Abstract
Suppose a region of spacetime is occupied by a large number N of identical particles. Suppose that they possess a wide variety of momenta and that there are so many of them that their spacetime and momentum distribution may be taken as effectively continuous
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- 1.
x 0 is not strictly a “time”. However, we shall assume here that hypersurfaces of constant x 0 are spacelike [which means \(\hbox{det}(g_{ij})>0\) and g 00 < 0] and that x 0 increases as one moves toward the future in any timelike direction.
- 2.
After we have this, we may let \(\Sigma\) be any connected hypersurface that intersects completely the support of the vector density j ?, cutting all its flow lines an odd number of times. Then, \(\Sigma\) need not be spacelike, even in part.
- 3.
Coordinate transformations can be shown to be canonical transformations, and hence the coordinate invariance of \(\hbox{d}^3{\user2{x}}\hbox{d}^3{\user2{p}}\) is a special case of this.
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© 2011 Springer-Verlag Berlin Heidelberg
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DeWitt, B., Christensen, S.M. (2011). Ensembles of Particles. In: Christensen, S. (eds) Bryce DeWitt's Lectures on Gravitation. Lecture Notes in Physics, vol 826. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-36911-0_7
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DOI: https://doi.org/10.1007/978-3-540-36911-0_7
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