Talk:Relativistic Euler equations

Untitled

Shouldn't $e$ be defined as $e=\rho c^{2}+e^{C}$, without the $\rho$ factor multiplying the internal energy? Or is $e^{C}$ the internal energy per unit mass? If it is per unit mass, that should be stated. (I'm new at this and I'm not confident enough in my comment to just enough to just edit the page.)

Clay Spence 15:52, 3 January 2007 (UTC)

Hi Clay. I think you're right. I would recommend, in general, that you be bold and edit the page. But in this case I've done it for you.

Best wishes, and really, do be bold: that way, we all gain. Robinh 20:21, 3 January 2007 (UTC)

My tensor maths isn't that strong at the moment, but I have reason to believe that there are a number of errors in this article and its use of tensor notation. Firstly, the 4-derivative has one lower index, mu, but the function T has two upper indices, mu and nu. Since we've got a covariant-contravariant vector product we should end up with an invariant equantity (if they're to obey the Lorentz transform in special relativity) however you have a free index - this means the product is be non-zero (indeed, you'll end up with a vector with 1 free index). Furthermore, the definition of T-mu,nu itself seems wrong. The left-hand side of the equation you have upper indices, the right-hand side you have lower indices which you cannot have by the definitions of contra- and co-variant vectors.

From other areas of physics where I have seen the continuity equation applied to situations, you define a 4-vector containing a "density" like term (that you connect with the temporal component of 4-derivative) and a "current" like term (which you connect with the spatial components of the 4-derivative); this 40current only needs 1 free index. -- Lateralis (16/1/07, 22:10 GMT)

I think most of this page is wrong. especially the un-index-balanced tensor equation which has already been mentioned. I think perhaps someone should either ammend or remove this page

The contraction of T (which has 2 indices) with one index (from the covariant derivative) gives a vector - true; but this vector doesn't have to be zero. From physics, we stipulate that it be zero (or, more precisely, the zero four-vector). I've fixed the positions of the indices in the equations that needed this fixing. The page should not be removed, as it is an essential topic in relativity. I do agree that the page does needs to be improved significantly. MP (talk) 19:21, 9 February 2007 (UTC)

That is fine MP, but there's still a problem with indecies, and I appreciate I am probably being a pedant. However, the partial derivative with one index, acting on a matrix with two indecies leaves a qauntity with 1 free index. In general, this would be a non-zero vector. If you were to stipulate that this result is 0 - and hence demand that all elements of the vector are 0 - then it needs stating explicitly for the sake of completeness and clarity that the result is a vector with all elements identically 0. Otherwise it looks like an index has just been obliterated. -- Lateralis (15/2/07, 13:13 GMT)

Lateralis: The expression

D_μ T^μν = 0

is a correct and valid equation and is commonly seen in relativity. It means

∑_(μ=0,...,3) D_μ T^μν = 0 for ν = 0,1,2,3.

The expression D_μ T^μν is called a "divergence", just as D_μ X^μ is a divergence, but I would fully agree that its intuitive meaning is harder to understand. 188.154.129.67 (talk) 08:57, 29 June 2018 (UTC)

Lateralis, while I see your logic that setting a vector equal to a scalar is wrong, it is extremely common to use the symbol "0" for the zero vector, or a zero tensor, for that matter, whose components are all zero as you say. Just to make sure I'm not mis-remembering, I checked in Misner, Thorne, and Wheeler, and in Landau and Lifschitz, and they express exactly the equation you mention in exactly this form. The 0 vector is special, and has a meaning independent of the coordinate system.Clay Spence (talk) 01:46, 30 June 2018 (UTC)

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Nearly worthless article

1) There is not enough information defining the terms

u^μ, e, p, n,

to make a well-posed equation. As a result, it's impossible to learn from this article what the equation says, even for an expert in differential geometry and relativity. You have to look somewhere else.

2) It's worth pointing out that u has length 1 and is timelike, and that e, p, and n are scalar functions on spacetime.

But there have to be "constitutive relations" between e, p, and n. For example, p is a function of e or vice versa, called the equation of state. This relation is a fixed property of fluid.

Also, the internal energy e is a function of the number density n -- at least if there is a Lagrangian, or the fluid is "isentropic", or the fluid is "barotropic" (barotropic gas = a gas whose pressure depends only on its density). I'm not sure how these conditions relate, but if they hold, then the function e(n) ultimately determines the equation of state.

From one random source on the net: "The Bianchi identities imply D_μ T^μν = 0, and this equation, together with an equation of state, determines the motion of the fluid."

3) Indeed, by projecting the equation D_μ T^μν = 0 onto u^μ and onto the 3-space perpendicular to u^μ, where T^μν has the diagonal form given in the article, one obtains an energy conservation law and a momentum conservation law. Together these are the relativistic Euler equations, though some authors appear to use this designation only for the momentum equation(?). If there are appropriate relations between e, p, n, then these equations determine the motion of the fluid.

4) What about the Lagrangian? That makes the above relationships much easier to grasp.

188.154.129.67 (talk) 22:54, 1 July 2018 (UTC)

Some sources:

Hawking and Ellis has a 2-page section on the relativistic Euler equation (pp 69-70) which is a challenge to read but not impossible.

J. Olsthoorn, "Relativistic Fluid Dynamics," University of Waterloo, http://mathreview.uwaterloo.ca/archive/voli/2/olsthoorn.pdf (15pp).

J.W. van Holten,Relativistic fluid dynamics (34pp)

S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, 1972, pp 47-57,127-130,241-243.

N. Andersson, G.L. Comer, Relativistic fluid dynamics: physics for many different scales, 2006 (76pp)

188.154.129.67 (talk) 13:17, 2 July 2018 (UTC)

The of

Hi, I haven't studied physics for more than 40 years since I was 16. But I reckon something is missing from the sentence "This says that the [?] of the system is dominated by the rest energy of the fluid in question." any suggestions? ϢereSpielChequers 19:21, 1 November 2020 (UTC)