Einstein's Field Equation Stress-Energy

[Ich]

Hi,

as I try to understand Einstein's Field Equations, I realize that I don't understand the meaning of its components.
I gather that Rtt=Ttt+Txx+Tyy+Tzz is the dominating parameter in cosmological models. (Sorry, I also have problems with tex.)
Ttt means "mass energy density". Is this "relativistic mass" per unit volume?
Txx etc. means pressure. Is this momentum flow per unit area? The net flow should be 0 (in a comoving frame), how do I calculate Txx.
Example:
A constant matter density gives Rtt=Ttt=rho in the comoving frame.
What if all that matter annihilates? What remains constant, Rtt or Ttt?
Rtt=rho+3*rho/3 or
Rtt=rho/2+3*rho/6?
Is Rtt equal to some total energy density?

Thanks.

 

[pervect]

Hi,

as I try to understand Einstein's Field Equations, I realize that I don't understand the meaning of its components.
I gather that Rtt=Ttt+Txx+Tyy+Tzz is the dominating parameter in cosmological models. (Sorry, I also have problems with tex.)
Ttt means "mass energy density". Is this "relativistic mass" per unit volume?

I don't like "relativistic mass" much, but by most defintions of the term it's equivalent to energy. This means that the answer according to standard defintions is "yes".

It's really better, though, to say that T_tt is the energy density. This includes all sources of energy, including the energy in rest mass (mc^2).

Txx etc. means pressure. Is this momentum flow per unit area?

It can be regarded as the flow of the x component of the x momentum in the x direction, as per http://math.ucr.edu/home/baez/gr/outline2.html

It can also be regarded more simply in the comoving frame as the pressure.

The net flow should be 0 (in a comoving frame), how do I calculate Txx.

It's not the total flow, the total flow of momentum is T_tx.

I find it simpler to regard T_xx in the comoving frame as the pressure in the comoving frame. (This simple approach doesn't work well in non-comoving frames, which is why Baez formulates it in the manner above). While Baez's defintion may be a bit confusing, consider a pressure P in some cube. It's transporting negative x momentum in the -x direction, and positive x momentum in the +x direction. The minus signs cancel each other out, and the net effect of the pressure is transport of the x-momentum in the x direction.

If Baez's approach is too confusing, always work in the comvoing frame, and think of T_xx as the pressure. If you must work in a non-comvoving frame, tensors transform in a well-defined manner. You can thus use the tensor transformation rules to assign a meaning to T_xx (and the other components) in a non-comoving frame.

Note that, using the "pressure in comoving frame" approach, pressure really is a rank 2 tensor in the general case. In ideal fluids, though, this pressure is always isotropic, which makes things a lot simpler (there is only one P, which appears on all the diagonal terms of the stress-energy tensor).

In the general case, the pressure tensor can be diagonalized by aligning the axes correctly.

 

[pmb_phy]

Hi,

as I try to understand Einstein's Field Equations, I realize that I don't understand the meaning of its components.
I gather that Rtt=Ttt+Txx+Tyy+Tzz is the dominating parameter in cosmological models. (Sorry, I also have problems with tex.)
Ttt means "mass energy density". Is this "relativistic mass" per unit volume?

Not in call cases. "relativistic mass density" equals the ratio of momentum density to speed and this will not always be equal to T⁰⁰. Only when there is no stress is are they gaurenteed to be the same.

Pete

 

[pervect]

Not in call cases. "relativistic mass density" equals the ratio of momentum density to speed and this will not always be equal to T⁰⁰. Only when there is no stress is are they gaurenteed to be the same.

Pete

Note that this disagrees with what I said earlier - this is an old argument between Pete and I about the defintion of "relativistic mass".

http://math.ucr.edu/home/baez/physics/Relativity/SR/mass.html is one source that supports my position. For more sources that I've found that support my view see

https://www.physicsforums.com/showpost.php?p=1096850&postcount=28

Neither of us has succeeded in convincing the other, and no third party has volunteered to arbitrate the dispute over the definition of "relativistic mass".

As far as I'm concerned, issues regarding the definition of the term are basically just another reason to avoid the term "relativistic mass" which is IMO depreciated (see the FAQ entry, the first URL I posted), and stick with "energy density".

Hopefully this relatively minor disagreement won't get in the way of the main point, which is that T_00, being the total energy, includes terms of the form mc^2, where m is the rest mass or invariant mass.

