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Why ##n^\mu T_{\mu\nu}## is called the pressure?

by ccnu
Tags: ##nmu, called, pressure, tmunu##
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ccnu
#1
Aug11-14, 09:39 PM
P: 9
I read in the textbook which says that, according to the usual definition the absolute value of ##n^\mu T_{\mu\nu}## is just the pressure. ##T_{\mu\nu}## is the energy-momentum tensor and ##n^\mu## is a four dimensional normal vector.
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ChrisVer
#2
Aug12-14, 05:11 PM
P: 1,045
As given I don't see that this holds in general... For example take a perfect fluid for which it can be written in rest frame:
[itex]T_{\mu \nu} = diag( \rho, p , p , p ) [/itex]
Then:
[itex] n^{\mu}T_{\mu \nu} = n^{0}T_{00} + n^{ii}T_{ii} = n^{0} \rho + (n^{11}+n^{22}+n^{33}) p[/itex]

Which doesn't have to be equal to the pressure... If though you choose [itex]n^{0}=0[/itex] (so it's not just any 4-dim unit vector) things can get better.

Can you give your reference?
ccnu
#3
Aug12-14, 08:24 PM
P: 9
Quote Quote by ChrisVer View Post
If though you choose [itex]n^{0}=0[/itex] (so it's not just any 4-dim unit vector) things can get better.
Oh~~ yes, I forgot to say that ##n^0 = 0##, and ##n^\mu## is pure spatial. Thanks a lot!

Orodruin
#4
Aug13-14, 01:17 AM
P: 680
Why ##n^\mu T_{\mu\nu}## is called the pressure?

Chris, your LHS has a free index and your RHS does not.

In a general frame you would have
$$
n^\mu T_{\mu\nu} = n^\mu \left((\rho+p) u_\mu u_\nu - p g_{\mu\nu}\right)
= (\rho+p) u_\nu (n \cdot u) - p n_\nu,
$$
where u is the 4-velocity of the fluid.
Meir Achuz
#5
Aug13-14, 09:53 AM
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P: 2,026
Quote Quote by ccnu View Post
I read in the textbook which says that, according to the usual definition the absolute value of ##n^\mu T_{\mu\nu}## is just the pressure. ##T_{\mu\nu}## is the energy-momentum tensor and ##n^\mu## is a four dimensional normal vector.
The association of ##n^\mu T_{\mu\nu}## with the pressure at a point follows from the fact that its integral over a closed surface equals the rate of change of momentum within the closed surface.
There are simple examples where ##n^\mu T_{\mu\nu}## is NOT the pressure at a point on the surface. For instance the pressure on a dielectric slab due to a point charge a distance d from the slab is not ##n^\mu T_{\mu\nu}##.
ChrisVer
#6
Aug13-14, 02:29 PM
P: 1,045
Quote Quote by Orodruin View Post
Chris, your LHS has a free index and your RHS does not.

In a general frame you would have
$$
n^\mu T_{\mu\nu} = n^\mu \left((\rho+p) u_\mu u_\nu - p g_{\mu\nu}\right)
= (\rho+p) u_\nu (n \cdot u) - p n_\nu,
$$
where u is the 4-velocity of the fluid.
oops sorry... yes the correct form would have to be:
[itex]n^{i} T_{i \mu} \equiv n_{i} p[/itex]
Orodruin
#7
Aug13-14, 03:17 PM
P: 680
Quote Quote by ChrisVer View Post
oops sorry... yes the correct form would have to be:
[itex]n^{i} T_{i \mu} \equiv n_{i} p[/itex]
[itex]n^{i} T_{i \mu} \equiv n_{\color{Red}\mu} p[/itex]



Assuming ##n^0 = 0## and that we are in the rest frame of the fluid.
ChrisVer
#8
Aug13-14, 04:18 PM
P: 1,045
the n0 is zero
and also mu=i for the expression not to be zero...

For the rest frame yes, I just corrected the expression I gave in my previous post
Orodruin
#9
Aug13-14, 05:09 PM
P: 680
Even if the expression is non-zero only for spatial ##\mu##, you must still have the same free indices on both sides of your equality. In your case you have i as a summation index on one side and as a free index on the other. I can guess what you mean because I know what you are aiming for, but formally it does not make sense.


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