Ideal fluids and general relativity

[accdd]

Why in general relativity do we need the physics of perfect fluids?

 

[Orodruin]

You don't need them for GR per se, but they are useful for modelling things.

 

[Ibix]

They are useful models of stellar interiors and the universe at large scales, both of which need GR to describe them. So, as Orodruin says, you don't need them for GR but you do need a GR-friendly model of perfect fluids for quite a lot of the things you want to learn GR for.

 

[PeroK]

Why in general relativity do we need the physics of perfect fluids?

Because some physicists like to go for a pint of beer after a hard day's theorising.

 

[Ibix]

Because some physicists like to go for a pint of beer after a hard day's theorising.

"A mathematician is a machine for turning coffee into theorems", as a double-espresso drinking mathematician friend once told me.

 

[martinbn]

"A mathematician is a machine for turning coffee into theorems", as a double-espresso drinking mathematician friend once told me.

Ken Ribet got a math book he didn't need, went to the local used book store, sold it, and on the way back bought himself a cup of coffee. Then realized he had just turned theorems into coffee.

 

[PeroK]

Ken Ribet got a math book he didn't need, went to the local used book store, sold it, and on the way back bought himself a cup of coffee. Then realized he had just turned theorems into coffee.

You'd be lucky to get a coffee for the money you'd get for an unwanted maths book.

 

[martinbn]

You'd be lucky to get a coffee for the money you'd get for an unwanted maths book.

The story goes back to the 80s I think, but I wasn't clear. He didn't want the book or didn't have a use for it, it may have been wanted in general. It may have been a book he has.

 

[Orodruin]

I buy coffee for the royalties from my book. 😇

 

[cianfa72]

Btw, is there any useful application of perfect fluids in GR as a 'physical realization' of coordinate systems ?

 

[vanhees71]

A perfect fluid is just one of the most simple models to describe matter. In cosmology you get a perfect fluid when using the FLRW metric to describe the large-scale coarse-grained situation. The equations of motion get closed by just choosing a simple equation of state (usually "cold/non-relativistic" matter, "radiation/relativistic matter", and "dark energy").

Far more interesting are neutron stars. Also here you just take a perfect fluid and you need an equation of state to solve the Tolman-Oppenheimer-Volkff (TOV) equations, and this is of utmost interest also for heavy-ion physicist as myself, because one can learn to some extent from the observed mass-radius relations, "how stiff" the equation of state of nuclear matter at the typical densities of neutron stars might be. It's still somewhat a puzzle, particularly in connection with the contribution of hypernucleons (baryons containing strange quarks) and with the question, whether in the interior of the neutron stars a quark phase (maybe even a color-supercondutor phase) might form or not.

In addition with the advent of gravitational-wave detectors and the possibility for multi-messenger astronomy of neutron-star mergers, one has even more constraints to figure out this equation of state of strongly-interacting matter. Another hope is that with heavy-ion collisions at not so high collision energies we get additional information in this part of the QCD phase diagram, because the reached (net-baryon) densities in the hot and dense fireballs are comparable to that of neutron stars, though in a more isospin-symmetric state.

BTW: also the "fireballs" created in heavy-ion collisions are well described by relativistic hydrodynamics with a shear-viscosity-over-entropy-density ratio close to the lower bound of $1/4 \pi$. This has lead to lot of developments of relativistic dissipative hydrodynamics. Most recently also the hydrodynamics of spinning matter and also relativistic dissipative magnetohydrodynamics has come into the focus and is thus heavily investicated in current research. The latter is of course also highly interesting for neutron star physics again, particularly in the context of the possible huge magnetic fields involved there (magnetars or possible very heavy neutron stars from neutron-star mergers).

 

[Orodruin]

The equations of motion get closed by just choosing a simple equation of state (usually "cold/non-relativistic" matter, "radiation/relativistic matter", and "dark energy").

Just to point out that the reverse is also true, given the expansion history you can figure out how the relation between pressure and energy density has evolved historically as fixing the metric fixes the stress-energy tensor.

 

[vanhees71]

Yes, indeed, and this history is pretty well known by the measurements of the cosmic-microwave-background fluctuations and the redshift-distance relation for type Ia supernovae, telling us about the "matter content" of the universe, of which only 4% or so is known "baryonic matter" the rest is "dark matter" (23%) and "dark energy" (73%) ;-).

 

[Ibix]

Btw, is there any useful application of perfect fluids in GR as a 'physical realization' of coordinate systems ?

Certainly in the case of pressureless dust, since there exists a coordinate system in which the fluid isn't moving even on a micro level. FLRW spacetime is an example. In a fluid with pressure there can be a coordinate system where there's no bulk motion (may also be one with no motion at all), so that works too. However, I think once you get to fluids with convection then you can't really use the notion anymore.

 

[pervect]

Why in general relativity do we need the physics of perfect fluids?

They're basically a simple approximation, that's usually sufficient.

For instance, the wiki on perfect fluids states:

In physics, a perfect fluid is a fluid that can be completely characterized by its rest frame mass density and isotoropic pressure.

Real fluids are "sticky" and contain (and conduct) heat. Perfect fluids are idealized models in which these possibilities are neglected.

The refienments of non-isotropic pressure (for instance, the internal stresses in a non-spherical object), or the transfer of heat usually don't have significant effects on the solution to Einstein's equations at the scales that GR is usually needed.

Matter just isn't strong enough to have a noticably non-spherical planet, to take a simple example of why it's usually a good approximation. This isn't very precise, it'd be worth thinking about any possible exceptions, this is just a general comment.

 

[vanhees71]

Well, whether or not ideal fluid dynamics is a good approximation for a given physical situation involving "fluids" is a question of time scales. It is a good approximation if the relaxation time to (local) thermal equilibrium is small compared to the typical time scales involved in the dynamics of the system.

In relativistic heavy-ion collisions, which is one of the prime applications of relativistic fluid dynamics, at high enough collision energies, it was a great surprise when around 2001 for the first time the first data from the Relativistic Heavy Ion Collider (RHIC) at BNL showed that the description of the rapidly expanding fireball of strongly interacting matter created in the collision with ideal relativistic hydrodynamics really could quantitatively describe the observed hadronic spectra, particularly the "elliptic flow", $v_2$, which indicates that the initial spatial asymmetry in non-central collisions are translated via the corresponding pressure gradients to the anisotropy of the flow of hadrons as expected from ideal fluid dynamics. It implies a very rapid relaxation to local thermal equilibrium and thus implying a very strongly coupled medium.

Today, more then 20 years later, triggered by these observerations, the development of relativistic viscous fluid dynamics has pretty much advanced. As is well known for decades, the first-order gradient expansion (aka relativistic Navier-Stokes equations) leads to acausalities as any parabolic-type equation like all kinds of diffusion equations. As is known since the fundamental work by Israel and Stewart this is cured by going to (at least) second-order hydrodynamics, implementing a finite relaxation time. Today, various types of dissipative relativistic hydrodynamics is derived from relativistic kinetic theory using, e.g., the method of moments.

A recent nice textbook is

P. Romatschke, U. Romatschke, Relativistic Fluid Dynamics In and Out of Equilibrium, Cambridge University Press (2019)
https://doi.org/10.1017/9781108651998