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I've been working through Bernard Schutz's book on GR and have run into some confusion in chapter 4 problem 20 part b. In this chapter, the stress-energy tensor for a general fluid was introduced and was used to derive the general conservation law for energy/momentum, where we found that $$T^{\alpha \beta}{ }_{, \beta}= 0$$

Schutz wants us to show that if the energy and momentum of a body are not conserved, for example because the body interacts with some other system, then we can define a nonzero relativistic force four-vector ##F^{\alpha}## which is defined by $$T^{\alpha \beta}{ }_{, \beta}= F^{\alpha}$$

To get this equation, it appears that we would have to add an extra term to the equation for conservation of energy, which is $$ \frac {\partial T^{0 0}} {\partial t} = -T^{0 i}{ }_{,i} + F^{0}$$ Without the extra term, the equation is ## \frac {\partial T^{0 0}} {\partial t} = -T^{0 i}{ }_{,i}## which says that the rate of change of energy per unit volume is equal to the net influx of energy per unit time per unit volume of the fluid element in question. In this case, I don't see what adding ## F^{0} ## really adds to this equation since I assumed that any influx of energy would be given by the term ## -T^{0 i}{ }_{,i} ##. If I'm looking at some fluid element within the fluid, and the body of fluid collides with some other body, changing it's total energy and momentum, then any change in energy of the fluid element would be due to local interactions with it's neighboring elements and the influx of energy would be given by ## -T^{0 i}{ }_{,i} ##.

Analogously, with the equation for momentum, we have $$ \frac {\partial T^{i 0}} {\partial t} = -T^{i j}{ }_{,j} + F^{i} $$

Again, without the term ##F^{i}##, this equation says that the time rate of change of the ith direction of momentum per unit volume of the fluid element is equal to the net influx of ith direction of momentum per unit time per unit volume of the fluid element. It appears that ##F^{i}## is just the ith component of the net external force. But it seems that any influx of momentum is given by ##-T^{i j}{ }_{,j}##, so as before, I'm not sure what ##F^{i}## adds. But, I see that combining these equations would give the equation Schutz wanted us to derive.

The only resolution I've come up with is that ## -T^{0 i}{ }_{,i} ## and ##-T^{i j}{ }_{,j}## only describe the net influx of energy and momentum, respectively, independent of any external source, and so we have to tack on the extra terms describing the contributions of the external source. These terms then only describe the interactions of the fluid element with neighboring fluid elements. But as I described, it seems that any external influence on the system would be conveyed by internal fluid elements on the fluid element in question. At the same time, I see that the total energy and momentum added to the system would have to be accounted for, and from that perspective, the added terms, ##F^{\alpha}## make sense to me. I am getting confused when I'm applying this to a fluid element within the system, which was the perspective from which Schutz derived ##T^{\alpha \beta}{ }_{, \beta}= 0##. I'm not sure what I'm misunderstanding. I appreciate any help!

Schutz wants us to show that if the energy and momentum of a body are not conserved, for example because the body interacts with some other system, then we can define a nonzero relativistic force four-vector ##F^{\alpha}## which is defined by $$T^{\alpha \beta}{ }_{, \beta}= F^{\alpha}$$

To get this equation, it appears that we would have to add an extra term to the equation for conservation of energy, which is $$ \frac {\partial T^{0 0}} {\partial t} = -T^{0 i}{ }_{,i} + F^{0}$$ Without the extra term, the equation is ## \frac {\partial T^{0 0}} {\partial t} = -T^{0 i}{ }_{,i}## which says that the rate of change of energy per unit volume is equal to the net influx of energy per unit time per unit volume of the fluid element in question. In this case, I don't see what adding ## F^{0} ## really adds to this equation since I assumed that any influx of energy would be given by the term ## -T^{0 i}{ }_{,i} ##. If I'm looking at some fluid element within the fluid, and the body of fluid collides with some other body, changing it's total energy and momentum, then any change in energy of the fluid element would be due to local interactions with it's neighboring elements and the influx of energy would be given by ## -T^{0 i}{ }_{,i} ##.

Analogously, with the equation for momentum, we have $$ \frac {\partial T^{i 0}} {\partial t} = -T^{i j}{ }_{,j} + F^{i} $$

Again, without the term ##F^{i}##, this equation says that the time rate of change of the ith direction of momentum per unit volume of the fluid element is equal to the net influx of ith direction of momentum per unit time per unit volume of the fluid element. It appears that ##F^{i}## is just the ith component of the net external force. But it seems that any influx of momentum is given by ##-T^{i j}{ }_{,j}##, so as before, I'm not sure what ##F^{i}## adds. But, I see that combining these equations would give the equation Schutz wanted us to derive.

The only resolution I've come up with is that ## -T^{0 i}{ }_{,i} ## and ##-T^{i j}{ }_{,j}## only describe the net influx of energy and momentum, respectively, independent of any external source, and so we have to tack on the extra terms describing the contributions of the external source. These terms then only describe the interactions of the fluid element with neighboring fluid elements. But as I described, it seems that any external influence on the system would be conveyed by internal fluid elements on the fluid element in question. At the same time, I see that the total energy and momentum added to the system would have to be accounted for, and from that perspective, the added terms, ##F^{\alpha}## make sense to me. I am getting confused when I'm applying this to a fluid element within the system, which was the perspective from which Schutz derived ##T^{\alpha \beta}{ }_{, \beta}= 0##. I'm not sure what I'm misunderstanding. I appreciate any help!

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