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Re: [Phys-l] power, energy flow, vectors, tensors, et cetera.

Regarding:

Conservation of energy (and momentum) can be expressed as the
divergence of the stress-energy tensor:

0 = ∇i Tij [1]

As always, there is an implicit sum over i=0,1,2,3.

The j=0 piece of equation [1] expresses conservation of energy.

The j=1,2,3 pieces express conservation of the various components
of momentum.

The j=0 piece says that the power is equal to -1 times the 3D
(spacelike) divergence of the flow of energy.

Um, that (j= 0 piece) is not the power. That's the time rate of change of the energy *density*. The power is the *spatial integral* of the j=0 piece or of the remaining 3-divergence (of the corresponding Poynting-like energy flux) integrated over the interior of an open region in space. The power is really the time rate at which the energy is changing in the interior region, or the rate at which work is being done across the boundary of the region between the interior and the exterior. The power for these situations (where the energy and momentum can be regarded as infinitely divisible extensive quantities) is defined only for each such specified *region* of space. More generally, the power is defined for a *system* as the time rate at which work is being done on/by the system/surroundings, or the rate at which the internal energy of the system is changing with time. But for fluidlike/field theoretic descriptions a system is typically specified by the region of space containing some 'stuff'to be specially considered, and the exterior of the spatial region is the system's surroundings.

Confusing a power with time rate of energy density change is sort of like confusing a heat capacity with a specific heat, or a resistance with a resitivity. One is defined for a specified finite sized system or object, and the other is defined differentially for the stuff or material out of which the particular system is made.

David Bowman