 

[Mentz114]

Grammar !

...this is an old argument between Pete and I ...

should be

...this is an old argument between Pete and ME ...

Luckily your physics is better than your grammar.

Yes, I am a pedant.

 

[pervect]

Grammar !

Luckily your physics is better than your grammar.

You ain't kidding :-)

 

[Ich]

Thank you, pervect!

So, in my example, if all matter would annihilate, Ttt would remain constant (=rho) and there would Pressure Txx=rho/3 etc. ?
Is therefore the statement "Rtt equals some total energy density" wrong, as we expect energy to be conserved?
Is it correct then, considering the entire universe and ignoring lambda, that its expansion would decelerate twice as fast (I hope you know wht I mean) after annihilating all matter?
Obviously, if we could annihilate a star, it would not double its gravitation. I guess there are some contributions of the boundary, if we confine the photon gas? IIRC, you did mention such things.
Is it ok to call a decelerating lamba a positive energy density?

Lots of questions...

 

[pervect]

Thank you, pervect!

So, in my example, if all matter would annihilate, Ttt would remain constant (=rho) and there would Pressure Txx=rho/3 etc. ?

Yes.

Is therefore the statement "Rtt equals some total energy density" wrong, as we expect energy to be conserved?

R_tt can be regarded in a couple of ways. It can be regarded as the second derivative of the volume of a ball of comoving coffee grounds, per Baez's http://math.ucr.edu/home/baez/gr/outline2.html

R_tt is closely related, but not quite the same as, one particular measure of energy, the Komar energy. (It's often called the Komar mass, which is just the Komar energy / c^2). This concept of energy is not completely general, however, it applies only to stationary or static systems.

I've written a few wikipedia articles about this

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

talks about some of the concepts of mass in GR, including the Komar mass.

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

goes into more detail on the Komar mass.

To get the the contribution of R_tt to the Komar mass of a body, you have to multiply R_tt by another factor, the square root of the metric coefficient g_tt, which can be regarded as the "redshift factor" at that location.

But we still haven't gotten to the heart of the matter. So let's take an example closely related to yours. This is closely related to an example I worked out in the first wiki article.

http://en.wikipedia.org/wiki/Mass_i...simple_examples_of_mass_in_general_relativity

Suppose we have a very strong steel sphere. Inside this we have matter and anti-matter.

The matter and anti-matter annihilate. This causes the matter m in the sphere to become energy. The average energy density inside the sphere T_00 stays constant during this process. The average pressure, T_xx, T_yy, and T_zz each go from 0 to some high value (1/3 T_00, as I recall).

What happens to the (Komar) mass of the system, and its gravitational field? We have the same energy, and we've added pressure. So it looks like it should go up (double when we work out the details). But when we actually consider the complete system, including the pressure vessel, the answer is that nothing happens to the mass. The reason that the answer is nothing is that while the average pressure inside the sphere is high, there is a counterbalancing average tension in the walls of the container that exactly counterbalances it. (A tension is just a negative pressure. It subtracts from the Komar mass, just as pressure adds to it).

The walls of the container are needed to keep the system "stationary". So this answer is a bit long, but you can see that while R_00, even when corrected by the appropriate redshift factor to become the Komar mass, is not a general measure of energy in GR. It does work correctly for a stationary system, but it wouldn't work right if we didn't have the pressure vessel there.

If the Komar mass doesn't work, is there some other formulation that does? To a limited extent, the answer is yes.

One of the most general defintions for a conserved energy exists is the ADM energy, which applies to any asymptotically flat space-time. The Komar mass is simpler to compute, but the ADM mass is more general. While it is more general, the ADM mass is not completely general. The ADM mass requires an asymptotically flat space-time. (Note that any isolated system surrounded by an infinite vacuum region will have this property of asymptotic flatness).

If you have a totally arbitrary space-time, there is no truly general defintion of conserved energy in GR. (see the wikipedia article I quoted for more details, and some of its references, such as the sci.physics.faq entry on energy in GR http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html)The problem with cosmology is that standard cosmologies aren't asymptotically flat.

The observable universe is assumed to be some part of a very large (possibly infinite) system, for instance, so it isn't required to have (and in fact doesn't have) the property of asymptotic flatness needed to have a conserved ADM energy.

