacuum solution and the radiation dominated solution, and the $\Lambda$CDM model fits very well, but that has been fitted to the data using Inflation, Dark Matter and Dark Energy, none of which have been independently verified in laboratory physics.
(Though the Bullet Cluster independently verifies DM astronomically)

The GP-B geodetic result has been a learning experience for me and as Ich said SCC now postdicts the GR result.
[If I may give an explanation for anybody interested] The Jordan conformal frame of the theory (as indeed with BD) can be cast in two forms, the original effective form in which the second derivative of the scalar field is convoluted with terms of the metric, and the true form in which they are taken out. In the true form of SCC, (which turns out to be my original 1982 theory) photons do not follow the geodesics of the metric, whereas they do in the effective form of the gravitational field equation. I had used the effective form of the equations to predict the original geodetic precession, whereas in the true form the prediction is as in GR. So GP-B has taught me that in SCC the true has to be used for matter - e.g. gyroscopes and the effective form for photons. The new paper has been submitted for publication and is on the physics arXiv here.

Garth

 

[bcrowell]

Having studied the Will review and Kreuzer 1968 and Will 1975, my understanding is that basically the PPN parameter that measures the coupling of pressure to gravitational fields is extremely tightly constrained by experiment to be within about 10^-8 of its GR value of zero. It's a little bit complicated because PPN has a lot of different parameters, and you don't typically constrain a single parameter with a single observation. My current understanding is written up here: http://www.lightandmatter.com/html_books/genrel/ch08/ch08.html#Section8.1 (subsection 8.1.2, at "Experimental tests").