In Hawking-Ellis Book(1973) "The large scale structure of space-time" p69-p70, they derive the energy-momentum tensor for perfect fluid by lagrangian formulation. They imply if ##D## is a sufficiently small compact region, one can represent a congruence by a diffeomorphism ##\gamma: [a,b]\times N\rightarrow D## where ##[a,b]## is some closed interval of ##R^1## and ##N## is some 3-dimensional manifold with boundary. The tangent vector of ##\gamma## is ##W=(\partial/\partial t)_{\gamma}##. The Lagrangian is taken to be $$L=-\rho(1+\epsilon)$$ and the action ##I## is required to be stationary when the flow lines are varied and ##\rho## is adjusted to keep ##j^a## conserved where ##\rho## is a function and ##\epsilon## is the elastic potential as a function of ##\rho##. A variation of the flow lines is a differentiable map ##\alpha: (-\delta, \delta)\times[a, b]\times N\rightarrow D## such that $$\alpha(0, [a,b],N)=\gamma([a,b],N).$$ They say "Then it follows that $$\Delta W=L_{K}W$$ where the vector ##K## is ##K=(\partial/\partial u)_{\alpha}##."(adsbygoogle = window.adsbygoogle || []).push({});

I'm curious this equation is correct. I guess ##\Delta W## means its components is ##(\partial W^i/\partial u)|_{u=0}## in their book. However r.h.s components are calculated as follows.$$(L_{K}W)^i=\frac{\partial W^i}{\partial x^j}K^j-\frac{\partial K^i}{\partial x^j}W^j=\frac{\partial W^i}{\partial u}-\frac{\partial K^i}{\partial t}$$ So I wonder $$(\Delta W)^i=\frac{\partial W^i}{\partial u}\neq (L_{K}W)^i=\frac{\partial W^i}{\partial u}-\frac{\partial K^i}{\partial t}.$$ ##(\partial K^i/\partial t)=0?## Will you tell me where I am wrong?

This pdf file is Eur. Phys. J. H paper by S. Hawking. See page 19. But I'm sorry my notation is little different.

https://epjti.epj.org/images/stories/news/2014/10.1140--epjh--e2014-50013-6.pdf

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# I Variation of perfect fluid and Lie derivative

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