By P. W Bridgman
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Extra info for A sophisticate's primer of relativity
For such a distribution, for example, a gas in thermal equilibrium, we may thus write the energy–momentum tensor as T= T i i where T are the energy–momentum tensors of its uniform components. i Even if we do not have particle-number conservation, the conservation of four-momentum implies that the energy–momentum tensor for a nonuniform distribution is still well defined. Note that the weak and strong energy conditions still hold, and that for massless particles we have tr T = 0. 14) i i mρ0 is the average mass density of the gas.
Here V and A are the instantaneous three-velocity and three-acceleration of the particle with respect to an observer with four-velocity e 0 . Since x¨ · x˙ = 0, these equations give γ˙ = −γ 4 A · V . 28) For small three-velocity we have γ ≈ 1, and if we also have small threeacceleration, this equation gives γ˙ ≈ 0. 27) thus reduce to x˙ = e 0 + V and x¨ = A. 29) where F (v ) is the electric part of the field (with respect to the observer) and m and e are the rest mass and charge of the particle.
2). 2. e. of the state of motion of an observer whose world line passes whose world line passes through through p. p) lies inside N( p), and each null ray of N( p) defines a unique point on his celestial sphere. This means that there are a sphere’s worth of null rays through p and, given any two observers who instantaneously occupy p, there exists a one-to-one correspondence between points on their celestial spheres where two points correspond if they define the same null ray. By the principle of relativity there are no preferred null rays through p, and hence, at least locally, the set of null rays through p will be spherically symmetric with respect to any observer instantaneously occupying p.
A sophisticate's primer of relativity by P. W Bridgman