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I am trying to teach myself general relativity and am working through the text 'a first course in general relativity' by Bernard F Schutz. So far I have made slow but consistent progress but I am perplexed by a couple of things in the fluid mechanics part where it derives the consequences of the laws of conservation of energy and momentum for a 'perfect fluid' (fluid with no heat conduction or viscosity).

In Schutz the conservation laws are expressed as

[itex]

T^{\alpha \beta},_{\beta} = [(\rho+p)U^{\alpha}U^\beta+p\eta^{\alpha\beta}],_\beta = 0

[/itex] (4.39)

where T is the stress-energy tensor of the fluid, [itex]\rho[/itex] is mass/energy density, p is pressure, U is four-velocity of the infinitesimal fluid element under consideration and [itex]\alpha[/itex] and [itex]\beta[/itex] are indices in Minkowski space hence can have values 0,1,2,3. [itex]\eta[/itex] denotes components of the metric tensor. A subscript preceded by a comma indicates partial differentiation with respect to the coordinate represented by that subscript.

Schutz says 'first let us assume that [itex](nU^\beta),_beta = 0[/itex] '

(where n is the particle density of the fluid, ie number of particles per unit volume)

and then uses this assumption in the derivations that follow, but does not give any explanation of why this assumption is made or what its physical significance is. Nor does he appear to subsequently relax this assumption, as one might expect he would given his use of the words 'FIRST let us assume'.

Can anyone explain what is the justification and the meaning of this assumption?

Secondly, Schutz derives the following formula for a perfect fluid, valid for [itex]\alpha[/itex] = 0,1,2,3:

[itex]nU^\beta(U^\alpha(\rho+p)/n) ,_\beta+p,_\beta\eta^{\alpha\beta} = 0[/itex] (4.45).

He then observes that, in the MCRF (momentarily co-moving reference frame of the infinitesimal fluid element), for [itex]\alpha[/itex] = 1,2,3 (ie the three spatial dimensions), [itex]U^\alpha=0[/itex] and hence the above formula can be written:

[itex](\rho+p)U^i,_{\beta} U^\beta+p,_{\beta} \eta^{i\beta}=0[/itex] (4.52)

This is true for i =1,2,3 but not for i=0 as [itex]U^0=1[/itex] in the MCRF. (Schutz uses Roman letters instead of Greek when only considering the three spatial dimensions).

Now comes the step I don't understand. Schutz says

'Lowering the index i makes this easier to read and changes nothing. Since [itex]\eta_i^\beta=\delta_i^\beta[/itex] we get:

[itex](\rho+p)U_i,_\beta U^\beta +p,_i=0[/itex] (4.53)'

The trouble is that, to lower the index i, you need to multiply equation 4.52 by [itex]\eta_{ji}[/itex] and sum over i = 0,1,2,3. But the equation is only true for i = 1,2,3, not 0.

So how is it possible to derive the last equation (which is required for what follows later) without cheating?

Thank you very much to anyone who can help with this.

Andrew

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# Problem with fluid mechanics part of GR derivation

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