 

[Ich]

Thanks, it becomes clearer.
Should I post my questions about expanison of the universe and definition of "Energy density" in the complete universe, including lambda, in the cosmology section? After all, it comes down to GR.

 

[pmb_phy]

Neither of us has succeeded in convincing the other, and no third party has volunteered to arbitrate the dispute over the definition of "relativistic mass".

This phenomena is known as having a difference of opinion and thus there is no reason that I can see why we'd want to have the same opinion. I' not very suree that you understand my position but se la vie!

Hopefully this relatively minor disagreement won't get in the way of the main point, which is that T_00, being the total energy, includes terms of the form mc^2, where m is the rest mass or invariant mass.

If there are certain stress components then they will contribute to the inertia of the body. It will even contrtibute to the rest mass of the body. I mentioned this because we're talking about densities here and densitiess don't transform in the same was as an isolated system

Pete

 

[Ich]

Does this have any bearing on my OP?

 

[pmb_phy]

Does this have any bearing on my OP?

Absolutely. Otherwise I woudn't have mentioned it.

Pete

 

[Ich]

Could you explain?
Does it mean that you count some components of T (other than Ttt) among "relativistic mass"?

 

[pmb_phy]

Could you explain?
Does it mean that you count some components of T (other than Ttt) among "relativistic mass"?

take a look at a this web page I created on this topic.

http://www.geocities.com/physics_world/sr/inertial_energy_vs_mass.htm

Eq. (2) is the momentum density. To obtain the mass density divide that by the velocity. You'll see that you don't end up with something proportional to T⁰⁰. However that term will disappear when you calculate the entire mass of the whole/isolated system.

Pete

 

[pervect]

The energy-momentum of an object with finite volume is not in general a covariant entity, see for instance http://arxiv.org/abs/physics/0505004. (The total energy-momentum of an object with finite volume is covariant if the object is an isolated system, but in general we are interested in non-isolated systems).

The energy momentum density, i.e. the stress-energy tensor, is always a covariant entity, however. (Roughly speaking, this is because it is a density, so it takes the limit as the volume approaches zero, where there is no issue.)

So the correct way to work out the momentum of a non-isolated body is to perform the appropriate integral of the stress energy tensor.

One finds that because of the Lorentz transform, pressure in one frame can convert to energy or momentum in another, just as for the simpler rank 1 tensor, energy can convert to momentum, or space into time.

If one does a "boost" of a stationary ball of fluid with a density of \rho and an isotropic pressure of P, one finds the momentum of such a fluid element in the boosted frame as being

u \gamma^2 (\left \rho + P \right)

where u is the 4-velocity of the boost (and \gamma = 1/\sqrt{1-|u|^2}).

See for instance Rindler, Introduction to special relativity, pg 132, where this is worked out. Rindler actually works out the more complex case of an anisotropic pressure, with all the T^ij in the stress-energy tensor differnet.

Note that Rindler uses non-geometric units, so he replaces P with P/c^2

There are several important points to be made, in my opinion.

1) The pressure does contribute to the momentum in the direction of motion, via the Lorentz transform.

2) The contribution is not of the form \rho + 3P. In fact, only the component of the pressure term in the direction of motion contributes to the momentum density, hence \rho + P

3) In general, momentum is a vector that doesn't point in the same direction as velocity. Therefore in general the concept of momentum / velocity is a tensor, not a scalar. The previous quantites mentioned (invariant mass and E/c^2 which is often called relativistic mass) were scalars, while the ratio of momentum / velocity is not a scalar.

 

[pervect]

Thanks, it becomes clearer.
Should I post my questions about expanison of the universe and definition of "Energy density" in the complete universe, including lambda, in the cosmology section? After all, it comes down to GR.

If it's mainly a theoretical question about GR, post it here. If it's more related to observation, you'd probably have better luck in the cosmology forum.

 

[pervect]

Does this have any bearing on my OP?

Dunno :-). If you're not interested in this particular issue, I'm not really that interested in rehashing it with Pete. It does relate to the general issue of mass and its various defintions. Some authors, for instance Max Jammer, have observed that "mass is a mess". This is not because we don't have any defiintions for mass, in my opinion it's because we have too many and they're all slightly different.

It also relates as to how seriously one can treat \rho + 3P, i.e. R_00, as some sort of energy density.