I Energy Momentum Tensor: Real Physics & Cosmology

[dsaun777]

The energy momentum tensor and its correlation to reimannian curvature is fascinating to think about. How much are the components and which of the components are taken into consideration when doing real physics. I suppose astrophysicists and cosmologists would be the main group of scientists that use such calculations routinely. When or how are they actually used experimentally to verify and make predictions? I am aware that the dominant figure in the energy tensor is density, but are factors like stress, flux etc key figures in doing real cosmology? Or are they more for computing black hole or galactic scale type curvatures.

 

[Orodruin]

The standard model of cosmology (the ΛCDM model) assumes that the universe is homogeneous and isotropic and filled with one or more ideal fluids. Such fluids can be described (in the frame where the universe is homogeneous and isotropic) by an energy density and a pressure. Different types of fluids have different relations between energy density and pressure, parametrised by the equation of state $p = w\rho$. The value of $w$ for regular matter is zero because the pressure, as you mentioned, is much lower than the energy density. However, this is not true for all types of fluids. For example, a photon gas has $w = 1/3$ and a cosmological constant corresponds to $w = -1$. Since the universe was once dominated by radiation and today is going into a dark energy dominated phase, cases where the pressure is relevant to the evolution of the universe are certainly of importance to cosmology.

Or are they more for computing black hole or galactic scale type curvatures.

Uncharged black holes are vacuum solutions to the EFEs. The stress-energy tensor in those spacetimes is identically equal to zero.

 

[Ibix]

See also https://www.physicsforums.com/threads/stress-energy-tensor-of-a-rotating-rod.981927/ (if my curiosity counts as "real physics").

 

[pervect]

The energy momentum tensor and its correlation to reimannian curvature is fascinating to think about. How much are the components and which of the components are taken into consideration when doing real physics. I suppose astrophysicists and cosmologists would be the main group of scientists that use such calculations routinely. When or how are they actually used experimentally to verify and make predictions? I am aware that the dominant figure in the energy tensor is density, but are factors like stress, flux etc key figures in doing real cosmology? Or are they more for computing black hole or galactic scale type curvatures.

I'm not quite sure what you're asking. The components of the stress energy tensor are the densities of energy, momentum, and pressure. All of these are can be and are used experimentally. The pressure term, as it includes some dynamic pressures, might take a little study to understand, the other terms are pretty basic.

One of the important uses of the stress-energy tensor is it tells you how all these components transform relativistically. It's necessary to use the stress-energy tensor, or a mathematically equivalent formulation, to write relativsitically covariant physics involving these quantities. So, if one is trying to do an experiment involving relativistic entities, it'll be needed if you are measuring any of energy, momentum, or pressure. You might be able to get away without it if you have a relativistic system that's a point particle and doesn't have any volume.

 

[dsaun777]

I'm not quite sure what you're asking. The components of the stress energy tensor are the densities of energy, momentum, and pressure. All of these are can be and are used experimentally. The pressure term, as it includes some dynamic pressures, might take a little study to understand, the other terms are pretty basic.

One of the important uses of the stress-energy tensor is it tells you how all these components transform relativistically. It's necessary to use the stress-energy tensor, or a mathematically equivalent formulation, to write relativsitically covariant physics involving these quantities. So, if one is trying to do an experiment involving relativistic entities, it'll be needed if you are measuring any of energy, momentum, or pressure. You might be able to get away without it if you have a relativistic system that's a point particle and doesn't have any volume.

I'm asking when do these components show up collectively and are used to calculate spacetime relativity effects like curvature. What are some of the instruments involved in measuring these components to get a corresponding reimann curvature, basically applied GR.

 

[Ibix]

I'm asking when do these components show up collectively and are used to calculate spacetime relativity effects like curvature. What are some of the instruments involved in measuring these components to get a corresponding reimann curvature, basically applied GR.

There seem to be two separate questions here. The components of the stress-energy tensor are used any time you solve the Einstein Field Equations. In a trivial sense you are even using one component of it if you use Newtonian gravity, and you are using it even if the components are all identically zero, as they are in vacuum solutions.

Separate from this is the question of how you determine the values of the components of the tensor (or, strictly, contractions of the tensor with various combinations of basis vectors). There are many different ways, depending on what you are measuring. For example in FLRW spacetime we can use telescopes (plus a certain amount of modelling to back out mass from luminosity) to measure stellar densities.

 

[pervect]

I'm asking when do these components show up collectively and are used to calculate spacetime relativity effects like curvature. What are some of the instruments involved in measuring these components to get a corresponding reimann curvature, basically applied GR.

I'm going to assume you are interested in solar system planetary and solar masses, as the question is very general.

For the solar system case, my understanding is that they are derived as a best fit to an epemeris, i.e. observations of the orbits. One also needs an associated value of G to report these masses in standard units, this is a relatively low precision measurement I believe.

For another astronomical realm, LIGO 's black hole inspirals, the value for the mass is derived from the frequency of the chirp signal.

So the observations are fit in the framework of a theory, it's not like we are able to accurately find the density of the Earth and integrate it's mass from the density measurements, for example. Details vary, but generally the mass is inferred from observations of orbits for the solar system.

Wiki seems to agrees with my understanding, for instance https://en.wikipedia.org/w/index.php?title=Planetary_mass&oldid=918747186

GR introduces only minor corrections in the case of the solar system.

 

[PAllen]

As far as I understand, all the best current ephemeris use a GR equation of motion for all major solar system bodies. However, it is not based on the field equations or stress energy tensor or curvature. Instead, they are based on the post Newtonian approximation, using the first order of corrections to Newtonian motion. The basic equation used is given here:

https://en.wikipedia.org/wiki/Einstein–Infeld–Hoffmann_equations