I do not understand stress energy tensor for fluids

[StenEdeback]

Homework Statement

I do not understand stress energy tensor for fluids

Relevant Equations

Goldstein's "Classical Mechanics" third edition equation 13.78

I do private studies on my own for fun and right now I read about relativistic field theory as a preparation for later studies of quantum field theory.
I simply do not understand where equation 13.78 in Goldstein's "Classical Mechanics" third edition comes from. Please explain.
Please also recommend books about advanced fluid physics. I have Faith A Morrison's "An introduction to fluid mechanics", but it does not treat relativistic stress energy tensors.
Please also recommend books about classical relativistic field theory which are a bit more detailed than Goldstein's rather brief treatment.
I would be very grateful for help. Physics Forums is my only mentor.

 

[malawi_glenn]

@StenEdeback can you at least post the equation?

 

[milkism]

Homework Statement: I do not understand stress energy tensor for fluids
Relevant Equations: Goldstein's "Classical Mechanics" third edition equation 13.78

I do private studies on my own for fun and right now I read about relativistic field theory as a preparation for later studies of quantum field theory.
I simply do not understand where equation 13.78 in Goldstein's "Classical Mechanics" third edition comes from. Please explain.
Please also recommend books about advanced fluid physics. I have Faith A Morrison's "An introduction to fluid mechanics", but it does not treat relativistic stress energy tensors.
Please also recommend books about classical relativistic field theory which are a bit more detailed than Goldstein's rather brief treatment.
I would be very grateful for help. Physics Forums is my only mentor.

Hey, nice to see a fellow self studier! I remember using Goldstein to study introduction to classical mechanics. Here is the equation, for malawi_glenn and those who are interested:

 

[vanhees71]

The idea is that by definition an ideal fluid is a fluid, for which Pascal's rule holds. In the relativistic realm you have to be careful, in which reference frame Pascal's rule should hold, and that's oviously the (local) rest frame of the fluid (cell), where all the intensive thermodynamical quantities are defined. According to Pascal's rule the stress tensor is $\mathrm{diag}(P,P,P)$, where $P$ is the pressure (as measured in the rest frame and thus by definition the pressure in all frames). In relativity you need to extent this 3D notion of the stress tensor to the appropraite 4D tensor, which is the energy-momentum-stress tensor. So we need $T^{00}$, $T^{0j}$, and obviously in the local rest frame $T^{0j}=0$, and $T^{00}=\rho$ is the energy density in the fluid restframe (which you can with some right call the "mass density" although it's a bit misleading in the relativistic context).

Now in the local restframe of the fluid cell the normalized four-velocity is $(1,0,0,0)$, and thus the expression $(T^{\mu \nu})=\mathrm{diag}(\rho,P,P,P)$ in the rest frame can be written as $T^{\mu \nu}=\rho u^{\mu} u^{\mu} + P (g^{\mu \nu}+u^{\mu} u^{\nu})$, where obviously the east-coast convention of the metric has been used, i.e., $g^{\mu \nu}=\mathrm{diag}(-1,1,1,1)$, and that's indeed (13.78). Since this is written in a manifestly covariant way, it holds in any (inertial) reference frame.

Note that in the rest of the book they use the west-coast convention. It's again very sloppy and confusing. In general, I'd not recommend the 3rd edition of Goldstein's book, which was distorted by some authors who intended to modernize the book :-(. BTW the much better 2nd edition, which is really by Goldstein, can also not be recommended for its treatment of relativity, because there it uses the $\mathrm{i} c t$ convention.

For a very good intro into special-relativistic fluid dynamics, which consistently works in the west-coast convention, see

https://arxiv.org/abs/0708.2433
https://doi.org/10.1088/0143-0807/29/2/010

There the ideal-fluid energy-momentum tensor is Eq. (28) (with the correct signs in the west-coast convention, where $(g_{\mu \nu})=\mathrm{diag}(1,-1,-1,-1)$ and thus $T^{\mu \nu} = (\rho+P) u^{\mu} u^{\mu}-P g^{\mu \nu}$ ;-)).

 

[StenEdeback]

The idea is that by definition an ideal fluid is a fluid, for which Pascal's rule holds. In the relativistic realm you have to be careful, in which reference frame Pascal's rule should hold, and that's oviously the (local) rest frame of the fluid (cell), where all the intensive thermodynamical quantities are defined. According to Pascal's rule the stress tensor is $\mathrm{diag}(P,P,P)$, where $P$ is the pressure (as measured in the rest frame and thus by definition the pressure in all frames). In relativity you need to extent this 3D notion of the stress tensor to the appropraite 4D tensor, which is the energy-momentum-stress tensor. So we need $T^{00}$, $T^{0j}$, and obviously in the local rest frame $T^{0j}=0$, and $T^{00}=\rho$ is the energy density in the fluid restframe (which you can with some right call the "mass density" although it's a bit misleading in the relativistic context).

Now in the local restframe of the fluid cell the normalized four-velocity is $(1,0,0,0)$, and thus the expression $(T^{\mu \nu})=\mathrm{diag}(\rho,P,P,P)$ in the rest frame can be written as $T^{\mu \nu}=\rho u^{\mu} u^{\mu} + P (g^{\mu \nu}+u^{\mu} u^{\nu})$, where obviously the east-coast convention of the metric has been used, i.e., $g^{\mu \nu}=\mathrm{diag}(-1,1,1,1)$, and that's indeed (13.78). Since this is written in a manifestly covariant way, it holds in any (inertial) reference frame.

Note that in the rest of the book they use the west-coast convention. It's again very sloppy and confusing. In general, I'd not recommend the 3rd edition of Goldstein's book, which was distorted by some authors who intended to modernize the book :-(. BTW the much better 2nd edition, which is really by Goldstein, can also not be recommended for its treatment of relativity, because there it uses the $\mathrm{i} c t$ convention.

For a very good intro into special-relativistic fluid dynamics, which consistently works in the west-coast convention, see

https://arxiv.org/abs/0708.2433
https://doi.org/10.1088/0143-0807/29/2/010

There the ideal-fluid energy-momentum tensor is Eq. (28) (with the correct signs in the west-coast convention, where $(g_{\mu \nu})=\mathrm{diag}(1,-1,-1,-1)$ and thus $T^{\mu \nu} = (\rho+P) u^{\mu} u^{\mu}-P g^{\mu \nu}$ ;-)).

Thank you vanhees71! Your answer has shed some light in my foggy mind. It is so good to have the possibility to put questions to Physics Forums. I think that most of my bewilderedness will now fade away for the moment. Thanks again!

 

[George Jones]

I'd not recommend the 3rd edition of Goldstein's book, which was distorted by some authors who intended to modernize the book :-(. BTW the much better 2nd edition, which is really by Goldstein, can also not be recommended for its treatment of relativity, because there it uses the $\mathrm{i} c t$ convention.

As an undergrad I took 2 classical mechanics that used the 2nd edition of Goldstein as a text, and all of Goldstein was followed fairly closely, except for the relativity section. The courses were taught by a general relativist, and his presentation of relativity was fairly geometrical, using Minkowki inner products to solve problems when possible.

 

[vanhees71]

The 2nd edition is rightfully a classic, and the use of the $\mathrm{i} c t$ metric was quite common in this time. Even my alltime favorite textbooks by Sommerfeld commit that sin.

However the 3rd edition is an insult against Goldstein. Why the heck, don't the authors publish their "modernization" under his name?